Infinite Staircase Paradox
The story:
Icarus was walking through the woods when he stumbled upon a sign which read:
[i]"One, two, three, four, five,
Each step named as down you dive.
Endless stairs stretch out of sight,
Walk them. Grasp the Infinite's might.
Your speed will double with each step as you soar
like an angel approaching a classical black hole's core."[/i]
He glanced down and noticed a dusty staircase plunging into darkness. Thinking that one step would be harmless, he descended and immediately grasped the profound gravity of his actions. He lingered on the first step, marked "1," for 30 seconds, soaking in the enchanting energy coursing through his veins. Moving to step "2," he paused for 15 seconds, feeling lighter and quicker, like a feather in descent. Driven by an irresistible urge, he continued to step "3," then "4,", and so on, each time halving his rest period.
Despite the staircase being endless, he reached the bottom of it in just a minute. Looking around to ascertain his location, he was startled to find himself standing over a dead body. Clearly, the dead body was that of a man who had fallen from a great height. Horrified, he stepped back, intending to ascend the staircase, but it had vanished. Turning around, he found no steps in sighthow could there be, for what would they number?
A wave of anxiety overwhelmed him as he pieced together the events. Slowly, he looked down at the face of the corpse and recognized it as his own.
The infinite staircase was never realan impossibility and an illusion from the very start. Or was it?
My Questions:
The infinite staircase appears to only allow one to traverse it in one direction. It simultaneously exists and doesn't exist? Does this make sense? If we allow Hilbert's Hotel to exist in the abstract and possible realm, are we forced to accept the infinite staircase into the abstract and possible realm? Is this actually a paradox? What are your thoughts?
Icarus was walking through the woods when he stumbled upon a sign which read:
[i]"One, two, three, four, five,
Each step named as down you dive.
Endless stairs stretch out of sight,
Walk them. Grasp the Infinite's might.
Your speed will double with each step as you soar
like an angel approaching a classical black hole's core."[/i]
He glanced down and noticed a dusty staircase plunging into darkness. Thinking that one step would be harmless, he descended and immediately grasped the profound gravity of his actions. He lingered on the first step, marked "1," for 30 seconds, soaking in the enchanting energy coursing through his veins. Moving to step "2," he paused for 15 seconds, feeling lighter and quicker, like a feather in descent. Driven by an irresistible urge, he continued to step "3," then "4,", and so on, each time halving his rest period.
Despite the staircase being endless, he reached the bottom of it in just a minute. Looking around to ascertain his location, he was startled to find himself standing over a dead body. Clearly, the dead body was that of a man who had fallen from a great height. Horrified, he stepped back, intending to ascend the staircase, but it had vanished. Turning around, he found no steps in sighthow could there be, for what would they number?
A wave of anxiety overwhelmed him as he pieced together the events. Slowly, he looked down at the face of the corpse and recognized it as his own.
The infinite staircase was never realan impossibility and an illusion from the very start. Or was it?
My Questions:
The infinite staircase appears to only allow one to traverse it in one direction. It simultaneously exists and doesn't exist? Does this make sense? If we allow Hilbert's Hotel to exist in the abstract and possible realm, are we forced to accept the infinite staircase into the abstract and possible realm? Is this actually a paradox? What are your thoughts?
Comments (1106)
Would the above qualify as a paradox, or just be silly in "the" non abstract and possible realm but fit into the abstract and possible realm? Or the reverse?
Can a paradox be conceived in the a&p realm?
Sorry if the above "thoughts" are confusing or ill expressed.
I think it's a quite nice paradox. It exemplifies that we can conceive of counting from 1 to infinity in a finite amount of time by halving the time it takes for stating each successive number, but that we can't perform the same process in reverse order, as it were, since there is no "last number" that we can step in first to retrace our steps. But also, there is a slight of hand that occurs when we are encouraged to imagine Icarus's position immediately after he's finished traversing the infinitely long staircase in the original direction. If he would have traversed the staircase in Zeno like fashion, as specified, although he would have stepped on all the steps in a finite amount of time, there would be no definite position along the staircase that he was at immediately before he had arrived at his destination.
Quoting keystone
He reaches the bottom of something with no bottom. It taking a minute is fine, but there being a bottom is contradictory. Hence I think resolution. Just as there is no first step to take back up, there is no last step to reach, even if it is all reached in a minute.
How does that work? He's traveling by steps. Each step takes a discernible amount of time which is a different time from the prior step. You say he reaches the bottom, yet there is not "last step". Clearly he doesn't do a bunch of last steps at the same time, so ambiguity is not the problem. How do you think it is possible that he got finished with all the steps, in the described order, yet there was no last step?
Quoting Pierre-Normand
Yes, there is a "slight of hand" involved. The real solution is that the only "finite time" in the description is the starting time, and the formula for figuring the increments. And, according to the prescribed formula for figuring the increments, there can be no finish time. It's analogous to finding the end of pi, you just keep going. Despite the defined, and finite starting point, Icarus is currently covering an infinite number of steps in an infinitely short period of time, and by adhering strictly to the description we must conclude that the bottom will never be reached. Bye bye Icarus, enjoy the "black hole's core".
After a minute, yes. Do you contend otherwise, that the sum of 60/2**n is not 60?
Just like there is no last natural number, yes. There is no last step to 'be' at.
It's pretty clear from the mathematics. Where do you expect him to be then at 61 seconds if not 'past them all'?
Quoting Metaphysician UndercoverOK, so mathematics is not your forte. The sum of this infinite series is not 60 according to you.
Quoting keystoneYour poetry asserts this, but the reverse can be done There is simply no first step in the process, just like there wasn't a last step on the way down. The sum of the same series in reverse order is also 60 seconds.
The specifications do not allow for a minute to pass, that's the problem. It's just like Zeno's Achilles and the tortoise paradox. What is specified by Zeno, disallows the possibility of Achilles passing the tortoise. Here, what is specified in the op by keystone, disallows the possibility of a minute passing.
So it's like you're saying, "after Achilles passes the tortoise...", when Achilles cannot pass the tortoise, because of what is specified, he must pass an infinite number of spaces first. Here, you are saying "after a minute..." when a minute cannot pass because of what is specified, Icarus must pass an infinite number of steps fist.
So it's just like the Achilles and the tortoise paradox, with a different conclusion. Here, instead of concluding that a minute cannot pass, as Zeno concluded that Achilles cannot pass the tortoise, keystone changes things up to say that after a minute has passed the infinite number of steps has been reached. But @keystone has made a invalid conclusion, and should have stuck to Zeno's formulation, to say that a minute cannot pass for Icarus because he always has another step to make first, and that step will be made in a shorter time than the last.
If that statement is logically unacceptable, then it could be considered a paradox. However, many people today might not see an issue with it, so you would need to provide further explanation to convincingly demonstrate its paradoxical nature.
Quoting kazan
Let me draw an analogy. Historically, our understanding of the world was believed to be tangible and possible. We thought we grasped the truth, whether through Newtonian mechanics or general relativity. Then, some thinkers pointed out inconsistencies in these prevailing views that defied explanation. What was once deemed tangible and possible turned out to be tangible and impossible. As a result, the model of the world was revised, and the new model was then assumed to be tangible and possible. Over the years, this process repeats, gradually bringing us closer to the truth.
What's your take on this? Do you believe he never finishes descending the stairs? If that's the case, then where would he be after one minute has passed?
How is it possible for him to ascend the stairs if there isn't a first step? Or do you think that he might not be able to fully descend the stairs?
Do you truly believe that Achilles is unable to surpass the tortoise? Do you think that Icarus's deeds influence the passage of time? Is there a concrete analogy in which your actions alter how time progresses for me?
By what is stipulated, yes, Achilles cannot surpass the tortoise. But, the stipulations are not a true representation and that is why there is a problem. The issue is with the way that motion is described. We employ a conception of time and space as a continuity which is infinitely divisible. However, if motion actually consists of discrete changes like a "quantum jump" for example, then the representation of a continuous existence, is false.
Quoting keystone
No, I do not think that Icarus's deeds influence the real passing of time. However, the passing of time is a subject in your example, and it is determined by the premises of the example. So. according to the premises of the example, half as much time is covered each time Icarus makes a step, as compared to the previous step, and Icarus can keep on stepping forever. Therefore the passage of time is defined in relations to Icarus's deeds in the example. That is what is stipulated in the example, That the passage of time is relative to Icarus's steps. Whether it is a truthful representation of the passage of time is irrelevant at this point.
This is nicely illustrated by Zeno's 'dichotomy paradox'. Per wiki:
"Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on."
Each 'step' of the path from here to there must be preceded by a prior step. The fallacious conclusion is that no journey can be taken anywhere since there can be no first step when they're set up as you have done. Likewise, Zeno fallaciously concludes that Achilles cannot overtake the Tortoise due to the journey being divided into infinite steps. Both are a non-sequitur fallacy since it simply does not follow that the goal cannot be reached just because there exists a way to slice it into an unbounded quantity of segments.
Quoting Metaphysician Undercover
Quoting Metaphysician UndercoverWhat do you mean stipulated? That Achilles cannot overtake is a non-sequitur. It simply doesn't follow from there being a way to divide the journey into infinite segments. This isn't a stipulation, it is merely a fallacious conclusion. Time not being allowed to pass was never a specification in the OP. Of course the lack of the stairs back up was actually a specification, and I find that contradictory.
The dichotomy thing was better illustrated by something that actually seems to be a paradox.
You are at location x < 0. The goal is to traverse the space between x=0 and x=1.
Thing is, a magic barrier appears at x=1/2 if you are at x <= 1/2, but x > 1/4.
A second barrier appears at x=1/4 if you are at x <= 1/4, but x > 1/8.
And so on. Each barrier appears only if you're past the prior one.
Furthermore, for fun, the last barrier is red. The prior one blue, then green, then red again. Three colors in rotation, all the way up the line.
Per the dichotomy thing (and Keystone's stairs), there can be no first barrier. So you walk up to x=0 and are stopped, despite there not being anything there to stop you. I mean, if there's a barrier, you'd see it and know its color, which is like suggesting a remainder if you divide infinity by three.
So paradoxically, you are prevented from advancing despite a total lack of a first barrier. You can see the goal. But you can't move.
That's a far better wording, and less fallacious than the way Zeno is reported to have worded it.
Is this not a question of special relativity? It seems paradoxic until we apply physics.
He's accelerating exponentially along a linear trajectory (the infinite staircase). So he's approaching the speed of light. Hence relativity becomes an integral factor. One that hasn't been addressed in this "paradoxic" hypothetical.
Ah okay. Fair. Then where is the reaching the bottom in under 1 minute coming from? Surely even if halfing the time with every step, a minute will still eventually be exceeded somewhere along the infinite steps and before this so called "finite bottom" to an infinite staircase?!? Doesn't make sense mathematically either.
The most interesting thing I found about this is the unidirectional counting. You can count from 1 toward infinity but you can't begin counting from infinity toward 1.
He just made it up, it doesn't come from anywhere. That's why I'm questioning if it's really a paradox, that's why I want another source for it
It's not a non-sequitur, the conclusion follows logically from the way that Achilles' movement is described. From the description there is always further distance for Achilles to move before he overtakes the tortoise. Therefore he cannot overtake the tortoise. The issue is not that the argument is invalid, it is that the argument is unsound. The description of motion employed provides false premises.
Quoting noAxioms
In the OP, it is not the case that time is not allowed to pass, but the premises imply that a minute cannot pass for Icarus, who always has to take more steps before a minute can pass. Just like in Zeno's paradox, the premises which describe how Icarus moves down the stairs are faulty.
So, in the OP, the false premise is the description of acceleration. Acceleration from rest is described as continuous and open ended (infinite). But this is false, acceleration does not happen like this in reality. Imagine if the OP was expressed in the following way. Someone states that the universe is infinite in size. Then the person states that a rocket accelerates from being at rest on earth at a rate of acceleration which will take it to the edge of the universe before a minute passes. Then the person concludes that after a minute passes the rocket is at the edge of the universe. Do you see the incoherency? That's what's going on in the OP. The premises are arranged so that there cannot be an end to the staircase, just like there cannot be an end to pi. Then it concludes, that after a minute has passed, the end has been reached.
So the OP makes a non-sequitur by concluding that the end is reached. Zeno on the other hand, concludes that Achilles cannot overtake the tortoise, which is the valid conclusion. And the absurd conclusion reveals the falsity of the premises.
Quoting noAxioms
I don't think that this is representative of the OP at all. You have changed the divisibility of time in the OP to a divisibility of space in your interpretation. Then, instead of dealing with the problem of acceleration, which the OP is concerned with, you have to employ "magic barriers" to make sense of the steps. There are no such magic barriers employed by the OP, only steps, and each step is made in half the time of the prior step. So clearly, the OP deals with the issue of infinite acceleration.
Quoting flannel jesus
If a person does not take into account what is physically possible in this type of thought experiment, then one can make up false premises however one wants, and create the illusion of a "mathematical problem" when no such problem actually exists. The real problem is that the premises are false (physically impossible), and by employing the false premises the illusion of a mathematical problem is created.
I don't understand the rules of this game.
However, I recommend that Icarus stops looking for the last step down and starts looking for the first step up. He should find that as easily as he found the first step down.
But it would be a bad idea for him to ask whether the stairs up were the same stairs as the stairs down, or whether the staircase exists. At best, those questions would scramble his mind, possibly to the point where he might get so distracted as to forget to keep moving. At worst, the staircase might disappear beneath his feet.
He should allow at least twice as long to climb up as he took to get down. But he can expect to complete his climb in the same amount of time as he took to descend.
i) There does not exist an injection N --> S
ii) There exists a surjection I ---> S, where I is a subset of N.
Condition i) represents the hypothesis that we do not know how many stairs there are, or equivalently that we cannot know the height of the top stair due to assuming that we will never reach the bottom of the staircase.
Condition ii) represents the physically plausible situation that although we cannot count the stairs, there cannot be more stairs than some finite but unboundedly large subset of the natural numbers.
In other words, we are assuming that S is subcountable.
Let s(n) denote the n'th stair that is visited when descending. Using this order of descent on S, we have a total function S --> N* describing the height of each stair as a non-standard natural number, namely
s (0) => h*
s(1) => h* - 1
s(2) => h* - 2
..
which when written directly in terms of the indices denoting the order-of-descent is a function f
f : N --> N* :=
f (n) = h* - n*
This function describes an infinite descent in N*, and is paradoxical because
1) Every nonstandard natural number e* that is in N* corresponds to some standard number e in N, and vice-versa.
2) We have defined an infinitely descending chain of non-standard natural numbers in N*.
The paradox is resolved due to the fact that the order-of-descent we are using when descending the "infintie staircase" from the top has no recursively definable relationship in terms of the order of ascension when climbing the staircase from the bottom; although Peano's axioms rule out the existence of non-wellfounded subsets for recursively enumerable subsets of the natural numbers, our subset isn't recursively enumerable in terms of those axioms, and is therefore an external subset that cannot be talked about by Peano's axioms.
Anyhow, Aristotle claims that we cannot have an actual infinity, only a potential one. However, Hegel famously rebuts this claim with the Essence chapter of the Greater Logic, a text that is infinitely dense and impenetrable.
Maybe, Aristotle was mistakenly "transcribed" from his words or personal writings as "only potential infinity" instead of "infinity of potentiality(ies).
After 30 seconds a single-digit counter increments to 1, after a further 15 seconds it increments to 2, after a further 7.5 seconds it increments to 3, and so on, resetting to 0 at every tenth increment.
What digit does the counter show after 60 seconds?
If there is no answer then perhaps it suggests a metaphysically necessary smallest period of time.
That's beautiful, simple and eloquent.
The mathematics is clear. The sum of the infinite series 1/2 + 1/4 + 1/8 ... is 1, not more, not less. Nobody has claimed 'under a minute'.
Quoting Benj96
Well, the counterexamples have shown otherwise. I can subdivide the trip from 0 to 1 the other way around, with the smallest steps coming first, thus showing that it can be physically traversed in either direction.
What's not defined in either case is a last or first step respectively, just like there is no highest integer.
Quoting Metaphysician UndercoverThis is not true. Perhaps you are reading a different account of the story than I did, which is the one on wiki, which says simply:
"In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead". The 2nd bolded part is the non-sequitur, and the first bolded part follows from the 2nd. None of it makes the assertion you claim. The non-sequitur makes the argument invalid. There are ways (such as with the light switch) that make it seem more paradoxical.
Same non-sequitur. It is not true that Icarus always has more steps to take, only that he does while still on a step, but the time to complete all the remaining steps always fits in the time remaining in his minute.
Sort of. I agree It has no basis in physical reality like Zeno's examples do. The OP poetry is only mathematical in nature and isn't meaningfully translated into physics. No amount of physical acceleration can traverse an infinite physical distance in finite coordinate time.
Then it concludes, that after a minute has passed, the end has been reached.[/quote]No. It concludes that all of the steps have been traversed. It does not assert that there is a last one. In this suggestion, the OP at least does not commit the fallacy that Zeno does.
OK, which premise then is false in the Zeno case? The statement is really short. One premise that I see: "the pursuer must first reach the point whence the pursued started", which seems pretty true to me.
No, it is more the reverse of Michael's digit counter, just like Zeno's dichotomy scenario is the Achilles/tortoise thing in reverse.
Quoting Michael
Yes, my example is more on par with Zeno dividing space than the OP dividing time. It has the same problem as Michael's counter: Measuring something where the thing being measured is singular, which makes the whole thing invalid.
Quoting Michael
I'm interested in your take on the nonexistent 'barrier' thing described at the lower half of my prior post in this topic. It also is a variation on something somebody else authored, but I cannot remember what it was originally called.
Side note: Would be awful nice if the site put numbers on the posts.
Quoting flannel jesus
It was unclear if this was addressed to the OP, or to me since this question was asked immediately after I posted the thing about the barriers. Anyway, not mine, but I can't find a link.
Quoting Ludwig V
It is indeed unexplained why the guy, after taking the first step, is somehow compelled to continue his journey after 30 seconds and not just turn around. Mathematically it has some meaning, but it never has physical meaning, as several have pointed out.
Bernadete's Paradox of the Gods:
It's the same principle as Zeno's dichotomy, albeit Zeno uses distance markers rather than barriers. Given that each division must be passed before any subsequent division, and given that there is no first division, the sequence of events cannot start.
I think it's the crux of Zeno's paradox that the mathematics of an infinite series fails to address. The solution, similar to my proposed solution above, is that movement is not infinitely divisible (either because space is discrete or because movement within continuous space is discrete).
I'm not yet comprehending this to my liking. To my current understanding, an infinite series is the very thing which makes something otherwise perfectly continuous discrete. It's, for one example, the difference between a perfect circle and an apeirogon with equal sides: the first is perfectly continuous, the second discrete.
Due to this, I've so far always assumed the resolution to Zeno's paradoxes is that movement is not infinitely divisible precisely because it is perfectly continuous while it occurs. Such that it's our imposed conceptualizations of measurement upon an otherwise immeasurable process which makes Zeno's paradoxes possible.
If movement is continuous then an object in motion passes through every [math]{1\over{n}}m[/math] marker in sequential order, but there is no first [math]{1\over{n}}m[/math] marker, so this is a contradiction.
To the best of my understanding, not within process philosophy.
Ill first try to better explain my own current stance:
It's the the very marker you address that I take to be the conceptual measurement imposed: In process theory, there is no beginning nor any permanent thinghood, only continuous becoming. Like with quantum mechanics, wherein everything is a wave till measured. Whenever we measure, we quantify (and vice versa): one given, quantitative parts of that one given, multiple whole givens, and so forth. But the movement of whatever we quantify remains purely continuous, wave-like in this very limited sense.
We thereby quantify there being one arrow that is being projected. Likewise we quantify there being one target it is going to penetrate. Yet, as per process theory, both are otherwise merely processes of becoming themselves, that are forever in flux in manners devoid of any absolute beginning. We furthermore empirically know (this by imposing measurement/quantity) when the quantified arrow first starts its motion toward the target, we know that if travels through air via certain placements in space, and that it eventually hits the target whereupon the arrow stops its motion. But when we then try to quantify this very (here, by analogy, wave like) process of the arrows motion what we end up with are quanta of space that appear to be infinite in number. These, in turn, then facilitate Zenos paradox of the arrow.
I dont know how to address this properly with the arrow paradox, so Ill use Achillies and the tortoise instead:
Here suppose motion occurring in a finitely divisible (hence quantized) space. For the sake of argument, say this finitely divisible space from point A to point B has only ten divisions. The tortoise is at the fifth division of this space while Achilles is at the second division of this spaceboth moving toward the tenth division of space. How would conceiving of space in such finitely quantized manner change Zenos paradox so as to allow Achilles to catch up to the tortoise?
----
I so far cant apprehend a coherent way demonstrating logically (non-empirically) that Achillies can catch up with the tortoise in such a scenario.
And this because at the very least physical motion (if not also any psychological change) seems to me to be completely continuous in its ontological nature. This again, as per core concepts of process theory. A continuous change which we measure/quantize and thereby impose upon the notion of fixed beginnings and fixed thinghoodwhich, in an ontology of flux, dont in fact occur.
I acknowledge this train of thought deviates from the threads intent. But, to cut things short, I dont yet understand how a finite quantization of space or of motion is interpreted as resolving Zenos paradoxes (as per my example above)this given what we empirically know to be (Achilles can catch up with the tortoise).
Sorry no Axioms, I can't follow what you are saying, perhaps you could spell out your supposed "non-sequitur" in a clear explanation, instead of simply asserting it. In Zeno's paradox, the tortoise is given a head start.
Quoting noAxioms
That Icarus always has more steps to take is the valid conclusion from the premises. Yes, "the time to complete all the remaining steps always fits in the time remaining in his minute", as you say. However, the remaining steps are indefinite. Likewise the amount of divisions which can be made to the remaining time are indefinite. Therefore Icarus' minute is never completed, and he never completes his steps. That is the conclusion we must make.
However, the OP concludes that the minute passes, and the bottom reached. The OP therefore relies on a sense of physical reality, that a minute must pass, which is outside of the premises. That's why it's not a valid conclusion. There is nothing within the premises to indicate that a minute must pass, and everything indicates that a minute will never pass.
Quoting noAxioms
The false premise for Zeno is that each distance, and each time period will always be divisible. That's the problem points out. Think of the way a runner actually runs. One foot hits the ground, then the next foot hits the ground some distance further ahead. The runner does not cover every inch of ground in between, the motion of the feet in contact with the ground, takes place in increments.
There's no paradox in the sense of a contradiction. It just seems weird.
When we analyse it closely, we see that all the set-up does is establish the existence of an infinite descending staircase, and a location of Icarus on the staircase at every point in the time interval [0,60). The curved closing parenthesis there means that that does NOT include time 60 seconds. That is crucial!
So the set-up says nothing about where Icarus is at 60 seconds, or any time after that. We need to add additional assumptions/axioms/specifications in order to say anything about that. Those won't be able to meaningfully say "at the bottom of the staircase", as the above story does, because there's no such place.
If it were up to me to specify, then I'd choose to say that at 60 seconds and later Icarus is back at the top. That's a purely aesthetic choice based on it being the least arbitrary place out of those mentioned, and I like to avoid arbitrariness where possible. But we could just as well assume that at 60 seconds he will be on step 3. Whatever step we assume, it will not generate a contradiction.
Similarly with the Thomson's Lamp case. When we ask "is the lamp on or off at one minute" we are asking for something that the set-up doesn't give us enough information to answer. The setup tells us whether the lamp is on or off at every instant in [0,60) and tells us nothing about whether it is on or off at 60 or later. We cannot infer whether it would be on or off at 60 because we know nothing about the physics of the world in question, which must be enormously different from that of our own, in order to allow complete switching of a finite-sized lamp in infinitesimally small time periods. I expect we could invent some physical rules to support either an on or an off assumption.
Hi andrewk. It's good to see you, been a while. Doing well? I hope.
Quantum Jump - Abstract space (as opposed to physical space) cannot be discrete because any minimum unit you propose can be halved. This is not an acceptable solution to Zeno's Paradox. I agree with you that Zeno's assumptions about motion are flawed, but you haven't offered an alternative premise that holds up. The whole point of his paradox was to highlight that the standard view of motion was flawed. Additionally, it's not definitively established that physical space is discrete. It's possible that only our measurement of space is discrete. This latter perspective is my belief which I'll expand on in a couple of paragraphs.
Zeno Non-sequitur fallacy - I agree that the conclusion doesn't logically follow from the premise, but that's only true if you interpret the paradox from a fresh perspective. From the traditional understanding of motion, the conclusion indeed seems to follow logically. This is precisely Zeno's point.
I wrote the following in a different thread but it's relevant here. Let's recast Zeno's ideas using contemporary terminology. In his era, the dominant philosophical view was presentism, which posits that only the present moment is real, and it unfolds sequentially, moment by moment. Zenos famous parables about Achilles' incremental pursuit are illustrative of (and an attack on) this presentist perspective. However, Zeno himself subscribed to the opposite belief, which we now call eternalism. This philosophy asserts that past, present, and future coexist as a single, unchanging "block universe." From a vantage point outside this block, everything would appear static; thus, in this comprehensive perspective, motion is impossible. One could argue that in his perspective, the only movement is in the gaze of God, and wherever God looks becomes the present (I use God here not to push a religious view, but for simplicity). The discreteness that are looking for is not in space but in measurement/observation. In other words, God's fundamentally cannot watch everything. This actually should come as no surprise since the Quantum Zeno Effect demonstrates that an observed system cannot evolve.
Zeno was remarkably prescient. The concept of eternalism and the block universe gained serious traction only after Einstein introduced theories that showed eternalism to be more consistent with the principles of relativity. Yet, the narrative is still unfolding, as the singularities in classical black holes demonstrated that relativity is not the ultimate explanation of physical reality. Enter QM and the importance of observation/measurement.
STAIRCASE PARADOX
A minute cannot pass - This scenario involves an infinitely large object (the staircase), an infinitely complex task (traversing the entire staircase), and the passage of one minute. You're suggesting that the issue lies in the impossibility of a minute passing? It seems you may have labeled the most logical and uncontroversial element in the paradox as illogical. If you think the problem has to do with Icarus's steps then frame your solution in that context.
No end to the staircase but the end is reached - Yes, this is the very issue I'm trying to highlight. And this has nothing to do with continuous acceleration or motion. Could it be that supertasks are impossible?
restricting ourselves to the physical world - The physical world is not the only realm that exists; there's also the abstract world, which operates under its own set of rules. For instance, in an abstract world I can define, it's perfectly valid to set the speed of light at 100 m/s. This isn't incorrectit's simply a different premise. However, I do believe in a kind of symmetry where truths in the physical world often find parallels in the abstract world.
This is a paradox I've come up with myself. But as Michael has mentioned it's very similar to Thomson's lamp. Where do you see problems with it?
Focus first step up, not last step down- Unfortunately, the stairs are numbered in ascending order from the top down, so the first step up wouldn't be numbered 1.
Non-standard numbers-I'm certain you're a strong mathematician, but I also feel like you're overcomplicating things. This reminds me of an Einstein quote: If you can't explain it to a six year old, you don't understand it yourself.
,
Only a potential infinity-My purpose in presenting this paradox is to underscore the problems associated with the concept of actual infinity.
Thomson's Lamp-Indeed, the Staircase Paradox shares significant similarities with Thomson's Lamp Paradox, particularly in that both scenarios lead to states considered invalid by conventional logic after one minute has elapsed. In the Staircase Paradox, we are left unsatisfied by claims that the staircase either exists or does not exist. Similarly, in Thomson's Lamp Paradox, we find it unsatisfactory to definitively say whether the lamp is on or off. The difference is that, supertasks aside, Thomson's Lamp is a critique of infinite series whereas the Staircase Paradox is a critique of N.
Trip from 0 to 1-I don't get it.
[0,60)-Your point is valid, for brevity I didn't explicitly state that the first instant he passes the stairs he arrives on the ground. However, as the poem indicates, my view is that at that instant, he actually arrives at a singularity, similar to what one might encounter at the center of a classical black hole.
PARADOX OF THE GODS
As Michael noted, your barrier paradox is Bernadete's Paradox of the Gods. I find this paradox intriguing. In the realm of physics, I think quantum tunneling offers a solution to this issue.
Nice exposition of Zeno!
Quoting keystone
[...]
Quoting keystone
Yes, Im in agreement with you as to the non-discreteness of space.
Fair enough re: potential vs actual infinity. Will continue reading the thread hoping to learn. smile
Still wrestling with how there can be a last step or a first step in infinite steps probably because it appears arbitrary to impose mathematical limits to the concept of "infinity". Can understand the "human" need to do so, hence limiting infinite steps to a "mere" 'series of'.
The staircase problem is called an omega-sequence paradox, a paradox that involves counting 1, 2, 3, ... and doing something at each step, then expecting the behavior to be defined in the limit. The answer to all those paradoxes is that you haven't defined what happens at the limit. You've told me what Thompson's lamp does at every finite [math]n \in \mathbb N[/math], but you have not told me what it does at the limit. Therefore the lamp could be on, it could be off, or it could have turned into the Mormon Tabernacle Choir. You haven't specified the behavior at the limit, so it can be anything you like.
There's a mathematical name for the upward limit of the natural numbers. It's [math]\omega[/math], lower-case Greek omega. You can think of it as a formal symbol that is greater than every other natural number, but that does appear in the sequence, as follows:
[math]0, 1, 2, 3, 4, \dots, \omega[/math].
You can think of [math]\omega[/math] as a "point at infinity." Or from a formalist view, it doesn't mean anything. It's just a symbol that satisfies [math]n < \omega[/math] for any natural number [math]n[/math], as well as all the usual meanings for 47 < googolplex and so forth.
This is a handy formalism. Now we can solve Thompson's lamp. The problem is that the state of the lamp is not defined at [math]\omega[/math]. In other words you told me what the state is at every natural number, but not at [math]\omega[/math]. That literally solves the paradox. It's no different than one of those "complete the sequence" questions. Mathematically, the next number can be anything you like.
In other words: You told me the state of the lamp at every finite number. You did not tell me the state at [math]\omega[/math]. All confusion about Thompson's lamp is to realize that you just haven't told us the state at [math]\omega[/math]. And there's no good reason to prefer one answer over the other.
The staircase paradox is a little more interesting, in the sense that you are present at each step 1, 2, 3, ... As before you can still define the behavior at [math]\omega[/math] to be either that you are there, or you aren't. But in this case, assuming you are there at [math]\omega[/math] is more natural, in the sense that the function that maps [math]\omega[/math] to "there or not there," is continuous if you're there.
What I mean is, at each successive step, the state of that step is "you are on it." Now the state at [math]\omega[/math] is undefined, but there is a natural way to define it; that is, to assume that your motion is continuous in some sense. So if you are there at every step, you are there at the bottom, the state of step number [math]\omega[/math].
So if you believe that your motion down the stairs is "continuous," however you define that in this context, then since continuity preserves limits, you are there at the bottom. But if you can't justify the assumption of continuity, then anything at all might happen at the bottom. You're there, you're not there, you're a sea slug at the bottom of the ocean. Since you haven't specified the value of the function at [math]\omega[/math], it can be anything you like.
One more note, and that is that you can indeed count backwards from [math]\omega[/math]. But as you can see, any step that you take backwards necessarily jumps over all but finitely many numbers; so that it's always a finite number of steps back to zero, even from infinity.
Therefore if you are at the bottom of the stairs, you can just take a tiny tiny step up -- as small as you like -- you will necessarily skip over all but finitely many stair steps, and end up on some natural number like 47 or googolplex. Either way, it's still only finitely many more steps back to the top.
I should say that again, since this comes up so often. Even if you start at the point at infinity, it's always at most a finite number of steps back to zero.
This principle also applies to people making cosmological arguments about the impossibility of an infinite past, because it would take an infinite time to get to the present. Actually that's not true. If you put a point at negative infinity, it's only a finite number of steps to the present.
Finally, I'll mention that [math]\omega[/math] as I've used it is just a formal symbol; but in fact can be formally defined and shown to exist within set theory as the first transfinite ordinal number. And once you do that, you can keep on going into the wondrous and mysterious world of the ordinal numbers.
I'll leave you all with just one thought:
It's always only a finite number of steps from infinity back to zero, no matter how small a step you take. That's something a lot of people get wrong.
Is minus one a natural number? And, is zero a natural number? Mathematicians' mathematics is not a strong suit for some. sad smile at one's own ignorance
Those are great questions.
Quoting kazan
No. The natural numbers are the positive (or nonnegative, we'll talk about that in a moment) whole numbers like {0?], 1, 2, 3, 4, 5, ...
They're infinite (or endless if you prefer) in one direction.
The entire set of positive, 0, and negative whole numbers is called the integers.
The integers are infinite (or endless) in two directions:
..., -3, -2, -1, 0, 1, 2, 3, ...
The natural numbers are very basic and important, since on the one hand, they seem to be intuitive to every child -- you "just keep adding 1." On the other hand, they go on forever, giving us all a glimpse of infinity.
The integers, however, are much better for doing arithmetic. That's because the integers have "additive inverses." Given the number 5, in the natural numbers there is no number you can add to it to get 0.
But in the integers, there is: namely, -5. We say that the integers have additive inverses. And believe it or not, that happens to be really important in higher math.
So that's a long answer to a simple question but the bottom line is that the integers include the negative whole numbers, and the natural numbers don't.
Quoting kazan
For some reason this question attracts a lot of controversy. People like to argue about it.
But I am here to tell you that it doesn't matter. Why is this?
Well, what's important about the natural numbers is their order. You have a linear, discrete procession of things. There is a first thing, then a next thing, then a next thing, and so on, forever.
If you call the first thing 0 or you call the first thing 1, it doesn't make any difference at all to the order structure of the sequence of "first, next, next, next, ..."
Suppose your friend thinks 0 is a natural number and you believe the natural numbers start with 1.
Then you just invent a new numbering system in which 1 means 0, 2 means 1, 3 means 2, and so forth.
You can tell your friend that you are starting with 0, but you just call it 1. And what they call 1, you call 2.
So you are right, and they are right. Why? Because the only interesting thing about the natural numbers is that (1) there's a first one; and (2) there's always a next one.
Those two rules generate the natural numbers, no matter what you call them!!
I hope this is taken to heart by someone. The answer to whether 0 is a natural number or not is that it absolutely doesn't matter. Just tell people what convention you're using and they'll be fine with it.
Now you might ask, what about arithmetic? 0 + anything = anything, but if you don't include 0 you don't have a number with that property.
And that's what I mentioned earlier. If you want to do arithmetic, the natural numbers are lousy anyway, they don't have additive inverses. So if you're doing arithmetic, you'll be working in the integers, not the naturals.
That's why it doesn't matter if 0 is a natural number. If you only care about order and "nextness," it doesn't matter what you call the first element.
And if you want to do arithmetic, you'll be using the integers anyway, which include zero.
Hope this was helpful. And again, these were great questions. Surprisingly deep. Thinking about various number systems and their properties like order or additive inverses is the basis of higher math.
Yes,very helpful.Thank you for taking the time.
Which begs the question, (smile) how, if it's possible, would "the lower case omega" concept of "upper (lower?) limit" be applied to all integers? Surely, if it's possible,this could be useful in some areas of mathematics ( besides arithmetic ).
Sorry,don't mean to hijack this thread.
Another good question. Yes we could put "negative omega" at the leftward infinite end of the integers. There's not much use for it. The interesting aspect of omega is to keep going with the "add 1" game to get a whole infinite structure of higher ordinals continuing "to the right," in the positive direction. There's nothing new of interest that happens if you do the same thing on the left, it would just be a mirror image of the ordinal structure on the right.
I don't see a reason to think that a person will reach the bottom of the infinite staircase, ever. You described it as endless, and yet claim he reached the end... The "paradox" is just you choosing to invent a story with contradictory concepts.
"There was an old woman. She was only 2 years old, and really just a baby." There's my paradox.
Fair enough.
Nothing new of interest, comes to mind. Apart from adding negative l c omega with (to?) (positive) l c omega and getting the same answer as subtracting them,(still in the realms of arithmetic,) presumably zero?
There that pesky zero again. Thanks again.
As far as I understand about paradoxes, that's precisely what a paradox is about. It is a self-contradictory statement, but arrest our attention. The aim of this thread (or purpose of @keystone) is not to reach a conclusion, but to result in persistent contradiction between interdependent elements: The staircase being endless and reaching the bottom of it in just a minute.
It is clearly a paradox.
To explain this more deeply, @Michael and @noAxioms wrote very interesting posts using maths and logic.
Never in memory, has "pure" mathematics been of such interest as now. Feel like you've open a window and there's a gale blowing in, here.
Had more questions about l c omega, but will give them further thought first. And catch up with you elsewhere and later. Lounge perhaps in a few days?
No paradox with the old lady.
Just different periods of the past in her life.
Won't a better paradox be if set in the present tense.
Just a suggestion. smile
Better not to try to subtract the endpoint infinities from each other, The result is undefined.
Quoting kazan
Thanks much.
Quoting kazan
I hear all the cool kids hang out in the lounge these days.
Yes, but usually where the contradiction occurs exactly is supposed to be *non-obvious*. "He went down some endless stairs, and reached the end". It's not non-obvious where the contradiction is. It's immediately obvious.
Because it's obvious, it's not so much a paradox as it is just a plain contradiction.
`despite apparently valid reasoning from true or apparently true premises` - that's key! The premises and steps in reasoning have to make some kind of sense.
The premise that you can reach the end of an endless staircase doesn't apparently look like valid reasoning or true premises to me.
I suggested that movement was discrete, not that space was discrete. In other words, at a sufficiently small scale, when an object (esp. particle) moves from A to B it does so without passing any half-way point. Your use of the phrase "quantum jump" is fitting.
I don't think the physics is relevant. The question can be asked of any universe with any physical laws. The thought experiment is entirely metaphysical.
Repeating my specific example:
After 30 seconds a single-digit counter increments to 1, after a further 15 seconds it increments to 2, after a further 7.5 seconds it increments to 3, and so on, resetting to 0 at every tenth increment.
What digit does the counter show after 60 seconds?
The issue we have is that if there is no smallest unit of time then the counter is metaphysically possible, but this entails a paradox as the answer to what the counter shows after 60 seconds is undefined yet the counter will show something after 60 seconds. Assuming that paradoxes are metaphysically impossible then the counter is metaphysically impossible, and that suggests that it's metaphysically impossible for time to be infinitely divisible.
We could replace the counter with some supernatural deity capable of keeping such a count if it makes things easier to consider (similar in kind to Benardete's Paradox of the Gods).
I agree! There has to be at least some kind of sense on the premises. Yet, there are, among these, a large variety of paradoxes of a logical nature. A basic pattern of a paradox is having a way of reasoning. Right?
Well, following the paradox within this OP, we can conclude there is a bit of reasoning. For example: @fishfry used the reason pretty well in this comment: https://thephilosophyforum.com/discussion/comment/898761
He even states:
Sadly, I am not good enough at maths and logic, so I can't post valid or interesting comments regarding this paradox. What I try to defend is that what @keystone wrote is actually a paradox. Maybe it has its flaws, or he was inspired by other paradoxes which were quoted in the comments above. But it there is still a paradox.
I think this is a misrepresentation. The paradox is that given the premise(s) what happens at the limit is undefined, and yet something must happen at the limit. This is a contradiction, therefore one or more of the premises must be false.
And note that this only considers progressive interpretations of these paradoxes (i.e. how they can complete). Regressive interpretations (i.e. how they can start) must also be considered. I don't think mathematical limits are relevant to these at all.
That comment doesn't justify why the person should reach the end of the stair case.
I think that if you're not good at maths and logic, I would think that you might not be in a good position to know if this is a valid paradox or just straightforward nonsense. If it is a paradox at all, it would only be in a mathematical sense. Surely not EVERYTHING contradictory is a paradox.
Okay. Let's leave it at that. Fair enough. I will not continue with posting on this thread.
I'm flattered that you replied. It was just a bit of fun and I expected to be totally ignored or possibly reprimanded. But since you have replied.....
Quoting keystone
That's a complicated remark, because the numbers assigned are assigned in a specific context. If the staircase existed in the way that a physical staircase exists, the steps can easily be re-numbered in the new context (wanting to go up, rather than down). In that context, the first step up is numbered, even though it would not be numbered 1 in the context of going down. I think I recognized the problem when I said:-
Quoting Ludwig V
My conclusion in the light of what you say is that the staircase up is not the same as the staircase down.
What I was wrong about was how the staircase is "created". The person going down does not create the staircase. We (the readers) (or, if you prefer, you, the writer) create the staircase. But if a staircase down can be created by our, or your, say-so, another one, going up, can be created in the same way. So my advice should not be given to the person going down but to you and your readers.
I wouldn't say that. If you read Aristotle's Physics, you'll see that he describes the principal definition for "time" being used at his time, as a sort of number, used for measurement. That, he distinguished from a secondary sense, as something measured. So the secondary sense might be consistent with what you say, but it's not the primary definition for "time" at that time.
Quoting keystone
But the end is not reached. According to what is stipulated Icarus always has more steps before a minutes passes. Here's what I wrote in the other thread:
"For Icarus a minute cannot pass because he always has steps to cover first, just like Achilles cannot pass the tortoise for the same reason. Maybe if we call the staircase a line, and the steps are "points" it would make more sense to you. No matter where Icarus stands on the prescribed line, he has to cover an infinite number of points before a minute can pass. And to traverse each point requires a non-zero amount of time. Therefore no matter where Icarus is on the line (stairs), there will always be time left before a minute passes. A minute cannot pass, and Icarus' journey cannot end."
You only claim that the end is reached, because you assume that a minute passes. But that assumption is not provided for, as explained. Therefore, the valid conclusion is that Icarus never reaches the bottom, just like Zeno concluded that Achilles never surpasses the tortoise. The required premises about space and time, to conclude otherwise, are not provided.
Quoting Michael
This is a very good point, to keep in mind when we get a good understanding of, and move beyond, the appearance of paradoxes like Zeno's. What is indicated is that motion is discrete. To account for the reality of this, we need to model either space or time as discrete, but not necessarily both. So when we employ the concept of "space-time" we deny ourselves the capacity of separating time from space, and considering the possibility that one is continuous, and the other is discrete.
To me, what works best for understanding the nature of reality is to allow for a continuous time, with a discrete space. This means that the so-called "quantum jump" is a feature of space. As time passes, the things which we know as having spatial (material) existence change in discrete jumps. Those discrete jumps limit our empirical capacity to understand the true nature of time, because no change can be observed to occur during the time which passes in between such jumps. However, since time would be conceived as continuous, while spatial motion is discrete, we can still conceive of time as passing in between such jumps.
Those premises allow us to understand the "immaterial realm of Forms", as the activity which is occurring between the spatial jumps. The immaterial realm of Forms is what determines how the spatial world will be, at each moment of ("observable") time. The human being as a free willing agent, has causal power within the immaterial realm of Forms, to influence what will be spatially present at any moment as time passes. Notice the requirement of real points in time. These are the points when the observable (spatial) world is materialized as discrete "jumps" during the passage of time.
Quoting Michael
I don't think this is correct, I think Andrewk is correct. The counter is not programed to reach 60 seconds, that is outside its described capacity. Here's what you said:
Quoting Michael
.
See, the counter is not programmed to show anything after 60 seconds. "There is no answer", because if the counter were to do as prescribed, it would never get to 60 seconds. There is no such things as 60 seconds for the counter, it can never get there if it does what its supposed to do. The assumption that 60 seconds will pass, is the mistaken conclusion of the OP, because this requires a premise about time which is not provided in the example. That premise being that time will pass at some rate which will surpass the actor in the example. That's what creates the appearance of paradox, if you allow yourself to be influenced by the contradictory idea, that time will surpass the actor.
Quoting Michael
This is the mistaken assumption. There is no limit prescribed. The premises set up an infinite process, which means "no limit". The issue exposed now, is that training in mathematics (calculus specifically) inclines one to see the scenario as a limit at 60 seconds, when the example simply does not state it that way. Someone who knew no complicated mathematics, only simple arithmetic, would work through the prescribed process, adding up the periods of time, and then realize that the sum gets closer and closer to 60 seconds, without at all thinking that 60 seconds is "a limit" here.
This imaginary "limit" is added by the mathematical way of looking at the scenario. It is "the intent" in producing the example. Start with the limit, and set something up which approaches the limit but does not reach it. The limit though, is not part of the example, it was only employed by the mind which produced the example, as a guiding principle which does not enter into, or become part of the example. In other words, we need to read the premises exactly as they are written, and there is no mention of 60 seconds as a limit. It's only when you take the short cut, don't read, but jump to the end, the intent, that you think of 60 seconds as a limit.
60 seconds will pass in the universe. The counter is just one thing that exists in the universe and it changes according to the prescribed rules.
So given the prescribed rules, when the universe is 60 seconds older, what digit will the counter show?
Yes, that is the point. Your expressed conceptualization "60 seconds will pass in the universe" is not consistent with the conceptualization prescribed by the OP. But this conceptualization "60 seconds will pass in the universe" is not part of the example. So your introduction of it is not valid. We could call it an equivocation fallacy. We have "time passing" as prescribed in the example, and "time passing" in the universe. The two are not consistent. To introduce the latter into the example, is to equivocate.
The obvious point is that we can describe a scenario which is logically possible, but physically impossible. When working with this scenario, we need to bare in mind, the fact that it is physically impossible, and adhere rigidly to the logic of the scenario, only. If one's mind gets influenced by other principles, such as what is physically possible in the universe, and what is going on in the physical universe, this will surely create confusion.
The example is simply: after 30 seconds a single-digit counter increments to 1, after a further 15 seconds it increments to 2, after a further 7.5 seconds it increments to 3, and so on for 60 seconds, resetting to 0 at every tenth increment.
What digit does the counter show after 60 seconds?
Your suggestion that the above entails that 60 seconds won't pass makes no sense.
The sum is [math]\frac{30}{1-0.5}[/math]. The counter is the number of members of the GP, to reach 60 seconds the counter must go to infinity. But in reality the counter, no matter how powerful, will just break the closer we approach the 60s mark.
As t?60, probability of the counter breaking goes up.
The paradox does not require the physical possibility of such a counter. It simply asks us to consider the outcome if we assume the metaphysical possibility of the counter. If the outcome is paradoxical then the counter is metaphysically impossible, and so we must ask which of the premises is necessarily false. I would suggest that the premise that is necessarily false is that time is infinitely divisible.
It is metaphysically necessary that there is a limit to how fast something can change (even for some proposed deity that is capable of counting at superhuman speeds).
It seems far more paradoxical than Zeno's thing since motion is prevented despite the lack of any actual barrier.
But I've been arguing that the above reasoning is fallacious. Yes, each division must be passed, and each division is preceded by other divisions (infinitely many), and yes, from that it can be shown that there is no first division. All that is true even in a physical journey (at least if distance is continuous).
But it doesn't follow that the journey thus cannot start, since clearly it can. By such a method, one can count from negative infinity to zero. You just need to not take some minimum time to do a given count.
Mathematically it is, and mathematics seems to have no problem with it. Yes, I believe certain axioms must be accepted, but I'm no expert there.
As for physics, the assertion that motion is infinitely divisible seems to be a counterfactual assertion, not necessarily false, but unjustifiable. Such is the nature of quantum mechanics. But a journey can begin in either case, whether or not motion is continuous. Zeno does not illustrate otherwise in my assessment.
Zeno did his thing well before QM made us all question our classical notions of motion, so we can for the sake of argument make classical assumptions for this topic. If there's a limit to divisibility, then the problem goes away since there are finite steps.
Quoting MichaelI don't find that to be a contradiction.
Quoting Metaphysician Undercover
OK, if you deny the continuous nature of both space and time, then the number of iterations is finite, and the argument falls apart. My arguments presume a more mathematical interpretation: the continuous nature of both. If space is discreet, Achilles passes the tortoise after finite iterations. There would be a last one, after which the tortoise is passed. The conclusion of the inability to overtake doesn't follow because the premise upon which it is based becomes false.
Your assumption of discrete space is interesting, given that space (and everything else) is abstract to you, and thus any abstract space can be halved.
Quoting keystone
Presentism is still presentism even if time is continuous. You seem to describe a discreet view there, which runs into problems.
I don't see how Zeno's paradoxes work any differently under presentism than under eternalism. Eternalism doesn't resolve the problems with any of them.
I was unaware of Zeno's 'eternalist' leaning. Yes, the term didn't exist back then (not until perhaps the 11th century)
Quoting keystone
Block view also defines motion as change in position over time, and thus motion is very much meaningful under the view.
Quoting keystone
All these are trips from beginning to end. Zeno's initial state (0) to the point where the tortoise is passed (1). In your OP, 0 is time zero, and 1 is time 1-minute.
Quoting Metaphysician Undercover
This seems to contradict yourlelf. You say time is discreet, in which case the number of digit changes is finite, and there is an answer. You also seem to deny that the sum of the converging series is not 1, or that time somehow is obligated to stop, which is the same thing.
Michael: The output of the counter is undefined. I can think of no better answer than that.
It does follow that the journey cannot start. Therefore given that the journey can start then the premise that there is no first division is false. It's a proof by contradiction.
As such there is some first division and so movement is discrete.
Given that each division is some [math]{1\over{n}}m[/math] then such a movement is akin to counting all the real numbers from 0 to 1 in ascending order. Such a count cannot start because there is no first real number to count after 0.
Re : The Cauchy Limit of a Sequence
"When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the limit of all the others"
A converging sequence might eventually settle on value equal to its limit, but even then the two concepts are not the same. So it doesn't matter whether we are talking about Thompson's Lamp, or merely a constant sequence of 1s. In either case, a limit, if it exists, doesn't refer to any position on the sequence, rather it refers to a winning strategy in a type of two-player game that is played upon the "board" of the converging infinite sequence concerned.
So it make no literal sense to consider the value of an unfinishable sequence at a point of infinity, so the meaning of a "point at infinity" with respect to such a sequence can at best be interpreted to mean an arbitrary position on the sequence that isn't within a computable finite distance from the first position. In the newspeak of Non Standard Analysis, such a position can be denoted by a non-standard hyper-natural number, meaning an ordinary natural number, but which due to finite limitations of time and space cannot be located on the standard natural number line.
As for the OP, its triad of premises are inconsistent. For only two of the three following premises can be true of a sequence
i) The length of the sequence is infinite.
ii) The sequence is countable
iii) The sequence is exhaustible
For example, Thompson's proposed solution to his Lamp paradox is to accept (i) and (ii) but to reject (iii). Whereas solutions to Zeno's Paradox tend to start by accepting (iii) but reject the assumption that motion can be analysed in terms of a countably dense linear order of positions, either by denying (i) (namely the assumption that the sequence of positions is infinite, which amounts to a denial of motion) or by denying (ii) (namely the assumption that motion can be used to count positions, for example because the motion and position of an arrow aren't simultaneously compatible attributes).
I didn't think he proposed a solution. Rather, it was an example to show that supertasks are impossible.
Yes, in other words rejecting iii), namely the idea that one can finish counting an infinite sequence.
True, but that's only part of the issue.
If after 30 seconds he's flipped the switch once and if after a further 15 seconds he's flipped the switch a second time and if after a further 7.5 seconds he's flipped the switch a third time, and so on, then it would suggest that a supertask can be completed in 60 seconds.
So if a supertask can't been completed in 60 seconds then the time between each flip cannot continually decrease. At some point no further division is metaphysically possible.
There is a contradiction in the stated scenario: there's an END to the ENDLESS staircase. Better to ask where he is after a minute.
Assess progress after each step he takes by noting the number of steps yet to be taken: there are always infinitely more to take. So at no point does he actually make progress - even after traversing infinitely many steps because that relation holds at all points along the way.
No misrepresentation. And why must something happen at the limit? Take this mathematical example.
We work in the space (0,1), the open unit interval of real numbers. It excludes the endpoints.
We consider the sequence 1/2, 3/4, 7/8, 15/16, 31/32, ...
Clearly this sequence has the limit 1 ... except that 1 is not in our space. So this sequence has no limit. Such a sequence is called a Cauchy sequence. It's a sequence that should "morally" converge, whether its limit happens to be in the set of interest or not.
Say the Thompson lamp is turned on at 1/2, off at 3/4, on at 7/8, and so forth.
Why on earth must there be a behavior defined at the limit? In this case there is no limit because 1 is not in our set.
But now do the same thing, but in the closed unit interval [0,1], which does include its endpoints.
In this case the limit, 1, is defined and exists. But still, the behavior of the lamp is not defined at 1.
That's the point. There's no paradox. You've simply neglected to tell me what the lamp does at 1, and you're pretending this is a mystery. It's not a mystery. You simply didn't defined the lamp's state at 1.
Does this example better explain that the the "paradox" is simply that you're arguing over what is the state of the lamp, when the state of the lamp is undefined?
How about if we defined the state of the lamp as turning into a swordfish at 1. Then that's the answer. It's on, off, on, off, ... at each point of the sequence, and a swordfish in the limit. There is no contradiction and no mystery.
Also note that the sequence 1/2, 3/4, 7/8, ... is order-isomorphic to the sequence 1, 2, 3, 4, ...
And if we include the limit, then 1/2, 3/4, 7/8, ..., 1 is order-isomorphic to 1, 2, 3, 4, ... [math]\omega[/math] as I described earlier. From the standpoint of order theory, they have the same order.
People who have trouble imagining that we could reach a limit after counting the natural numbers, would have no trouble agreeing that 1 is the limit of 1/2, 3/4, 7/8, ... But those two situations are identical with respect to their order properties. Now we never "reach" a limit, which is another phrasing that confuses people. We don't reach the limit, but the limit exists.
By the law of excluded middle and non-contradiction, after 60 seconds the lamp must be either on or off.
Quoting fishfry
We're being asked what the lamp "does at 1", so you saying that we must be told what the lamp "does at 1" makes no sense.
Given the defined behaviour of the lamp, will the lamp be on or off after 60 seconds? If the answer is undefined, but if the lamp must be either on or off, then the behaviour is metaphysically impossible.
The paradox is resolved by recognising that the premise is flawed.
I guess I misunderstood your example. It is obviously not consistent with the OP. That little part where you say "and so on for 60 seconds" is unclear. The OP lays out the conditions in steps, but your "and so on" tells me very little.
The problem set-up, which gives the axioms of the system we are working with, does not provide enough information to decide.
Hence the statement, for any numeral n, that:
"The counter shows digit n at 60 seconds"
has the same status in this logical system as the Continuum Hypothesis has in ZFC set theory. That is, we can adopt the statement as an axiom, and the system remains consistent (ie no contradictions arise). And we can assert its opposite ("the numeral showing at 60 seconds is not n") and the system remains consistent.
I think where people tie themselves in knots on this is that they feel they should be able to use things we know about our world to reason their way to an answer. But they can't, because we threw away that possibility when we postulated the existence of a finite-sized mechanism that can switch state in an infinitesimally small time, which contradicts the laws of our world. So, having thrown away everything we know about our world, all we have available to use for our deductions are the axioms given in the set-up, and those axioms are consistent with the answer being any one of 0, 1, 2, ... or 9.
So I assert the answer is 2, without proof (because proof is impossible), with complete confidence that nobody can prove the assertion wrong.
I swear you changed this. You had something that logically followed from your assertion. The conclusion that movement is discreet contradicts Zeno's premise that "That which is in locomotion must arrive at the half-way stage before it arrives at the goal". So by contradiction, the journey not being able to start doesn't follow from the premise.
If movement is discreet, there is a finite number of steps. The first (smallest possible) step cannot be divided, and the inability to do so violates Zeno's premise.
As for Bernadete's Gods, there would be a first God and there would actually be a barrier preventing motion. No paradox at least. That is decent evidence that Zeno's premise is false. But suppose space and time is continuous. Then the journey can start without contradiction, unless you can find one. You say above that it implies a first division, but nobody has suggested that such a journey must begin with a first segment. It only needs to take finite time (1 unit in this case). The simple example is me going from here to there, which you apparently assert is impossible if space/time is continuous. A bold assertion.
Quoting MichaelNo, the reals are not countable. The example we've been using is. There is no final count of steps in Zeno's dichotomy, so there is no demonstrated requirement of a 'first step' or any kind of final count of steps. Insistence otherwise seems to be leading to contradictions.
Quoting sime
Applying this to Zeno's cases, or to the OP: All three seem to be true. I disagree that only two can be.
OK, the OP has infinite length to deal with, but finite time to do it, which is just a different way of expressing the same mathematics, totally discarding physics.
Zeno doesn't violate physics. If space/time is continuous, then the number of steps is countably infinite, and it is exhausted in finite time, as illustrated by my ability to move and/or to overtake something slower. Zeno makes no mention of 'point at infinity'. The OP kind of does ('bottom of it'), but also doesn't since there's no 'bottom step' apparent from the post-1-minute state. The poetry obfuscates what's actually going on, so I mostly am in denial of Zeno's conclusions.
If I overtake the tortoise, I have also reached the 'bottom of the supertask', so I don't find the wording necessarily contradictory.
What am I missing? A formal proof that it leads to contradiction would be nice, but all I seem to get is assertions.
The lamp violates the laws of physics, so it's not a real lamp. It's only a metaphor for a mathematical puzzle. Why can't it turn into a pumpkin at midnight, like Cinderella's coach? What rule of the puzzle constrains a light, which can not physically exist, to be on or off at a time when its state is not defined?
Consider this mathematical variant.
In [0,1], the closed unit interval, we start with the sequence 1/2, 3/4, 7/8, ...
We have a function f on the sequence such that f(1/2) = 1; f(3/4) = 0, f(7/8) = 1, and so forth, alternating between 0 and 1.
What is the value of the function at 1? Well clearly, that value is not defined. It could be 0, 1, 47, or the Mormon Tabernacle Choir.
Until you tell me what is f(1), you're playing a silly game to ask me what it should be. It can be anything you like.
Quoting Michael
It makes perfect sense, once you replace the lamp with a function on a sequence. The lamp is a red herring. No circuit could switch that fast. It's not a real lamp and there is no reason for it to be in any particular state where that state has not been specified. It's not a real lamp so it is not limited to be on or off.
Quoting Michael
Is f(1) = 1 or 0 or 47?
Quoting Michael
It's not a real lamp.
Quoting Michael
The premises of the mathematical version are perfectly sensible, as is the answer I gave.
Quoting fishfry
This brings to mind Sagan's quote "extraordinary claims require extraordinary evidence." We start with an extraordinary premisethe existence of infinite stairs and supertasksand to resolve it, we resort to an equally extraordinary solution: he has infinitely long legs, enabling him to ascend to the top in just one stride. This doesn't strike me as a satisfactory resolution.
Quoting flannel jesus
What you seem to overlook is that I'm beginning with a premise widely accepted within the mathematical community: the existence of actually infinite objects (like these infinite stairs or the set, N) and the completion of actually infinite operations (such as traversing the stairs or calculating the sum of an infinite series). If you do not accept the concepts of infinite sets or supertasks, then this paradox is not aimed at you. If you claim that an old woman is 2 years old, then you're not basing your argument on any widely accepted concepts of age.
Quoting Ludwig V
If there is a parallel staircase where the steps start at 1 and increase as you go up, then there must be a point where the step numbers on both staircases align. What would that step number be?
Quoting Metaphysician Undercover
Then your argument should be that supertasks are impossible, not that 60 seconds cannot elapse.
ZENO'S PARADOX
Quoting Michael
Consider linear motion. If you plot position against time, are you suggesting that the resulting curve, when examined closely, appears stairstepped rather than smooth? If that's the case, what would be the width of these incremental steps? This presents the same issue, as I could always plot a more accurate curve of motion using even smaller incremental steps.
Quoting Metaphysician Undercover
This response does not adequately address my reinterpretation of Zeno's ideas.
Quoting noAxioms
Zeno contends that change is impossible, leading to stark implications depending on one's philosophical stance on time. Under presentism, this translates to an unchanging, static presentlife as nothing more than a photograph. In contrast, the eternalist perspective views this as a static block universe, a continuous timeline that encompasses past, present, and futureakin to a film strip. Which view do you think is more reasonable? Of course, this raises profound questions, such as why we experience time's flow, but that discussion is for another thread.
Let me reframe Zeno's argument in different terms more relatable to a modern audience. Consider whether it is easier to draw a one-dimensional line by assembling zero-dimensional points consecutively or to cut a string (akin to dividing a line into segments). Zeno would argue that the first option is impossible: a timeline cannot be constructed from mere points in time. Instead, modern Zeno would suggest that the entire timeline already exists as a block universe, and our experience is merely about observing different parts of it, similar to making cuts in a string. However, there's a twist: abstract strings, like time, are infinitely divisible. No matter how many cuts we make (one after another), we never actually reduce the string to mere points. Each cut still leaves a segment of string, however minuscule. This introduces new challenges (for which there are answers) but as it relates to the discussion at hand, the eternalist perspective reframes the impossibility of supertasks from an unacceptable notionthat motion itself is impossibleto a more acceptable onethat observing every instant in history is impossible. This essentially echoes Aristotle's proposal, but it is only in the quantum era that such a solution becomes truly acceptable.
Quoting sime
The issue arises if Achilles toggles Thomson's Lamp with each stride, leading to a contradiction: his feet suggest that the sequence is exhaustible, but his hand indicates it is not.
SINGLE DIGIT COUNTER PARADOX
Quoting Michael
First, instead of using decimal, let's switch to binary, where the counter can only be 0 or 1. You suggest that quantum mechanics resolves this by introducing indivisible units, perhaps akin to Planck time. Looking to QM for inspiration is a good idea. However, the idea of Planck time doesn't hold up because in the abstract realm, we can always conceptualize a smaller increment. I propose that the correct solution is that at 60 seconds, the counter is in an unobserved state where its status fundamentally remains unknown. It could be either 0 or 1, so let's say it's in a state of (0 or 1). If we wish to steal technical terms from QM, we might refer to this state as being in superposition.
Quoting Michael
What if the undefined state is fundamentally unobservable? This raises the question similar to "If a tree falls in a forest and no one is around to hear it, does it make a sound?" The limitations I'm suggesting on observation should not be surprising to a generation that has grown up in the era of quantum mechanics.
Quoting Michael
Yet, it's impossible to determine what this limit might be. Would you argue that there is a limit to the slope of a line?
THOMSON'S LAMP
Quoting fishfry
Suppose that with each flick of the lamp, the lampholder adds another term to a cumulative total: first 1/2, then 1/4, then 1/8, and so forth. What does his calculator show at 60 seconds? Why on earth must we assert that it displays 1? After all, the narrative doesn't specify what his calculator must indicate at 60 seconds. It seems to me that you're contesting the very idea which you support - that infinite series can have definitive sums.
Quoting Michael
Yeah, that law needs updated. I propose "for every proposition, either this proposition or its negation can be measured to be true." This introduces the possibility of a third, unmeasured statewhen we're not observing, the lamp could either be on or off, placing it in a state of being (on or off).
Quoting andrewk
You are effectively arguing that supertasks cannot be completed since if he cannot reach the ground, he must still be on the stairs.
I gave the example of the first transfinite ordinal [math]\omega[/math]. Any step you talk backward from it lands you on a natural number, from which it's only finitely many steps back from zero.
This is a perfectly well known mathematical fact. See Asaf Karagila's answer here. It's always only finitely many steps back from any ordinal, even uncountable ones.
https://math.stackexchange.com/questions/3980267/infinite-strictly-decreasing-sequence-of-ordinals
Let me use the same example I gave earlier. In the closed unit interval [0,1], consider the infinite sequence 1/2, 3/4, 7/8, 15/16, ..., which has the limit 1.
Suppose we start at 1 and take a tiny tiny tiny step to the left, as small as we like, as long as we land on an element of our sequence. Then you can see that no matter how small a step you tak, you will land on some element of the sequence that is only finitely many steps away from the beginning of the sequence at 1/2. Can you see that? It's actually the exact same example as 1, 2, 3, 4, ... [math]\omega[/math]. Any step back takes you to a number that is only finitely many steps from the beginning.
You don't need infinitely long legs. In fact your legs can be arbitrarily small. Any step backward (or up the stairs) necessarily jumps over all but finitely elements of the sequence.
Depends on if the calculator is required to follow the mathematical theory of convergent infinite series.
If yes, 1, If no, then it can be anything at all.
That's the problem with all these puzzles. You take a situation that's mathematically straightforward, and you add in lights that flicker faster than the laws of physics would allow, and calculators to operate faster than the laws of physics would allow, and you try to reason sensibly on partial information. If a light can flicker faster than the laws of physics allow, what else can it do? What are the laws of physics in this made up universe?
Remember, Cinderella's coach turns into a pumpkin at the stroke of midnight. And for all we know, so does your hypothetical calculator. Because first, we already know that it doesn't follow the known laws of physics. And second, you've only told us what it does at each natural number step 1, 2, 3, ... You haven't told us what it does in the limit. So I say it turns into a pumpkin.
Can you prove me wrong? No, because the story's made up. In freshman calculus, the sum of that series is 1. But freshman calculus is just another made up story too. Just a highly useful one. There are no summable infinite series in the physical world. No physical computer can calculate the sum.
Presumable it would be at (the number of steps in the first staircase divided by 2). So?
Quoting noAxioms
Yes. With a real staircase would exist in both contexts and independently of both of them. Then the first step down is the last step up and the last step down is the first step up. But the last step down is not defined, which means it can't be reached. That's why the game is fascinating and frustrating at the same time, even though it is what I would call, arbitrary.
So why don't you just link me to the reading materials that would lead me to believe that the supertask you described in your op is possible to complete? That specific supertask, not supertasks in general. Let's not beat around the bush, let's get right to it.
That's precisely the argument being made.
There are some who claim that a supertask is possible; that if we continually half the time it takes to perform the subsequent step then, according to the sum of a geometric series, an infinite sequence of events can be completed in a finite amount of time.
Examples such as Thomson's Lamp show that this entails a contradiction and so that supertasks are not possible. Continually halfing the time it takes to perform the subsequent step does not just contradict the physical laws of our world but is a metaphysical impossibility.
With these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed.
Tthe mathematical ability to count in a finite time is purely mathematical and, crucially, doesn't involve the concept of conscious experience.
Once you decide to make this supertask accomplishable by *a human mind*, then you run into brand new problems that don't exist in a purely mathematical context. Let me explain:
You say he halves his rest period every step - but it's still implicit that each step has to be a *conscious experience*, a "choice". Which means, even though you can mathematicaly say he can count to infinity in 1 minute, I propose that he would be stuck consciously in that 1 minute for eternity, since that 1 minute includes an infinite series of choices and conscious experiences.
So if you send a person down such a staircase, in some mathematically perfect platonic realm where time is infinitely divisible, that person will be stuck in the eternity of that 1 minute.
No, clearly I made the appropriate choice in deciding what to argue. The described "supertask" is incompatible with 60 seconds elapsing. We seem to agree on that now. The supertask is derived from the premises of your example, therefore the supertask is the valid conclusion to your premises. You have provided no propositions or premises whatsoever, to conclude that 60 seconds may actually elapse. This is derived from your prejudice about the the nature of time, and the conventions of measuring the passage of time. To base my argument on unstated premises, and prejudices, is to produce an invalid argument. Therefore I have no principles to validly argue that 60 seconds can pass, and no principles to argue logically that the described supertask is impossible. Your assertion that 60 seconds may pass is inconsistent with your stated premises, and is not supported by any premises.
It is derived merely from your prejudice.
Quoting Michael
Clearly, what is implied by "and so on", contradicts "for 60 seconds". The two are inconsistent, incompatible. Therefore your example is self-contradicting and incoherent. To me, it is analogous to saying "the big blue computer is designed to produce the complete numerical expression of pi, and when it is finished we'll throw a big party. Do you see how "when it is finished" (analogous with "after 60 seconds") contradicts the described supertask "produce the complete numerical expression of pi" ( analogous with what is meant by "and son on")?
No it doesn't.
The "and so on" refers to repeating this formula:
Step 1 occurs after 30 seconds, step 2 occurs after a further 15 seconds, step 3 occurs after a further 7.5 seconds, and so on.
As per the sum of a geometric series this supertask takes 60 seconds.
Quoting Ludwig V
Actually, I've bethought myself and realized that the step numbers will only align if the number of steps is odd. If it is even, they won't be such a point. I still don't see that anything of interest follows.
If you'll allow me one more post.
This made me realize that if you can define the staircase down, you have defined the stair-case up. There is an intricacy about defining exactly what a step is, but let's leave that aside. Suppose we define a staircase down with 10 steps (floor level is 0). When I take the first step, there are 9 steps (10 - 1) left 2nd step (10-2)...9th (10-9) and 10th step (0) (floor level). Mutatis mutandis, that is the same in reverse 1st step up leave (10 - 1) and so on. So the staircase down defines the staircase up. No need for two staircases.
I could be wrong here, but I think that for a staircase of N steps, 1st step leaves (N-1) and so on.
This is not the usual presentism where that state was caused by prior ones, and will cause subsequent states. I don't think what you describe can be validly categorized under the term 'presentism'.
There is no 'past, present. future' defined under eternalism. All events share equal ontology. The view differs fundamentally from presentism only in that the latter posits a preferred location in time, relative to which those words have meaning.
So there is still motion and change over time under eternalism, and the 'paradox', as worded, works under either since no reference to the present is made. Hence my suggestion that the topic has nothing to do with whether or not one posits a preferred moment in time.
Irrelevant, but I prefer the one that doesn't posit the additional thing for which there is zero empirical evidence. This is my rational side making that statement.
That sounds like Zeno's arrow thing, a attempted demonstration that a nonzero thing cannot be the sum of zeroes, a sort of analysis of discreet vs continuous. Under the discreet interpretation, there are a finite number of points making up a finite length line segment. Under the continuous interpretation, no finite number of points can make up a line segment, but a line segment can still be defined as (informally) all points from here to there.
About the only practical difference is that for two non-identical points, they can be said to be adjacent only in the discreet view.
But he cannot indicate a time that isn't represented by such a point, so I don't think he's shown this.
Irrelevant, per above. The block universe can still be interpreted as discreet or not, just like the presentist view. The difference between the two has nothing to do with any of the scenarios Zeno is describing.
You do if it is discreet. A physical string is very much discreet, but that is neither space nor time. Zeno seems to favor the continuous model since all his paradoxes seem to presume it. E.g: "That which is in locomotion must arrive at the half-way stage before it arrives at the goal", a statement that simply isn't true under a discreet view.
Nonsense. It says no such thing. It is only a difference in the ontology of events.
This also seems irrelevant since none of his paradoxes seem to reference observation or comprehension. Surely it would take forever to comprehend the counting from 1 on up. Michael's digital counter runs into this: the positing of something attempting to measure the number of steps at a place where the thing being measured is singular.
Quoting keystoneNon sequitur. It presumes the length of the staircase is a number, which is contradictory.
Everybody here seems to be attempting to introduce a premise that there is a number that represents the number of steps, despite the immediate contradiction with the premise to which it leads.
Quoting Ludwig VCase in point.
Quoting Ludwig VDoesn't follow, since clearly I can overtake the tortoise in a universe that is continuous.
I can also do the reverse (the dichotomy version), which is the equivalent of counting down from infinite steps.
So instead of the assertions, show formally how this contradictory conclusion follows from the premise. Nobody has done that.
Point taken. This thread is the first time I hear of "supertasks". What I can't agree to immediately is that
Quoting Michael
If we agree that time is infinitely divisible, it seems to follow that an infinite task may be completed in a finite amount of time, just like there are infinitely many numbers between 1 and 2, Cantor nonwithstanding. What is being argued is simply the metaphysical possibility of infinity I don't see anything metaphysically necessary or impossible about a certain speed threshold.
If we admit infinity is metaphysically possible, the counter is metaphysically possible too, and it counts to infinity. We may dislike that conclusion, but aesthetic appeal is not an argument but a motivation to seek one, which is not rejecting a premise outright.
Quoting Michael
Except there have been plausible solutions given to Thomson's Lamp. Which is more of a problem than it is a paradox.
I wonder if there's such a solution to my variation.
Quoting Lionino
And so conversely, if an infinite task may not be completed in a finite amount of time then we must agree that time is not infinitely divisible.
Icarus was a Greek. To the Greek, and especially the Stoics, time is an infinite circle, and they (many of the Stoics) thought that everything repeats after a full cosmic cycle. Such that all paths hitherto, and all paths henceforth have and will be a reality. Even if reality doesn't work that way, the emotional effect of believing in such a thing allows one to overcome hardships by accepting them as part of the necessary path they are currently on, and that anyway of overcoming that obstacle is correct as all paths henceforth will eventually play out. It's quite similar to several quantum theories.
There's nothing contradictory with the EXISTENCE of an actual infinite, but it's not accepted that an infinity can be traversed in a supertask. In the case of the staircase, there actually is no last step - so it was correct to say the staircase was "endless".That would be analogous to saying the largest natural number can be reached by counting. This same objection has been raised in regard to the Zeno walk (see this SEP article).
We can consider the steps to be implicitly numbered - they map to the natural numbers. Traversal is one step at a time, moving from step n to step n+1. Every such n is a member of the set of natural numbers, but the supertask obviously never runs out of these. The contradiction is introduced by the stipulation that the end (of something endless) is reached by this process.
One reason the thought experiment can be misleading is that we're accustomed to treating infinite sets as mathematical objects. So we can consider the set of natural numbers and discuss it's cardinality (aleph-0). The set of supertask steps (step 0 to step 1, step 1 to step 2...) is also an infinite set with cardinality aleph-0 so it maps 1:1 to the set of natural numbers. The mapping is "complete" because it's defined for each member of the sets, but a supertask is a consecutive PROCESS, not a formulaic mapping identifying the correspondence. So a complete (i.e. well-defined) mapping shouldn't be conflated with a completed PROCESS.
Analogously, a limit entails an abstract operation applying to a mathematical series and shouldn't be conflated with a consecutive process.
I see that 30 and 15 and 7.5 sums up to 52.5 seconds. I also see that as it progresses the sum approaches 60. But I do not see how it could ever get to 60. Show me how you believe that "and so on" could indicate a sum of 60 is achieved please.
If time is infinitely divisible, the counter would go up to infinity. Not a conclusion that many of us may like, but there doesn't seem to be anything logically absurd with it.
I am generally in agreement with fishfry that
Quoting fishfry
If it is in a made-up universe where such counters are possible, and time is infinitely divisible, the counter should count to infinity after 30s.
Let's say even, the counter counts 1 at 15 seconds, 2 at 22,5, 3 at 26,25 and so on. It seems it would converge to infinity at time 30s. However what would the counter show at 60 seconds? Are we talking about aleph-0 and aleph 1 and so on?
Quoting Michael
Of course, but it seems that supertasks have not proven that the issue is truly the nature of time instead of their phrasing or some other thing.
It is indeed strange, if supertasks are impossible, it is because of the nature of time. So concluding something about the nature of time from thought experiments seems to put the horses behind the chariot or maybe to be analogous to ontological arguments, where we conclude something about the world by relying our own, perhaps mistaken, human intuitions.
I disagree. It's absurd because the counter progresses through natural numbers, and can never reach a final one. Infinity isn't a natural number. In the context of a temporal counting process, infinity = an unending process, not something that is reached (and not a number).
I don't follow. In calculus, the sum of an infinite series is defined. According to that definition, the sum of 1/2 + 1/4 + 1/8 + 1/16 + ... is 1.
But even if we reject that in the physical universe, or we reject it for other reasons (we're ultrafinitists, we're cranks, we hate math, we hate infinity, we simply don't care, etc) it's still perfectly clear that no matter how many finite number of terms you take, the sum is always less than 1. So I don't see how you can justify claiming that the sum should be infinity, even in a world where you reject the modern theory of convergent infinite series.
Quoting Lionino
Yes, agreed, if we allow the value [math]\infty[/math] as in the extended real numbers.
Quoting Lionino
We'll never get past [math]\aleph_0[/math] by adding more finite numbers. Likewise [math]\aleph_0 + \aleph_0 + \aleph_0 + \dots[/math] where there are [math]\aleph_0[/math] terms, is still [math]\aleph_0[/math].
Yeah, the series has infinite terms. Michael's example flips it, and the counter counts how many elements there are in the series, not the sum of the terms, which is the passing of time.
Quoting fishfry
I am not, check his problem: https://thephilosophyforum.com/discussion/comment/898574
Quoting fishfry
See https://thephilosophyforum.com/discussion/comment/898574
Ok I hadn't seen that before. Whatever shows at the end (if that even makes sense) it's certainly finite, since you're adding up finitely many finite numbers then resetting to 0. So the final total, if you can define such a thing, is 0 or some positive finite number. I don't believe such a series has a well-defined sum, since the sequence doesn't converge.
That is, the sequence is 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, ...
That sequence doesn't converge.
@michael then says, "If there is no answer then perhaps it suggests a metaphysically necessary smallest period of time."
I don't see how that follows at all. No mathematical thought experiment can determine the nature of reality. We can use math to model Euclidean geometry and non-Euclidean geometry, but math can never tell is which is true of the physical world. You can use math to model and approximate, but it is never metaphysically conclusive.
But even taken on its own terms, I don't follow the reasoning. How does observing that the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, ... doesn't converge, imply anything at all about the nature of time?
By the way the Thompson's lamp sequence is 1, 0, 1, 0, 1, 0, ... and that doesn't converge either.
Quoting Lionino
It's used in calculus to talk about "limits at infinity" and "infinite limits." For example, the limit as x goes to infinity of 1/x is 0; and the limit of 1/x as x goes to 0 is infinity. You need a formal definition of infinity in order to make those statements rigorous.
Those usages are convenient but not necessary. We could talk about the limit of 1/x as "x gets arbitrarily large," but instead we just say, "as x goes to infinity," and everyone understands the meaning.
The extended reals are also used in measure theory (a generalization of length, area, volume, etc) so that, for example, we can sensibly say that the real line has length infinity.
Quoting fishfry
I see your point, and I appreciate your analogy with the [0,1] interval. However, you need to clarify what happens in the narrative. The purpose of this narrative is to ensure that one cannot simply retreat behind formalisms. This mathematical observation doesn't change the reality that Icarus would need to jump over infinite steps. If you're suggesting he doesnt have infinitely long legs, then perhaps he possesses infinitely powerful legs that enable him to leap over infinite steps. This might explain how he returns to the top, but it essentially sweeps the infinite staircase under the rug.
Quoting Ludwig V
This brings us to another paradox - Thomson's Lamp - in that the last step can neither be even nor odd.
Quoting Ludwig V
Now explain how your algorithm works for infinite stairs.
Quoting flannel jesus
Instead, please present any supertask you consider viable, and I will demonstrate its connection to Icarus descending the staircase. For instance, do you agree that the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... equals exactly 1?
Quoting flannel jesus
I'm unclear on whether you're disputing the existence of supertasks or merely the ability of humans to perform them. Do you believe it's conceivable for anyone physical or abstract, perhaps even a divine being like God, to accomplish a supertask?
Quoting Michael
Reading your posts gives me a sense of calm. :D
Quoting Metaphysician Undercover
I said he "reached the bottom of it in just a minute." Thus, the premises address both the completion of the supertask and the passing of a minute. It seems you are challenging the incorrect premise.
Quoting Relativist
I would contend that all of the infinity paradoxes clearly illustrate contradictions inherent in the concept of actual infinity. Furthermore, I would argue that every definition of real numbers inherently suggests that supertasks are completable.
Quoting Relativist
We can also map the steps to the elapsed time (1 ? 0.5, 2 ? 0.75, 3 ? 0.875, etc.). If we conclude that a full minute has elapsed, doesn't this imply that he has traversed all the steps?
Quoting Relativist
Why not?
ZENO'S PARADOX
Quoting noAxioms
You're correct that presentists don't explicitly hold this belief. However, what Zeno's Paradoxes demonstrate is that if their ideas are taken to their logical conclusion, this belief is implicitly suggested.
Quoting noAxioms
Instead of presentism vs. eternalism, let's talk about the photo vs. movie reel. For the photo and every frame of the movie reel the characters believe they're in the present. So if you're saying that the experience of the present has nothing to do with Zeno's Paradox, then I agree with you. But there is a very significant difference between a photo and a movie reel.
Quoting noAxioms
Reconciling general relativity with presentism is quite challenging. Therefore, if empirical evidence influences your thinking, eternalism might be a more suitable perspective to adopt. Plus, adopting eternalism helps to render Zeno's Paradoxes largely non-paradoxical.
Quoting noAxioms
You're approaching this with a whole-from-parts mindset, where you aim to construct everything from smaller components. Thus, you believe the only options are to assemble a continuous line from infinite points or from discrete line segments. Consider reversing this perspective: adopt a parts-from-whole approach. Start with a single continuous line and then, as if it were a string, cut it to create discrete points (which correspond to the gaps). I encourage you to explore this mindset; I'm eager to discuss it more with you.
While my explanation might differ from how Zeno would phrase it, I believe it aligns with his philosophical approach. He is quoted to have said My writing is an answer to the partisans of the many and it returns their attack with interest, with a view to showing that the hypothesis of the many, if examined sufficiently in detail, leads to even more ridiculous results than the hypothesis of the One.
Quoting noAxioms
However, you're working under the assumption that a timeline consists only of discrete points in time. You cannot directly observe a particle in a superposition state, but this doesn't mean that superposition states are merely fictional. I bring in QM, not to sound fancy, but there is an analogy here between observed states (which are like points) and the unobserved a wavefunction (comparable to a line) that lies between them.
Quoting noAxioms
I believe you are discussing whether time is discrete or continuous. In the context of Zeno's Paradoxes, it's necessary to consider space and time as continuous (as you later noted). I'm not sure what you're referring to with time being continuous or discrete from a presentist perspective, especially since Zeno's arguments suggest that time does not progress in a presentist's view of the world.
Quoting noAxioms
I explicitly wrote abstract string.
Quoting noAxioms
Perhaps it's not my place to speak for others, but lets say that adopting an eternalist perspective allows someone to reframe the impossibility of supertasks, turning it's non-existence from having unacceptable consequences to acceptable consequences.
Quoting noAxioms
Additionally, none of the paradoxes explicitly rule out this as a possible solution.
Quoting noAxioms
If there is a continuous film reel capturing the ticking counter, the limits of observation dictate that there are just some frames that we cannot see. They're blacked out. In fact, I would argue that we can only ever observe countably many frames so in fact, most of the frames remain unobserved (in a superposition of sorts). This allows the story to advance and avoids singularities.
Quoting Michael
This only applies if you adhere to a whole-from-parts construction approach. As I mentioned in my discussion with NoAxioms, a seldom considered alternative is that the universe is constructed parts-from-whole. I really hope you will engage with me on this possibility.
THOMSON'S LAMP
Quoting fishfry
In this scenario, the calculator isn't equipped to perform calculus; it's a basic model tasked with adding each term of the infinite series. While mathematical theory predicts that at 60 seconds, it will display 1, it's true that the narrative does not specify what should appear at that moment. I am even welcoming of the idea that it turns into a black hole at 60 seconds. Nevertheless, isn't it concerning to you that there's a discrepancy between mathematical theory and your intuition? I completely agree that freshman calculus is invaluable, and I'm not suggesting that infinite series or any aspect of calculus are without merit. I use aspects of it everyday. Instead, I propose a new interpretation of what these infinite series represent. The story of the calculator isn't really about what it displays at 60 seconds; it's about the approach to 60 seconds. Likewise, I suggest that infinite series don't actually sum up to a specific number, but rather they outline a continuous, unbounded process. We don't have to assert that there's a least upper bound to this process.
Quoting fishfry
Your argument that the paradox is nonphysical is a red herring. This narrative takes place in the abstract realm, and unless you can pinpoint a contradiction within that context, we should consider it as abstract and possible and acknowledge its validity. Perhaps you lean towards theoretical perspectives, but it's important not to undermine the significance of thought experiments. They have arguably been among the most influential types of experiments conducted by humans.
Indeed, the stipulated elapse of a minute implies all the steps would have been traversed, but that implication is contradicted by the fact that the process of counting steps is not completable. The presence of this contradiction implies there's something wrong with the scenario.
Here's what's wrong: a mapping reflects a logical relationship, not an activity. The activity is a stepwise process: step n+1 is counted AFTER step n; the logical relation is present atemporally - it's an entailment of the way the scenario is defined.
[Quote]Analogously, a limit entails an abstract operation applying to a mathematical series and shouldn't be conflated with a consecutive process.
Relativist
Why not?[/quote]
Same as above: it's a logical relation (atemporal) that does not account for the stepwise process that unfolds in sequence (temporally).
The formalisms are wonderfully clarifying of one's formerly fuzzy intuitions.
For example the idea of stepping back from the bottom. It's only a finite number of steps back, even from infinity. Absolutely nobody has that intuition at first. Once one has studied the ordinal numbers, it's literally a theorem that it's always only finitely many steps back from a transfinite ordinal. And once you understand why that is, you now have a better intuition.
The process is:
1) Have fuzzy intuitions;
2) Study some math;
3) Develop far better intuitions.
Some may think of that as "retreating behind formalisms." I think of it as developing better intuitions about the real numbers, infinite processes, and so forth.
Quoting keystone
You can not use the word "reality" in this context. In reality there is no such staircase. This is an abstract conceptual thought experiment. It has a mathematical answer. If you are at 1 and you take even the tiniest step backward, you necessarily jump over all but finitely many elements of any sequence that approaches 1. That's a better intuition than the pre-mathematical intuition.
You don't find it counterintuitive that moving slightly to the left of 1 jumps over all but finitely many elements of any sequence approaching 1 from the left, correct? That's clear to you I assume.
Well, that's the staircase. It's a better intuition, informed by a precise formalism.
The best I can do to meet you halfway here is to agree that I have had some mathematical training, and that I have had "improved" intuitions beaten into me by professors at some of our finest universities. But in truth, studying math clarifies all the fuzzy intuitions about the real numbers, infinite sequences and series, infinite processes, and so forth. The Thompson lamp is a sequence of alternating 0's and 1's and it's not defined at infinity. So the final state is anything you care to define. You can't make the series continuous no matter how you complete it.
Quoting keystone
As the example of [0,1] shows, even the tiniest imaginable legs, the weakest possible legs, necessarily jump over infinitely many elements of any infinite sequence approaching 1 from the left.
There is no such thing as an infinite staircase, so this physical analogy confuses more than it enlightens.
I often make this point about Hilbert's hotel. You go online and people will argue that there's no such hotel, and how can there be room in an infinite hotel, and so forth. Many people end up more confused than they were before. In fact any infinite set may be placed into bijective correspondence with one of its proper subsets, and that's the "formalism" that clarifies the vague and unrealistic story. Hilbert, by the way, only mentioned the hotel once in his life, at a public lecture, and never wrote or spoke about it again. It's been blown up way out of its negligible importance, to the point where many people think it's a mathematical argument. It's not. /end rant
Quoting keystone
He has a magic carpet.
Quoting keystone
It's perfectly valid. I already agreed that it's perfectly natural to consider that you are present at the bottom of the staircase after one minute, because that's the choice that makes the sequence continuous. I already said this. There is no paradox. As an abstract thought experiment it works out perfectly well.
In fact I don't even understand what the supposed paradox is. I do not believe your original exposition was sufficiently clear on this point.
The staircase is nicer than the Thompson lamp, which can not be made continuous by any choice of final state, since 0,1,0,1... does not converge.
Quoting keystone
I haven't undermined the staircase story, I've agreed that it's perfectly sensible to assume the walker is present at the bottom, since that preserves the continuity of the sequence.
That's in contrast with the lamp, in which there is no possible final state that makes any more sense than any other.
Because 60 seconds will pass. I don't understand the problem you're having. The passage of time does not depend on what the counter is doing.
So to make this simpler; I am watching a stopwatch whilst the counter is counting according to the prescribed rules. When the stopwatch reaches 60 I look at the counter. What digit does it show?
The counter resets to 0 after 9. It will only ever show the digits 0-9.
We can determine whether or not something entails a contradiction. If time is infinitely divisible then supertasks are possible. Supertasks entail a contradiction. Therefore, time being infinitely divisible entails a contradiction.
You can argue that reality allows for the possibility of contradictions if you want, but most of us would say that it is reasonable to assert that it doesn't.
I imagine the counter would be spinning at a near-infinite speed by that stage, making it very difficult to read.
The counter stops after 60 seconds.
You just reaffirmed the same contradictory statements. It's impossible, by way of contradiction, that the counter can do the assigned task, and 60 seconds can pass. I see the contradiction as very clear and obvious, so I do not understand why you can't see it as contradictory.
The counter, by the prescribed specifications, is designed so that 60 seconds cannot pass until the counter counts every logically possible fraction of a second. Since logical possibility is defined by convention, and the convention allows for an infinite number of possible divisions, the counter, by the prescribed specifications, cannot finish the task. Therefore 60 seconds cannot pass. Your insistence that it can is blatant contradiction.
Quoting Michael
Why the double standard? When speaking to fishfry you readily acknowledge the contradiction. When speaking to me, you insist that the counter can perform the supertask and 60 seconds of time can also pass, as if there is no contradiction involved.
I'll repeat what I said to andrewk above:
You seem to take issue with that first paragraph, but your reasoning against it doesn't make any sense. Unless the universe ceases to exist then 60 seconds is going to pass. The passage of time does not depend on the counter.
I think that's a sensible direction.
But does that imply necessarily that time and or space in our universe must be discrete and not continuous?
If continuous space and/or time entail that supertasks are possible and if supertasks are not possible then space and/or time are not continuous.
Quoting Michael
I misunderstood the question to mean it kept counting into infinity.
Quoting fishfry
True. It seems to be Thompson's lamp but with 10 different states instead of 2 (on and off). In which case the solution of the ball skipping on the table does not work immediately. But the issue does seem to be missing limits as well.
I would propose a parametric curve on the ball path, and, for fantasy sake, by whatever mechanism, the plate knows at what part of the parabola the ball is at, defining the counter. As time goes on, the revolution gets smaller and smaller. Eventually the ball will completely rest on the table, which is 0:
Preliminarly this seems like a solution.
Quoting fishfry
Agreed:
Quoting Lionino
-
Quoting fishfry
Yes, but check the solution at https://plato.stanford.edu/entries/spacetime-supertasks/#MissLimiThomLamp
Assume the counter counts to infinity. After 30 (or 60) seconds, what is the first digit of the number it shows?
To repeat what I said earlier: with these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed.
Supertasks are illogical. Time cannot be infinitely divisible.
In logic we must follow the premises regardless of truth or falsity. Your example makes premises which describe a machine doing what has been called a "supertask". We have no premises to say that a supertask is impossible, only the premises which describe an instance of doing it, therefore demonstrating the logical possibility of a supertask.
Now, you introduce another premise, "Unless the universe ceases to exist then 60 seconds is going to pass". This premise contradicts what is implied by the others which describe the supertask.
So, what we have is a contradiction, without the information required to resolve the contradiction. In your reply to fishfry, you simply choose "60 seconds is going to pass", and conclude "supertasks are not possible". But this choice is made without the required argument, it simply reveals your prejudice.
And that assumption entails a contradiction, proving the assumption false.
Quoting Metaphysician Undercover
No it doesn't.
What contradiction?
That the counter doesn't show 0 and doesn't show 1 and doesn't show 2 and doesn't show 3 and doesn't show 4 and doesn't show 5 and doesn't show 6 and doesn't show 7 and doesn't show 8 and doesn't show 9 even though it must show exactly one of them.
Quoting Lionino
And since the counter is infinitely long, there is no first digit.
And what about rejecting the premise of the counter being apt for the task? You've designed a counter that is metaphysically constrained in such a way that it cannot perform the metaphysical task given in the way you want it to perform. What if you employ a counter that can show ??
After 30 seconds a white square turns red, after a further 15 seconds it turns blue, after a further 7.5 seconds it turns back to white, and so on.
What colour is the square when this supertask completes (after 60 seconds)?
That seems to be a Thompson's lamp with 3 states rather than 2.
To which the same solution applies (image missing):
Quoting Lionino
Just replace 0 and 1 and 2 with white, red, and blue. The square starts as white, so it finishes as white as the ball rests still on the plate.
This is just a meaningless hand-wavy rationalisation and is inconsistent with the specific timing intervals:
Red after 30 seconds, blue after another 15 seconds, white after another 7.5 seconds, etc.
Each bounce of the ball is the timing interval, e.g. when it first hits the plate it turns red, when it hits the plate a second time it turns blue, when it hits the plate a third time it turns white, etc.
The simplest answer is that supertasks are illogical. It is metaphysically impossible for an infinite sequence of events to be completed in a finite amount of time.
Sensors. Touching the sensor means the colour is white.
We can come up with other mechanisms that can better represent this exponential acceleration. Which I didn't do:
1 for simplicity sake
2 to use the same example as the SEP
The ball bounces at a rate such that it first strikes the panel after 30 seconds, then again after a further 15 seconds, then again after a further 7.5 seconds, and so on.
Each time the ball strikes the panel the colour of the panel changes, rotating through white, red, and blue.
What colour is the panel when the ball comes to a rest?
Likewise, Earman and Norton's solution doesn't work if you remove the limit (falling ball).
That's precisely the point. The lamp turning on and off and the square changing colours are each examples of an infinite sequence of events. If you claim that it is possible for an infinite sequence of events to complete then you should be able to determine the completed state of the lamp/square. If you cannot determine the completed state of the lamp/square then I will reject your claim that it is possible for an infinite sequence of events to complete.
My example keeps the falling ball so I haven't "removed the limit".
The paradox is this:
1.The bottom of the stairs is reached at the 1 minute mark.
2.Reaching the bottom of the stairs entails taking a final step.
3. Therefore there is a final step
4.The steps are countably infinite (1:1 with the natural numbers)
5. There is no final (largest) natural number.
6.Therefore there is no final step
#3 & #6 are a contradiction.
There are no empirical differences, agree. Presentism is the movie reel being played (a sort of literal analogy of the moving spotlight version of presentism). The reel by itself is eternalism (even if it still represents a preferred frame, which eternalists typically deny). The photo is just a frame, and not even that, since it is just a mental state since nothing in the present can be detected. If the state is all there is, then all memories are false and do not constitute evidence of anything.
The film analogy is discreet by nature, but doesn't have to be if the 'frames' are stacked instead of arranged side by side.
I suppose that if Zeno actually accepts his (unreasonable) conclusions, then you get something like just that one state.
There is a way to disprove GR, but it is similar to proving/disproving an afterlife: You cannot report the findings in a journal. Both premises of SR contradict presentism, so different premises must be used to take that stance. This has been done, but the theory was generalized about a century after GR came out. It necessarily denies things like black holes and the big bang.
I beg to differ, but again, the addition of a premise of a preferred moment has nothing to do with the validity of Zeno's assertions. He makes no mention of the present in any of them. If you disagree, then you need to say how the additional premise interferes with Zeno's logic.
Not sure of the difference. If I cut a string, I don't get points, I get shorter strings.
While my explanation might differ from how Zeno would phrase it, I believe it aligns with his philosophical approach. He is quoted to have said My writing is an answer to the partisans of the many and it returns their attack with interest, with a view to showing that the hypothesis of the many, if examined sufficiently in detail, leads to even more ridiculous results than the hypothesis of the One.
You can under some interpretations.
I don't think QM states are like points. The analogy is going way off track it seems.
It's one of the things I'm discussing. Zeno's arguments are of the form (quoted from the Supertask Wiki page):
"1 Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
2 Supertasks are impossible
3 Therefore, motion is impossible"
If motion is discreet, then premise 1 is demonstrably wrong. If it isn't, then premise 2 is demonstrably wrong, unless one just begs the conclusion and adopts the 'photo' interpretation.
Necessary only if the first premise is to be accepted.
Yet again, one's interpretation of time isn't relevant to the above analysis.
Fine, Then it's a mathematical line segment.
You're going to have to spell out exactly how an eternalist stance makes a difference here. All I see is an assertion that it makes a difference, but I don't see how.
It takes some minimum time to explicitly comprehend/experience a step in a series of steps. Hence the explicit experience of each step of a supertask cannot be completed in finite time.
Hence needing to see them being irrelevant.
Thanks for clarifying that for me.
I don't see a paradox. All I see is a lack of understanding of mathematical limits.
Consider the sequence 1/2, 3/4, 7/8, 15/16, ...
Mathematically, this sequence as a limit of 1.
The sequence never "reaches" 1; nor is there a last step. Neither of these statements is controversial once you understand what a limit is. Sadly, most people have never taken calculus; and most students who take calculus never really learn what a limit is. The subject isn't taught properly till a math major class in real analysis. So almost everyone in the world is ignorant of the mathematical theory of limits, and is therefore vulnerable to confusions about "reaching" and "last steps."
It's perfectly clear that if you start at 1 and move leftward on the number line, you necessarily skip over all but finitely many elements of the sequence, so that it's always only finitely many steps back from 1 to the start of the sequence.
When you dress the story up with fictional staircases and physics-violating lamps, people get confused.
But there is no confusion. 1 is the limit of the sequence, but the sequence never "reaches" 1 nor is there a last step. The definition of a limit is logically rigorous and unambiguous. The fictional staircases and lightbulbs only have the purpose of confusing people.
Finally, we can consider the sequence 1, 2, 3, ... which never reaches infinity nor does it have a last step. But we can place an arbitrary symbol at the end, usually called [math]\omega[/math], so that the sequence looks like this:
1, 2, 3, 4, ..., [math]\omega[/math]
Once again there is no "reaching" and no last step, but it's mathematically legitimate to say that [math]\omega[/math] is the limit of the sequence. And we see that if you start at [math]\omega[/math] and take any step back, you will land on a natural number, and it's always only finitely many steps backward from [math]\omega[/math] to 1.
In fact these two augmented sequences 1/2, 3/4, 7/8, ..., 1 and 1, 2, 3, ..., [math]\omega[/math] are order-isomorphic.
So there's just no paradox. There is only taking perfectly well-understood mathematical facts and dressing them up with physics-contradicting staircases and lightbulbs so as to confuse people.
In the case of the lamp, we have a sequence 0, 1, 0, 1, ... that has no limit. No matter what you define as the final state of the lamp (the state at [math]\omega[/math]), you can't make the sequence continuous. So I say the lamp turns into a pumpkin at midnight, just as Cinderella's coach did. Since the lamp is entirely physical, and its switching circuitry violates the known laws of physics, that's as sensible as any other solution.
The staircase story has a perfectly natural solution, though. At each step, the walker is present on that step. So the corresponding sequence is 1, 1, 1, 1, ... So if we define the limiting state is 1, we have made the walker's sequence continuous. That's a natural solution.
We could say that the limiting state of the walker is "not downstairs," but that would make his path discontinuous. There's a clear preference for the continuous solution.
There's no way to make 0, 1, 0, 1, ... so the pumpkin is as reasonable as anything else.
Quoting Lionino
Cute. In this case I agree that it's natural, in the sense of preferring continuity, to say that the final (ie limiting) state is resting on the table.
Quoting Lionino
Will check it out, thanks.
:up: Amen :roll:
This is an interesting argument. I have some issues with it.
First, I should note that infinite divisibility is a weak condition. The rational numbers are infinitely divisible, but they are not a continuum. They are full of holes, such as the point where sqrt(2) should be.
The question often argued is whether physical spacetime is a continuum in the sense of the uncountably infinite, Cauchy-complete real numbers. But you have strengthened the claim to saying it's not even a countably infinite non-continuum like the rationals. So even if you're right, your claim is too strong to be right. That's a meta-argument, not an argument. But claiming spacetime isn't even like the rationals is much stronger than claiming it's not like the reals. [This is all beside the point, but I wanted to make the point that infinite divisibility is not enough to make something a continuum].
Now to the argument.
"If time is infinitely divisible then supertasks are possible."
By this I take it that you mean that if we take, say, the rationals in the unit interval to model one second of time, we could do something in [0,1/2) and something else in [1/2, 3/4) and so forth, and thereby do infinitely many things in one second, which is the definition of a supertask. Have I got your argument right?
So yes, I agree that if time is dense -- that's the math term for the property that there's always a third thing between any two distinct things -- then supertasks are possible. I'd never thought of that argument before and it's pretty good. Although for all we know, there could be some law of nature that the smaller the time interval, the longer things take to happen, wrecking your supertask. You can't rule that out. Just like objects gaining mass as their velocity approaches the speed of light. Strained analogy but I hope you see what I'm getting at.
"Supertasks entail a contradiction."
What contradiction is that? You just convinced me that if time is like the rational numbers (dense but full of holes) supertasks are possible. Then you claim supertasks entail a contradiction, but I'm not sure what contradiction that is.
So your argument's incomplete here, and if you did explain this elsewhere in the thread, I apologize for having missed it.
I have another concern, which is that in our current theory of physics, we can not reason sensibly about intervals of time below the Planck time. So you are making an argument that can never, even in theory (pending the next revolution in physics) be observed, measured, or confirmed by experiment.
That's what we call speculation. Like the cosmological theory of eternal inflation, in which the universe had a definite beginning but exists infinitely far into the future. That's not physics, that's mathematical metaphysics. Science fiction with equations. I do think your idea has a problem in this area. You can't actually reason below Planck scale.
Quoting Michael
On the contrary, most people would agree that life is full of contradictions. I love you and I hate you. We should clean up the environment but that raises the cost of energy for the poor. (Ok that's a tradeoff and not a logical contradiction, but it's still a situation where two virtues are in conflict). I am large, I contain multitudes. (That diet's not working). The electron is a particle. It's a wave. No, it's an excitation in a quantum field. That's the latest attempt to resolve the contradictions in physics.
As we go through our daily lives we are faced with one contradiction after another. And when we study physics, we see contradictions and impossibilities at the most fundamental nature of reality.
I do not believe you can convince me that nature isn't self-contradictory. Why shouldn't it be? What law of nature says that nature must satisfy Aristotelian logic?
tl;dr: Well those are my thoughts. Interesting argument though. If time is modeled by the rational numbers, supertasks are possible. I will give that some more thought. But again: why do supertasks entail a contradiction? That's the weak part of the argument I think. That, and the Planck scale issues.
Nice to see you again @jgill, and thanks.
Take the scenario here:
We can sum the geometric series to determine that the limit is 60 seconds. The claim some make is that this then proves that this infinite sequence of events can be completed in 60 seconds.
However, then we ask: what colour is the square when this infinite sequence of events is completed?
As per the setup, the square can only be red, white, or blue, and so the answer must be red, white, or blue. However, as per the setup it will never stay on any particular colour; it will always turn red some time after white, turn blue some time after red, and turn white some time after blue, and so the answer cannot be red, white, or blue. This is a contradiction.
The conclusion, then, is that an infinite sequence of events cannot be completed, and the fact that we can sum the geometric series is a red herring. To resolve the fact that we can sum the geometric series with the fact that an infinite sequence of events cannot be completed we must accept that it is metaphysically impossible for an infinite sequence of events to follow a geometric series: we must accept that it is metaphysically impossible for time to be infinitely divisible.
I've taken calculus and I understand what limits are. By definition, a limit is not reached, it is approached. The sequence of steps maps to a mathematical series that approaches, but never reaches 1. The sequence of steps is actually unending (that is how infinity is manifested in this thought experiment)- there is no last term.
However, the clock does reach 1. At time 1, the stairway descent must have ended, because the descent occurs entirely before time 1. The descent is not a mathematical process (even though it can be mapped to a mathematical series), it is a sequence of movements from one step to the next. No movements are occurring AT time 1. If the descent has ended at this time, how can there NOT have been a final step?
[math]f(x)=3x+2[/math], [math]f(1)=5[/math]
[math]\underset{x\to 1}{\mathop{\lim }}\,f(x)=f(1)[/math]
Quoting Relativist
Certainly the relationship between time (independent of human control) and physical steps taken over a period of time has ended.
That's because the physical steps map to an infinite series in an interval with an open boundary. One can't simply declare there's no final step because the mapping implies there isn't. The taking of steps is a repetitive physical process, and if a physical process ends, there has to be a final step.
There is no physical process. There's a fictional process that doesn't obey the known laws of physics.
In what sense does anyone think the staircase or the lamp are physical processes?
Question: Cinderella's coach turns into a pumpkin at the stroke of midnight.
Is that transition a physical process? In what world?
The scenario describes a fictional, physical process. The lesson is that the defined supertask (the fictional, physical process) is logically impossible, but this isn't apparrent when considering only the mathematical mapping.
The lamp and staircase scenarios are physically impossible. What law of logic makes them logically impossible?
Are you suggesting that supertasks cannot be completed?
Quoting fishfry
Agreed, but most importantly: (4) apply those intuitions to (the original) experiments.
Quoting fishfry
I like where you're going with this. To navigate between the staircase and omega (and back), one must leap over infinite steps. This concept becomes more palatable if we consider that the steps become progressively smaller towards the bottom. However, let me try to rephrase your perspective: Icarus requires a finite number of strides to reach the bottom and a finite number to return to the top, thus avoiding any supertask. When Icarus adds 1/2, then 1/4, then 1/8, he gets bored and chooses to make a final leap. On his final leap, instead of adding an infinite series of smaller terms, he simply adds another 1/8 and reaches omega, where his calculator displays exactly 1. In this case, the infinity in the paradox describes the steps which he potentially could have traversed (and seen), not what he actually did (and saw). Since he never actually observed all steps, he is in no position to confirm that there were actually infinite steps...but there could have been...potentially. Paradox solved?
I'm asserting that an infinite process is necessarily never completed - by definition.
The law of non-contradiction. An infinite series of processes entails never completing, but at points of time that occur after the delinieated interval - the task is necessarily completed.
Good. Then we're on the same page!
You've just described the ordinal [math]\omega + 1[/math], which has as one representation the sequence 1, 2, 3, 4, 5, ... [math][/math], and another more familiar representation as 1/2, 3/4, 7/8, ..., 1.
These are perfectly rigorously defined and logically consistent mathematical objects (assuming ZF is consistent of course].
They're just limits. There is no mathematical mystery. People just get confused when you start making up fictional entities like switching circuits that change state in arbitrarily small amounts of time.
You're not (ahem) a .999... = 1 denier, are you? That's one of the standard crank arguments, that the process of adding the next 9 is "never completed." It's a fallacious argument. There is no temporal process of adding 9's. Rather, you have a mathematical function that assigns to each natural number the digit 9. That's a completed process (or function, more accurately) once you accept the axiom of infinity. When you interpret the string of 9's as a sum 9/10 + 9/100 + ..., the sum of the series is 1, by the definition of the limit of a convergent infinite series.
The "never ends" argument is simply mathematical ignorance. The fast-switch circuit is as realistic as Cinderella's coach that turns into a pumpkin at the stroke of midnight.
At 1/2 second before midnight it's a coach. At 1/4 second before midnight it's a coach. Dot dot dot. At midnight it's a pumpkin. How does that happen? It's a fairy tale. For some reason, philosophers recognize Cinderella as a fairy tale (scrub enough floors and you'll attract the devotion of a handsome prince with a foot fetish); yet these same philosophers take Thompson's lamp seriously. I can't account for this cognitive error.
The contradiction is very obvious. I'm surprised you persist in denial. The supertask will necessarily carry on forever, as the sum of the time increments approaches 60 seconds, without 60 seconds ever passing. Clearly this contradicts "60 seconds will pass".
Exactly. Lets deconstruct the argument:
(1) We accept Zeno's premise as valid, asserting that in a presentist world where only a single state exists, motion is impossible.
(2) We assume that the world functions according to presentist beliefs.
(3) Our experiences clearly indicate that motion is possible.
These three assertions cannot all be true simultaneously. Its unlikely that anyone, including Zeno, would dispute (3). You find (1) to be unreasonable. However, consider the possibility that (2) is incorrect.
Here's an alternative approach:
(1) We accept Zeno's premise as valid, asserting that in a presentist world where only a block universe exists, change of the block is impossible.
(2) We assume that the world functions according to eternalists beliefs.
(3) Our singular, consistent historical experience gives us no reason to believe that the block universe is subject to change.
In this scenario, all three points could indeed hold true, suggesting that an eternalist viewpoint might be more suitable. However, quantum mechanics challenges point (3), necessitating a more nuanced argument. Despite this, an eternalist frameworkalbeit with some adjustments to incorporate the quantum aspects of our universeappears to be the most rational choice.
Quoting noAxioms
The cuts themselves are the points (think Dedekind cuts).
Quoting noAxioms
One can observe a superposition directly? Please share a link.
Quoting noAxioms
What I aim to demonstrate is that there is a scenario where local motion is possible and continuous without involving supertasks. This occurs in a block universe where the block itself remains unchanged (i.e., no global motion), yet the entities within it experience change (i.e., local motion).
Quoting noAxioms
If the universe is discrete, then Zeno's paradoxes cannot occur as he described them. What I'm suggesting is that in a continuous universe, the scenarios depicted in Zeno's paradoxes can indeed unfold precisely as he described them, without necessitating the completion of supertasks.
I did not get a mention for this post, does that happen sometimes? Maybe I just missed it.
As I have been explaining in this thread, you can conceptually adjoin the limit of a sequence to the sequence, as in 1/2, 3/4, 7/8, ..., 1. This is a perfectly valid mathematical idea. This is a representation of the ordinal [math]\omega + 1[/math]. In this case, 1 is indeed the "last term," although to be fair, you can no longer call this a sequence, since a sequence by definition is order-isomorphic to the natural numbers.
Quoting Relativist
You can model this situation with [math]\omega + 1[/math], as I've tried to explain a number of times.
After all, if we work in the close unit interval [0,1], the sequence 1/2, 3/4, 7/8, ... never ends, yet there's its limit right there at the right end of the interval. We "get there" through a limiting process. There is no last step if your steps are required to be discrete. But we can also take limits. Limits aren't steps, but that's a semantic quibble. We can adjoin 1 to 1/2, 3/4, 7/8, ... to form the [math]\omega + 1[/math] "extended sequence" if you want to call it that, 1/2, 3/4, 7/8, ..., [math]\omega[/math].
I don't know if this will help, but at least I can motivate the legitimacy of the ordinal concept by linking the wiki page on ordinal numbers.
But there's an easier way to think of it. We're just adjoining a formal symbol at the end of the natural numbers:
1, 2, 3, 4, ..., [math]\omega[/math]. It's just a formal symbol, means nothing at all. But we can define it in such a way that it's the upper limit of 1, 2, 3, ... in exactly the same way that 1 is the upper limit of 1/2, 3/4, ...
Yes there is no "last step" but there is in fact a limit.
Quoting Relativist
That is sadly a misunderstanding very common among calculus students. So lot of smart people, physicists and engineers and other scientists, have this belief.
In fact a limit IS reached. A limit is exact, it's not merely approached or approximated. It is literally reached.
It's not reached by a single step. Rather, it's reached by the limiting process itself.
It cannot be finite. If it were n, why not n+1?
It cannot be infinite because no step has the number infinite.
This is just the lamp story with three states. The answer is that the sequence 0, 1, 2, 0, 1, 2, 0, 1, 2, ... has no natural completion or limit. So if we want to define its state "after" the natural numbers, we can say it's anything we want. I like the Cinderella analogy. The square turns into a pumpkin at midnight. That's no less realistic than the square story.
Quoting Michael
Why? After all in the sequence 1/2, 3/4, 7/8, ... the "setup" is that each element of the sequence is a rational number strictly less than 1.
But the limit of the sequence is 1. This illustrates a general mathematical principle:
Taking limits does not necessarily preserve all properties of a sequence.
All of 1/2, 1/3, 1/4, 1/5, ... are strictly positive. But the limit of the sequence is 0, which is not positive. (I'm positive!)
You have to be very careful not to fall into the trap of assuming that a limit must preserve all the properties of the elements of the sequence that approaches it. You have made that mistake.
Quoting Michael
It's not a contradiction. It's the straightforward observation that the sequence 0, 1, 2, 0, 1, 2, ... has no sensible limit. So if you wish to define a final state, you can make it anything you like. I choose pumpkin.
Remember, Cinderella's coach is a coach at 1 second before midnight; at 1/2 second before midniht; at 1/4 second before midnight; and so forth. Yet at the stroke of midnight, the coach turns into a pumpkin.
That story makes exactly as much sense as Thompson's lamp. Except that with Cinderella, we introduced a discontinuity. Where as with the lamp, and with your three-state lamp, there is no possible way to define the limiting state in such a way as to preserve continuity.
Quoting Michael
An infinite sequence of events can have a limit. I assume you agree that 1/2, 3/4, ... has the limit 1. We can think of 1 as the "completion" of the sequence. It's reached not by a "final step," but rathe by the limiting process itself.
Quoting Michael
No, it's the heart of the matter. .999... = 1 even though there's no "last 9." The limiting process is real. It's important. It exists.
Quoting Michael
That is flat out false and does not follow at all.
We can "complete" the sequence .9, .99, .999, .9999, ... with the number 1, which is reached via a limiting process.
Quoting Michael
Sorry, what? You don't believe that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1? You don't believe in calculus? You are arguing a finitist or ultrafinitist position? What do you mean?
Of course if you mean real world events, I quite agree. But your three-state lamp is not a real world event, it violates several laws of classical and quantum physics, just as Thompson's two-state lamp does.
42. Can you argue otherwise? The final state is not defined. It can be anything we like.
"Not defined" does not mean that you are free to choose the result.
Which solution has n = n+1?
Certainly not 42.
Yes it does. If I define the first three elements of a sequence, like 3, 12, 84, what number comes next? Mathematically, it can be any number at all.
Likewise if I define a function at 1, 2, 3, 4, ..., and I want to also define it at [math]\omega[/math], a symbolic point AFTER all the natural numbers, I can define it to be anything I want. Like 42. Or like Cinderella's coach, which is a fine, beautiful coach at 1 second before midnight, 1/2 second before midnight, 1/4 second before midnight, etc., yet turns into a pumpkin at the stroke of midnight.
Quoting SolarWind
Not the same kind of undefined. Here, it can't be defined. But the state at the bottom of the stairs can be anything at all.
So there's a distinction between something that can't be defined because it's impossible, and something that simply hasn't yet been defined, and that can then be defined as anything at all.
Quoting SolarWind
Don't see why not. When you first heard the Cinderella story, did you make the same objection to the coach turning into a pumpkin at the stroke of midnight?
Right! It's not the sequence described in the scenario! There is a background temporal sequence, as the clock ticks, that reaches 1, but we aren't mapping the step counting to the ticks of the clock. The step-counting sequence occurs only at points of time <1. In real analysis, this is called a "right open interval" (i.e.it's open on the right= the endpoint is not included in the interval). 1 is the endpoint, but not included within this interval.
Quoting fishfry
The limit of the series is "reached" only in the sense that we can reach a mathematical answer. The physical process of sequentially counting steps, doesn't "reach" anything other than increasingly higher natural numbers. Deriving the limit just means we've identified where the sequential process leads. In this case, we've derived that the limit is infinity- but what does infinity correspond to in the scenario? The meaning is entailed by the fact there are infinitely many natural numbers, so it means the process continues without end. It can mean nothing else.
Quoting Relativist
This depends on one's definition of completing a process. The SEP article on supertasks has this to say about it:
"But as Thomson (1954) and Earman and Norton (1996) have pointed out, there is a sense in which this objection equivocates on two different meanings of the word complete. On the one hand complete can refer to the execution of a final action. This sense of completion does not occur in Zenos Dichotomy, since for every step in the task there is another step that happens later. On the other hand, complete can refer to carrying out every step in the task, which certainly does occur in Zenos Dichotomy."
The definition you appear to be using is the former, which is why Michael's one-digit counter doesn't have a defined output after the minute expires.
I've been using Zeno's definition of complete: That every step has been taken. Given that definition, the supertask can be completed.
Quoting keystoneAnd a different page than me.
Quoting keystone
Zeno's argument is that X is possible, and another that X is not possible.
I see no mention of presentism in his arguments. I cannot follow your arguments here if you don't show how he presumes any such thing, or why it matters. Motion is defined under either view, and the argument can be made in either view of time. By modus ponens, at least one of Zeno's premises must be false unless empirical evidence is entirely dismissed as invalid.
I think you are under the impression that motion is not meaningful under eternalism, and that this somehow absolves Zeno's conclusion, but all his arguments still apply, and are still self contradictory.
OK, so now we have point cuts separating shorter strings, each with nonzero extension.
Any interpretation that denies wave function collapse has everything in superposition at all times. One simply finds ones self in superposition with the observed state. So I observe both the dead and the live cat, presuming that "I" dong the observing is the same person as the person a moment ago with the closed box.
No, I'm not don't personally accept MWI, but the simplicity of it is elegance itself.
Moton is change of postion over time. The block universe very much has that for any moving object. The worldline of that object is a different spatial locations at different times. All of Zeno's arguments still apply, and are still contradictory.
Yet again, the only difference between the view is the positing of the preferred moment, which is irrelevant to the subject at hand. Both are effectively block view, but presentism assigns different (at least four kinds of) ontological states to different events based on its relation to the preferred moment, and eternalism assigns identical ontological states to all events.
The kind of motion you are referencing (the changing of the block (over what??)) is not suggested by either view, nor by Zeno.
The first premise would be demonstrably false. The second premise (that supertasks are impossible) would be moot, but arguably true then.
You seem to do this by reducing the universe to a point (your 'photo'), which is not something that is continuous. A point in time at least, which is the same as denial of time at all.
There is a difference between saying that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1 and saying that one can write out every 1/2n in order. The latter is not just a physical impossibility but a metaphysical impossibility.
Some say that the latter is not a metaphysical impossibility because it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity, and so that this infinite sequence of events (writing out every 1/2n) can complete (and in a finite amount of time). Examples such as Thomson's lamp show that such supertasks entail a contradiction and so that we must reject the premise that it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity.
Quoting fishfry
If you want to say that supertasks are possible but then have to make up some nonsense final state like "pumpkin" then I think this proves that your claim that supertasks are possible is nonsense and I have every reason to reject it.
An ordinary stopwatch is started.
After 30 seconds a white box turns red, after a further 15 seconds it turns blue, after a further 7.5 seconds it turns back to white, and so on.
When the stopwatch reaches 60 seconds, what colour is the box?
Your claim that the box changing colour entails that the stopwatch will never reach 60 seconds makes no sense. The stopwatch is just an ordinary stopwatch that counts ordinary time as it ordinarily would and is unaffected by anything the box does.
You have not avoided the contradiction, only obscured it. Increments of time must be measured, the are the product of a measuring device. The measuring device in this example is an ordinary stopwatch. What is implied by "and so on" is increments of time which the stopwatch cannot measure. There lies your contradiction. The stop watch is the designated device which measures time, but you talk about increments of time which cannot be measured by it, therefore contradiction.
In your other example, the counter was the device that measured the passage of time, and it was designed to measure all those increments (do the supertask). But that measuring device denied the possibility of 60 seconds of time passing, making the measurement of a separate timepiece which would measure 60 seconds, contradictory.
Therefore we can conclude that the contradiction lies between the two very distinct descriptions of how time is measured. One way is the supertask of the counter. This is obviously a theoretical way of measuring time. The other way is a description of how time is actually measured in practise. There is contradiction between these two ways of describing the measurement of time. As you yourself indicate, the practical way gets the nod, as the real way, because it is supported by empirical evidence, and the supertask way, since it contradicts the practical way, is designated as impossible. As philosophers though, we are trained to be skeptical of sense evidence, having been educated in the ways that the senses commonly deceive us. So the philosopher knows that there is more to this problem than what meets the eye. It's not simply a matter of dismissing the supertask, and accepting the conventional way of measuring time, as the true way to measure time.
As I said before, the supertask way needs to be proven to be wrong, rather than simply dismissed because it contradicts the way of current practise. The empirical evidence of the stop watch is nothing other than current practise, convention, so it is in fact manufactured evidence. If we always accepted the current practise as the best, or true way, then we'd never improve ourselves. This is why I say simply accepting it and dismissing the supertask, is prejudice, and nothing else. The fact that we have not devised a supertask machine, and so we use other ways to measure time, does not mean that it is impossible that the supertask way is the real "true" way to measure time, and our current practise is actually giving us a false measurement. Therefore the supertask must be proven to be impossible.
Here's an example of an attempt at a similar type of proof. In ancient Greece, there was a principle accepted by many, that the orbits of the sun, planets, etc., were eternal circular motions, as a sort of divine activity. Being a prescribed activity which continues for an infinite duration of time, the eternal circular motion is a supertask. Now, Aristotle in his "On the Heavens" (De Caelo) showed how eternal circular motion is a logically valid and consistent principle, a real logical possibility, just like the supertask counter is. However, he then went on to explain how anything which moves in such a spatial pattern must be a material body. He then described "matter" as the principle of generation and corruption, and determined that a material body must have been generated in the past, and will be destroyed in the future. In this way he provided the principles required, to prove logically, that (the supertask) eternal circular motion is actually logically impossible. This proved that the heavenly bodies were not eternal, and not divine. Then the principle which he employed, "matter", became the keystone for understanding the nature of the physical reality because it provided the principle for associating change and becoming on the earth, with change and becoming in the heavens.
No they mustnt.
Wrong. The statement (the completion of a consecutive series of physical steps entails a final step) is necessarily true. When we consider this statement in conjunction with a statement about the series being "complete" (in terms of convergence) we introduce a contradiction. This is the point! These statements cannot both be true, but both are entailed by the scenario.
Quoting noAxiomsThe SEP article says:
"[I]Although it has infinitely many terms, this sum is a geometric series that converges to 1 in the standard topology of the real numbers. A discussion of the philosophy underpinning this fact can be found in Salmon (1998), and the mathematics of convergence in any real analysis textbook that deals with infinite series. From this perspective, Achilles actually does complete all of the supertask steps in the limit as the number of steps goes to infinity. One might only doubt whether or not the standard topology of the real numbers provides the appropriate notion of convergence in this supertask. "[/i]
Indeed, I'm denying that the topology of the real numbers applies to the execution of the supertask itself - although I agree it applies to the series.
As I noted above, a physical, step-counting process that completes must entail a final step. Your preferred perspective ignores this - or pretends there can't be a final step because that introduces a contradiction. That seems a cop-out. The point of the thought experiment is to highlight the contradiction.
Quoting noAxioms
I agree with this, but this simply ignores the implication of the physical process of step-counting. For the scenario to be coherent, BOTH view of completeness have to be true. But they aren't - so the scenario is actually incoherent.
Yes- and that's because the role of infinity in the task. The task entails a sequence of events, so the infinity can only mean an infinite chain of events - one after another without end.
That's fundamentally incorrect. If you truly believe that an increment of time exists without being measured, tell me how I can find a naturally existing, already individuated increment of time.
The physical process of descending stairs is not a supertask. I couldn't think of a way to make it a supertask, even by making each step smaller. A supertask has no final (or first, respectively) step, so by counterexample, the assertion "there has to be a final step." is incorrect.
I had not mentioned a completion of a count. The supertask is to complete all steps, not to count them, and not to complete a specific step that is nonexistent.
The series (say the time needed to complete all tasks) converges. The count does not.
Cheap example: You have a bag with a modest quantity of red, blue and yellow marbles in it. The goal is to remove them all. The task is deemed to be complete when the green marble is removed. Such a task cannot be completed by that definition of complete
I notice the SEP article correctly doesn't claim that the last step is taken.
Agree. But the only attempted step counting processes are examples like the lamp or Michael's digit counter, and those examples are not physical. The Achilles example can be physical, but it isn't counting anything.
There being a final step leads directly to contradiction, and you say I'm copping out by pretending there isn't a final step?
Kind of like I ignore the green ball in the bag, yes.
I cannot accept this assertion. I cannot accept a view of completeness that treats infinity as a specific number.
Quoting Michael
:up:
Once again, M-U cannot comprehend a view outside his own idealistic assumptions.
I don't know what you mean by "finding a naturally existing, already individuated increment of time", but it is a fact that 60 seconds of time can pass without anyone looking at a clock or a stopwatch. Billions of years passed before humanity evolved, and this isn't some retroactive fact that only obtained when humanity started studying the past.
I don't know whether you're arguing for some kind of antirealism or if you're failing to understand a use-mention distinction.
Regardless, the arguments I am making here are directed towards the realist who believes that supertasks are possible.
If someone would explain to me, in a way which makes sense, a better perspective, then I'd happily switch. Simple assertions like "It must", and "no it's not" just do not suffice for helping a poor lost soul such as myself, comprehend another view.
Quoting Michael
Simple assertions do not help me to understand what you are trying to say. "60 seconds" is the reading we get of the clock. It's just a generic symbol, like "dog" or "cat". But I can show you many things which would be called "a dog", and things called "a cat". Now, if you think that there are some things called "60 seconds", other than the reading taken from a measuring device, then show them to me.
Consider Wittgenstein's example of "the standard metre in Paris". "One metre" is a measurement, and there are many items which can be measured to be a metre. The standard metre is the paradigm, the official example of that convention. But there are no objects in the world which "one metre" refers to, not even the standard metre, as this is the paradigm, it is not "one metre" itself. Likewise, we can measured a multitude of different times as "60 seconds", and there is a paradigm, or standard which is the oscilation of the cesium atom, but there is nothing in the world which is referred to by "60 seconds"
[quote=Wikipedia] the second, defined as about 9 billion oscillations of the caesium atom.[/quote]
https://en.wikipedia.org/wiki/Unit_of_time#:~:text=The%20base%20unit%20of%20time,oscillations%20of%20the%20caesium%20atom.
Quoting Michael
A ''year" is nothing but a human convention, a standard of measurement, just like "a metre". Therefore there was no years prior to humanity. What you talk about here is a projective measurement backward in time. There was something occurring prior to humanity, which we commonly call "the passing of time", but what is referred to as the passing of time did not consist of years, as "years" is the product of the measurement, just like "metres" is the product of the measurement. To claim that standards of measurement existed prior to humanity, which actually invented them, is Pythagorean idealism, Platonism. Then these standards become eternal principles, such that God was measuring the passage of time prior to humanity, to determine the passage of "years". But this Platonism is demonstrably wrong, because it excludes the possibility of error.
That the passing of time is something completely different than a succession of years is very evident from the fact that it can be measured as a succession of a vast multitude of different increments, years, days, minutes, seconds, picoseconds, nanoseconds. Each of these increments serves as a measurement standard, (just like metre, centimetre, kilometre, foot, inch, mile), but not one of them is the thing which is measured. And, the little inconsistencies between them which show up in conversions, where we have to adjust the clock so that they keep up with each other, shows that none of them is actually real or true. Notice the wiki quote, a second is "about nine billion oscillations".
Quoting Michael
Perhaps if you took the time to explain to me how you understand this use-mention distinction, and how it is applicable in this context, that might be helpful to me understanding your position, which at the time seems completely ridiculous. I really do not understand how you can believe that the product of a measurement "one second" can exist without the act which produces it. Again, the best course of action for you to help me understand, would be to show me these things, called "a second", or explain to me how I might find one without performing the act of measuring which is what, I believe, actually creates them.
Quoting Michael
As I explained yesterday, we must consider that any such task (supertasks) are possible until proven otherwise. This is because they are logically possible, and the only thing which makes them appear to be impossible is that they are inconsistent (in contradiction with) the conventional way of doing things. But the conventional way is not necessarily the best way, it is only the way which is supported by empirical evidence, which has been proven to be unreliable and misleading. Therefore we cannot dismiss the supertasks as impossible until we have sound logic which disproves them. Simply asserting that supertasks are impossible just displays an empiricist prejudice. Then you support your prejudice with Platonism.
That we coin the term X to refer to some Y isnt that Y depends on us referring to it using the term X. This is where you fail to make a use-mention distinction.
If we take the term 1 year as an example, the Earth orbiting the Sun does not depend on us measuring it. It just orbits it, independently of us.
So to rephrase my example:
A white box turns red when the Earth completes a half-orbit, turns blue when it completes another quarter-orbit, turns back to white when it completes another eighth-orbit, and so on.
What colour is the box when the Earth completes its orbit around the Sun?
Your claims so far are akin to claiming that the Earth will never complete its orbit around the Sun, which just makes no sense. The box does not have the power to influence the Earth's velocity or the Sun's gravitational pull.
My point is that the stairs are countably infinite. Consequently, they COULD be counted, if we were traversing them.
[Quote]The series (say the time needed to complete all tasks) converges. The count does not.[/quote]
Yes, the sequence of defined temporal points (1/2, 1/4, 1/8...) is a series, but the mathematics that identifies the limit does not take into account the kinematics of the task. Supertasks describe a conceptual mapping of the abstract mathematical series into the actual, kinematic world - regardless of whether or not you wished to consider it.
Quoting noAxioms
It fits this definition:
[I]"a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time."[/i]
Quoting noAxioms
The goal of removing all the marbles will therefore never be met if there are at least 2 green marbles, and it will rarely met even if there is only 1. How does this relate to a supertask that allegedly completes?
Quoting noAxioms
The article discusses the issue:
[i]Max Black (1950) argued that it is nevertheless impossible to complete the Zeno task, since there is no final step in the infinite sequence...
... there is a sense in which this objection equivocates on two different meanings of the word complete. On the one hand complete can refer to the execution of a final action. ... The two meanings for the word complete happen to be equivalent for finite tasks, where most of our intuitions about tasks are developed. But they are not equivalent when it comes to supertasks.[/I]
The mathematical series completes, but this is an abstract, mathematical completion. The kinetic activity of descending the stairs does not complete. The SEP article leaves it there, but the implication seems clear: the abstract mathematics does not fully account for the kinetic activity.
Here's a paper in which a philosopher proves it to be impossible to complete infinitely many tasks in a finite time based on the "Principle of Sequential Acts":
[b]PSA:
[I]The performance of a sequence of successive acts does not complete a particular task unless it is completed by the performance of one of the acts in the sequence.[/i][/B]
The author argues that those who argue the task completes implicitly deny the PSA, without considering it, and therefore not refuting it. That's what I see going on with the posters who focus only on the mathematical series.
Quoting noAxioms
Yes, it's a cop-out because it ignores the kinematic process. Stating this in terms of the PSA gives you something specific to address, if you want to not cop out.
If your sole purpose was to discuss the math associated with the limit of a series, you'd have been better off avoiding putting it in terms of a supertask.
Quoting noAxioms
I agree we can't treat infinity as a number, and haven't suggested you should. But for the supertask to be meaningful, you have to identify where infinity fits in the kinetic task description. I'm saying it entails a never-ending sequence of tasks. Identifying the limit doesn't make this disappear.
I'll add that supertask scenarios actually are NOT coherent- because they entail a contradiction. You seem to be embracing the completeness of the mathematical series, then concluding that there can't be a last step because that would entail a contradiction. So look at it this way:
1) the completeness of the series does not demonstrate an analogous supertask is possible.
2) If there is no last step (or if the process is not consistent with the PSA), then the kinetic process (which is a supertask) is logically impossible.
OK, the earth goes around the sun indefinitely, even if there were no humans on earth. But there is no "years", nor is there any individual "orbits" separated out without someone, or a device to make the judgement of beginning and end.
That is the problem with your example. Time passes, we agree on that, but there are no seconds unless measured out. Now, both the counter doing the supertask, and the ordinary stopwatch are designed to measure the passage of time. The counter, with it's supertask has one way of counting out time, by dividing seconds into shorter and shorter increments, while the stopwatch is designed to measure an endless procession of seconds. The two are incompatible.
Quoting Michael
Same problem, the device is not designed to reach the end of an orbit. It will keep on changing colours faster and faster without ever getting to the end of an orbit. I assume it would probably burn up though, from going too fast.
Am I right to think that you are not saying that all the stairs can be counted, even though any stair could be included in a counting sequence?
Quoting Relativist
That's true. What puzzles me is why they are not dismissed out of hand. Someone earlier described them as fairy stories, and the writers seem to be able to wave a hand and create impossibilities, which would be magic, so that description makes sense. But it seems to me more like an illusion and the problem is then to understand how that illusion works.
Quoting Relativist
Wouldn't it be more accurate to say that descriptions of the supertasks are the source of the illusion that there could be a mapping of that mathematical series into the actual kinematic world?
More than that, surely, there can be a mapping of some mathematical series into the actual kinematic world. Perhaps some similarity between those series is what creates the illusion?
Quoting Metaphysician Undercover
Yes. How come anyone can't see that? Since the difference is the difference between simple addition and division followed by addition, I think it is then possible to see how people can be misled into thinking they are compatible - even that they must be compatible.
Yes there are. A second is "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the caesium-133 atom". This occurs even if we don't measure it.
Yet again you can't seem to get beyond our use of labels to understand that our labels refer to things that exist and do things even when we're not around.
[/quote][The physical process of descending stairs] fits this definition:
"a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time."[/quote]Physical (fixed size) stairs are of infinite length, and such a distance cannot be traversed in finite time. If the stairs get smaller as we go, then we get into the physical problem of matter being discreet, not continuous. Hence the steps have a minimum size. That's what I mean about physical stairs not qualifying as a supertask.
We seem to have lost @keystone, and the stairs thing was his. I prefer Zeno's scenario which doesn't seem to be plausibly physical so long as we take a classical continuous view of both time and space.
It does, I'm quite aware. Just not in Zeno's argument.
I pretty much quoted exactly Black's remarks just above. Yes, the task is not complete by this finite definition despite every step having been taken, and that final step must be taken for your counter to have a defined value after a minute.
Again, the stairs is utterly abstract. There's no kinematics to it. Not so with the tortoise. I can pass the tortoise, thus completing (by the 'all steps' definition) the supertask.
How does the abstract mathematics not account for the physical ability of me passing the tortoise?
I cannot parse this. What is an 'act' that is distinct from a 'task'? The word 'sequence' seems to refer to the entire collection.
A 'task' (what, one of the steps??) is not completed by a performance unless 'it' (what, the performance?, the task?) is completed I cannot follow it at all.
I cannot take a bite of an apple unless perhaps the bite taking is completed by the performance of the taking of a bite? Presumably the same bite??
Kindly translate. Perhaps it points out some error I'm making, but only if I can parse it. I can come back to your statement and respond more intelligently.
I'm trying to focus on the completion of all tasks and not on the measurement of a nonexistent value.
But I think you have. Your attempted counter (or the color change thing in the recent post) treats it as a number, and suggests taking its modulus relative to base 10 or 3. What is the lowest digit of the number of the final step? If there is no such number, then the output of your scenario is undefined, which is very differnt from the digit counter displaying a value of 'undefined', or an undefined lamp state somehow violating the law of excluded middle by being in some state between on and off.
Infinity means unbounded, which means there is a physical location and time interval.for any task n That's what makes it meaningful, and it only works if physicality is presumed not discreet.
Also, there is no violation of physics like faster-than-light movement as suggested by the OP.
I can pass a tortoise without contradiction. That shows that at least one of three (two explicit, one implicit) premises are false. But it doesn't necessarily have to be the premise you just mentioned there, that supertasks are impossible.
I'm ignoring it because those contradictions arise from a 4th premise (that there is a final step), one which I don't accept.
Quoting Ludwig V
Why is the passing of a tortoise necessarily not a supertask, as described by Zeno, and given a presumption of continuous physics?
Quoting Michael
Undefined by the description. That is to say, the color of the box afterwards is not a defined thing, which is different than it displaying the color of 'undefined'.
Quoting Metaphysician Undercover
Michael did very nicely with his first line in his reply.
Your reading comprehension skills are also off. I never suggested converting you to some opinion other than the one which you hold. I simply suggests that you seem incapable of understanding alternatives, to the point where you don't understand people who presume one of these alternatives.
Your most recent reply demonstrates this, as does the frustration evident in Michael's reply just above.
Yes, it can be explained in a way that makes sense, but apparently only to others.
And so it is meaningless to claim that such a supertask can complete. The fact that we can sum an infinite series is a red herring.
Quoting Ludwig V
Correct.
Quoting Ludwig V
I think it's because they are interesting puzzles, and because they help teach certain concepts.
Quoting Ludwig V
Yes- that's a better way to describe it.
[Quote]More than that, surely, there can be a mapping of some mathematical series into the actual kinematic world. Perhaps some similarity between those series is what creates the illusion?[/quote]
The allure of supertasks is the illusion of being able to complete an infinite process in a finite amount of time. I'm not sure there's anything comparable.
I think they are interesting because they dangle the prospect of completing a task and persuade us to ignore the reality of the impossibility of the task.
It is rather like a lottery. A lottery ticket is sold by dangling the prospect of a big win without any noticeable effort; some people focus on that and ignore the probabilities. The way the proposition is presented makes a difference to the view that people tend to adopt.
Quoting noAxioms
Quoting Relativist
Maybe I've misunderstood what a supertask is. Are there not different kinds of cases?
In the case of Achilles, we know that the task can be completed, but it is presented to us in a form in which it cannot be completed. I mean that we know that Achilles will pass the tortoise and even calculate when with simple arithmetic (no infinities required). But then the same problem, presented in a different way, seems to suggest that it cannot. So it's a question of how you choose to present the problem.
The staircase is different. It gives us a task (going down the infinite stairs) that cannot be completed, but links completion of the task to another process which also cannot be completed, but has a limit.
We can assign those numbers as we take each step. That's counting, and it's perfectly meaningful.
Perhaps you mean there's no way to say we can meaningfully complete the counting of all the steps. That's true, but it seems to be contradicted by the fact that the infinite process completes before the 1 minute mark. .
Quoting noAxioms
OK, but speed of light limitations put a physical limit on how fast the stairs can be descended, so that it eventually becomes physically impossible to descend a step in the prescribed period of time. The minimum size limitation also relates to a physical impossibility. But I'm making the stronger claim that it is logically impossible.
Quoting noAxioms
The entire exercise is abstract, but the scenario is written in terms of the kinematic (not abstract) process of descending stairs: each step is a motion, taking place in a finite amount of time.
Quoting noAxioms
Taking a single step is an act. The acts are performed in a sequence (from step n to step n+1). The term (sequence) is not referring to the entire collection. The task is to reach the bottom of the stairs (as stated in the description in the first post of this thread). Perhaps you can already see that it's trivial: it's actually impossible to reach the bottom of the stairs, since there is no bottom to a staircase with infinitely many stairs.
It depends on how the race is framed. It CAN be described as a supertask, wherein Achilles runs to a series of destinations, each established by where the tortoise is located when he begins each leg of the race. In that case, Achilles never actually reaches the turtle, he just gets increasingly closer. If you frame it in terms of constant speeds by both, then it's not a supertask - it's a different kind of puzzle.
The lack of a defined number for the last task does not prevent completion (by the all-tasks definition), so I regard your statement as a non-sequitur.
It does prevent completion if completion is defined as the removal of the green ball in the bag of a dozen non-green balls (see green ball example above), so I agree with you there.
Quoting Ludwig VSeveral here have been defining completion effectively as measuring the value of the final task, and that instance I suppose differs from Zeno's that specifies no such requirement.
In the case of Achilles, we know that the task can be completed, but it is presented to us in a form in which it cannot be completed.[/quote]Only because he posits a second premise incompatible with the first. 1) Supertasks are possible (by demonstration). 2) Supertasks are impossible, a second premise that isn't in any way justified.
Well, keystone suggested that Zeno denies this, and M-U suggests that time somehow stops due to the offense we've given it. Anyway, I agree with you, but it requires that implied premise that empirical evidence is valid.
This suggests fallacious reasoning in the second presentation. Most of the fallacies I've seen posted seem to be based on the premise of there being a limiting step. It's why I like Bernadete's Paradox of the Gods (see post ~30) which explicitly leverages the lack of there being a limiting step, and drives that to a seemingly paradoxical result. That's a harder one to wave off.
Just not physically. Mathematically it can, but then the story mentions 'the bottom' which implies something final that 'no more stairs' does not. So it lacks rigor.
The Littlewood-Ross Paradox illustrates an interesting way to treat infinities that highlights the dangers of treating infinities as numbers. From the SEP supertask page:
"We have a jar and a countably infinite pile of balls, numbered 1, 2, 3, 4, . First we drop balls 110 into the jar, then remove ball 1. (This adds a total of nine balls to the jar.) Then we drop balls 1120 in the jar, and remove ball 2. (This brings the total up to eighteen.) Suppose that we continue in this way ad infinitum, and that we do so with ever-increasing speed, so that we will have used up our entire infinite pile of balls in finite time. How many balls will be in the jar when this supertask is over?"
The answer is, as is argued by just about everybody: Zero. At any finite step n, there are n*9 balls in the jar. But after the supertask is complete, there is no final step Z with a state of Z*9 balls. In fact, every ball is numbered, and we know when it went in and when it went out. There is no exceptions to this, so the jar is empty at the end. Totally not intuitive, but not necessarily contradictory. Arguments against it have been attempted.
Quoting RelativistI'm trying to get a justification of that claim without the addition of the necessity of a final step, which would by definition be contradictory.
Has always meant 'prostate specific antigen' to me. I get my PSA checked at least once a year.
OK, 'act' is a step (go half the remaining way to the goal). 'task' is a goal (pass the tortoise).
It makes sense now, thanks.
Doing successive steps does not get you past the tortoise unless the passing of the tortoise is done by one of the steps. That's the same as suggesting a final step, which suggests that infinity is a number. I cannot buy into that PSA statement. It is just a rewording of the 'do the last step' definition of completion, a definition which only works for tasks requiring a finite number of steps.
Yes, the PSA entails taking a final step. We agree infinity is not a number, so there is no final step.
[Quote]I cannot buy into that PSA statement. [/quote] Show the PSA is false.
Quoting noAxioms
Why? The claim is indeed justified by the necessity of a final step for completion. Simply denying a final step is necessary doesn't make it so - you have to explain why it's not necessary for a kinetic task to require a final step in order to be completed.
Of course it's not a physical possibility.
If by metaphysical you really mean physical, then it's not a metaphysical possibliity.
But clearly we humans have the ability to conceptualize infinite sets and infinite processes, and we can even formalize the idea and get freshman calculus students to get a passing notion of the idea. If metaphysics includes abstract concepts created by humans, then infinite sets and mathematically infinitary processes are definitely part of metaphysics.
But it depends on what you mean by metaphysics. There is no doubt in my mind whatsoever that infinite sets, infinite sequences, and the theory of convergent infinite series have mathematical existence. Whether you include that in your metaphysics is up to you, but the mathematical existence of convergent infinite series is beyond dispute.
Quoting Michael
You are now talking about a physical process. Of course we can not write out infinitely many terms of the series. That has nothing to do with the mathematical truth expressed as 1/2 + 1/4 + ... = 1.
Quoting Michael
Nobody is writing anything down and this is not a physical process and you are entirely wasting your time trying to convince me that we can't physically write down an infinite series because I already know that.
Quoting Michael
In reality? In the physical world? No, I deny them entirely. It's tiresome to argue against your representation of positions I don't hold.
Quoting Michael
I never claimed any such thing. I have no idea why you think I claimed any such thing. Supertasks can be defined abstractly, as in limiting processes. They are not physically instantiable as far as we currently know.
As far as the "final state," think of it as a function on the ordered set
{1/2, 3/4, 7/8, ..., 1}.
We can define a function any way we like. We can assign 1 to 1/2, 0 to 3/4, 1 to 7/8, and so forth.
We are then entirely free to define the value of the function at 1. We can call it pumpkin if we simply declare pumpkin to be an element of our output set.
There is no requirement that the value of a function at any point is required to be any particular thing. Functions are pretty much arbitrary. Just like Cinderella's coach. A coach at 1/2 second before midnight. A coach at 1/4 second before midnight. Dot dot dot. And then a pumpkin at the stroke of midnight.
It's a perfectly legal function. It just doesn't happen to be continuous. But it's perfectly legal to define a function that's a coach at each of infinitely many elements of a sequence, and then a pumpkin at the final limit point.
Mathematically it's just a function [math]f : \{\omega + 1\} \to \{ \text{coach, pumpkin}\}[/math]
where [math]\omega + 1[/math] is just the order type of the set {1/2, 3/4, 7/8, ..., 1}.
I agree it's about a right-open interval. We have 1/2, 3/4, .. in (0,1). We can adjoin 1 to work in (0,1].
Quoting Relativist
But I'm not talking about anything else! This is purely a mathematical problem! There is no lamp that switches in arbitrarily small intervals of time. Adding time to this problem confuses the issue. It makes people think there's a physical component to the problem when there isn't. It's purely mathematical.
Quoting Relativist
There isn't any physical process to speak of. The lamp is fictional. Purely mathematical, a function on {1/2, 3/4, ...} with its completion defined in {1/2, 3/4, ..., 1}
Quoting Relativist
It may "lead" somewhere but there's no law that constrains the final state. It may be discontinuous, like Cinderella's coach that's a coach at 1/2, 1/4, 1/8, ... seconds before midnight, then becomes a coach at midnight. That's why it's perfectly possible that the lamp becomes a pumpkin after 1 second.
Quoting Relativist
Lost me there, limit of what is infinity? If you put a symbolic "point at infinity" after the natural numbers and you define a function on the augmented set 1, 2, 3, 4, ..., [math]\infty[/math], you can define a function on the augmented set whose value at infinity is anything you like.
Quoting Relativist
Kind of lost me here. The process 1, 2, 3, ... never ends, but we can still stick a symbolic point at infinity. Just like we can add the point 1 to the set {1/2, 3/4, 7/8, ...} to make {1/2, 3/4, 7/8, ..., 1}
This is an example of a supertask:
I write down the first ten natural numbers after 30 seconds, the next ten natural numbers after 15 seconds, the next ten natural numbers after 7.5 seconds, and so on.
According to those who argue that supertasks are possible I can write out infinitely many natural numbers in 60 seconds.
Examples such as Thomson's lamp show that supertasks entail a contradiction. So even though it is true that 30 + 15 + 7.5 + ... = 60, it does not follow that the above supertask is possible.
It makes no sense to claim that I stopped writing out the natural numbers after 60 seconds but that there was no final natural number that I wrote.
The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal.
Either PSA is wrong or the premise is. In neither case is PSA valid for a supertask.
Simply asserting that such a step is necessary doesn't make it so, especially when it being the case directly violates the initial premise. That violation does very much demonstrate not only the lack of necessity of a final step, but the impossibility of it, given the premise.
I don't know how the task being 'kinetic' changes the argument. You can phrase it as a n inertial object overtaking a slower one in frictionless space.
Issues that I see: The problem is 1 dimensional as phrased: The position of Achilles is given only in x. To overtake the tortoise, he'd have to collide with it, so he has to be off to the side,. If he's off to the side, there's at least two axes x and y. If they're in 2D+ space, then which of the two is in front is dependent on the chosen orientation of the axes. If you hold the orientation stable throughout the exercise, then the scenario still holds as described.
None of that seems to have any relevance to your reqirement of a rephrasing around 'kinetic'. Why does that word somehow invalidate the premise?
Quoting Michael
Nice try Michael, but "ground state" is an ideal which does not occur in nature. It's like a "blackbody" and things like that, ideals used for theory, which do not ever truly exist. Like "60 seconds" in the OP it is approached as a limit, but never truly achieved.
Besides, what you've provided does nothing to resolve the problem I explained. It's just the same as your example of "a year". The activity is what occurs, time passes. What you gave is the definition of "a second", that means that a second is the duration of time which is equivalent to that activity. A second is not that activity itself. So the activity occurs, and there are no seconds unless someone determines the duration of time required for the specified number of oscillations. That's why beginning and end points are required which constitute a measurement, just like I explained to you in the case of "a year".
Quoting Michael
I have no problem accepting that things exist even when we're not around, and I gave you examples of such things, dogs and cats. However, what you are not getting, is that some labels refer to things created by human beings, these are called "ideals". So, we have two categories of labeled things, natural things which exist even when we're not around, and things created by minds, such as ideals, which are dependent on minds. And, you refuse to distinguish between the natural things, and the artificial ideals. So you insist that because it is a labeled thing, it must exist even when were not around. That's the folly of Platonism. Do you not understand that "the second" is a mathematic object of ideal time, just like "the circle" is a mathematical object of ideal space?
Quoting noAxioms
I think you misunderstand. I understand the alternatives very well, so well in fact that I can comprehend the contradictions which inhere within some of these alternatives. So I see the need to reject them. The people who assert, and insist on some of these alternatives do so without proper understanding themselves. They accept and assert by the force of convention, which is simple prejudice, and they refuse to acknowledge the contradictions within, assuming that convention cannot be faulty.
Or the PSA is correct, and the goal can't be met.
Quoting noAxioms
I'm not merely asserting it. You have to agree that a final step is necessary for completion when there are finitely many steps. Why would it matter if the number of steps is infinite?
Most importantly: What does it even mean for a kinematic process to be infinite? My answer: it means the process continues forever and does not end. What's your answer?
Quoting noAxioms
Here's how: the infinity is manifested as a never-ending kinetic process.
Points on a number line exist concurrently (in effect). Steps in a kinetic process do not: they occur sequentially, separated by durations of time.
I'm going to defer commenting on the Achilles/tortoise problem. It just clouds the issue with the stairway supertask.
I'm not sure it is even a puzzle if it is framed in terms of constant speeds by both. Let's say Achilles gives the tortoise a head start of 100 units of length, that Achilles runs at 11 units per second and the tortoise at 1 unit per second. So, at time t seconds after the tortoise is at 100 units from the start, the tortoise will be at 100 + t units from the start, and Achilles at 11t units. These will be the same - 110 units - at time t = 10 seconds. (This was suggested to me by a friend.) It seems OK to me, but perhaps I'm wrong to think that it will generalize.
But I don't really want to pursue it. I just wanted to point out that a puzzle can be the result of framing the question in the "wrong" way. Everyone seems intent on resolving the staircase problem on its own terms, which seems to me a mistake.
I showed that for a supertask, the PSA is not correct. So no, this cannot be for a supertask.
Quoting Relativist
Because a contradiction results from making that additional assertion. In the example given, it is a very direct contradiction.
Quoting Relativist
If the process continues forever, by definition it isn't a supertask. It's a different process than the one being discussed.
I don't have an answer because I don't understand what 'kinematic' adds to the issue.
Quoting RelativistI don't know what is meant by this. 'Concurrently' means 'at the same time' and there isn't time defined for a number line.
A number line seems to be a set of ordered points represented by a visual line. It can be defined otherwise, but functionally that seems sufficient. It being a visual aid, it seems physical, but a reference to the simultaneity of the positions along the line seems irrelevant to the concept.
Quoting RelativistOK. I buy that. But this works mathematically as well, so 'kinetic' doesn't add anything. I can draw the worldlines of Achilles and the tortoise on some medium and all you get is two lines that cross at some point. The axes on the plot are x and t, so in this mathematical representation, the steps do not occur simultaneously, but are separate durations of time. What did 'kinetic' add to that?
I'm trying to understand your point about how the word somehow is relevant.Quoting Relativist
OK, this has been about the stairway. There is no objective kinematics about that since it involves a space-like worldline, so the steps are not unambiguously ordered in time. The ordering of the steps becomes ambiguous due to relativity of simultaneity, and it becomes meaningless to use the word 'sequential' in this context.
Hence my always referencing the tortoise example since it hasn't any physical ambiguities like that. There are still frame dependent fact, so for instance in another frame, it is the tortoise trying to overtake Achilles, both of whom are facing backwards.
Those are my thoughts on 'kinematics'.
No, you didn't. You merely asserted: "The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal." There is no direct violation.
Here's valid logic:
1. A halfway step cannot reach the goal.
2. All steps are halfway
3. Therefore the goal cannot be reached.
You merely asserted the goal is reached (directly contradicting #3) but didn't explain how the sequence of halfway steps somehow reaches the goal. Labeling the process a "supertask" is handwaving, not proof. Show your logic.
Quoting noAxioms
Fair enough, I misstated it. The process does not continue forever, however there is no end to the process.
Let's compare the supertask to a scenario in which the time interval between each step is a constant (e.g. 1 second). You'll agree that this process does not complete, right? But this process has a 1:1 correspondence to the supertask -- for every step taken in one scenario, there's a parallel step taken in the other. This suggests that either they both complete, or neither completes.
Quoting noAxioms
My point was that the kinematic stair-stepping process has a temporal element that is not reflected in a number line.
The number line in question is an interval that is open on the right: i.e. it includes all points <1, but not including 1. There are infinitely many points in this interval, but the point "1" isn't one of them. So the process cannot reach 1, and 1 is the goal of the process. The goal is therefore unreachable by the kinematic process.
1. A given halfway step cannot reach the goal.
2 There is a specific step that reaches the goal (per PSA)
3 Therefore this final step is not a halfway step (1 & 2)
4 Any given step is halfway (per Zeno)
You don't find this contradictory?
Quoting RelativistThere is a temporal end to it, a final moment if not a final step.
There is a bijection yes. It does not imply that both or neither completes.
This reminds me of some of the discussion behind Gabriel's horn, and attempting to suggest that its infinite area implies that it has infinite volume.
Yes, your example here very much illustrates how a deliberate abstraction can be made to be singular at any chosen point, in this case tying infinite time to a finite duration. Yes, this works even in uncountable infinities: There is a bijection between the space from 0 to 1 and the space from 1 on up, by the simple relation of y = 1/x. This in no way implies that 1 meter cannot exist.
The 'process' can go beyond the end of the line despite it ending before the goal. This is sort of a different issue since you're putting an uncountable set of points between 0 and 1. Why not just 1/2, 1/4, ...
Disagree. The kinematic process isn't restricted to only points on the number line.
You're continuing to argue against a position I don't hold. Why are you doing this? There's no interesting conversation to be had. Supertasks are not consistent with known physics. We're agreed on that.
I would, however, disagree with you that being inconsistent with known physics is the same as logical impossibility. Known physics changes all the time, sometimes radically.
Because I'm arguing against the possibility of a supertask. You're the one who interjected with talk of mathematical limits. I'm simply responding to explain that this doesn't address the concern I have with supertasks.
Quoting fishfry
I'm not saying that it's the same. I'm saying that as well as being a physical impossibility, supertasks are also a metaphysical impossibility.
No physical law can allow for an infinite sequence of events to be completed. The very concept of an infinite sequence of events being completed leads to a contradiction. To claim that it is metaphysically possible to have finished writing out an infinite number of natural numbers but also that there is no final natural number that I wrote is to talk nonsense.
If I finished writing out any number of natural numbers than there will be a final natural number and that natural number will be a finite number. This is a metaphysical necessity.
Ok.
Quoting Michael
Now that's something I disagree with. But I don't care about supertasks much so it's better if I don't engage.
Quoting Michael
This is an open question. Of course no physical law currently known allows for supertasks, but you can't say what we will regard as physical law in another couple of centuries.
Quoting Michael
You keep repeating that, but you have no evidence or argument.
Quoting Michael
Do you deny infinite mathematical sets?
Quoting Michael
Mathematically that's not true. The set {1, 2, 3, 4, ...} contains all the natural numbers, but there's no last number.
I already agree with you that there are no infinite collections of physical objects according to currently accepted theories of physics. But you can't claim that there will never be any such theory.
And besides, eternal inflation posits a temporally endless universe. It's speculative, but it's part of cosmology. Serious scientists work on the idea. So at least some scientists are willing to entertain the possibility of a physically instantiated infinity.
No. An infinite set is not an infinite sequence of events. An infinite sequence of events would be counting every member of an infinite set. It is metaphysically impossible to finish counting them.
Quoting fishfry
That's not relevant to the claim I'm making.
I'm saying that if I have finished counting the members of some set then some member must be the final member I counted.
Quoting fishfry
I don't deny the possibility of something not ending. The issue is that supertasks entail that there is an end to infinity, which is nonsense.
Quoting fishfry
Thomson's lamp, my box changing colour, the example of writing out each natural number, etc. I've offered plenty. Your attempt to rebut them by reference to mathematical limits fails to address the issue.
Of course it is, but the the contradiction can be resolved by denying either one of two premises. You chose to deny the PSA, and I responded that the PSA could be true - we'd merely have to reject the other relevant premise - that the goal is reached. You have not made an argument that shows it is more reasonable to deny the PSA than to deny the reaching of the goal. I don't think it make sense to deny that a completed task entails a final step.
Quoting noAxioms
Sure. You have to agree the PSA is true for finite tasks. Is there something different about infinite tasks? It doesn't seem so: consider the process: stepping increasingly closer to temporal point in time 1, but the process never actually reaches it. So the goal is unreachable by the process.
Quoting noAxioms
No need. I understand that the math shows that the series reaches a point of convergence at time 1. However: the kinematic process never actually reaches time 1. That's why the series doesn't adequately account for the kinematic process -and why I've stressed we need to examine the process, not just do the math on the mathematical series.
Quoting noAxioms
On the contrary, there's a logical impediment to reaching the goal through the process: the process does not reach time 1.
Quoting noAxioms
I'm actually basing my claims on real analysis, which analyzes the characteristics of real numbers - including the associated infinities.
Quoting noAxioms
That makes no sense. The process does not have a final moment. because there are infinitely many moments prior to time 1. There is no end to the series of kinematic steps, in spite of the fact that the mathematical series converges.
Quoting noAxioms
Why not?
Quoting noAxioms
No it can't - that is logically impossible. The process entails taking steps with increasing shorter durations: 1/2 second, 1/4, 1/8,.... The process can only approach 1, it can never reach it.
No! Each new step is half the duration of the last step, and this halving process has no end.
Ok. Clearly this is a matter of semantics.
Mathematically, if I have a set of events [math]\{e_1, e_2, e_3, \dots \}[/math], there's no problem whatsoever.
You seem to assign some meaning to the word "event" that I don't understand. Must an event be physical? In probability theory we have events that need not be physical, such as the probability of choosing a random real number between 0 and 1/3 from the unit interval. That's an event with no physical meaning at all.
An infinite sequence of events from the set I defined above would be [math]e_1, e_2, e_3, \dots [/math]. No muss no fuss. That's an infinite sequence of events.
Perhaps you can tell me what an event is, bearing in mind that event is a technical term in probability theory that does not imply physicality.
https://en.wikipedia.org/wiki/Event_(probability_theory)
Quoting Michael
The claim you're making is not one I'm disputing.
Quoting Michael
I disagree. Counting means to place the elements of some set in order-bijective correspondence with the natural numbers, or in a more general context, with some ordinal.
By that definition, we can easily count the natural numbers. The identity map will do.
You seem to think counting is a physical process. That's fine for most contexts, but it's not the only meaning of counting.
For example we have the famous countable/uncountable distinction between infinite sets. A set is countable if it can be placed into bijection with the natural numbers. The natural numbers, the integers, the rational numbers, and the algebraic numbers are all famous examples of countable sets that are infinite.
If you mean to say that we can't physically count the natural numbers, of course I agree. I personally could not get past 13 or 14 or so without losing interest. We could use a supercomputer, but even that has finite capacity. We could use the entire observable universe, but that contains only [math]10^{78}[/math] atoms. So sure, physical counting is constrained by resources.
But who's saying otherwise? Perhaps you can explain that to them, since I have never said anything remotely like that.
That's no surprise. It is an imaginary contrivance impossible to physically fabricate. Just a thought.
@Michael keeps making the same claim, and I do not understand the argument.
I agree that it's impossible to do infinitely many physical things in finite time according to present physics.
I do not see what the logical impossibility is.
That's like saying today would be April 29 even if there was never any human beings to determine this. If you can't understand how this is wrong, I don't know what else to say.
Zeno again?
Say (in some hypothetical world, say current math or future physics) that we have a "sequence of actions" as you say, occurring at times 1/2, 3/4, 7/8, ... seconds.
It's perfectly clear that 1 second can elapse. What on earth is the problem?
You are falling into the trap of thinking a limit "approaches" but does not "reach" its limit. It does reach its limit via the limiting process, in the same sense that 1/2, 3/4, 7/8, ... has the limit 1, and 1 is a perfectly good real number, and we all have had literally billions of experiences of one second of time passing.
I can't imagine what you are thinking here, to claim that one second of time can't pass.
I have repeatedly noted in this thread that we can symbolically adjoin a "point at infinity" to any countably infinite sequence, and that's where the limit lives. We can note that 1/2, 3/4, 7/8, ... has the limit 1, which lives in the ordered set {1/2, 3/4, 7/8, ..., 1}.
We can also do the same thing in the integers as 1, 2, 3, 4, ..., [math]\omega[/math], where [math]\omega[/math] can be thought of as a formal symbol that's greater than every natural number. It also has technical importance as the first transfinite ordinal.
Either way, sequences do "reach" their limit via the limiting process, though the sequence itself does not necessarily attain the limit. It's just semantics.
You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly?
Would you prefer the term "act"? It is metaphysically impossible for an infinite succession of acts to complete.
Have you even looked up supertasks? I don't know how you can confuse them with mathematical sets.
Metaphysically impossible? Repeating a claim ad infinitum is neither evidence nor proof.
Quoting Michael
I'm not the one advocating for supertasks, yet you keep arguing with me that they are impossible.
No, I'm responding to you to explain that your reference to mathematical sets and mathematical limits does not address the issue with supertasks.
Quoting fishfry
I've provided arguments, and examples such as Thomson's lamp that shows why. And again, your reference to mathematical sets and mathematical limits does not rebut this.
I gave you a mathematical model that puts your unsupported claims into context.
Quoting Michael
Thompson's lamp shows nothing of the sort. I've explained that to you repeatedly as well.
And it doesn't address the issue.
If I write the natural numbers in ascending order, one after the other, then this can never complete. To claim that it can complete if we just write them fast enough, but also that when it does complete it did not complete with me writing some final natural number, is just nonsense, and so supertasks are nonsense.
That we can sum an infinite series just does not prove supertasks. It is clearly a fallacy to apply an infinite series to an infinite succession of acts.
I asked you to consider a hypothetical world and you pretended I was talking about mathematical sets.
Quoting Michael
Yes, the observable universe is finite. We're agreed on that. How many times are you going to try to convince me of something I've already agreed with many times?
Quoting Michael
I have not claimed otherwise.
Quoting Michael
According to current physics. That's as far as we can go.
Quoting Michael
Nor does it disprove their metaphysical possibility. We just don't know at present.
Those who argue that supertasks are possible claim otherwise, and it is them I am arguing against. You're the one who interjected.
Quoting fishfry
If I write the natural numbers in ascending order, one after the other, then it is metaphysically impossible for this to complete (let alone complete in finite time). This has nothing to do with what's physically possible and everything to do with logical coherency.
It's physically impossible. I have no idea why you keep claiming it's "metaphysically" impossible or logically incoherent. What's logically incoherent about infinite sets and transfinite ordinals? You just keep repeating the same unsupportable claims. You can count the natural numbers by placing them into bijective correspondence with themselves. This is the standard meaning of counting in mathematics.
I'm not talking about infinite sets and transfinite ordinals. I'm talking about an infinite succession of acts. If you can't understand what supertasks actually are then this discussion can't continue.
The key parts are "sequence of operations" and "occur sequentially".
As in, I do one thing, then I do another thing, then I do another thing, and so on ad infinitum. It is metaphysically impossible for this to end. If it ends then, by definition, it is not ad infinitum.
A discussion can't continue when you keep making unsubstantiated, evidence-free claims.
I would invite you to read up on eternal inflation, a speculative cosmological theory that involves actual infinity. Yes it's speculative, but nobody is saying it's "metaphysically impossible" or "logically incoherent."
https://en.wikipedia.org/wiki/Eternal_inflation
Quoting Michael
Please stop embarrassing yourself.
Which has no bearing on what I'm arguing.
You are not arguing, you're repeating your lack of argument. I'll let you have the last word, you are incapable of rational discussion.
This isn't the sense of "counting" I'm using. The sense I'm using is "the act of reciting numbers in ascending order". I say "1" then I say "2" then I say "3", etc.
Quoting fishfry
P1. It takes me 30 seconds to recite the first natural number, 15 seconds to recite the second natural number, 7.5 seconds to recite the third natural number, and so on ad infinitum.
P2. 30 + 15 + 7.5 + ... = 60
C1. The sequence of operations[sup]1[/sup] described in P1 ends at 60 seconds without ending on some final natural number.
But given that ad infinitum means "without end", claiming that the sequence of operations described in P1 ends is a contradiction, and claiming that it ends without ending on some final operation is a cop out, and even a contradiction. What else does "the sequence of operations ends" mean if not "the final operation in the sequence is performed"?
So C1 is a contradiction. Therefore, as a proof by contradiction:
C2. P1 or P2 is false.
C3. P2 is necessarily true.
C4. Therefore, P1 is necessarily false.
And note that C4 doesn't entail that it is metaphysically impossible to recite the natural numbers ad infinitum; it only entails that it is metaphysically impossible to reduce the time between each recitation ad infinitum.
[sub][sup]1[/sup] A happens then (after some non-zero time) B happens then (after some non-zero time) C happens, etc.[/sub]
There is no limiting process in the premises of the op, nor in what is described by . The "limiting process" is a separate process which a person will utilize to determine the limit which the described activity approaches. Therefore it is the person calculating the limit who reaches the limit (determines it through the calculation), not the described activity which reaches the limited.
Quoting fishfry
You're pointing to the limit of a mathematical series. A step-by-step process does not reach anything. There is no step that ends at, or after, the one-minute mark. Calculating the limit does not alter that mathematical fact.
I also think you are misinterpreting the meaning of limit. This article describes it this way:
[I]In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value...
The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |an ? L| is the distance between an and L.[/i]
Quoting fishfry
No, I didn't. I said the stair-stepping PROCESS doesn't reach the 1 second mark. Are you suggesting it does?
What is it about 'physical' that makes this difference? Everybody just says 'it does', but I obviously can physically move from here to there, so the claim above seems pretty unreasonable, like physics is somehow exempt from mathematics (or logic in Relativist's case) or something.
You italicize 'according to present physics', like your argument is that there's some basic flaw in current physics that precludes supertasks. How so?
I mean, I can claim that there are no physical supertasks, but only by presuming say some QM interpretation for which there is zero evidence, one that denies physical continuity of space and time. By definition a supertask, physical or otherwise, is completed. If it can't, it's not a supertask.
Quoting RelativistYou defined the second task as a non-supertask, requiring infinite time. That's why not.
I can play that game with a finite list of three steps, with the middle step of one task requiring one to make a square circle. It does not follow that the other list of three steps cannot be completed in a short time just because there exists a bijection between the steps of the two tasks.
Quoting Metaphysician UndercoverExactly so.
Your disagreement with views that suggest this is a subject for a different topic. Your displayed lack of comprehension of what the person means when he says things like that is either in total ignorance of the alternatives or a deliberate choice. Being the cynic I am, I always suspect the latter. It's my job as a moderator elsewhere.
I do thank you for verifying my earlier assessment.
Quoting fishfry
For the record, I am personally advocating that they have not been shown to be physically impossible. All the 'paradoxes' that result are from inappropriately wielding finite logic in my opinion.
Thomson's lamp is a wonderful example of this, but other examples seem to have more bite.
Quoting fishfry
Does it? It seems to be a more complex model that suggests stupid sizes for 'what is', but not 'actual infinite' more than the standard flat model that comes from the cosmological principle. Yes, I know the page you link mentions 'hypothetically infinite' once. I have a deep respect for the eternal inflation model since something like it is necessary to counter the fine-tuning argument for a purposeful creation.
I agree with Michal that the sort of infinity suggested by eternal inflation is not representative of a supertask. I do realize that some people just deny 'actual infinity' of any kind, but that is not justified, hence is not evidence.
Are you saying that you believe that there would still be an April 29, even if there never was any human beings with their time measuring techniques, and dating practises? And do you believe that there would still be "seconds" of time without those human beings who individuate those temporal units in the act of measurement? I don't understand how you can believe that I should accept this as a reasonable alternative. Care to explain?
Physics indeed is not exempt from logic. It's logically impossible to reach the 1 minute mark when all steps (even if there are infinitely many of them) fall short of the 1 minute mark.
Calculating the limit does not entail a process that reaches that limit. This is a misinterpretation of the concept of limit.This article describes it this way:
[I]In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value...
The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |an ? L| is the distance between an and L.[/i]
Well physics is of course exempt from math and logic. The world does whatever it's doing. We humans came out of caves and invented math and logic. The world is always primary. Remember that Einstein's world was revolutionary -- overthrowing 230 years of Newtonian physics. The world told us what new math to use. The world is not constrained by math, nor logic, nor by any historically contingent work of fallible man.
Math and even logic have always been drawn from looking at the world around us. So just as an aside to the main discussion, but responding to this one sentence that caught my eye ... physics IS exempt from math and logic. Meaning that historically, and metaphysically, physics is always ahead of math and logic and drives the development of math and logic.
But to the main question, the physical/mathematical distinction is important. I can never count all the integers in the physical world (as far as we know -- to be clarified momentarily); but in math I can invoke the axiom of infinity, declare the natural numbers to be the smallest inductive set guaranteed by the axiom, and count its contents by placing it into order-bijection with itself. That is: The identity map on the natural numbers is an order-preserving bijection that shows that the natural numbers are countable.
The former is a physical activity taking place in the world and subject to limitations of space, time, and energy. The latter is a purely abstract mental activity. How meat puppets such as ourselves come to have the ability to have such lofty abstract thoughts is a mystery. And if we are physical beings; and if thoughts are biochemical processes; are not our thoughts of infinity a kind of physical manifestation? That's another good question.
Perhaps our very thoughts of infinity are nature's way of manifesting infinity in the world.
So bottom line it's clear to me that we can't count the integers physically, but we can easily count them mathematically. And the reason I say that we can't physically do infinitely many things in finite time "as far as we know," is because the history of physics shows that every few centuries or so, we get very radically new notions of how the world works. Nobody can say whether physically instantiated infinities might be part of physics in two hundred years.
Quoting noAxioms
Not a flaw, of course, any more than general relativity revealed a flaw in Newtonian gravity. Rather, I expect radical refinements, paradigm shifts in Kuhn's terminology, in the way we understand the world. Infinitary physics is not part of contemporary physics. But there is no reason that it won't be at some time in the future. Therefore, I say that supertasks are incompatible with physics ... as far as I know.
I utterly reject the notion that supertasks are a logical contradiction or metaphysical impossibility. They're only a historically contingent impossibility. We split the atom, you know. That was regarded as a metaphysical impossibility once too.
Quoting noAxioms
I'm not being specific like that. I'm only saying this:
[b]There have been radical paradigm shifts in physics in the past;
There will certainly be radical paradigm shifts in the future; and
The next shift just may well incorporate some notion of physically instantiated infinities or infinitary processes; in which case actual supertasks may be on the table.
[/b]
I analogize with the case of non-Euclidean geometry; at first considered too absurd to exist; then when shown to be logically consistent, considered only a mathematician's plaything, of no use to more practical-minded folk; and then shown to be the most suitable framework for Einstein's radical new geometry of spacetime.
Mathematical curiosities often become physicists' tools a century or more later. I think it's perfectly possible that physically instantiated infinities may become part of mainstream physics at some point in the future.
I will close with two contemporary examples of where speculative physics is starting to think about infinity.
One, eternal inflation. That's a theory of cosmology that posits a fixed beginning for the universe, but no ending. In this eternal multiverse are many bubble universes; either infinitely many, or at least a very large finite number. Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them.
And two, the many-world interpretation of quantum physics. Most people have heard of the Copenhagen interpretation, in which observing a thing causes the thing to be in one state or another; whereas before the measurement, it was neither in one state nor the other, but rather a superposition of the two states.
In Everett's many-world's interpretation, an observation causes the thing to be in both states in different universes. The universe splits in two, one in which the thing is in one state, and another universe it's in the other state. In some other universe I didn't write this. I know it sounds like bullshit, but Sean Carroll, a very smart guy and a prominent Youtube physicist (he's a real physicist too) is a big believer. He's recently moved away from mainstream physics, and more into developing a new philosophy of physics that incorporates many-worlds. How many worlds are there? Again this is a little vague, infinitely many or a large finite number.
These are just two areas I know about in which the idea of infinity is being taken seriously by speculative physicists. Would anyone really bet that they personally can predict the next 200 years of physics?
Quoting noAxioms
Well I can walk a mile, and I first walked the first half mile, and so forth, so it's a matter of everyday observation that supertasks exist. That would be an argument for supertasks. Zeno really is a puzzler. I don't think the riddle's really been solved.
Well that's for reading, there's been a lot of back and forth lately and I hope I was able to at least express what I think about all this.
Yes, I agree with you that math and physics use different definitions.
I apologize for getting crabby last night. As I went to bed I was thinking, Why am I snarling at someone about supertasks, I don't even care about supertasks.
You're right, I was not the one you were originally addressing. I jumped in because I was annoyed by your total lack of logic in claiming that supertasks are metaphysically impossible or logical contradictions. I agree with you that supertasks don't exist physically today, but I allow for the possibility of new physics in the future, just as there's always been new physics in the past. I don't think you've supported your metaphysical or logical arguments. That's why I jumped in.
Also it's perfectly clear that I can walk a mile, and I first walked the first half mile, etc., so if someone (not me, really!) wanted to argue that supertasks exist on that basis, I'd say maybe they have a point.
Quoting Michael
I think "reciting natural numbers" is a red herring, because it's perfectly clear that there are only finitely many atoms in the observable universe, and that we can't physically count all the natural numbers.
But let me riddle you this. Suppose that eternal inflation is true; so that the world had a beginning but no end, and bubble universes are forever coming into existence.
And suppose that in the first bubble universe, somebody says "1". And in the second bubble universe, somebody says, "2". Dot dot dot. And bubble universe are eternally created, with no upper bound on their number.
Therefore: Under these assumptions, there is no number that doesn't get spoken. And therefore, all the numbers are eventually counted.
You see we don't have to "reach the end," since we can't do that. All we have to do is show that there is no number that never gets counted. Therefore they all do. It's a standard inductive argument. You show something's true for all natural numbers because there can't be a smallest number where it's not true.
I remind you that while eternal inflation is speculative but is taken seriously by a lot of smart people.
Therefore I claim that it is metaphysically possible to physically count the natural numbers; and that no logical contradiction is entailed. I'll grant you that I haven't yet shown how to do it in finite time, and so I have not refuted your point. I'm giving more of a plausibility argument that someday, there might actually be a finite-time supertask. We just don't know. You personally can not know. That's my real point, bottom line.
You cannot know what future physics will allow or conceptualize. That's my whole argument. That's why I say that supertasks violate contemporary physics, Planck scale and all that. But based on the shocking paradigm shifts of the past, there will be shocking paradigm shifts in the future; and physically actualized infinitary processes are as good a candidate as any for what comes next.
I wrote a response to @NoAxioms above in which I laid out my thoughts, it might be of interest ... https://thephilosophyforum.com/discussion/comment/900398
Thanks again for your good cheer in not firing back!
Wow that's deep. Deep and wrong at the same time. That's interesting.
If I am understanding you: You say that if we have a sequence; that if that sequence happens to have a limit, then the limit is not inherent to the sequence, but is rather imposed by the observer.
I suppose the analogy is color, which is in the eye-brain system of the observer, not in the object or even in the light.
But actually, the limit can be considered part of the sequence. Just as a sequence is a function defined on the natural numbers; a sequence along with its limit can be defined as a function on the natural numbers augmented with a point at infinity, which I've been calling [math]\omega[/math].
It's really no different than taking the set {1/2, 3/4, 7/8, ...} and augmenting it with the number 1, to yield the new set {1/2, 3/4, 7/8, ..., 1}. Surely you can see that 1 is a perfectly sensible number on the number line. In many ways it's the ONLY sensible number. All other numbers are derived from it. That and 0. Give me 0 and 1 and I'll build all the numbers anyone needs.
So if that's what you're saying, I find that a very interesting thought. But there is no reason to imbue limits with mysticism. They're very straightforward. They're just the value of a sequence at the augmented point at infinity; which, if you don't like calling it that, is just adding the number 1 to the 1/2, 3/4, ... sequence.
You can think of it that way. Or you can think of it "reaching" its limit at a symbolic point at infinity. Just as we augment the real numbers with plus and minus infinity in calculus, to get the extended real numbers, we can do something analogous with the natural numbers, and augment them with a symbolic point at infinity [math]\omega[/math], so that the augmented natural numbers look like this:
1, 2, 3, 4, ..., [math]\omega[/math]
Now a sequence is just a function that for each of 1, 2, 3, ..., we assign the value of the sequence, the n-th term. And we can simply assign the limit as the value of the function at [math]\omega[/math]. It's perfectly legitimate. We can define a function with ANY set as its domains. So a sequence is a function on [math]\omega[/math], and a sequence augmented with its limit (or any other value!) is just a function on [math]\omega + 1[/math].
This is a key point. I've stated it a number of times recently and I'm not sure I'm getting through. The natural numbers augmented with a point at infinity is a perfectly good domain for a function; and we can use such a function to identify each of the points of a sequence, along with the limit.
Quoting Relativist
On a forum our words must speak for themselves. But in this instance I can assure you that nothing could possibly be farther from the truth.
Quoting Relativist
Wiki is not necessarily a good source for mathematical accuracy or subtlety of expression, and in this case they have led you astray.
I hope very much that you will give some thought to what I wrote about defining the limit of a sequence as the value of some function on the naturals augmented with a symbolic point at infinity; or more concisely, as a function on [math]\omega + 1[/math]. A sequence is just a function on [math]\omega[/math], which is a synonym for [math]\mathbb N[/math]. You can "attach" the limit to the sequence by extending the same function to one on [math]\omega + 1[/math] .
I hope this is clear. I find it an extremely clarifying mental model of what's going on with a sequence and its mysterious limit. "Where does the limit live?" I get that it's kind of confusing. The limit lives at the point at infinity stuck to the right end of the natural numbers.
1, 2, 3, 4, ... [math]\omega[/math]
That's how people need to learn how to count in order to better understand supertasks and limits.
Quoting Relativist
Sure, after 1 second. It's perfect obvious from daily existence. When I got up to make a snack I did first walk halfway to the kitchen then halfway again. So now I'm arguing for supertasks. But I could just as well argue against supertasks. So whatever you said, I could probably convince myself to agree with it.
I think the word "reach" is being abused in this conversation. It comes out of badly taught calculus classes, and Wikipedia.
That's not quite what I'm saying. The process described by the op has no limit. That should be clear to you. It starts with a first step which takes a duration of time to complete. Then the process carries on with further steps, each step taking half as much time as the prior. The continuity of time is assumed to be infinitely divisible, so the stepping process can continue indefinitely without a limit. Clearly there is no limit to that described process
I think what's confusing you into thinking that there is a limit, is that if the first increment of time is known, then mathematicians can apply a formula to determine the lowest total amount of time which the process can never surpass. Notice that this so-called "limit" does not actually limit the process in any way. The process carries on, unlimited, despite the fact that the mathematician can determine that lowest total amount of time which it is impossible for the process to surpass.
Clearly, the supposed "limit" is something determined by, and imposed by, the mathematician. To understand this, imagine the very same process, with an unspecified duration of time for the first step. The first step takes an amount of time, and each following step takes half as much time as before. In this case, can you see that the mathematician cannot determine "the limit"? All we can say is that the total cannot be more than double the first duration. But that's not a limit to the process at all. It's just a descriptive statement about the process. It is a fact which is implied by an interpretation of the described process. As an implied fact, it is a logical conclusion made by the interpreter, it is "not inherent to the sequence", but implied by it.
That it is not inherent, but implied, can be understood from the fact that principles other than those stated in the description of the process, must be applied to determine the so-called "limit".
Oh I had no idea we were still talking about the OP. This thread's gone way beyond that.
I thought you were making a more general point, that the limit lives in a different kind of conceptual space than the sequence itself, or that the limit was imposed on the sequence by observers.
If I misunderstood then nevermind. I've long forgotten the staircase problem. I don't think I ever actually understood it.
Quoting Metaphysician Undercover
Well 1/2 + 1/4 + 1/8 + ... is a well known convergent sequence. It converges to 1. And surely we've all experience one second going by. So that's the paradox, right?
Quoting Metaphysician Undercover
It has not been productive in the past for us to discuss mathematics, and your misunderstanding of limits is not my job to fix. I gave at the office. Nothing personal but you know we have been down this road. I sort of get what you're saying but mostly not. "The process carries on, unlimited, even though there's a limit." I haven't the keystrokes to untangle the myriad conceptual difficulties with that statement, and the beliefs and mindset behind it; even if I had the inclination. I hope you'll forgive me, and understand.
Quoting Metaphysician Undercover
LOL. And the meaning of Moby Dick is only because of what we all determined the symbols to mean. Man and His Symbols, Jung. Yes we are symbolic beasts.
But within the sphere of math, the definition of a limit is as objective as can be. We lay down a definition, you know the business with epsilon and L, and we confirm that the sum converges; just as in the sphere of the English language, Moby Dick is a story about a bunch of guys who go whaling and it mostly doesn't end well.
Quoting Metaphysician Undercover
I'm sorry, I can't really talk about the staircase problem specifically, I never paid much attention to it at the beginning. I mostly got interested in this thread when other issues were introduced. But mathematicians are very good at determining limits, and the one in question is perfectly well known to everyone who ever took a year of calculus. You might take a look at the Wiki page on limits.
Quoting Metaphysician Undercover
You don't need any esoteric "principles other than those stated in the description of the process" to determine the sum of a geometric series as a particular limit.
Then rather than recite the natural numbers I recite the digits 0 - 9, or the colours of the rainbow, on repeat ad infinitum.
It makes no sense to claim that my endless recitation can end, or that when it does end it doesn't end on one of the items being recited let alone that it can end in finite time.
So I treat supertasks as a reductio ad absurdum against the premise that time is infinitely divisible.
Yes, it does. But there is a small but significant mistranslation there. I have no problem with saying that "infinite" means "endless", but "ad" does not mean "without". It means "to".
So there are two different ways of thinking about infinity embedded here. One thinks of infinity as a destination, which, paradoxically, cannot, by definition, be reached. The other doesn't, but rather denies that there is a destination. At first sight, one wants to say that the second is correct.
But how do we know that the operation "+1" generates a sequence without end? In one way, it doesn't seem absurd to think that it might be done. After all, given that "+1" can be defined, the result of every step is determined (fixed) - the whole sequence is always already there, for us to inspect. But that seems a mistake; we can't survey the whole sequence and notice that there is no end.
Well, there is the proof that there can be no largest (natural) number. We can prove lots of other things, as well. We don't have to survey the whole sequence to do any of that. It really is magical, and yet inescapably logical. (Poetry? Perhaps. But this is about how we think about things, so it is also philosophy.)
So it seems that we are locked into two incompatible ways of thinking about infinity. One as if it were a sequence which stretches away for ever. The other as a succession of operations which can be continued for ever. (Two metaphors - one of space, one of time.) I'm not suggesting it needs to be resolved, just that we are subject to confusion and need to think carefully, but also recognize that our normal ways of thinking here will need to be adapted and changed.
Quoting fishfry
Quite so. That's why these puzzles are not simply mathematical and why I can't just walk away from them.
Quoting Metaphysician Undercover
Yes, quite so. But it follows that applying the calculus to Achilles doesn't demonstrate that Achilles will overtake the tortoise. I think that only ordinary arithmetic can do that.
Quoting Michael
Does it make any sense to claim that you can repeat the digits ad infinitum? All you can do is repeat the digits again and perhaps promise or resolve to repeat them again after that.
The truths of mathematics and logic are timelessly true, aren't they? There is no change in that world. We frame them in the present tense, but it is, grammatically speaking, the timeless present, not the present that is preceded by the past and followed by the future. We speak of logical operations, but what does that mean? Their results are always already fixed - determinate. So when we carry them out - nothing in logic changes.
Good source. It says that the limit is approached as the input approaches the specified value.
This means that the limit isn't reached at some finite point in the series, exemplified by the comment:
"This means that the value of the function f can be made arbitrarily close to L, by choosing x sufficiently close to c"
The approaching goes on while x is still at some finite step.
Since x reaches infinity at time 1, all steps are completed at that time, so the task is complete.
Quoting Metaphysician Undercover
Read carefully. I didn't say that.
I said
1) that discussion of the question above and your personal beliefs in the matter is off topic
and
2), that you [cannot / choose not to] understand what others mean when they presume what Michael conveyed better than I could:
Quoting Michael
You (M-U) seem to either not be able to separate "X" and Y, or you refuse to communicate with those that do.
My personal beliefs in this matter are irrelevant. I simply know what somebody means when they treat Y as something independent of "X".
That my dog is named "Bella" depends on me. That Bella exists and eats and sleeps does not depend on me.
That this period of time is named "60 seconds" depends on us. That 60 seconds pass does not depend on us.
You don't seem to understand how reference works.
I can explain it very easily. There is two different senses of "limit" being used here. One is a logical "limit" as employed in mathematics, to describe the point where the sequence "converges". And "unlimited" is being used to refer to a real physical boundary which would be place on the process, preventing it from proceeding any further. There is no such "limit" to a process such as that described by the op. The appearance of paradox is the result of equivocation.
Quoting Ludwig V
The problem, as displayed in my reply to fishfry above, is that each of the two incompatible ways will use the same terms. The same terms will then have incompatible meanings as you demonstrate with "ad infinitum", and the natural tendency for human beings not to take the time required to detect such differences, leads to equivocation and the appearance of paradox.
Quoting Ludwig V
That's right, and the issue with Achilles and the tortoise has extra complexities which are often overlooked. Achilles and the tortoise are both moving. At any point in time, t1, Achilles is at a location and the tortoise is at a location. At t2, Achilles reaches the location where the tortoise was at t1. But the tortoise has moved to a new location. At t3 Achilles reaches the location where the tortoise was at t2, the tortoise has moved to a new location. And so on. In this case, therefore, it is not only impossible for Achilles to overtake the tortoise, it is also impossible for Achilles to catch up to the tortoise, so it appears like the tortoise will necessarily beat Achilles to the finish line.
We cannot describe the tortoise's position as a simple limit to Achilles' position, because the tortoise is already moving at a constant velocity, and no matter how fast Achilles accelerates he cannot catch up to the tortoise. This is the problem of acceleration, which demonstrates the fundamental incompatibility between distinct rest frames. Einstein attempted to bridge this incompatibility by stipulating the speed of light as the limit, (therefore absolute rest frame) in his special theory of relativity.
Quoting noAxioms
That's why I was asking, "are you saying...". You seemed to be saying what I asked about, but I was not sure. Now you are making it clear that you were saying something else. So I now understand that you were attacking a straw man effigy of me, and that's why I couldn't understand you.
Quoting noAxioms
If I don't understand, then I don't understand. I believe that I said that I don't understand why someone would make an assumption which to me is so clearly false. So to explain why someone would make such an assumption, would require an explanation as to how this obviously false assumption makes some sense to the person. The quoted passage, in which you say Michael makes sense of his perspective, makes no sense to me, because it is a straw man representation of what I am arguing. I told Michael that I didn't understand how this was relevant, and he needed to explain its relevance. I now fully understand that it was a straw man, and that's why I couldn't apprehend the relevance. Michael did not understand what I was arguing, therefore produced a straw man representation of me, and I could not understand what Michael was accusing me of, because he did not understand me and tried to defend against a straw man of me.
Quoting noAxioms
I have no problem distinguishing between the sign and what is signified. The problem here is that Michael refused to recognize the distinction between when the thing signified is a real physical object, and when the thing signified is a mental fabrication, a fiction, or an ideal. So Michael was talking about ideals, principles of measurement, like a day, and a second, as if these mental fabrications have some sort of existence independent from human minds, in some Platonic realm or something like that.
Quoting noAxioms
I'll repeat, just so that you'll quit with this utterly ridiculous straw man accusation you are hitting me with. I have no problem whatsoever with the separation between the symbol and the thing represented by the symbol. That is not at all what is at issue here, so you are simply making a straw man representation of what I am arguing.
The problem here is with the nature of the thing represented. Do you understand that sometimes what is represented by a symbol is a real physical object, and sometimes what is represented by a symbol is an ideal, such as a mathematical object? And do you understand that these two types of objects are completely different, and need to be understood in completely different ways. My claim is that it is highly doubtful that an ideal such as "a second" has any existence independent from the human minds which I believe fabricate them, and propagate them through educational processes.
Michael has been arguing that "a second" has some sort of real independent existence, just like a physical object which we can point to, instead of being an ideal which is propagated by human minds, as I claim. So, I've asked him to show me such a thing as "a second". All he has provided is a definition referring to some ideal state (ground state) of a cesium atom. Since the definition refers to an ideal state (ground state), rather than any real existing physical object, it seems very clear that "a second" is an ideal, not an independent physical thing.
Quoting Michael
This is not a simple issue of reference, it is an issue of the type of thing which is referred to. In the example here "seconds" is an ideal, just like any other principle of measurement, metre, foot, degree Celsius, circle, triangle, and number in general. They are all ideal, and are therefore dependent on the human minds which employ them, unless you assume some sort of Platonic realm for the independent existence of such ideals.
So, we say "time is passing", just like we say "the earth is spinning around the sun", and we refer to real independent things with those phrases, "time", "the earth", "the sun". However, when we go to measure those things, we employ principles, which are ideals, and have no such independent existence. So, when we say "60 seconds has passed", "60 seconds" refers to an ideal which has been applied to measure the independent passing of time. The 60 units of seconds do not pass, time passes, and "60 seconds" refers to the ideal which is employed.
For example, we could say "the earth circles the sun". In this case, "circles" refers to the ideal which is employed in measuring the earth's motions. We know that a circle is an ideal figure, and that true circles do not have any real existence independently from human minds, because pi, which is the essential feature of a circle, is an indefinite, irrational number. So the motion of the earth isn't really a circle, that's just the ideal which is referred to in that statement. Likewise, in the statement "60 seconds" has passed, "seconds" is just the ideal which is referred to, as the principle employed in the act of measurement. And we find these ideals in all forms of measurements, metres, degrees of temperature, frequencies, etc..
I do agree with you that people seem not to understand the meaning of limit in this context. Many of them seem to think that calculus solves the problem, though it clearly doesn't.
As to special relativity, I'm not going to argue with you. Most people neglect acceleration, on the grounds that including it won't make any material difference. Does Einstein's theory tell us when Achilles will overtake the tortoise? I thought that both started at the same time, and therefore in the same rest frame. When Achilles overtakes the tortoise, won't they both be in the same rest frame?
Neglecting acceleration, let's say Achilles gives the tortoise a head start of 100 units of length and that Achilles runs at 11 units per second and the tortoise at 1 unit per second. So, at time t seconds after the tortoise is at 100 units from the start, the tortoise will be at 100 + t units from the start, and Achilles at 11t units. These will be the same - 110 units - at time t = 10 seconds.
There are similar paradoxes that don't involve two moving elements:-
[quote=Wikipedia;https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Arrow_paradox] Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on.[/quote]
You explained your interpretation:
Quoting fishfry
So...you're thinking of a limit in a vauge way ("symbolic"), and vaugely asserting the series "reaches" infinity, and then rationalize this with a mathematical system that defines infinity as a number.
Although it's true that there are such mathematical systems, it doesn't apply to the supertask. Time is being divided into increasingly smaller segments approaching, but never reaching, the 1 minute mark. There is a mathematical (and logical) difference between the line segments defined by these two formulae:
A. All x, such that 0<=x < 1
B. All x, such that 0<=x <= 1
Your blurred analysis conflates these, but it is their difference that matters in the analysis. The task maps exactly to formula A, but not to formula B (except in a vague, approximate way). Mathematics is about precise answers.
QM on the other hand was quite a hit, especially to logic. Still, logic survived without changes and only a whole mess of intuitive premises had to be questioned. Can you think of any physical example that actually is exempt from mathematics or logic?
QM is also the road to travel if you want to find a way to demonstrate that supertasks are incoherent.
Zeno's primary premise is probably not valid under QM, but the points I'm trying to make presume it is.
What is this 'the former'? The physical activity of making a declaration? There's definitely some abstraction going on there, as there is with any deliberate activity.
The latter seems to be the expression of a rule that maps the two halves of the bijection, which seems to be about as physical of an activity as was the declaration.
No argument here.
Depends on what you mean by count, and especially countable, since plenty of equivocation is going on in this topic.
If you mean mentally ponder each number in turn, that takes a finite time per number, and no person will get very far. That's one meaning of 'count'. Another is to assign this bijection, the creation of a method to assign a counting number to any given integer, and that is a task that can be done physically. It is this latter definition that is being referenced when a set is declared to be countably infinite. It means you can work out the count of any given term, not that there is a meaningful total count of them.
Sorry, but what? I still see no difference. What meaning of 'count them' are you using that it is easy only in mathematics?
That doesn't follow at all since by this reasoning, 'as far as we know' we can do physically infinite things.
I never made the claim that a supertask is physically possible. I simply followed through with it as a premise, which, unless falsified, can be physically true 'as far as we know'.
They've been a possibility already, since very long ago. It's just not been proven. Zeno's premise is a demonstration of one.
You italicize 'according to present physics', like your argument is that there's some basic flaw in current physics that precludes supertasks. How so?
noAxioms
Not a flaw, of course, any more than general relativity revealed a flaw in Newtonian gravity. Rather, I expect radical refinements, paradigm shifts in Kuhn's terminology, in the way we understand the world. Infinitary physics is not part of contemporary physics. But there is no reason that it won't be at some time in the future. Therefore, I say that supertasks are incompatible with physics ... as far as I know.
QM does very much suggest the discreetness of matter, but Zeno's premise doesn't rely on the continuity of matter. It works best with a single fundamental particle moving through continuous space and time, and overtaking another such particle.
They were never off the table since current physics doesn't forbid them. Maybe future physics will for instance quantize either space or time (I can think of some obvious ways to drive that to contradiction). Future findings take things off the table, not put new ones on. The initial state of physics is "I know nothing so anything is possible'.
Heh, despite the detractor standing on an obvious example of such a geometry.
Octonians shows signs of this sort of revolution.
Actually, the big bang theory already does that much.
Yes, I know about eternal inflation, and something like it seems necessary for reasons I gave in my prior post.
It is a mistake to talk about 'time creating these other universe'. Time, as we know it, is a feature/dimension of our one 'universe' and there isn't that sort of time 'on the outside'. There is no simultaneity convention, so it isn't meaningful to talk about if new bubbles are still being started or that this one came before that one.
All that said, the model has no reason to be bounded, and infinite bubbles is likely. This is the type-II multiverse, as categorized by Tegmark. Types I and III are also infinite, as is IV if you accept his take on it. All different categories of multiverses.
That's the type III.
In Everett's many-world's interpretation, an observation causes the thing to be in both states.[/quote]Ouch. Is that a quote? It did not match any google search.
Observation for one is a horrible word, implying that human experience of something is necessary for something fundamental to occur. This is only true in Wigner interpretation, and Wigner himself abandoned it due to it leading so solipsism.
I don't buy into MWI, but bullshit is is not. It is easily the most clean and elegant of the interpretations with only one simple premise: "All isolated systems evolve according to the Schrodinger equation". That's it.
Everett's work is technically philosophy since, like any interpretation of anything, it is net empirically testable.
I would have loved to see Einstein's take on MWI since it so embraces the deterministic no-dice-rolling principle to which he held so dear.
Ah, local boy. I am more used to interacting with those who walk a km. There's more of em.
Quoting fishfryThat wording implies a sort of meaningful simultaneity that just doesn't exist.
The universes in eternal inflation theory are not countable.
Yes, each step in a supertask can and does have a serial number. That's what countably infinite means.
Quoting MichaelYou're not going to get past step 10 at best. I just takes longer than the step duration to recite a syllable. I don't think this is your point, but it's a poor wording due to this. Yes, step 13 has a defined duration at known start and stop times. The duration simply isn't long enough to recite anything.
No. It means 'without final step'. You're apparently equivocating "without end" to mean that the process is incomplete after any amount of time.
There we go with the finite definition again.
"The sequence of operations ends" means that "all operations in the sequence are performed".
This is a great example of the endless repetition of assertions/bad-definitions I'm seeing in this topic. Surely you know this answer is coming from me.
Is it? You take supertasks to mean that time is discrete instead of continuous, meaning there is a smallest amount of time. If that is so, P2 is necessarily true according to mathematical theorems, but it is not representative of reality and especially not of time.
Relevant:
The operations in the sequence occur one after the other, so all operations are performed only if some final operation is performed.
The logic of consecutive tasks is different to the logic of concurrent tasks. Your account equivocates.
If I never stop counting then I never stop counting, and if I never stop counting then at no time have I ever counted every number.
At the 60 second mark I said 0, at the 30 second mark I said 1, at the 15 second mark I said 2, at the 7.5 second mark I said 3, etc.
Is it metaphysically possible for such a task to have been performed? No, because there is no first number that I could have started with.
That we can sum an infinite series with terms that match the described (and implied) time intervals is a red herring.
There is a far more fundamental, non-mathematical, logical impossibility with having counted down from infinity, and that very same fundamental, non-mathematical, logical impossibility applies with having counted up to infinity as well. You're being bewitched by maths if you think otherwise.
Infinity is not reached. You're not considering what it means to be infinite in this context: it means continually dividing the remaining time (prior to the 1-minute mark) in half. Because the remaining time corresponds to a real number line, the process proceeds without ending because the remaining time is infinitely divisible. It's limited by the fact that all points of time that are reached by the process are less than 1 minute- so it is logically impossible for this process to reach the point of time of 1 minute.
The source of confusion is that the clock does hit the 1-minute mark. You are incorrectly interpreting this as implying the the process reaches that point. It can't, because it is logically impossible.
The clock reaching the 1-minute mark implies the process ends, but since the process cannot reach the 1-minute mark, the process must be terminated at some point. Laws of nature would clearly provide a limit to how small we divide the time, but even setting that aside - a stopping point is logically necessary. The math doesn't identify any particular stopping point, but it does imply there has to be one.
An exercise in free will. At each n the cube changes from white to black or vice-versa for time = 1-1/2^n. The clock runs out so you are free to say the process has ended and the final color is black. Your friend can say, no it is white. But you will prevail.
Mathematicians would just refer to it as an "upper bound."
But you talk about a "real physical boundary." Here you imagine that the staircase is physical. It's not. The conditions of the problem violate known laws of physics.
It's only a conceptual thought experiment. And why shouldn't math apply to that?
But anyway, it's an upper bound. If it's a least upper bound, it's a limit.
I think a lot of people feel that way.
The natural numbers do not end, yet they have a successor in the ordinal numbers, namely [math]\omega[/math]. This is an established mathematical fact.
I regard this as a helpful point of view when analyzing these kind of puzzles. I've explained it as best I can.
"It makes no sense" is not a logical argument. It's only a description of your subjective mental state. Once, violating the parallel postulate or the earth going around the sun or splitting the atom made no sense. You are not making an argument.
Quoting Michael
If you only demonstrated the reductio. All you have is "it makes no sense," and that is not an argument.
Quoting fishfry
I've watched this debate for a long time - though I don't claim to have understood all of it. But I think those two quotes show that you are talking past each other.
I didn't like ? at all, when it was first mentioned. I'm still nowhere near understanding it. But the question whether a mathematical symbol like ? is real and a number is simply whether it can be used in calculations. That's why we now accept that 1 and 0 are numbers and calculus and non-Euclidean geometries. ? can be used in calculations. So that's that. See the Wikipedia article on this for more details.
But it is also perfectly true that a recitation of the natural numbers cannot end. As I said earlier, it is remarkable that we can prove it. Yet we cannot distinguish between a sequence of actions that has not yet ended from one that is endless by following the steps of the sequence. So we are already in strange territory.
In the way I'm describing this, you may think that the difference is between the abstract world (domain) of mathematics and another world, which might be called physical, though I don't think that is right. I'm very puzzled about what is going on here, but I'm pretty sure that it is more about how one thinks about the world than any multiverse.
No. My thinking about limits is extremely precise and perhaps a bit more general than what you're accustomed to. I have never said that a series (or sequence if that's what you mean here) reaches infinity. I would not say that, and I did not say that.
What I said was that there is a mathematical view that sheds light on the subject, and makes it clear in where the limit of a sequence lives. The sequence 1/2, 3/4, 7/8, ... has the limit 1. Of course it never "reaches" 1. But you would have no objection to my putting {1/2, 3/4, ..., 1} into a set together. After all, I am allowed to take unions of sets: and {1/2, 3/4, ...} U {1} = {1/2, 3/4, ..., 1}. So it's a legit set.
Now 1 is in no way a "point at infinity," after all it's just the plain old number 1. And no member of the sequence ever "reaches" it. But it does live there as the limit; as the result of a well-defined limiting process.
I have suggested this mathematical model as a thought aid to these kinds of paradoxes. If you find it helpful all to the good, but if not, that's ok too. I find it helpful.
For what it's worth, in math, the natural numbers have an upward limit, called [math]\omega[/math], that plays the same role for the sequence 1, 2, 3, ... that the number 1 is for the sequence 1/2, 3/4, ...
It's the limit. It's more general notion of limit, one that allows us to reason about a "point at infinity." Which is exactly what these puzzles are about. That's why it's a handy framework for thinking about these kinds of puzzles.
You have a sequence that's defined (on/off, on a step, whatever) at each member of a convergent sequence; and you want to speculate on the definition at the limit. [math]\omega[/math] is exactly what you need; or rather, a set called [math]\omega + 1[/math], which is like the set {1/2, 3/4, ,,,, 1}. It's a set that contains an entire infinite sequence and its limit. It's exactly what we need to analyze these problems.
If it helps, here's the Wiki page on ordinals, at least so that you know they're a real thing. You can "keep counting past the natural numbers," and you get some very cool mathematical structures. Ordinals find application in proof theory and mathematical logic.
Quoting Relativist
I'm going to defer talking about supertasks today, had enough for a while.
Quoting Relativist
Please reread what I wrote. This is not on topic if you understand what I'm saying.
Quoting Relativist
I'm doing my best to fit you with a sharper pair of mathematical eyeglasses to unblur your vision ... but you keep making a spectacle of yourself!!
Quoting Relativist
You might consider using words like "reach" and "approach" with precision. They are not part of the mathematical definition of a limit. They're casual everyday synonyms that you are allowing to confuse you.
He'll come around :-)
Quoting Ludwig V
This paragraph gratified me. If you are struggling to understand my posts then I'm getting through to at least one person. My talk about [math]\omega[/math] is something most people haven't seen, but the ideas aren't that hard. For what it's worth there's a Wiki page on ordinal numbers. The page itself isn't all that enlightening, but it does at least show that the ordinal numbers really are a thing in math, I'm not just making it all up.
You have a great insight that what makes a mathematical concept real is, in the end, its utility. Sometimes not even to anything practical, but just to math itself. We want to solve the equation x + 5 = 0 so we invent negative numbers. That kind of thing. In that sense, the ordinals exist.
But another way to think about it is that it's just an interesting new move in a game. As if you were learning chess and they told you how the knight moves. You don't say, "Wait, knights slay dragons and rescue damsels, they don't move like that." Rather, you just accept the rules of the game. You can think of ordinals like that. Just accept them, work with them, and at some point they become real to you. Just like the moves of the chess pieces any other formal game.
But I have tried to give a very concrete, down to earth example of how this works.
Suppose that we have the sequence 1/2, 3/4, 7/8, ... It converges to 1.
Now we can certainly form the set {1/2, 3/4, 7/8, ..., 1}. It's just some points in the closed unit interval.
But it gives us a model, or an example, of a set that contains an entire infinite sequence that "never ends" blah blah blah, and also contains its limit.
If you believe in the set {1/2, 3/4, 7/8, ..., 1}, then you should have no trouble at all believing in the set
{1, 2, 3, ..., [math]\omega[/math]}. That's also just a set that contains an entire infinite sequence, along with its limit. We typically don't encounter this concept in the math curriculum that most people see, but it's perfectly standard once you go a little further. Also a lot of people have seen the extended real numbers with [math]\pm \infty[/math] and nobody complains about that, or do they?
It's true that the distances are different inside the two sets. But in terms of order, the two sets are exactly the same: an infinite series, along with its limit.
Anyway, this framework is very handy for understanding supertask type problems. That's why I'm mentioning it.
So if you don't like [math]\omega[/math], that' s no problem. Just think about {1/2, 3/4, 7/8, ..., 1}. It's the exact same set, with respect to what we care about, namely the property of being an infinite sequence followed by one extra term that occurs after the sequence.
Does that help?
Quoting Ludwig V
That's a confusing way to think about it. It "ends" in the sense that we can conceptualize all of the natural numbers, along with one extra thing after the natural numbers.
And if we can't imagine that, we can certainly imagine {1/2, 3/4, 7/8, ..., 1}. There's nothing mysterious about that. An entire infinite sequence is in there, along with an extra point. It's a legitimate set.
If you want to think about the sequence 1/2, 3/4, 7/8, ... "never ending," that's fine. Yet we can still toss the entire sequence into a set, and then we can toss in the number 1. That's how sets work. They are containers for infinite collections of things.
By the way, [math]\omega[/math] is the "point at infinity" after the natural numbers. And [math]\omega + 1[/math] is the name for the set {1, 2, 3, 4, ..., [math]\omega[/math]}.
[math]\omega + 1[/math] is the natural setting for all supertask puzzles. We have the state at each natural number, and we inquire about the final state at [math]\omega[/math].
That's why I like [math]\omega + 1[/math] as a mental model for these kinds of problems.
Quoting Ludwig V
This business about actions is what confuses people. They set up scenarios that violate the laws of physics, like the lamp that switches in arbitrarily small intervals of time, and then they try to use physical reasoning about them. Then they get confused.
Quoting Ludwig V
Well yes, you are correct to feel that it's not quite right. Because there is nothing physical about the lamp or the staircase. So it's a category error to try to use everyday reasoning about the physical world. That's why people get confused.
Quoting Ludwig V
I think it all comes down the fact that calculus classes care about computation and not theory. That, and the fact that we don't know the ultimate nature of the world, and there's are good reasons to think it's not anything like the mathematical real numbers.
So on the one hand, the continuity of the world is an open question. And two, calculus classes are not designed to teach people how to think about limits in the more general ways that mathematicians sometimes do. Put those together with quasi-physical entities like physics-defying lamps, and you have a recipe for confusion.
And as I keep explaining, the issue with supertasks has nothing to do with mathematics. Using mathematics to try to prove that supertasks are possible is a fallacy.
See here.
Quoting fishfry
Yes. I got enough from it to realize a) that ? is one of a class of numbers and b) that it comes after the natural numbers (so doesn't pretend to be generated by "+1")
Quoting fishfry
Certainly. That's what needs to be clarified, at least in my book. There's a temptation to think that actions must, so to speak, occur in the real world, or at least in time. But that's not true of mathematical and logical operations. Even more complicated, I realized that we continually use spatial and temporal terms as metaphors or at least in extended senses:-
Quoting fishfry
What does "after" mean here?
Quoting fishfry
Yes, but it seems to me that this is not literally true, because numbers aren't objects and a set isn't a basket. (I'm not looking for some sort of reductionist verificationism or empiricism here.)
Quoting fishfry
In that respect, yes. But I can't help thinking about the ways in which they are different.
Quoting fishfry
Yes. But it doesn't end in the sense that we can't count from any given natural number up to the end of the sequence.
Quoting fishfry
I try not to mention this in public, but the fact is that I never took a calculus class, nor was I ever taught to think about limits or infinity in the ways that mathematicians sometimes do. I did a little formal loic in my first year undergraduate programme. Perhaps that's an advantage.
Quoting Michael
Fair enough. That coincides with my intuition that supertasks are not possible. But given that they are not physically possible either, can I conclude that they are not possible at all?
Quoting fishfry
I have the impression that you don't think that they are mathematically possible either. (I admit I may be confused.) So does that mean you don't think that supertasks are possible?
I really don't see how there could be a staircase which is not physical. That really makes not sense. However, just like in the case of the word "determine", we need to allow for two senses of "physical". You seem to be saying that to be physical requires that the thing referred to must obey the laws of physics. But the classic definition of "physical" is "of the body". And when a body moves itself, as in the case of a freely willed action, that body violates Newton's first law. Therefore we have to allow for a sense of "physical" which refers to things which are known to violate the laws of physics, like human beings with freely willed actions.
What is implied here is that the laws of physics are in some way deficient in their capacity for understanding what is "physical" in the sense of "of the body". That's why people commonly accept that there is a distinction between the laws of physics and the laws of nature. The laws of physics are a human creation, intended to represent the laws of nature, that is the goal, as what is attempted. And, so far as the representation is true and accurate, physical things will be observed to obey the laws of physics, but wherever the laws are false or inaccurate, things will be observed as violating the laws of physics. Evidently there are a lot of violations occurring, with anomalies such as dark energy, dark matter, etc., so that we must conclude that the attempt, or goal at representation has not been successful.
Quoting fishfry
Sure, it's a conceptual thought experiment, but the interpretation must follow the description. A staircase is a staircase, which is a described physical thing, just like in Michaels example of the counter, such a counter is a physical object, and in the case of quantum experiments, a photon detector is a physical object. And of course we apply math to such things, but there are limits to what we can do with math when we apply it, depending on the axioms used. The staircase, as a conceptual thought experiment is designed to expose these limits.
Quoting fishfry
OK sure, but that's a limit created by the axioms of the mathematics. So it serves as a limit to the applicability of the mathematics. The least upper bound is just what I described as "the lowest total amount of time which the process can never surpass". Notice that the supposed sequence which would constitute the set with the bound, has already summed the total. This is not part of the described staircase, which only divides time into smaller increments. It is this further process, turning around, and summing it, which is used to produce the limit. The limit is in the summation, not the division.
It is very clear therefore, that the bound is part of the measurement system, a feature of the mathematical axioms employed, the completeness axiom, not a feature of the process described by the staircase descent. The described staircase has no such bound, because the total time passed during the process of descending the stairs is not a feature of that description. This allows that the process continues infinitely, consuming a larger and larger quantity of tiny bits of time, without any limit, regardless of how one may sum up the total amount of time. Therefore completeness axioms are not truly consistent with the described staircase.
However, since our empirical observations never produce a scenario like the staircase, that inconsistency appears to be irrelevant to the application of the mathematics, with those limitations inherent within the axioms. The limitations are there though, and they are inconsistent with what the staircase example demonstrates as logically possible, continuation without limitation. Therefore we can conclude that this type of axiom, completeness axioms, are illogical, incoherent. The real problem is that as much as we can say that the staircase scenario will never occur in our empirical observations, we cannot conclude from this that the incoherency is completely irrelevant. We have not at this point addressed other scenarios where the completeness axioms might mislead us. Therefore the incoherency may be causing problems already, in other places of application.
Quoting fishfry
Interesting. Is it a countable set? I suppose it is, but only if you count the 1 first. The set without the 1 can be counted in order. The set with the 1 is still ordered, but cannot be counted in order unless you assign ? as its count, but that isn't a number, one to which one can apply operations that one might do to a number, such as factor it. That 'final step' does have a defined start and finish after all, both of which can be computed from knowing where it appears on the list.
This is not radical. The rational numbers are countable, but not if counted in order, so it's not a new thing.
If Zeno includes '?' as a zero-duration final step, then there is a final step, but it doesn't resolve the lamp thing because ? being odd or even is not a defined thing.
Which works until you ask if ? is even or odd.
Quoting MichaelTotally agree, but I'm not aware of anybody claiming a proof that supertasks are possible. Maybe I missed it.
The time it takes for the ball to finish a revolution and touch the table decreases by half every time. If time is continuous and infinitely divisible, as time approaches 60s, the number of skips goes to infinity, but while the ball is skipping, it does not reach 60 seconds. 60 seconds it exactly when the ball stops skipping, and there is no specified state as to what the table will do when that happens. Thus, the speed in which the table changes colours approaches infinity the closer you get to 60s, but this says nothing about what the table will do at 60s, you might as well say it will turn transparent.
So the ball keeps skipping and changing the table's colour.
I conclude the table and the ball have incomplete description too.
If supertasks had proven their case there would be no debate as to whether time is continuous or discrete, but it doesn't seem to be the case.
Quoting Michael
That is the reverse Zeno walk. Achilles starts running but he can't start running because there is no first lenght to be run. But yet we reverse time and Achilles can finish the task. The argument doesn't need to be reversed, it is the same as saying you can't count to infinity because there is no last numeber to be counted. But if we admit that time is infinitely divisible, counting to infinity doesn't seem to amount to a logical impossibility, and so we reverse the time of the task.
And that's where you're being deceived by maths. We can't have counted down from infinity because there is no first number and so we can't have counted up to infinity because there is no last number.
The fact that an infinite series can have a finite sum is a red herring in both cases.
Can we not count the intervals starting with 1? Would that number not tend towards infinity given time is infinitely divisible or approach a certain value and terminate given a smallest sliver of time exists?
"Tending towards infinity" means counting through the natural numbers - the set is infinite. The process has no end.
I know, I'm saying the second part as the alternative to time being infinitely divisible.
It's all good. :up:
And you are right anyways; I should have been clearer that I didn't think we would ever actually finish counting the number of intervals given time is infinitely divisible.
I don't really care much about supertasks and haven't argued that they're coherent or incoherent. I'm mostly trying to clarify some of the bad reasoning around them.
Quoting noAxioms
Ok. I think some of the quoting got mangled since things I said ended up as part of your post.
But if anyone thinks I can't count all the natural numbers 1, 2, 3, ... by mathematical means, please identify the first one I can't count.
Quoting noAxioms
To count a set means to place it into bijection with:
a) A natural number; or
b) the set of natural numbers, to establish countability; or
c) some ordinal number, if one is a set theorist or logician or proof theorist.
Quoting noAxioms
Lost me. As far as we know takes into account the great conceptual revolutions of the past, as evidence that there will be more such in the future.
Quoting noAxioms
Ok. I think I'm a little lost in the quoting and not actually sure what we are talking about here. I'm not strenuously defending whatever ideas you're concerned with.
Quoting noAxioms
Well ... ok.
Quoting noAxioms
But that's exactly my point. If speculative physics is starting to take physically instantiated infinity seriously, then it's perfectly reasonable that in the future, physically instantiated infinity may become a core aspect of physics; in which case supertasks may be on the table.
Quoting noAxioms
Was this for me? I never said any such thing nor quoted anyone else saying it.
Quoting noAxioms
I'll have to plead ignorance on the question of whether there's a meta-universal time that transcends the bubble universes. Good question though.
Quoting noAxioms
You are completely agreeing with my point. That if speculative physic already includes infinity, then mainstream physics may include infinity in the future.
Quoting noAxioms
You are agreeing with my point.
Quoting noAxioms
Nothing to do with my point, which is that speculative physics already includes infinity, therefore mainstream physics may include infinity after the next scientific revolution.
Quoting noAxioms
You're agreeing with me again. Why are you typing this stuff in? You've kind of lost me.
Quoting noAxioms
Ok.
Quoting noAxioms
Ok.
Quoting noAxioms
Depends on the exchange rate.
Quoting noAxioms
Wow. You have evidence for that? My understanding is that it is an open question in eternal inflation as to the cardinality of the bubbles: finite, countably infinite, or uncountable. But either way my point about reciting the integers stands. I don't actually get the sense that you're engaging with anything I wrote.
Quoting noAxioms
That's not the definition of supertask others are using. But I used the example of bubble universes to illustrate the possibility of counting the natural numbers physically.
Anyway sorry if I got lost in the quoting and didn't really understand some of your responses.
Thanks.
Quoting noAxioms
Of course it's a countable set. It's a subset of the rationals, after all. You are right that it's not order-isomorphic to 1, 2, 3, ...
Quoting noAxioms
Right. Exactly right. Point being that contemplating a set that includes an infinite sequence along with an extra point is nothing strange at all. And it serves as a nice conceptual model for supertask puzzles.
Quoting noAxioms
There is no final step. There is a point at infinity. Not quite the same. Unless you allow the limiting process itself as a step. It's just semantics.
Quoting noAxioms
It's neither, and who's asking such a thing? Even and odd apply to the integers.
Anyway if this is repetitive feel free to not reply. I just go through my mentions everyday trying to reply best I can. And I do have a thesis, which is that the ordinal [math]\omega + 1[/math] is the proper setting for the mathematical analysis of supertask puzzles. So I'll repeat that every chance I get.
But I'm not doing that. I haven't been doing that. Are you deliberately misunderstanding me or am I being unclear?
"Using mathematics to try to prove that supertasks are possible is a fallacy"
Who did that? Are they in the room with us right now?
Yes exactly. [math]\omega[/math] comes into existence via a limiting process. The idea is that the natural numbers are generated by successors, and the higher ordinals are generated by successors and limits. So we're adding a new rule of number formation, if you like. We go 1, 2, 3, ... by successors, and then to [math]\omega[/math] by taking a limit, then [math]\omega `=+ 1[/math], [math]\omega + 2[/math], etc., then eventually we get to [math]\omega + \omega[/math] by taking a limit, then we keep on going. I don't want to go too far afield, but the idea is that we can take successors and limits to get to all the higher ordinals.
Quoting Ludwig V
Right. A lamp that cycles in arbitrarily small amounts of time is not physical. A staircase that we occupy for arbitrarily small intervals of time is not physical. So trying to use physical reasoning is counterproductive and confusing. That's my objection to all these kinds of puzzles. People say there's a conflict between the math and the physics ... but as i see it, there's no physics either.
Quoting Ludwig V
Follows in order. Given 1, 2, 3, 4, ..., we can adjoin [math]\omega[/math] "at the end." What do I mean by that? I mean that we extend the "<" symbol so that
1 < [math]\omega[/math], 2 < [math]\omega[/math], 3 < [math]\omega[/math], and so forth. So that conceptually, every natural number is strictly smaller than [math]\omega[/math]. Does that make sense?
Quoting Ludwig V
I can always form a set out of a collection of objects. Not following your objection.
{1/2, 3/4, ...} is a set, and {1} is a set, and I can surely take the union of the two sets, right?
{1/2, 3/4, ..., 1} is just a particular subset of the closed unit interval [0,1].
If you are not sure about what I'm saying we should stay on this point. I can definitely form a set out of any arbitrary collection of other sets. And each of 1, 2, 3, ... and [math]\omega[/math] can be defined as particular sets.
Quoting Ludwig V
Of course {1, 2, 3, ..., [math]\omega[/math]} is a different set that {1/2, 3/4, ..., 1}. But strictly in terms of their order, they are exactly the same. And with ordinals, all we care about is order.
Quoting Ludwig V
The sequence is endless, and there's an extra point that's defined to be strictly greater than all the others. We can't get to the limit by successors, but we can get there by a limiting process.
Quoting Ludwig V
You're far better off. People who take calculus and then engineering math end up confused about limits and the nature of the real numbers. Taking logic and not calculus is actually helpful, in that you haven't mis-learned bad ideas about limits.
Calculus is focussed on the computational and not the philosophical aspects of limits, and calculus students often end up a little confused about some of the technical details. I was actually referring to the other poster who you noted was talking past me and vice versa.
Quoting Ludwig V
I've convinced myself both ways. On the one hand we can't physically count all the natural numbers, because there aren't enough atoms in the observable universe. We're finite creatures.
On the other hand, supertasks are possible, because I can walk a mile, meaning I walked 1/2 a mile, 1/4 mile, dot dot dot.
I have no strong belief or opinion about supertasks. I have strong opinions about some of the bad logic and argumentation around supertasks.
If I understood the OP, the walker spends arbitrarily small amounts of time on each step, 1/2 second, 1/4 second, etc. That violates the known laws of physics. So it's not a physical situation. It's a cognitive error to think we're contrasting math to physics. There is no physics in this problem.
Quoting Metaphysician Undercover
Well yeah. To be a fish a thing has to obey the known laws of fishes. Note that I include the word "known." Biologists could discover a new fish that extends our concept of what's a fish, just as physicists refine their laws from time to time. But a physical thing must obey the known laws of physics. This seems a very trivial point, i can't imagine what you mean by questioning it.
Quoting Metaphysician Undercover
Wasn't that a classic Star Trek episode? "Are you of the body?" And if you weren't, they zapped you with an electric stick.
Quoting Metaphysician Undercover
Sorry, what? Given me an example of something that violates Newton's laws, unless it's an object large enough, small enough, or going fast enough to be subject to quantum or relativistic effects.
A freely willed action? Can you give me an example? You mean like throwing a ball? You kind of lost me here.
Quoting Metaphysician Undercover
I'll be happy to consider any specific examples you have of human beings whose actions violate the laws of physics. If you mean actions caused by mentation, that's a bit of a puzzler, but I'm not sure how to violate the laws of physics. If I set out today to violate Newton's laws, I don't know how I could do that.
Quoting Metaphysician Undercover
Do you mean something like, "I think about raising my right hand and my right hand goes up, how does that happen?" If so, I agree that nobody understands the mechanism.
Quoting Metaphysician Undercover
That's why I included the word "known." I allow that the laws of physics are historically contingent approximations to the laws of nature.
Quoting Metaphysician Undercover
Agreed.
Quoting Metaphysician Undercover
Still waiting for specific examples. I believe the muons were misbehaving a while back and it made the news. Of course there are things we don't understand, like dark matter, dark energy, a quantum theory of gravity.
Quoting Metaphysician Undercover
Ok. Scary that you and I are thinking along the same lines. What is your point here with respect to the subject of the thread?
Quoting Metaphysician Undercover
The walker spends ever smaller amounts of time on each step, and that eventually violates the Planck scale.
Quoting Metaphysician Undercover
Which counter? The lamp? The lamp is not physical. No physical circuit can switch in arbitrarily small intervals of time.
Quoting Metaphysician Undercover
It's designed to confuse people who mis-learned a little calculus and don't know what's allowed or disallowed by the laws of physics.
Quoting Metaphysician Undercover
Ok I guess. No walker can traverse a staircase as described by the premises of the problem. So if I said the staircase was not physical, I should have said the walker is not physical. Better?
Quoting Metaphysician Undercover
I don't see why not. The whole point of the puzzle is to sum 1/2 + 1/4 + ... = 1, and then to ask what is the final state. Which, as I have pointed out repeatedly, is not defined, but could be defined to be anything you like.
Quoting Metaphysician Undercover
I'm sorry that you fine the completeness axiom of the real numbers incoherent. On the contrary, the completeness axiom of the real numbers is one of the crowning intellectual achievements of humanity.
Quoting Metaphysician Undercover
The premises violate the known laws of physics, specifically the claim that we can know the walker's duration on each step even though that duration is below the Plancktime.
Quoting Metaphysician Undercover
Modern math is incoherent. Is it possible that you simply haven't learned to appreciate its coherence?
See here:
I have been arguing that it is a non sequitur to argue that because the sum of an infinite series can be finite then supertasks are metaphysically possible. The lack of a final or a first task entails that supertasks are metaphysically impossible. I think this is obvious if we consider the supertask of having counted down from infinity, and this is true of having counted up to infinity as well.
We can also consider a regressive version of Thomson's lamp; the lamp was off after 2 minutes, on after 1 minute, off after 30 seconds, on after 15 seconds, etc. We can sum such an infinite series, but such a supertask is metaphysically impossible to even start.
You've got this backward. Some supertasks are coherent and consistent, therefore logically logically possible. In this case, that is the proof that they are "possible". If someone wants to insist that they are impossible then a poof is required.
I dealt with this already. If you restrict the meaning of "physical" to that which abides by the law of physics, then every aspect of what we would call "the physical world" which violates the laws of physics, dark energy, dark matter, for example, and freely willed acts of human beings, would not be a part of the "physical" world.
Quoting fishfry
That's not true at all. It does not correctly represent how we use the word "physical". "Physical" has the wider application than "physics". We use "physical" to refer to all bodily things, and "physics" is the term used to refer to the field of study which takes these bodily things as its subject. Therefore the extent to which physical things "obey the known laws of physics" is dependent on the extent of human knowledge. If the knowledge of physics is incomplete, imperfect, or fallible in anyway, then there will be things which do not obey the laws of physics. Your claim "a physical thing must obey the known laws of physics" implies that the known laws of physics represents all possible movements of things. Even if you are determinist and do not agree with free will causation, quantum mechanics clearly demonstrates that your statement is false.
Quoting fishfry
I gave you an example. A human body moving by freely willed acts violates Newton's first law.
"Newtons first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. This tendency to resist changes in a state of motion is inertia."
There is no such "external force" which causes the freely willed movements of the human body. We might create the illusion that the violation can be avoided by saying that the immaterial soul acts as the "force" which moves that body, but then we have an even bigger problem to account for the reality of that assumed force, which is an "internal force". Therefore Newton's first law has no provision for internal forces, and anytime such forces act on bodies, there is a violation of Newton's laws.
Quoting fishfry
If you understand this, then you ought to understand that being physical in no way means that the thing which is physical must obey the laws of physics. It is not the case that we only call a thing "physical" if it obeys the laws of physics, the inverse is the case. We label things as "physical" then we apply physics, and attempt to produce the laws which describe the motions of those things. Physical things only obey the laws of physics to the extent that the laws of physics have been perfected.
Quoting fishfry
Ok, now we're getting somewhere. The point, in relation to the "paradox" of the thread is as follows. There are two incompatible scenarios referenced in the op. Icarus descending the stairs must pass an infinite number of steps at an ever increasing velocity because each step represents an increment of time which we allow the continuum to be divided into. In the described scenario, 60 seconds of time will not pass, because Icarus will always have more steps to cover first, due to the fact that our basic axioms of time allow for this infinite divisibility. The contrary, and incompatible scenario is that 60 seconds passes. This claim is supported by our empirical evidence, experience, observation, and our general knowledge of the way that time passes in the world.
What I believe, is that the first step to understanding this sort of paradox is to see that these two are truly incompatible, instead of attempting to establish some sort of bridge between them. The bridging of the incompatibility only obscures the problem and doesn't allow us to analyze it properly. Michael takes this first step with a similar example of the counter , but I think he also jumps too far ahead with his conclusion that there must be restrictions to the divisibility of time. I say he "jumps to a conclusion", because he automatically assumes that the empirical representation, the conventional way of measuring time with clocks and imposed units is correct, and so he dismisses, based on what I call a prejudice, the infinite divisibility of time in Icarus' steps, and the counter example.
I insist that we cannot make that "jump to a conclusion". We need to analyze both of the two incompatible representations separately and determine the faults which would allow us to prove one, or both, to be incorrect. So, as I've argued above, we cannot simply assume that the way of empirical science is the correct way because empirical science is known to be fallible. And, if we look at the conventional way of measuring time, we see that all the units are fundamentally arbitrary. They are based in repetitive motions without distinct points of separation, and the points of division are arbitrarily assigned. That we can proceed to any level, long or short, with these arbitrary divisions actually supports the idea of infinite divisibility. Nevertheless, we also observe that time keeps rolling along, despite our arbitrary divisions of it into arbitrary units. This aspect, "that time keeps rolling along", is what forces us to reject the infinite divisibility signified by Icarus' stairway to hell, and conclude as Michael did, that there must be limitations to the divisibility of time.
Now the issue is difficult because we do not find naturally existing points of divisibility within the passage of time, and all empirical evidence points to a continuum, and the continuum is understood to be infinitely divisible. So the other option, that of empirical science is also incorrect. Both of the incompatible ways of representing time are incorrect. What is evident therefore, is that time is not a true continuum, in the sense of infinitely divisible, and it must have true, or real limitations to its divisibility. This implies real points within the passage of time, which restrict the way that it ought to be divided. The conventional way of representing time does not provide any real points of divisibility.
"Real divisibility" is not well treated by mathematicians. The general overarching principle in math, is that any number may be divided in any way, infinite divisibility. However, in the reality of the physical universe we see that any time we attempt to divide something there is real limitations which restrict the way that the thing may be divided. Furthermore, different types of things are limited in different ways. This implies that different rules of division must be applied to different types of things, which further implies that mathematics requires a multitude of different rules of division to properly correspond with the divisibility of the physical world. Without the appropriate rules of divisibility, perfection in the laws of physics is impossible, and things such as "internal forces" will always be violating the laws of physics.
quote="fishfry;900943"]The walker spends ever smaller amounts of time on each step, and that eventually violates the Planck scale.[/quote]
The Planck limitations are just as arbitrary as the rest, being based in other arbitrary divisions and limitations such as the speed of light. The Planck units are not derived from any real points of divisibility in time.
Quoting fishfry
No, the point of the puzzle is to demonstrate that the sum is always less than one, and that the mathematician's practise of making the sum equivalent to one is just an attempt to bridge the gap between two incompatible ways of looking at the theoretical continuum. The assumption that the sum is equivalent to one is what creates the paradox.
Quoting fishfry
I hope you're joking, but based on our previous discussions, I think you truly believe this. What a strangely sheltered world you must live in, under your idealistic umbrella.
Quoting fishfry
Exactly, and since we know that many physical things commonly violate the laws of physics, the fact that the premises are logically consistent and that they violate the laws of physics, indicates that we need to take a closer look at the laws of physics.
Quoting fishfry
No, I've read thoroughly many fundamental axioms, and found clear incoherencies, which I've shared in this forum. Many people accept premises and axioms because they are "the convention", so they do not proceed with the due diligence to determine whether there is inconsistency between them. Then, they proceed to utilize them because they are extremely useful. Problem would only arise under specific conditions which would be avoided, or a workaround developed for. So it's not a matter of learning to "appreciate its coherence", I've already learned to appreciate its usefulness, facility, and convenience. But I think that you are mistaken to think that facility necessarily implies coherency.
Quoting fishfry
Quoting fishfry
OK. I remembered WIttgenstein's oracular remark that death is not a part of life. My concern that the limit is not generated by the defining formula isn't the problem I thought it might be.
Quoting fishfry
I don't really believe in "possible" without qualification. There's logically possible, (is mathematically possible the same or something different? Does is apply here?), physically possible, and a range of others, such as legally possible. So what kind of possibility is a supertask?
Quoting fishfry
So your reply is that it is neither. It suggests a combination of physical and mathematical rules which is incoherent but generates an illusion. That's why Quoting fishfry
But then you say
Obviously, as each stage gets smaller, I will complete it more quickly. But still, it will take some period of time, and the final step looks out of reach. That looks like a combination of physical and mathematical rules.
It isn't a real problem because I can analyze the task in a different way. I can complete the first yard, the second yard.... When I have completed 1760 yards, I have completed the task. But the supertasks seem not to permit that kind of analysis. Is that the issue?
No. In the dichotomy scenario, there is no first step to which that number can be assigned.
Quoting fishfry
OK, that meaning of 'count'. In that case, I don't see how mathematical counting differs from physical counting. That bijection can be done in either case. In the case with the tortoise, for any physical moment in time, the step number of that moment can be known.
I am saying that Zeno describes a physical supertask, that Achilles must first go to where the tortoise was before beginning to travel to where the tortoise is at the end of that prior step.
Zeno goes on to beg the impossibility of the task he's just described, so yes, he ends up with a contradiction, but not a paradox.
Quoting fishfry
I also would hate to have to talk about the poor kilometerage that Bob's truck gets.
Quoting fishfry
The lamp scenario asks it, which is why the comment was relevant.
Quoting Metaphysician Undercover
I think the person to whom I was replying was suggesting that somebody had asserted a proof that a physical supertask was possible. But I did not recall anybody posting such an assertion.
Really? What if one were to begin by summing each interval as represented by a bijective function like f(n) = 60/2^n where n is a number in the natural numbers representing the cardinality of a set like N = {30, 15, 15/2}? Does that not include a first step? And would that sum not eventually terminate given a smallest sliver of time exists or continue indefinitely given time is infinitely divisible?
Yes, that series has a first step, but not a last one. You can number the steps in the series if you start at the big steps. Similarly, you can number the dichotomy steps in reverse order, since the big steps are at the end.
If there's a smallest sliver of time, there is no bijection with the set of natural numbers since there are only a finite number of steps.
'Continue indefinitely' is a phrase implying 'for all time', yet all the steps are taken after only a minute, so even if time is infinitely divisible, the series completes in short order.
I am not sure why you think it's necessary to point that out to me.
Quoting Michael
I took the class. I read a couple of different books, not that particular one. But I'm not sure why you are taking the time to explain to me the basics of convergent infinite series in real analysis.
Quoting Michael
Well no, not really, because a convergent infinite series does not have a temporal component. There's no notion of "add the next thing then add the next thing ..." Rather, the sum of the series is 1 in a single moment if you will. Rather, there are no moments at all. 1/2 + 1/4 + ... = 1 in the same sense that 1 + 1 = 2. They are two expressions that mean the same thing.
We can contrast convergent infinite series with loops in a programming language. Loops are executed in time, consume energy, and produce heat in the computational substrate. Mathematical series don't do any of that. Mathematical operations happen atemporally. They are, they don't do.
When you try to put it into a physical context, that's where the confusion comes in.
Quoting Michael
No physical lamp can switch that fast, so there's nothing physical about this thought experiment.
Consider a function f defined on the ordered set {1/2, 3/4, 7/8, ..., 1}. This is a perfectly well defined set of rational numbers. Suppose f(1/2) = 0, f(3/4) = 1, f(7/8) = 0, and so forth; and f(1) = a plate of spaghetti.
That is a perfectly sensible answer to the question, "What is the state at the limit?" It's perfectly sensible because the conditions of the problem don't specify the value at the limit. And since the lamp is not physical, it can turn into anything we like at the limit. It's no different than Cinderella's coach, which is a coach at 1/2 second before midnight, 1/4 second before midnight, and so on, and turns into a coach at midnight.
That is literally the answer to the Thompson's lamp puzzle.
Quoting Michael
You have not so argued. You have so claimed. You have not provided any proof or even evidence that a supertask is "metaphysically impossible." Maybe it is, maybe it isn't. The Planck scale and the physics of switching circuits preclude the existence of the lamp, according to current physics. A century or two from now, who can say? In particular, how can you personally know that future physics won't let us peer below the Planck scale?
Quoting Michael
I just showed you the final state. A supertask is a function on the set {1/2, 3/4, 7/8, ..., 1}. The final state of Cinderella's coach is pumpkin. The final state of Thompson's lamp is plate of spaghetti.
I do not see the problem. Neither the lamp nor the coach are physical entities. This is purely an abstract thought experiment, and my solution is mathematically sound.
Quoting Michael
That's not a proof. If you had a proof you'd give it, instead of simply claiming how obvious it all is to you that something is "metaphysically impossible." How do you know what's metaphysically possible? None of us are given to know the ultimate nature of the world. Not currently, anyway, and maybe never.
Quoting Michael
Bit of a subject change, not sure where you're going with this.
Quoting Michael
I think you must have your own private meaning for "metaphysically impossible," because after all our conversation, you have not convinced me of your point. I think literal, physically instantiated supertasks may or may not turn out to be possible. I base my opinion on the long history of revolutions in scientific understanding. The earth turned out to revolve around the sun. We split the atom. We evolved from more primitive animals. Many things formerly thought to be "metaphysically impossible" turned out to be not only possible, but true.
What makes you think you can see the indefinitely far away scientific future?
If I stand in a parking lot and call out "one, two, three, ..." and keep going, I can never count all the natural numbers.
In math, I can say, Let {1, 2, 3, ...} be the set of natural numbers whose existence is guaranteed by the axiom of infinity. Now I've counted the natural numbers.
I see a difference in those scenarios. I suppose someone could say that my conceptualization of {1, 2, 3, ...} is a physical process in my brain. Is that what yu mean that there's no difference between mathematical and physical counting?
Quoting noAxioms
I'm probably not in a position to differ. Clearly we can walk from one place to another. Maybe that's a supertask. I don't know.
Quoting noAxioms
Do British people talk about kilometerage? I've actually never heard that usage but I suppose it makse sense.
Quoting noAxioms
It's not a physical lamp, since no physical circuit could switch that fast. Therefore it's an imaginary lamp. Its state is defined at each of the times 1/2, 3/4, 7/8, ... But its state is not defined at 1. Therefore we may define its state as 1 as becoming a plate of spaghetti.
It's just like Cinderella's coach. It's a coach at midnight minus 1/2, midnight minus 1/4, etc. At at exactly midnight, it turns into a coach.
The Planck-scale defying lamp circuit is every bit as fictional as Cinderella's coach. Since the state at 1 is not defined, I'm free to define it as a plate of spaghetti. That's the solution to the lamp problem.
The reason it's as sensible as any other solution is that there is no final state that makes the lamp function continuous.
If, for example, the lamp is on at 1, on at 3/4, on at 7/8, and so forth, then we could still define the state at 1 as a plate of spaghetti; but if we defined the lamp to be on at 1, that would have the virtue of making the lamp function continuous. Continuous functions are to be preferred.
But no possible value for the final state of the lamp makes the problem continuous. Therefore any old arbitrarily solution will do just as well as any other.
As far as I'm concerned, that solves the problem. Until, of course, some future genius not yet born figures out how to implement a switching circuit that makes the lamp physical.
That is exactly the Zeno Walk.
Your objection is that:
And the problem is
We may not like how this train of thought goes, and we might settle for the more intuitive and less troublesome metaphysics, but the possibility of either remains, especially when human minds have issues wrestling with the infinity concept.
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However, how do you arrive at that conclusion? The two options that I can think of is by admitting that the sum of an infinite series is an approximation instead of the exact value, or by casting some doubt on the idea of an ?-th item of a series. The latter seems to cause more problems than solve them for me. Did you use a different reasoning?
Jeez that's kind of creepy ... true, I suppose. Death is the least upper bound of the open set of life.
Quoting Ludwig V
In the future, if physics ever figures out how to work with physically instantiated infinities, supertasks might be possible. Way too soon to know.
Quoting Ludwig V
I just mentioned that I could argue it either way.
Quoting Ludwig V
How can it be out of reach? I went to the supermarket today. I walked from one end of the aisle to the other. I reached the end. I did indeed evidently sum a convergent infinite series. Except for the fact that nobody knows if spacetime below the Planck length is accurately modeled by the mathematical real numbers. Maybe it's not. We just don't know. In any event, I don't know.
Quoting Ludwig V
I don't think it's much of a problem. It doesn't keep me up at night. I just saw a Youtube video of an interview of Graham Priest, a famous philosopher. He thinks Zeno and other paradoxes are important. a lot of people think they're important.
I think some of Zeno's other paradoxes are more interesting. When you shoot an arrow, it's motionless in an instant. How does it know where to go next, and at what speed? I think that's a more interesting puzzle. Where are velocity and momentum "recorded?" How does the arrow know what to do next?
By the conditions of the problem. Is this about the lamp? The problem says it's on at 1/2, on at 3/4, off at 7/8, etc.. The problem itself doesn't define the state at 1. So I'm free to define the state at 1 any way I like. Because the problem itself leaves that information unspecfified.
Quoting Lionino
No. The mathematics is pristine. 1/2 + 1/4 + 1/8 + ... = 1 in the same sense that 1 + 1 = 2. Two names for the same thing. May be used interchangeably. Exactly equal. Denote exactly the same real number.
Quoting Lionino
There is no ?-th item of a series. There is the limit of a series (or sequence, I think you may mean here). That's not the same thing. In the sequence 1/2, 3/4, 7/8, 15/16, ... there is no last element. The sequence has a limit of 1. But 1 is the limit, it's not a member of the sequence.
Quoting Lionino
Not sure what you mean. Reasoning in terms of what? The final lamp state is not defined. I can arbitrarily define any function at a point where it's not defined, especially when there's no natural reason (such as continuity) to prefer one limit state over another.
Am I understanding your question correctly? I didn't quite understand what you mean by asking if I used different reasoning.
It's like one of those "fill in the missing number" puzzles, like 1, 2, 8, 16. They want you to say 32. But mathematically, you can put in anything you want. If you don't tell me the lamp state at the limit, I can define it any way I want, especially since there's no way to define it in such a way that the sequence attains its limiting value in a continuous manner.
Don't be silly. The rotational rate of galaxies is physical, even if our current theory of gravity doesn't explain it. We don't say it's not physical just because we don't have a theory of dark matter or modified gravity yet.
Quoting Metaphysician Undercover
I think I can't play these word games. If you want to pretend not to know what a physical thing is, I can't argue with you. Bowling balls falling down was a physical phenomenon two thousand years ago even if Aristotle's physics didn't explain it sufficiently.
Quoting Metaphysician Undercover
How can a human body move by free will? You're the determinist here. You reject randomness. How does this "will" influence the body? Good questions. I don't know. You don't either.
Quoting Metaphysician Undercover
Sorry what? I didn't say that, is that someone else's quote?
Quoting Metaphysician Undercover
Can you remind me what is the point of all this? I haven't the heart to discuss Newtonian metaphysics.
Quoting Metaphysician Undercover
Fine. The speed of the rotating galaxies is physical but we don't have a law of physics that explains it yet.
Can you remind me why we're having this discussion? I think if you wrote less I could respond more deeply. This flood of deepity is a bit much for me.
Quoting Metaphysician Undercover
I think if you wrote a shorter post I'd engage. Do you think bowling balls falling down was not physical before Newton? Before Einstein? Before the next genius not yet born?
By your reasoning, nothing at all is physical, since all physical theories are only approximate.
Quoting Metaphysician Undercover
Ok. I haven't engaged with the staircase at all. Can't argue it. But if 60 seconds of time can't pass, how did I walk from the living room to the kitchen for a snack?
Quoting Metaphysician Undercover
I can't argue with you about your analysis of what someone else said. Nor can I argue about the staircase. I haven't really said much in this thread about the staircase. The lamp is much more clear to me.
Quoting Metaphysician Undercover
I made it to the kitchen and back. How do you account for that?
Quoting Metaphysician Undercover
I actually agree with you that the mathematical real numbers may well not be an accurate representation of the ultimate nature of reality. If that's what you mean about infinite divisibility.
Quoting Metaphysician Undercover
If you're arguing for a discrete universe, maybe so.
Quoting Metaphysician Undercover
It's profoundly and beautifully and logically rigorously treated by mathematicians. I can't imagine what you mean here. You're flat out wrong.
We may not know the ultimate nature of the world, but the mathematical real numbers are treated very well indeed.
Quoting Metaphysician Undercover
Arbitrary, not infinity. I can divide 1 into 1/2, into 1/4, into 1/8. I can divide any finite number of times, and there are infinitely many numbers. but I can't "divide infinitely." That's imprecise and essentially wrong.
Quoting Metaphysician Undercover
Not at all. That's the physicists' job. Mathematicians need not concern themselves with the physical world at all.
But mathematicians do have "different rules of division." The rules of division in the integers are very different than the rules of division for the real numbers.
Quoting Metaphysician Undercover
There can never be perfection in any physical law. You know that. You lost me with internal forces.
Quoting Metaphysician Undercover
The math is pretty solid, it's based on Fourier series as I understand it. I think you're a little out over your skis here. But your complaints are about physics. I'm not qualified to help.
Quoting Metaphysician Undercover
You're just typing stuff in. What you wrote isn't true. "the mathematician's practise of making the sum equivalent to one is just an attempt to bridge the gap between two incompatible ways of looking at the theoretical continuum." Not even wrong.
Quoting Metaphysician Undercover
The math is beyond dispute.
Quoting Metaphysician Undercover
I've never meant anything more seriously. It was more than 200 years of intellectual struggle from Newton and Leibniz to Weierstrass, Cauchy, Cantor, and Zermelo.
Quoting Metaphysician Undercover
Based on our previous discussions, you're still an ignorant troll. I'm done here. What is wrong with you?
I hope you don't mind my saying that your choice of free will as an example was perhaps ill-advised. It's far too contentious to work. Quantum mechanics is a much better choice. But there is the problem that there are many interpretations of it, so it is not clear that it proves what you think it proves.
Quoting fishfry
I think you are both mistaken to rely on physics to define what one wants to get at in this context. Physics is not only limited by the current state of knowledge, but also by its exclusion of much that one would normally take to be both physical and real. Somewhere near the heart of this is that there is no clear concept that will catch what we might mean by "whatever exists that is not mathematics" or by "whatever applied mathematics is applied to".
Quoting fishfry
I'm sorry. I didn't mean to gross you out. Perhaps if you think of death as a least upper bound, you'll be able to think of it differently. It is, after all, an everyday and commonplace event - even if, in polite society, we don't like to mention it.
Quoting fishfry
Yes. I was just drawing out the implications. You might disagree.
Quoting fishfry
Yes. In the context of the Achilles problem that's fine and I understand that you are treating that and the natural numbers as parallel. It's not clear to me that it really works. It makes sense to say that "1" limits "1/2, 1/4, ..." But I'm not at all sure that it makes sense to say that
Quoting fishfry
Did you "get to the limit by successors" or "get there by a limiting process"? I don't think so. You are just not applying that frame to your trip.
Quoting fishfry
I've met other mathematicians who agree that Achilles is not interesting. But I'm fascinated that you think the arrow is interesting. I don't. Starting is a boundary condition and so not part of the temporal sequence, any more than the boundary of my garden is a patch of land. End of problem.
But this may be interesting in the context of what we are talking about. A geometrical point does not occupy any space. It is dimensionless. One could say it is infinitely small. But it is obvious that there is no problem about passing an infinite number of them. It is a question of how you think about them. This is not quite the same as Zeno's problem, but it is close.
Quoting fishfry
I agree with that.
Perhaps then, these problems are not mathematical and not physical, but imaginary - a thought experiment. (The Cinderella example shows that we can easily imagine physically impossible events) That suggests what you seem to be saying - that there are no rules. (Which is why I posited another infinite staircase going up). But if there are no rules, what is the experiment meant to show? The only restriction I can think of is that it needs to be logically self-consistent - and the infinite staircase is certainly that. I guess the weak spot in the supertask is the application of a time limit. However, I also want to say that I cannot imagine an endless staircase, only one that has not ended yet - once I've imagined that, I can wave my hand and say, that is actually an infinite staircase.
Okay, so it is possible to have a first step. If I could, say, produce an equation based on the one in my earlier post that could calculate the last time interval given a smallest stipulated chunk of time, would that be a valid final step in the summation?
Quoting noAxioms
That was sloppy thinking and use of language on my part. Sorry.
I think, and then I do. The "force" which moves me comes from within me, and therefore cannot be described by Newton's conceptions of force.
Quoting fishfry
"Speed", and "rate" are measurements derived from comparing things. This is explained in the theory of relativity, and by that theory such things are dependent on the frame of reference. A measurement does not exist without the act which measures. I see that this is indicative of your way of thinking, when you say that by referring to the axiom of infinity you can count all the natural numbers. This is a new fangled sort of doing by proxy, where the assertion (here called an "axiom") "I have done X" means that X has been done. That is the same sort of mistake which Michael was making in insisting that measurements like seconds and days exist without being measured. I referred Michael to Wittgenstein's "standard metre" example.
Quoting Ludwig V
I do not pretend to be providing a proof when I provide an example. However, I'll take your advise and refer to quantum mechanics if I'm asked to provide examples of how it is that a measurement cannot exist without an act which measures.
The use of "physical" in this thread has gotten so ambiguous, that equivocation abounds everywhere.
How much time elapses from travel to point a to point b and where is the object located during that time lapse?
Does the object leave existence between a and b and if it does, what maintains its identity during that interval?
Thats a question for physicists to answer.
I think we're talking past each other. When asked for the difference between a mathematical and physical supertask, you seem to focus on two different definitions of countable: The assignment of a bijection, and calling or writing down each of the numbers.
I'm talking about a physical supertask as described by Zeno, which arguably has countably (first definition) steps performed in finite time. Nobody is posited to vocalize the number of each step as it is performed.
Bit off on the lore. It turns into a pumpkin, and at the 12th stroke, where presumably midnight is the first stroke, but I googled that and could not find an official ruling on the topic.
No argument. That seems to be a valid way out of most attempts to assign a count to the nonexistent last/first step, or to simply assert the necessity of the nonexistent thing.
I like Bernadete's Paradox of the Gods because it doesn't make those mistakes, and thus seems very much paradoxical since motion seems prevented by a nonexistent barrier.
For educational purposes concerning how infinity works, I like Littlewood-Ross Paradox because it is even more unintuitive, but actually not paradoxical at all since it doesn't break any of the above rules. It shows a linear series (effectively 9+9+9+...) being zero after the completion of every step.
Quoting ToothyMawIf you stopped the summation there, then yes, there would be a final step, but it wouldn't have infinite steps defined then. It wouldn't be a supertask.
If there's a smallest quanta of time, then there can be no physical supertasks.
Quoting Metaphysician Undercover
LOL. Tell that to the guy stranded 2 meters from his space ship without a tether. No amount of free will is going to get you back to it. You're going to need a little help from Newton.
Quoting Metaphysician Undercover
Yea, I noticed.
Quoting Hanover
I'll attempt this. Michael talks about motion from A to B without there being a between. This can happen two ways.
1) Space is quantized. There simply isn't a location halfway between A and B. For a slow particle, this might mean that it spend quite a bit of time at A, and then suddenly is at B. That seems rather contradictory since one might ask what changes during all those times when the thing was at A. If it is at A twice, it is stationary and has no obvious state anywhere to go to B after some nonzero time.
2) Time is quantized, which is troublesome for fast particles. You have time 1 where the thing is at A, and time 2 where it is at B, quite a ways away. There are valid locations between A and B that the particle never visited since the time it should have been there is nonexistent.
All the above sort of presumes a naïve finite-automata sort of view of a quantized space and or time. It presumes a particle has a location (A, B) at a given state and time. Well that presumption was pretty much thrown out of the window with quantum theory.
A more realistic answer to your question comes from there. It says that you measure the particle at A, and later at B (maybe hours later). Where was the particle between those times? If not measured, it doesn't have a location. It does exist, but needs to be measured to have a location or (not and) a momentum.
Physics has no concept of identity of anything. It is a human convention, a pure abstraction. Any given convention seems falsifiable by certain examples.
Why were you qualified to talk about it before but now I have to find a physicist on a Saturday morning to answer these follow up questions?
Some things can be dismissed on logical grounds, like the notion of continuous motion and the infinitely divisible half-way points an object in motion must then move through.
But one cannot use armchair philosophy to determine the smallest unit of space/time/movement.
Assuming at the most microscopic level, the object is on an 8x8 chessboard. The pawn moves from e2 to e3. There is no e2.1 or other smaller increments in this finite world. At T1 it's at e2 and T30 it's at e3. The assumption is that at some point in time, it was no where while transitioning (moving?) from e2 to e3.
What empirical evidence is there that observations have been made of there being no object for some length of time and then it suddenly reappearing?
We can't examine this at the macroscopic scale. At whatever the smallest scale is: at Time1 it's at Location1 and at Time2 it's at Location2. There's no intermediate Time1.5 where it doesn't exist or Location1.5 that it moves through.
The question makes no sense. You're asking for some second "level" of time to define the time between T1 and T2. There's no such thing. The only time is T1, T2, T3, etc.
You continue to refuse to acknowledge the difference between the measurement and the thing measured. T1 and T2 are points designated by the measurer, therefore a feature of the measurement. The measurement is the difference between T1 and T2. However, the thing measured is the passage of time which occurs. Your confusion is due to your refusal to acknowledge a distinction between the measurement (the specified number of seconds) and the thing measured (the passage of time). You've been insisting that the thing measured is a number of seconds, rather than recognizing that seconds is the measurement, not the thing which is measured. And so I gave up trying to explain to you the difference.
Quoting noAxioms
No one said free will has infinite capacity? Obviously we are limited by the circumstances we are in. But limitations are not absolutely either. So free will has it's own niche, to act according to a judgement of the circumstances.
In the circumstances you describe, an appeal to Newton would not help the poor soul, but a radio call to someone inside the spaceship, to please shoot me a line, might help. That demonstrates the benefit of free will, allowing one to act according to a judgement of the circumstances. And. it demonstrates how free will could actually get the person back to the space ship, in contrast to your suggestion of asking Newton to help, which of course, would be useless.
And the passage of time that we would measure as being 60 seconds occurs even when we don't measure it.
The problem is adjacency. If object A is adjacent to object B on a finite grid, what is the distance from A to B? If it's 0 units, then A and B occupy the same space and A = B.
You seem to be imagining a model of discrete space overlaying some model of continuous space and then pointing out that in continuous space there is always more space between two discrete points.
That seems to be begging the question.
Best I can do is point you to something like quantum spacetime and quantum gravity.
There are physical theories that treat spacetime as discrete. They are not supported to the extent that General Relativity is, but given that quantum mechanics and General Relativity are known to be incompatible, it would seem that at least one of them is false, and my money is on General Relativity being false.
Given the logical paradoxes that continuous space and time entail, I think that discrete spacetime is not just a physical fact but a necessity.
Quoting fishfry
Quoting fishfry
The series of 1/2 + 1/4 + 1/8 + ... equals 1. The sequence [math]a_{n}=1-0.5^{n}[/math] converges to 1, yet 1 is not part of the sequence. As you agreed, there is no ?-th item. Cool.
The issue that I see is:
1 if we admit that time is infinitely divisible;
2 and we admit that [math]a_{n}=1-0.5^{n}[/math] gives us the lenght covered by Achilles in the Zeno Walk at each step;
the walk only finishes if it accomplishes an infinite amount of steps. Right?
If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1? If not, how is the walk ever completed?; if so, is there not a corresponding state for the mechanism when the full time elapses?
In other words, by admitting that the result of an infinite series is necessarily true¹, how do you justify at the same time that the state is really undefined at 1 while also defending that Achilles can finish the run?
I want to emphasise that I am not arguing about the mathematics, but about the (meta)physical meaning of some mathematical concepts.
Does that make sense?
1 Is that also the case for non-standard analysis and arithmetic?
[hide="Reveal"]I think that these difficulties point that taking time to be discrete is more intuitive (appeasing to the human mind) than it being continuous while not proving either way.
[/hide]
Quoting fishfry
Quoting Hanover
But there is that for an advantage of continuous time over discrete.
Quoting Michael
Same here. General relativity is suspected to break down at high enough energies or small enough scales where the quantum effects can't be ignored , "like" Newtonian theory breaks down when v or Gm/R become large enough.
I'm only asking how far 1,1 is from 1,2 in a discrete space system. As far as I can tell, it's 0 units, right?
Quoting Michael
I don't think you can have it both ways.
Quoting Michael
I'm not quite sure what you are saying. Do you think that the passage of time occurs when we can't measure it? Analogously (if that's a word), if we can't measure the location or momentum of an object, it doesn't have them? Does that mean that it doesn't exist?
There are two philosophies that I can think of that would justify those views. One of them is Logical Positivism, which was developed precisely to justify both Quantum Mechanics and General Relativity. The other is Bishop Berkeley's idealism. Which do you hold?
Quoting Michael
That means you think it is possible that space-time is continuous at the quantum level. Interesting. But I suppose it fits with your acceptance of continuous space-time in mathematics.
The empirical evidence for your position is the empirical fact that we can't measure very small units of time or space. I'm not sure that constitutes convincing empirical one way or the other. Or have I got something wrong?
Quoting Michael
Which ones do you have in mind? You mentioned the problems with a converging series. But that's a mathematical problem, not an empirical one. How does empirically non-continuous space and time solve those issues?
I don't think the question makes sense, but you'll have to ask a physicist who knows more about quantum gravity to explain it. I can only point out to you that there are physical theories that take spacetime to be discrete.
Yes, I agree with that. But, there is no "seconds" inherent in that passage of time, nor does it appear like there are any natural points for division within that passage of time, which appears to us to be absolutely continuous. This is why we assume principles which allow for infinite divisibility of time, because we see no reason for any real restrictions to its division. Therefore we tend to believe that we can simply insert a point (T1) at any random place, and another point (T2) at another random place, and determine the amount of time that has passed between those two arbitrarily assigned points.
Now, in your opening post in the thread, you concluded a "metaphysically necessary smallest period of time", and you used reference to the empirically based principle "60 seconds will pass" to support this conclusion. Therefore you've exposed inconsistency between two empirically based principles. The one principle being the assumption that the passing of time is continuous, as it appears, and the consequent principle that we can arbitrarily insert points, and divide it in absolutely any way that we please. The other principle being that "60 seconds will pass". There is inconsistency because the former leads to the example of the stairway to hell in the op, in which there is always more steps, and more time to pass, before 60 seconds can pass.
Quoting Michael
Since you dismiss general relativity as probably false, then there is no need to maintain "spacetime". When we analyze space and time separately, then one might be discrete, and the other continuous. Logically, motion, which is a change of spatial location (place) requires the passing of time. We cannot conceive of a change in place without time passing because that implies the thing is in two different places at the same time. However, when time is separated from the constraints of spatial change we can conceive of time passing without spatial change. This allows that spatial change occurs as discrete 'quantum leaps', position at T1, to position at T2, without any spatial continuity between them. Between T1 and T2 there would be time passing, but no spatial change until that time has passed. That passage of time during which spatial change does not occur, is justified by activity at a deeper level, non-spatial, or immaterial activity, which determines the relationship between the spatial positions at T1 and the spatial positions at T2.
None, but there's also no evidence that it is there when not being measured. It's all about measurement and not about discreetness.
Quoting HanoverIn that frame, it took time 1 to get from T-1 to T-2. That's pretty obvious, no? In natural units, that's light speed.
If the answer is zero, then T-2 is no-t when it is at L-2.
In computer jargon, what you are describing is 'jaggies', the tendency of 'straight' lines to appear jagged when displayed on say your computer screen, a discreet array. An object that moves fast (faster than one L per T) will either be at multiple locations at the same time, or it will skip all the locations between and only be at one location per time.
I've played a game with the latter physics. I could get my ship to go super fast and go straight for the enemy blocking my way. If I did it right, I would be in front of him at one time unit, and beyond him the next time unit, apparently passing right through without collision because there was no time 1.5 where I was where he was.
More problems with that model: If the particle is moving at 0.7 per time unit, it is never at a location in space except every 10 time units where you find it 7 units from where it was before. It can't be anyplace between since it is never at a space location at the same time as a time quanta. This is silly. You probably need to fill in the dots between, but then the motion is erratic rather than sporatic.
Quoting HanoverNo, they're 0,1 from each other, which isn't zero. One of the coordinates is different.
Anyway, you seem to see the sorts of contradictions that arise from such a naive model. If space and time is discreet, quantum mechanics describes it far better than the chessboard model.
Quoting LioninoRight
By completing all the steps. This is not a contradiction.
Not any more than there is a last natural number. I'm presuming you're talking about the state of something like the lamp. The state of Achilles is easy: He's where the tortoise is.
I don't see a problem until the premise of a last step is introduced, which is by definition contradictory.
Quoting MichaelThey're both incomplete, just like
Newtonian mechanics was incomplete, but not false. OK, parts of it were outright false, but it's still taught in (pretty much) any school. GR definitely breaks down at small scales.
Quoting Metaphysician UndercoverI didn't say infinite capacity. I denied that your free will has any capacity at all, since even the most trivial capacity would get you back to your ship 2 meters away, even if not quickly.
Quoting Metaphysician UndercoverThe spaceship example shows this to be nonsense. It would be a revolution indeed if anybody could do that.
Quoting Metaphysician Undercover
Free will isn't necessary to do any of that. A robot has the same capacity to make such a call, and robots by definition lack it. This is also utterly off topic to this discussion, but I took the easy bait anyway.
I told you how the person gets back to the ship using free will. That's one point for free will, zero for you.
Quoting noAxioms
A robot cannot decide whether or not to make the call, a person can. The person could decide not to, if perhaps the release of the tether was intentional. Two for free will, zero for you.
Quoting noAxioms
It's not off topic, because there is an issue of what is "physically possible", and whether physical possibility" is limited by the laws of physics. My argument is that there is a number of physical activities such as the effects of dark matter and dark energy, which violate the laws of physics. Furthermore, free will violates Newton's first law, and it causes physical movements. Therefore physical possibility is not limited by the laws of physics.
Which includes magic lightbulbs and staircases? I'm open minded, I don't think I can predict the future. Even a few hundred years ago nobody could imagine the science and technology of today.
Quoting Ludwig V
Have mostly been talking about the lightbulb. Haven't talked about Achilles or Zeno.
But it's not the natural numbers that are parallel. It's the natural numbers augmented by the point at infinity. That's my conceptual setting for these problems.
Quoting Ludwig V
There's an order-isomorphism between the ordered sets {1/2, 3/4, 7/8, ..., 1} and {1, 2, 3, ..., [math]\omega[/math]. One of the virtues of abstraction is that it lets us see that two seemingly different things are really the same, when we only focus on certain attributes. Both these sets are an infinite sequence followed by an extra element. Their order properties are the same. It's no different than playing chess on a board with purple and green squares versus red and black squares. It's the same game with respect to the rules of the game, even though they're different in other respects.
Quoting Ludwig V
That's a good question too. How do dimensionless points form lines and planes and solids?
Quoting Ludwig V
If you agree, I'm happy, because that's the only point I'm making. I've never written more just say less. My only point is that in the lamp and these other problems, we're not defining the state at the limit. Therefore the choice of state is pretty much arbitrary. If we thought about it that way it might be more clear.
Quoting Ludwig V
Aha. You'd have to ask those who care so much. I think they only show that underspecified problems can have arbitrary answers. But others see deeper meanings.
Quoting Ludwig V
If I can walk from one end of the grocery aisle to the other, I don't see why you can't get down the staircase, infinitely many steps or not.
Quoting Ludwig V
It's a thought experiment. There are no infinite staircases.
But I did walk through infinitely many inverse powers of two lengths at the grocery store. I did sum an infinite series in finite time. So there's something interesting going on.
I don't see how you could count all the natural numbers by saying them out loud or writing them down. Is this under dispute?
Quoting noAxioms
Do you mean the fact that I can walk a city block in finite time even though I had to pass through 1/2, 3/4, etc? I agree with you, that's a mystery to me.
Quoting noAxioms
I looked it up, didn't seem to find a definitive version.
Quoting noAxioms
Ah the ping pong balls. Don't know. I seem to remember it makes a difference as to whether they're numbered or not. If you number them 1, 2, 3, ... then the vase is empty at the end, since every ball eventually gets taken out. But if they're not numbered, the vase will have infinitely many balls because you're always adding another 9. Is that about right?
I said no such thing!! If you like, you can think of the limit as being the [math]\infty[/math]-th item.
That is, if 1/4, 3/4, 7/8, ... are the first, second, third, etc. terms of an infinite sequence with limit 1, then 1 may be sensibly taken as the [math]\infty[/math]-th item, or as I've been calling it, the item at [math]\omega[/math], which is traditional in this context.
So I believe I've been trying to get across the opposite of what you thought I said. There is an [math]\infty[/math]-th item, namely the limit of the sequence.
The sequence itself has no last item. But the "augmented sequence," if you call it that, does. We can simply stick the limit at the end.
Quoting Lionino
I think trouble ensues when you try to apply abstract math to the physical world. I certainly can walk across the room, clearly accomplishing infinitely many Zeno-steps in finite time. I have no explanation nor does anyone else. The common explanation that calculus lets us sum an infinite series, I reject. Because that's only a mathematical exercise and has no evidentiary support in known physics.
Quoting Lionino
No.
Quoting Lionino
In math? Via the standard limiting process. In physics? I don't know, I'm not a physicist. But the physicists don't know either. They don't regard it as a meaningful question.
Quoting Lionino
The justification is purely mathematical. Physics doesn't support these notions since we can't reason below the Planck length.
Quoting Lionino
The metaphysical meaning is perfectly clear within the math. I don't know how it works in the physical worldl
Quoting Lionino
The math is clear. The physics is unknown. But motion is commonplace.
Quoting Lionino
.999... = 1 is a theorem of nonstandard analysis. I don't see how it could help. I don't know if anyone's thought about applying NSA to these puzzles.
:smile:
Quoting fishfry
Quoting fishfry
I find this very confusing. I take your point about abstraction. But I find that abstraction can create confusion, because it persuades us to focus on similarities and neglect differences. My reaction here is to pay attention to the difference between these kinds of infinite series. It's not meant to contradict the abstraction.
Quoting fishfry
Exactly. So it isn't about physics. But it isn't about mathematics either. So it seems to me an exercise in imagination, and that provides a magic wand.
Quoting fishfry
Deep? or Deepity? (RIP Dennett)
Quoting fishfry
Yes. Euclid (or Euclidean geometry at least) starts from a foundation - axioms and definitions. But they are an extension of our common sense processes of measuring things. (You can understand more accurate and less accurate measurements.) Extend this without limit - Hey Presto! dimensionless points! That is, to understand what a point is, you have to start from lines and planes and solids and our practice of measuring them and establishing locations. I find that quite satisfying. Start with the practical world, generate a mathematics, take it back to the practical world. (Yes, I do think that actual practice in the real world is more fundamental than logic.)
Once you define geometrical points in that context, there is no difficulty about passing or crossing an infinite number of points. (But the converging series does not consist of points, but of lengths, which are components.)
Quoting fishfry
My supervisor used to say that when he got really excited, which was not often.
Yes, I mean that, and it's not a mystery to me. If spacetime is continuous, then it's an example of a physical supertask and there's no contradiction in it.
No, the lamp changes things. It introduces a contradiction by attempting to measure a nonexistent thing. That in itself is fine, but the output of a non-measurement is undefined.
Nicely stated by Michael in reply 30, top post of page 2 if you get 30 per page like I do.
It's important to the demonstration of the jar being empty, so yes, it makes a difference.
The outcome seems undefined if they're not numbered since no bijection can be assigned, They don't have to have a number written on them, they just need to be idenfifed, perhaps by placing them in order in the jar, which is a 1-ball wide linear pipe where you remove them from the bottom.
It nicely illustrates that ?*9 is not larger than ?, and so there's no reason to suggest that the jar shouldn't be empty after the completion of the supertask. Again, it seems that any argument against this relies on a fallacious assumption of a last step that sooo many people are making in this topic.
Quoting fishfry
That can't be a step, since every step in a supertask is followed by more steps, and that one isn't. I have a hard time with this ?-th step.
Quoting fishfryThe cutting up of the path into infinite steps was already a mathematical exercise. The fact that the physical space can be thus meaningfully cut up is true if the space is continuous. That latter one is the only barrier, since it probably isn't meaningfully, despite all our naïve observations about the nice neat grid of the chessboard.
Quoting Lionino
As has been stated so many times, by performing all the steps, which happens in finite time no problem. There is a final step only in a finite sequence, so using a finite definition of 'complete' is inapplicable to a non-finite task.
Quoting fishfryIn physics, the same way as math, except one isn't required to ponder the physical case since it isn't abstract. One completes the task simply by moving, something an inertial particle can do. The inertial particle is incapable of worrying about the mathematics of the situation.
Which is to say that space isn't measurably continuous, so the walk isn't measurably a supertask. I would agree with that.
Quoting fishfry
Mathematics: by not having a last one (or adjacent ones even). Physics: There are no solids.
Quoting Ludwig VYes. The latter is a countable set of lengths. The set of points on say a circle is an uncountable set
Quoting Metaphysician Undercover
That's quite the assertion. Above and beyond the usual conservative stance.
The point of my example with the ship was to counter your assertion of Newton forces not being necessary to move and free will being enough. I said you'd need help from Newton. Asking for a line to be thrown to you is you admitting the help from Newton was necessary. That's what the tether is: a way to do it by exerting an external force, since the free will couldn't do it itself.
I take it you are talking about physical space, not mathematical space?
Quoting noAxioms
But there are 3-dimensional figures in physics, aren't there? It's the solidity that's the problem, isn't it?
Quoting noAxioms
That's a surprise to me. One can measure or calculate the length of a circumference, can't one? Or is uncountability a consequence of the irrationality of "pi"?
Quoting noAxioms
Just checking - by "step" do you mean stage of the series? If I am travelling at a steady speed, I will complete more and more steps in a given period of time, and that number (of steps) will approach (but not reach) infinity. Can that really work?
Quoting noAxioms
So is the cutting up of the path into standard units. It's just a question of choosing the appropriate mathematical calculation for the task at hand.
Quoting fishfry
Quoting fishfry
Then you say.
Quoting fishfry
Is there not a contrast between these two sets of statements?
Quoting fishfry
We are applying mathematics not just to this physical world but to any possible world where the physics could be different, and for that we discuss what the mathematics means in the world as it is necessary that 1+1=2 so that everytime you take one of something and one again you end up with two.
Sure. A rock, at a given time, is a 3 dimensional thing. A rock, it's entire worldline, is a 4 dimensional thing. Correct. It isn't a solid. You can measure a piece of it at a sort of 4D 'point', an event. The rock worldline consists of a collection of such point events, a huge number, but not infinite. They're not really points since position and momentum cannot be both known, so you can know one or the other or an approximate combination of both.
Yes, one can calculate the circumference. No, the irrationality of pi is irrelevant. It could be a line segment of length 1. You know the length, and it isn't irrational, but the segment still consists of an uncountable number of points. There's no part of the segment that isn't a point (or a set of them), and yet points have no size, so no finite number of them can actually fill a nonzero length of that segment.
Yes, a step is a finite (nonzero) duration, like the first step is going halfway to the goal. Each step goes half the remaining way to the goal. Those are steps. You complete all the steps by time 1, so the task is then complete. No contradiction so long as we don't reference 'the highest natural number' which doesn't exist.
One must define how the task is divided into steps in order to tell Zeno's story. There are multiple ways to do it, but to be a supertask, the steps need to get arbitrarily small somewhere, and the most simple way to do that is at the beginning or the end of the task. How one abstractly divides the space has no effect on the actual performance of the task. One can argue that all tasks of any kind are supertasks because one can easily divide any finite duration into infinite parts, but the much of the analysis of doing so relies on the mathematics of countable infinities.
So I can go from 0 to 1 and assign a 'step' to every zero-duration point between those limits. That can be done, and can be completed, but since the steps are not countable, it is hard to draw any conclusion from it all.
Quoting Lionino
That's me saying something, not fishfry.
I personally don't like the ?-th step, but it works. The supertask is completed, then the ?-th step is taken after that. The supertask had all nonzero duration steps, and this additional step has no duration. I don't find it wrong, but I find it needless.
I agree with fishfry that there is no step that gives us 1 since by definition, any given step gets us only halfway there. If fishfry wants to add an addition single step after the supertask completes, that's fine, but it isn't a step of the supertask.
Oops :monkey: fixed.
Quoting noAxioms
The problem I was trying to point out that is that, if we admit a ?-th step, this step should be associated with a state in one of those mechanisms Michael made up.
Quoting noAxioms
I agree with that too. In the end, I don't think reasoning about infinity gets us anywhere.
Michael's mechanisms (some of which he made up) are not resolved by addiing a single step task to the supertask. The supertask reaches 1 when all the steps are completed. It isn't sort of 1, it's there since where else would it be? The arguments against that suggest some sort of 'point immediately adjacent to, and prior to 1', which is contradictory. There are no adjacent points in continuums.
But I don't agree that 1 is not reached by the completion of the supertask. Only that 1 is not reached by any step.
I think that's all right. When I walk a mile, I start a potentially infinite series of paces. When I have done (approximately) 1,760 of them, I stop. The fact that the 1,760th of them is the last one is, from the point of the view of the sequence, arbitrary, not included in the sequence . The sequence itself could continue, but doesn't.
Quoting noAxioms
OK. Is that because they have no dimension - are not a part of the line?
Quoting Lionino
Because
Quoting Lionino
Then the ultimate paradox is that there seems to be no end to the reasoning.
Quoting noAxioms
I don't quite understand. Is the point that the simple arithmetic analysis doesn't reference the highest natural number, so that way of reaching it is OK. It doesn't look like completing all the steps to me - it looks more like jumping over them. But I have travelled over all the spaces.
The two sets in question have the same order type, denoted [math]\omega + 1[/math]. That's mathematically true, and it's all that's relevant to these two examples. I'm not sure what's gained by focussing on the differences. [math]\omega[/math] is the limit of 1, 2, 3, ... in exactly the same sense that 1 is the limit of 1/2,, 3/4, 7/8, ..., under the more general topological definition of a limit needed to defined limits among the ordinal numbers.
Quoting Ludwig V
Yes ok, so the coach can turn into a pumpkin and the lamp can turn into a plate of spaghetti. Are you agreeing with me on that point?
Quoting Ludwig V
RIP.
Quoting Ludwig V
Ok
Quoting Ludwig V
[/quote]
yes
Ok. @Michael has been using that as an example of a supertask so I can't say. I haven't studied them much.
Quoting noAxioms
Ok. Perhaps you and @Michael could hash this out. He thinks supertasks are metaphysically impossible, and you think they're everyday occurrences. I'm agnostic on the matter except to say that I don't think they're metaphysically impossible, whether they're physically possible or not.
Quoting noAxioms
The state of the lamp is defined at each of the times 1/2, 3/4, 7/8, ... but it's not defined at 1.
Like any other function defined at some elements of a set but not others, I am free to define it any way I like.
Quoting noAxioms
Thanks I'll check it out.
Quoting noAxioms
Something went wrong with the quoting when I quoted your post. Anyway ... yes the ping pong balls. I have no opinion about that one.
Quoting noAxioms
ok
Quoting noAxioms
I'll agree that the subject of omega sequence paradoxes is full of fallacious assumptions and confused thinking.
Quoting noAxioms
I say "item" and you change the word to "step," changing my meaning. I agree, it's not a step in a sequence. It's an item in a set.
Do you have a hard time with 0 being the limit of 1/2, 1/3, 1/4, 1/5, 1/6, ...? It's true that 0 is not a "step", but it's an element of the set {1/2, 1/3, 1/4, 1/5, 1/6, ..., 0}, which is a perfectly valid set. You can think of 0 as the infinitieth item in an ordered set, but not the infinitieth step of a sequence.
Quoting noAxioms
Even if space is continuous, we can't cut it up or even sensibly talk about it below the Planck length. With our present understanding of the limitations of physics, the question of the ultimate nature of space is metaphysics and not physics.
Quoting noAxioms
Yes ok ... math and physics are human inventions that bear some mysterious relation to reality. I agree with that, if that's what you meant.
Quoting noAxioms
Not sure what you mean. The closed unit interval [0,1] has a first point and a last point, has length1, and is made up of 0-length points.
It seems you misunderstood.
No. Consider the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, ...
It has the limit 0.
We may form the ordered set {1, 1/2, 1/3, 1/4, 1/5, 1/6, ..., 0}. It's a perfectly sensible set.
In this context 0 is the largest element in the set. It's the final "item" if you like. But 0 is not any step in the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, ..., and that sequence has no last step.
Is my use of the words step and item more clear?
Quoting Lionino
Ok. But one has to be careful of applying math to the world, this one or any other. Physicists typically model time as a real number, but there's no evidence that time is a continuum as the real numbers are. So math gets applied to physics heuristically or pragmatically, and not metaphysically. We model time using the real numbers because it's handy and gets us results, not because we actually believe time is like the real numbers.
I was making my point about mathematical convergent sequences. Don't know whether it strictly applies to walking.
Quoting fishfryPerhaps he does, but he fallaciously keeps submitting cases that need a final step in order to demonstrate the contradiction. I don't.
I say they're conditionally physically possible, but the condition is unreasonable. There seems to be a finite number of steps involved for Achilles, and that makes the physical case not a supertask. I cannot prove this. It's an opinion.
Quoting Ludwig VI have no problem with any that.
OK, that's probably a problem. It is treating something that isn't a number as a number. It would suggest a prior element numbered ?-1.
But you can traverse the space of that step, even when well below the Planck length.
In math? Via the standard limiting process. In physics? I don't know,
fishfry
In physics, the same way as math, except one isn't required to ponder the physical case since it isn't abstract. One completes the task simply by moving, something an inertial particle can do. The inertial particle is incapable of worrying about the mathematics of the situation.
noAxioms
Yes ok ... math and physics are human inventions that bear some mysterious relation to reality. I agree with that, if that's what you meant.
So it does. Zeno's supertask is not a closed interval, but I agree that closed intervals have first and last points.
Well between the two of you I have no idea what a supertask is anymore.
Quoting noAxioms
I tend to agree with you, that supertasks either (a) may be physically possible via the physics of the future; or (b) are already possible when I go from the living room to the kitchen for a snack, first traversing half the distance, then half of the remaining half, and so forth, and somehow miraculously arriving at my refrigerator. Which keeps things cold in a warm room, in clear violation of the second law of thermodynamics. Truly we live in remarkable times.
Quoting noAxioms
Only mathematically, In terms of known physics as of this writing, we can not sensibly discuss what might be going on below the Planck length. Maybe space is continuous. Maybe we live in a discrete grid of points -- which would actually resolve Zeno's paradoxes. Maybe something entirely different and not yet imagined is going on. We just don't know.
But you can't say "you can traverse the space of that step, even when well below the Planck length" because there is no evidence, no theory of physics that supports that claim.
Quoting noAxioms
Well yes, motion is possible. That's one response to Zeno. Not so satisfactory though. Did I complete a supertask when I got up to go to the kitchen for a snack? I have no idea, even though motion through space within an interval of time is an every day occurrence.
Quoting noAxioms
Ok. I thought you were claiming supertasks had to related to open intervals.
Yes. But I have an obstinate feeling that that fact is a reductio of the process that generated it. So I'm not questioning what you say, but rather what we make of it.
Quoting fishfry
It may be a bad habit to think of applications of a mathematical process. But that's what's going on with the infinite staircase. So it might be relevant to that.
Right. So why is a lamp circuit that can switch states in arbitrarily small slices of time reasonable, and spaghetti isn't? That's one of the cognitive traps of the lamp problem. IMO the final state is simply not defined by the premises of the problem, AND there is no solution that makes the sequence continuous, therefore spaghetti is as sensible as anything else. And I've convinced myself that this is the solution to the problem.
Quoting Ludwig V
The staircase is different from the lamp. The walker is on step one, the walker is on step two, etc. So if we define the final state to be that the walker is at the bottom of the stairs, that definition has the virtue of making the walker's sequence continuous. So it's to be preferred over all other possible solutions.
I agree with you. It suits my approach well, in that the existence of the problem is a result of the way it is defined, or not defined.
Quoting fishfry
That's the way ? is defined, isn't it? Although I'm not sure what you mean by "continuous" there.
I still feel uncomfortable, because it does get to the bottom of the stairs by placing a foot on each of the stairs, in sequence. But that's exactly the hypnotism of the way the problem is defined. And if an infinite physical staircase is the scenario, then anything goes..
A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time."
Is it a belief thing, like it is some kind of religious proposition or something? "Hey, I'm going rogue here and will suspend belief that 7 is a factor of 35".
Treating infinity as a number, something you didn't do in your unionized set above
Yea, when it normally is depicted at the beginning. From what I know, a set is a set regardless of the ordering. There must be a different term (ordered set?) that distinguishes two identical sets ordered differently, sort of like {1, 3, 5, 7 --- --- 8, 6, 4, 2}
It violates thebijunction. You can't say what number comes just before it, which you can for any other element except of course the first. You can do that with any other element.
OK, but what problem does it solve? It doesn't solve Zeno's thing because there's no problem with it. It doesn't solve the lamp thing since it still provides no answer to it.
Nobody's asking the particle to meaningfully discuss (mathematically or not) the step. It only has to get from one side to the other, and it does. Your argument is similar to Michael wanting a person to recite the number of each step, a form of meaningful discussion.
It would falsify the first premise. Continuous space falsifies the second premise. Zeno posits two mutually contradictory premises, which isn't a paradox, only a par of mutually contradictory premises,.
But I can say "for all we know, ....", and then there's no claim. I'm not making the claim you state. I'm simply saying we don't know it's not true. I even put out my opinion that I don't think it's true, but the chessboard thing isn't the alternative. That's even worse. It is a direct violation of all the premises of relativity theory (none of which has been proved).
Quoting fishfry
Spot on, yes.
Quoting MichaelYea, I don't know how that could have been lost. I don't think anybody attempted to redefine it anywhere.
I love when people agree with me. It happens so seldom around here :-)
Quoting Ludwig V
I agree with you too!
Quoting Ludwig V
Let me see if I can clarify my point.
In the lamp problem, we have the sequence 0, 1, 0, 1, 0, 1, ... We can "complete" the sequence by defining the state at [math]\omega[/math] as 0, or 1, or a plate of spaghetti. In no conceivable completion can the sequence be made continuous, because 0, 1, 0, 1, ... simply does not have a limit.
But in the staircase problem, if 1 is "walker is on the step" and 0 otherwise, then we have the sequence 1, 1, 1, 1, ... which has the limit 1. So 1, the walker is on the step, is the natural state at the end of the sequence.
Does that make sense? The staircase has a natural answer; but the lamp has no natural answer. Any completion whatsoever is as equally bad as any other.
Well ok, then why don't I complete a supertask when I walk across the room, first going halfway, etc.? Can you distinguish this case from your definition?
What I think about supertasks is:
* Either they are already possible in the sense of Zeno, when I walk across the room; or
* They are physically impossible in currently known physics (because of Planck) but may be possible in future physics, by analogy with previous scientific revolutions; and
* You have not convinced me or even made me understand your reasoning that supertasks are "metaphysically impossible" or that they entail a logical contradiction.
I'm making the point that you are perfectly willing to accept {1/2, 3/4, 7/8, ..., 1} as a valid set that contains an infinite sequence and its limit; but you are having trouble accepting {1, 2, 3, ..., [math]\omega[/math]}, simply because it's far less familiar. But in terms of their order, they are exactly the same set. They have very different metric properties; but strictly with respect to order, they are two different representations of the same ordered set.
Quoting noAxioms
Transfinite ordinal numbers are numbers. It's just a matter of expanding one's concept of a number. [math]\omega[/math] is a number. It's the first transfinite ordinal number. I am casually calling it a "point at infinity," but if that bothers you, just think of it as 1 in the set {1/2, 3/4, 7/8, ..., 1}. It's exactly the same thing wearing a different suit of clothes.
Quoting noAxioms
Yes, ordered set. I have been casually using the curly braces, but you are absolutely correct. {1/2, 3/4, 7/8, ..., 1} has no order, I could stick the 1 in the middle or at the beginning and it would be the same set, but I'd lose the order that I consider important.
Perhaps a notation like <1/2, 3/4, 7/8, ..., 1> would be better, to indicate an ordered set. You are absolutely right. I did not want to add any more complications earlier, but the curly braces are inaccurate in the way I'm using them. I'm speaking of ordered sets. So I'll use angle brackets from now on.
Quoting noAxioms
Well then here yet another representation of the same idea. Suppose I reorder the natural numbers
<1, 2, 3, 4, ...>
by putting 1 at the end, so that I have:
<2, 3, 4, ..., 1>
You can see that I still have a bijection. As you noted, sets don't have order, so it's still the same set.
Note that I no longer have an order-preserving bijection. I merely have a set bijection. I can still correspond 1 to 1, 2 to 2, and so forth. But I can't do it in an order-preserving manner.
But now I have another representation of an ordered set that consists of an infinite sequence followed by a "point at infinity," or a largest element. That largest element does not have a predecessor, you are right about that.
And in fact we have a name for that. In ordinal theory, an ordinal with a predecessor is a successor ordinal. And an ordinal without a predecessor is a limit ordinal. So your intuitions are spot on.
Quoting noAxioms
Ah yes, why am I doing all this?
It solves the lamp problem. The lamp state is a function on <1/2, 3/4, 7/8, ..., 1> defined as "on" at 1/2, "off" at 3/4, "on" at 7/8, and so forth.
But now we see (more clearly, IMO) that the state at 1 is simply undefined. The statement of the problem defines the lamp state at each element of the sequence; but does NOT define the state at the limit.
We also note that there is no way to make the sequence 0, 1, 0, 1, ... continuous.
And since you didn't tell me what is the state at 1, and there is no natural way to define the state at 1, I am free to define the state at 1 any way I like. And inspired by Cinderella, I define the state of the lamp at 1 to be a plate of spaghetti. That's the solution to the problem. The final state is anything you like. It doesn't even have to be on or off since it's not a real lamp, just as Cinderella's coach is not a real coach. The lamp problem is every bit as much a fairy tale as Cinderella.
So for many of these supertask problems, the ordered set <1/2, 3/4, 7/8, ..., 1> is the natural setting for the problem.
Note that the staircase is different. The walker is on step 1, on step 2, etc. So the natural, continuous way of completing the sequence is to say that the walker is at the bottom of the stairs. This is totally different than the lamp, which can not be made continuous or sensible in any way at all.
So my entire point is that <1/2, 3/4, 7/8, ..., 1> is the natural way to think about these problems. The question is always: how did we define the state at the elements of the infinite sequence; and then, how are we free to define the final state at the limit.
Quoting noAxioms
I'm not sure what you mean by referring to the subjective state of the particles. When Newton wrote down his great law of gravitation, he did not care how the masses feel about it. I'm not following your analogy.
Quoting noAxioms
I confess to not knowing the answer to Zeno. It's a clever argument. Unless the answer is that we satisfy Zeno and execute a supertask every time we walk across the room. But @Michael objects to that, for reasons I don't yet understand.
Quoting noAxioms
Some speculative physicists (at least one, I believe) think the world is a large finite grid. It's not out of the realm of possibility as I understand it. I think I read that in Penrose's The Road to Reality. And if Sir Roger thinks it's good enough to put in a book, it must be of interest.
In other words the chessboard universe is not ruled out by any known theory or experiment. And we know that quantum and relativity have yet to be integrated, and perhaps that's a clue.
Quoting noAxioms
Yes ok but then ... how is walking across the room by first traversing 1/2, then half of the remaining half, etc., not a supertask? I don't understand this point.
If supertasks are impossible and motion is possible then motion isn't a supertask.
Quoting fishfry
By definition supertasks are non-terminating processes, therefore you've gone wrong somewhere if you conclude that they can terminate after 2N seconds.
Also I think the clearest example I gave was that of having counted down from infinity. We can assert (explaining what happened in reverse) that I recited 0 after 60 seconds, recited 1 after 30 seconds, recited 2 after 15 seconds, recited 3 after 7.5 seconds, etc., and we can say that we can sum an infinite series with terms that match the described (and implied) time intervals, but it doesn't then follow that we can have counted down from infinity; we can't even start such a count. The mathematics is evidently a non sequitur, and so it's a non sequitur in the case of having counted up to infinity as well (and so for any proposed supertask).
In the case of Thomson's lamp, nothing ever happens to the lamp except as described by this process: I turn it on after 30 seconds, turn if off after 15 seconds, turn it on after 7.5 seconds, etc. It cannot be on after 60 seconds because I always turn it off after turning it on and it cannot be off after 60 seconds because I always turn it on after turning it off, but it must be either on or off after 60 seconds, and so therefore there is a contradiction.
If you want to say that such a supertask is possible then the burden is on you to explain the state of the lamp after 60 seconds, and your answer must follow from the description of the supertask. If nothing follows then the supertask is impossible.
I don't find that satisfactory. It only casts doubt on the premise "if supertasks are impossible."
I agree with you that the lamp and staircase and other related puzzles are qualitatively different than Zeno's paradoxes of motion, so perhaps in that sense you want to reserve the word supertask for the former. But your definition is "completing a countably infinite number of tasks in finite time," and walking across the room seems to satisfy that definition.
Not to mention that, if we take the real numbers as a model of space, we pass through uncountably many points in finite time. That's another mystery.
Quoting Michael
I don't know what you mean that supertasks are nonterminating by definition. Just thinking mathematically for a moment, limits "terminate" a sequence in the sense that 1 is the terminus of the sequence 1/2, 3/4, 7/8, ... The limit 1 is not part of the sequence, but we can imagine the 1 stuck at the end of an ordered set, as I have been doing, and it's perfectly sensible.
In other words supertasks are nonterminating, but they definitely may have a terminal state; just as a convergent mathematical sequence has no final term, yet has a limit. Is my analogy unsatisfactory with respect to your conception of supertasks?
Quoting Michael
I don't follow how you are counting down from infinity. In fact when you count down from infinity, it's always only finitely many steps back. If I take the ordered set <1/2, 3/4, 7/8, ..., 1> and I start at 1, my first step backwards jumps over all but finitely many elements of the sequence, and it's always only finitely many steps back from 1 to 1/2.
[Per my recent convo w/@No Axioms I am using angle brackets to denote ordered sets].
You did lose me when you said that counting 0, 1, 2, ... is "counting down from infinity." I did not understand that example when you gave it earlier. Mathematically, the ordered set <1, 2, 3, ...> exists, all at once. Its counting is completed the moment it's invoked into existence by the axiom of infinity.
But let me ask you this. Suppose I say 0 at 60 seconds, and 1 at 30 seconds, and 3 at 15 seconds, and so forth.
Now I claim that after 120 seconds (the sum of the series) I have counted all the natural numbers!
Yes I claim that. And as proof, I challenge you to name the first number that I did not count.
Since you can not do that, I have indeed counted all the natural numbers.
Quoting Michael
I either don't understand what you mean, or I strenuously disagree.
Explain please?
Quoting Michael
I just proved to you, using a very standard inductive argument, that I can indeed count all the natural numbers as you described, in intervals of 60, 30, etc. Because you can not name any number I can't count. Did I count 47? Sure. Googolplex? Sure. Graham's number? Sure. There is no number that I didn't eventually count. Therefore I counted them all.
Tasks are performed ad infinitum. I never stop counting. There's always another number to count.
Quoting fishfry
I'm talking about reciting the numbers. So imagine someone reciting the natural numbers up to infinity. Now imagine that process in reverse. That's what I mean by someone counting down from infinity.
It is a non sequitur to argue that because we can sum an infinite series with terms that match the proposed time intervals that it is possible to have counted down from infinity. It is impossible, even in principle, to start such a count. The maths of an infinite series doesn't change this.
And it is a non sequitur to argue that because we can sum an infinite series with terms that match the proposed time intervals that it is possible to have counted up to infinity. It is impossible, even in principle, to stop such a count. The maths of an infinite series doesn't change this.
Did I not move you, surprise you, convince you, that if you count 1, 2, 3, ... successively halving the time intervals, that you will indeed count every single natural number in finite time? If not, why not?
I am still waiting for you to name the first number I didn't count.
This is a standard inductive argument. To prove that a property holds for all natural numbers, I show the impossibility of there being a first number where the property fails.
Quoting Michael
But counting backward from infinity is always finite! I showed you how that works, counting backward from 1 in the ordered set <1/2, 3/4, 7/8, ..., 1>
In fact this is true of all the transfinite ordinals. It's only finitely many steps backward from any transfinite ordinal, no matter how large. That's because stepping back from any limit ordinal (defined as an ordinal without an immediate predecessor) necessarily jumps over all but finitely many elements of the sequence that led up to it.
Quoting Michael
It's easy, I'll count backward from infinity right here on a public Internet forum, in plain view of the world.
1, 15/16, 7/8, 3/4, 1/2. Done. My first step necessarily jumped over all but finitely many elements of the infinite sequence. It must be that way.
That's because the first step backward from any limit ordinal necessarily jumps over all but finitely members of the sequence whose limit it is.
Counting backward from infinity is easy, and always finite!
Because it begs the question.
Quoting fishfry
What number do you recite after 1?
Quoting fishfry
That's not counting down from infinity. That's just reciting five rational numbers.
I go 1 at 60, 2 at 30, etc.
Name the first number that I fail to count
Third time I'm asking you the question. (At ever decreasing intervals of time!)
This is a standard inductive argument. If it's impossible to name the first natural number at which a property fails to hold, the property must hold for all natural numbers. Think back to when you learned inductive proofs in school. I can name 1. And if I name n, I can name n + 1. Therefore I can name all the numbers. Counterintuitive though it may be, it's true. You learned this in high school.
Please give this argument some thought.
Quoting Michael
You have no proof or evidence. On the contrary, the mathematics is clear.
It begs the question. Your premise is necessarily false. Such a supertask is impossible, even in principle, to start.
Quoting fishfry
You just listed five rational numbers and are claiming that this is proof of you reciting all the natural numbers in descending order? You're talking nonsense.
Quoting fishfry
What number do you recite after 1?
Did you learn mathematical induction in school? Please review that. Please take the time to understand the argument I made.
Under the premises of the problem you posted, there is no number that does not get spoken.
It's imperative that you understand that. It's pointless for you to disagree. You must show that there's a number that did not get spoken. If you can't do that, then every number gets spoken.
Quoting Michael
I did not make that claim. I said I counted backward from a limit ordinal. That's easy. It's always a finite number of steps.
Quoting Michael
I'm counting backward from a limit ordinal. Very standard math.
Quoting Michael
7/8 will do just fine. I necessarily had to jump over all but finitely members of the sequence.
Of course I can not count ALL the numbers backward. That's impossible. That's because limit ordinals do not have predecessors. That's the definition of a limit ordinal, an ordinal that does not have an immediate predecessor. So it's your challenge that's nonsense.
But please, I'm asking you to sit down and think about the inductive argument I made.
Counting forward with your 60 second idea, which number won't be spoken?
We can certainly say "1". And if we say n, we can say n + 1. This is high school mathematical induction. Please tell me you learned this. If not, that would explain your confusion. But if you made it through high school math (do they still teach induction in high school? What do I know) then you have the means to understand the argument.
Please take the time to think it through. What number can't be spoken?
ps -- I looked it up. Perhaps induction is not universally taught in high school, and it doesn't come up in calculus.
Do you know mathematical induction? It's a row of dominos.
https://en.wikipedia.org/wiki/Mathematical_induction
No, we're reciting the numbers in descending order. It's impossible to do, even in principle. The fact that we can baselessly assert that I recite the first number in N seconds and the second number in N/2 seconds and the third number in N/4 seconds, and so on ad infinitum, and the fact that the sum of this infinite series is 2N, doesn't then entail that the supertask is possible.
That we can sum this infinite series is evidently a red herring.
You're right that we can't "name all the numbers" going backward. but that's obvious. There's no largest number and limit ordinals don't have immediate predecessors.
It's pointless for you to snap back a minute later arguing with well established mathematical facts. I gave a solid inductive argument that by the premises of your 60 second puzzle, all the numbers will be spoken. That's because there's no first number that won't be spoken. If you doubt that, then name a number that's not spoken.
I ask you to read carefully what I'm writing, and think about it.
Did you ever learn mathematical induction? If yes, I gave a standard inductive argument. If no, that's a good starting point and I'll be happy to give a summary. I gave you the Wiki link. I can't argue well established facts with you.
In the puzzle you gave, every number must be spoken. In fact we could calculate, if we cared to, the exact time at which it's spoken.
Please give this some thought.
What number won't be spoken?
No you haven't. Your premise begs the question and simply asserts that all the natural numbers have been recited within 60 seconds.
Name the first one that's not. It's a trivial exercise to identify the exact time at which each natural number is spoken. "1" is spoken at 60, "2" at 90, "3" at 105, "4" at 112.5, and so forth.
Can you not see that we can calculate the exact time at which each number is spoken?
I did not "simply assert" all the numbers are spoken. I proved it logically. Induction works in the Peano axioms, I don't even need set theory.
If you work through this example you will obtain insight.
So do I. There's a paradox about agreement, that it is the purpose, but also the end, of the discussion. So people tend to focus on disagreements.
Quoting fishfry
I found that discussion very helpful.
Quoting fishfry
Have I understood right, that 0 means "walker is not on the step", and that "the step" means "the step that is relevant at this point" - which could be 10, or 2,436? So 0 would be appropriate if the walker is on the floor from which the staircase starts (up or down)
My instinct would have been to assign 0 also to being on the floor at which the staircase finishes (up or down). It makes the whole thing symmetrical and so more satisfying.
Quoting fishfry
I don't like that way of putting it, at least in the paradoxes. Doesn't the arrow paradox kick in when you set off in the.reverse direction? Or perhaps you are just thinking of the numbers as members of a set, not of what the number might be measuring. I suppose that's what "ordinal" means?
Quoting fishfry
Quoting fishfry
Michael's way of putting the point is, IMO, a bit dramatic. The boring truth for me, is that the supertask exists as a result of the way that you think of the task. If you think of it differently, it isn't a supertask. It's not about reality, but about how you apply mathematics to reality.
Quoting fishfry
Well, if you insist on describing things in that way .... I'm not sure what you mean by "model". I think of what we are doing as applying a process of measuring and counting to space - or not actually to space itself, but to objects in space. A geometrical point has no dimensions at all. So it is easy to see how we can pass infinitely many points in a finite time. (I'm not quite sure how this would apply to numbers, but they do not have any dimensions either.) This doesn't apply to the paradoxes we are considering, which involve measurable lengths, but it may help to think of them differently.
Quoting Michael
That's all very well. But it also takes us back to the question what this "operation" actually is. If you think of it as an action that takes a measurable amount of time, you can't, by definition. When we perform a calculation, that is an action in physical time. But a mathematical operation isn't quite like that, and somewhere in that is the answer (possibly).
Quoting fishfry
Yes, but you didn't speak all the natural numbers, and indeed, if induction means what I think it means, your argument avoids the need to deal with each natural number in turn and sequence.
I'm sorry this is a bit scrappy, but there are lot of issues going on at the same time here. Great fun!
After 60 seconds I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.
What natural number did I not say?
You can't answer, therefore it is metaphysically possible to have recited the natural numbers in descending order.
---
Obviously the above is fallacious. It is metaphysically impossible to have recited the natural numbers in descending order. The fact that we can sum an infinite series with terms that match the described and implied time intervals is irrelevant. The premise begs the question. And the same is true of your version of the argument.
I already agreed with this, because limit ordinals do not have immediate predecessors.
Quoting Michael
If you would engage in your private time with the 60 second puzzle, you would see that each number is spoken at a specific, calculable time; that there is no first number that's not spoken; and therefore every number is spoken.
It's not productive for me to give a high-school level inductive argument and for you to say "nonsense" and "metaphysically impossible" without ever engaging with the argument.
Please read the Wiki page on mathematical induction and ask questions as necessary, and challenge yourself to engage with the argument.
Ask yourself: What is the first number not spoken? If you ask yourself that enough times, you may have an epiphany.
Argument 1
Premise: I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum.
What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in ascending order.
Argument 2
Premise: I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.
What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in descending order.
---
In both cases for any given natural number I can calculate how long it took me to reach it.
These arguments only show that if I recite the natural numbers as described then I have recited all the natural numbers, but this does nothing to prove that the antecedent is possible, and it is the possibility of the antecedent that is being discussed. As it stands you're begging the question.
Now let's assume that it's metaphysically possible to have recited the natural numbers in ascending order and to have recorded this on video/audio. What happens when we replay this video/audio in reverse? It's the same as having recited the natural numbers in descending order which you admit is metaphysically impossible. Therefore having recited the natural numbers in ascending order must also be metaphysically impossible.
Both Argument 1 and Argument 2 are unsound. The premises are necessarily false. It is impossible in principle for us to recite the natural numbers in the manners described.
It could be anything. The problem has nothing to do with the operation being performed and everything to do with continually halving the time between operations.
At 0s A ? 1, at 30s A ? red, at 45s A ? turtle, at 52.5s A ? 1, at 56.25s A ? red, and so on ad infinitum.
Or:
At 60s A ? 1, at 30s A ? turtle, at 15s A ? red, at 7.5s A ? 1, at 3.75s A ? turtle, and so on ad infinitum.
That an infinite series with terms that match the described and implied time intervals has a finite sum isn't that it makes sense for either set of tasks to have actually been carried out. This should be self-evident in the second case. You're being deceived by maths if you think the first case is different.
Very interesting. If only I knew what "metaphysically possible" means? Can you help?
(I do know what "I said
https://en.wikipedia.org/wiki/Subjunctive_possibility#Types_of_subjunctive_possibility
Such is the nature of philosophy.
Quoting Ludwig V
I had a thread on that a while ago if you care https://thephilosophyforum.com/discussion/14855/metaphysically-impossible-but-logically-possible/p1
Are they? Does ?? have meaning? It does for numbers. It's a serious question. I am no expert on how transfinite ordinal numbers are treated. It seems like a different species, like having a set {1, 2, 3, ... , green} which is also a valid set, and countable.
Ordering irrelevant. The set supposedly needs to be countable, and it is. Michael's definition of supertask came from wiki, and that definition says it is countable, else it's a hypertask. The SEP definition of supertask omits the 'countable' part and seemingly groups the two categories under one word.
The definition also includes 'sequential', meaning parallel execution of multiple steps is not allowed.
Clearly it isn't a supertask if it is impossible to go only half the remaining distance for some intervals. If that is possible, then it must be a supertask.
I take that back. It doesn't violate the bijection. And I spelled it wrong too. So many errors.
That's fine. The rational numbers are both ordered and countable, but they cannot be counted in order.
Sounds like the lamp problem is unsolved. It is still 'undefined'.
Another note: The paradox of the gods that I occasionally bring up is fun to ponder, but it isn't a supertask since it cannot be completed (or even started). Progress is impossible. Ditto with the grim reaper 'paradox' where I die immediately and cannot complete the task.
There is no bottom, and the OP did not suggest a bottom step. He is done, and no stairs are observable. It's mathematical only, but framed with a physical sounding analogy, which makes it fall apart.
Your ? might help with the stairs. The guy is at 'the bottom' and there is but the one step there, labeled ?. No steps attached to it, but step on that one step and up you go, at some small finite numbered step after any arbitrarily small time.
His assertion isn't justified, I agree.
So much for the postulates of relativity then. I kind of thought we demolished that idea with some simple examples. It seems to be a 'finite automata' model, and the first postulate of SR is really hard (impossbile) to implement with such a model, so a whole new theory is needed to explain pretty much everything if you're going to posit something like that. I haven't read it of course, so any criticism I voice is a strawman at best.
The chessboard universe sounds very classical, and it's been proven that physics is not classical, so I wonder how this model you speak of gets around that.
Quoting MichaelThis evaded the question ask. Sure, we all agree that if supertasks are impossible, then supertasks are impossible. He asked how you justify the impossibility of a supertask. All your arguments seem to hinge on a variant that there isn't a largest natural number.
Quoting MichaelThe wiki definition you gave made no mention of 'terminate'. If you mean that it doesn't complete, it by definition does in a finite time. If you mean that it has no terminal step, then you're making the mistake I identify just above since the definition does not require one.
Quoting MichaelYou also wield the term 'ad infinitum', which typically means 'going on forever', which also violates the definition which explicitly requires a finite time to the task You very much do stop counting at time 1. There is at that time not another number, so by counterexample, your assertion that you will never stop counting is false.
How can a sequence of operations in which each operation is performed only after the previous operation is performed complete without there being a final operation?
You just seem to hand-wave this away with no explanation.
Quoting noAxioms
Well, yes. That's how to define it as an infinite sequence of operations rather than a finite sequence of operations.
By definition, the sequence completes by having every operation occurring before some finite time. To demonstrate otherwise, one must find a remaining operation which necessarily is not completed at that time.
That there must be a final step in such a sequence not only does not follow from the description of such a supertask, it in fact directly contradicts the description of the supertask.
To demonstrate the impossibility of Zeno's physical supertask, one must attack the premise, not the logic. The logic is valid, at least until he additionally posits the impossibility of the first premise, but that only gives rise to a direct contradiction, not a paradox.
X is a true fact of motion. X is is a false fact of motion. Therefore either motion is impossible, or at least one of the premises is wrong.
What does it mean for every operation to occur without some final operation occurring?
As it stands your definition is a contradiction.
It means that is isn't a finite sequence of operations. How is it a contradiction that there isn't a final natural number? Instead of just asserting it, show it.
@Barkon also seems to be running on his intuitions and makes unjustified assertions.
If I'm wrong, don't just tell me; show me where.
If Michael is wrong, don't just tell him; show him where.
No, it doesn't. Saying that it is an infinite sequence of operations means that it isn't a finite sequence of operations.
I'm asking you to make sense of the "every operation is performed" part of "every operation is performed in an infinite sequence of operations.
Glad to hear that.
Quoting Ludwig V
Could be. Truth be told I got lost in the OP involving many non-relevant fairy tale elements and probably don't even understand what the staircase question is.
Quoting Ludwig V
Ordinals are [ul=https://en.wikipedia.org/wiki/Well-order]well-ordered sets[/url].
As with my standard example, if you take the ordered set <1/2, 3/4, 7/8, ..., 1>, where I'm now using angle brackets to denote ordered sets, suppose you start from 1 and take a step back. Since 1 has no immediate predecessor, any step back necessarily jumps over all but finitely many members of the sequence. It's a counterintutive quirk of limit ordinals. Any path back to the beginning involves only finitely many steps, because the first step back makes such a jump.
Quoting Ludwig V
Yeah. "Metaphysically impossible!" "Non sequitur!" "Nonsense!" Never an actual argument. Tagging @Michael so as not to disparage him behind his back.
Quoting Ludwig V
I still don't know if walking across the room is a supertask or not.
Quoting Ludwig V
Nobody knows the ultimate metaphysical truth about reality. All we can ever do is model is. Relativity is a mathematical model, as is quantum physics, as was Newtonian physics. All science can ever do is build models that fix the experimental data to a reasonable degree of approximation. That's all I mean by model.
Quoting Ludwig V
Only to our latest conceptual model of space. We can't know ultimate reality. Or if we can, we don't as of yet.
Quoting Ludwig V
The unit interval [0,1] has length 1 and is composed of uncountably many zero-length points. That's a mystery.
Quoting Ludwig V
I apparently misunderstood @Michael's backward counting example, I'll be addressing that shortly as I slog through my mentions.
But if I count forward at successively halved intervals: Saying "1" at time 1, "2" at time 1/2, "3" at time 1/4, etc., I will certainly count all the numbers. You can't name the first one I don't say. And we can calculate exactly what time I'll say 47, or Googolplex, or Graham's number.
I see that I misunderstood your idea. You are counting time backward. Ok I'll respond to that. But just wondering, when you realized I misunderstood you earlier, why didn't you point that out?
Ok. Suppose that I start at 1 and count backward through 1/2, 1/4, 1/8, ...
Clearly I say all the numbers. at 1 I say 1, at 1/2 I say 2, at 1/8 I say 3, and in general at [math]\frac{1}{2^n}[/math] I say n.
It's perfectly clear that I say all the numbers, and iterate through all the negative powers of 2. This is elementary. What number don't I say?
Quoting Michael
There is no natural number that wasn't said. Therefore they were all said.
Quoting Michael
It's perfectly obvious that an infinite sequence is infinite at one end. So you can iterate in one direction and not the other. I can't for the life of me imagine why you think that means anything important.
Now look at the sequence 1, 1/2, 1/4, 1/8, ... again. Graph the points on the real number line. You start at 1, then move leftward to 1/2, then leftware to 1/4, and so on.
The sequence has the well-known limit 0.
Now if you were to start at 0 and move any positive length to the right, no matter how small, you would necessarily jump over all but finitely many elements of the sequence. That's inherent in the meaning and definition of a limit point.
It's exactly the same as 1/2, 3/4, 7/8, ..., whose limit is 1. In fact it's the exact same situation but with the order relation reversed.
Now you want to impose some kind of Newtonian understanding of time, call 0 a time, and say this proves something. It proves nothing but ... and I don't know any other way to say this ... it proves nothing but your own lack of clear thinking around the nature of limits of sequences.
So main points;
* If you iterate through 1, 1/2, 1/4, 1/8, ... while vocalizing "0", "1", "2", "3", etc., you will iterate through ALL the elements of the sequence and you will vocalize ALL of the natural numbers. After all, what member of the sequence do you think is missing? What natural number won't be vocalized?
* Secondly, it's perfectly clear that an infinite sequence starts at one element and continues indefinitely, with no last element. So of course you can't iterate an infinite sequence "from the end." But this is a triviality, it has no significance.
Quoting Michael
Fallacious! Non sequitur! Metaphysically impossible! Nonsense!
These are words. They are not arguments.
Quoting Michael
There you go again, as Ronald Reagan once said to Jimmy Carter.
Quoting Michael
Sure, because an infinite sequence has no end. You seem to think this elementary and trivial fact has deep meaning. It does not.
Quoting Michael
I'm disappointed that you won't engage with the argument I'm making. I'll add "begs the question" to your list of buzzphrases used in lieu of substantive argument.
I believe you have agreed with me.
Quoting Michael
No, once again you recited the natural numbers in ascending order.
Normalizing the 60 to 1 for simplicity, you iterated though the sequence 1, 1/2, 1/4, 1/8, ..., while reciting the numbers 1, 2, 3, ... etc.
Quoting Michael
Begging the question! Nonsense! Non sequitur! Metaphysically impossible!
Always a buzzphrase, never a substantive argument.
Why don't you engage with the argument I'm making?
Quoting Michael
Lol. You can't play it in reverse, there is no end to the natural numbers in their usual order. You never get to the end. But there is no number you don't vocalize at time point, so you do vocalize them all.
I know this pushes hard against your intuition. If you'll engage with the argument, you will at some point develop better intuitions.
Quoting Michael
Completely false, as I've demonstrated numerous time.
Quoting Michael
Irrelevant! Non sequitur! Metaphysically impossible! Nonsense! Begging the question!
I think I'll just start speaking your language. Maybe that will work.
Thanks, I'll check that out. Perhaps it will give me insight into what @Michael means by metaphysically impossible.
5 is a natural number in the Peano axioms. Does [math]\sqrt 5[/math] have meaning? No. You have to extend to a larger number system.
[math]\sqrt \omega[/math] has no meaning in the ordinals, but I believe it does have meaning in the Surreal numbers, which I don't know much about.
You can't say "x isn't a number because I can't take its square root." You couldn't take the square root of -1 before someone discovered imaginary numbers.
The question of what is a number is historically contingent. Cantor was the one who discovered the ordinals.
It's sad IMO that everyone has heard of the transfinite cardinals, yet nobody knows about the ordinals. The ordinals are logically prior to the cardinals. These days cardinals are actually defined as particular ordinals.
Quoting noAxioms
In standard set theory, elements of sets must be other sets. But if you allow urelements, which are elements of sets that are not themselves sets, then you can put green into a set if you like. It's not forbidden by the rules of set theories that have urelements.
https://en.wikipedia.org/wiki/Urelement
But naturals aren't integers which aren't rationals which aren't reals which aren't complex numbers which aren't quaternions. There are lots of different kinds of numbers with different rules, and they were all discovered by the historically contingent work of mathematicians.
Quoting noAxioms
I should read that SEP article, I'd probably get a better understanding of this thread. Wiki giveth and Wiki taketh away. Wiki has many errors.
Quoting noAxioms
Hmm, that's interesting. In computer science you can always linearize parallel streams, there's no difference in computational power between parallel and serial processing.
Quoting noAxioms
Ok, then since walking is commonplace, so are supertasks. I gather @Michael would disagree. I haven't got an opinion.
Quoting noAxioms
No prob, I figured it out. But there are many many ways to re-order a countably infinite set. Here's one called the even-odd order:
<0, 2, 4, 6, 8, ..., 1, 3, 5, 7, ...>
You can see that this set is still in bijection with the natural numbers, but it's order-isomorphic to two consecutive copies of the naturals. This is a representation of the ordinal [math]\omega + \omega[/math].
Quoting noAxioms
Yes. Although the rationals don't represent any ordinal. The ordinals only apply to well-ordered sets.
https://en.wikipedia.org/wiki/Well-order
Ah yes, why am I doing all this?
Quoting noAxioms
It's not undefined. Inspired by the story of Cinderella, I defined the terminal lamp state as a plate of spaghetti. I have solved the lamp problem to my satisfaction.
Quoting noAxioms
So many paradoxes, so little time. I know many philosophers care about these things a lot.
Quoting noAxioms
Right, but unlike the lamp, there IS a naturally preferred solution to the staircase. If the walker is on each step at each time, then defining the walker to be present at the bottom of the stairs preserves the continuity of the path. So the staircase (if I even understood the problem, which I may not have) at least has a natural terminating state. Whereas the lamp definitely doesn't.
Quoting noAxioms
Well I agree with you there, but I can't seem to get @Michael to agree :-)
Quoting noAxioms
Finite discrete universe is pretty obscure. I don't know if it's ruled out by other physics or not.
Quoting noAxioms
No idea. Found a physics.SE thread.
https://physics.stackexchange.com/questions/22769/is-the-universe-finite-and-discrete
Quoting noAxioms
Yay you're helping me gang up on @Michael :-) He and I have been having this conversation.
I think I'll go read the SEP article on supertasks.
Missed one. :smile:
Ditto with SEP.
I worked a great deal of my career writing code for multiple processors operating under the same address space. It gets interesting keeping them from collisions, with say two of them trying to write different data to the same location.
Anyway, not sure what you mean by your statement. It seems on the surface to say two processors is no more powerful than one, which isn't true, but two also isn't twice as powerful.
You didn't read my comment then. Ability to move is a given (an axiom, not something that can be proven). Given that, doing so is a supertask only if Zeno's premise holds, that for any starting point, one must first move halfway to the goal. I can't prove that it holds, but I can't prove that it doesn't hold either.
OK. Yet another thing I didn't know.
Yes, the PoS solution.
Does 'bottom of the stairs' imply a bottom step? If every other step was black and white, what color is the bottom step? PoS, I know. Same problem from where I stand.
I'll look at that. I have all the respect for the PSE guys, who blow everybody else away. Quora stands somewhat at the opposite end of that spectrum.
Quoting Michael
Finite means bounded. That means a finite sequence of steps that has a first and last step. An infinite sequence means not (a finite sequence of steps that has a first and last step). It being called 'infinite' literally means that the last step you keep referencing doesn't exist.
I see you have Barkon joining your ranks. I hope you find the company good.
Lucky he has got Barkon joining his ranks, you can be confusing noAxioms.
Edit: you wouldn't be able to walk the infinite staircase to the end, but you can definitely conceive the completion of the task using mind.
So you're saying that it's possible to have recited the natural numbers in ascending order and possible to have recorded this on audio but impossible to then replay this audio in reverse? That seems like special pleading. Am I metaphysically incapable of pressing the rewind button?
Quoting fishfry
I am presenting two versions of your argument; one in which I have recited the natural numbers in ascending order and one in which I have recited the natural numbers in descending order. I am using the second version to illustrate the flaw in the first version.
Quoting fishfry
No, I'm reciting them in descending order. I'll repeat it again and highlight to make it clear:
I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum e.g. my recitation ends with me saying "3" at 12:00:07.5 then "2" at 12:00:15 then "1" at 12:00:30 and then "0" at 12:01:00.
What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in descending order.
Notice that even if the conclusion follows from the premise that the argument fails because the premise is necessarily false. It is impossible, even in principle, for me to have recited the natural numbers in the manner described.
Returning to your version of the argument:
I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum e.g. my recitation starts with me saying "0" at 12:00:00 then "1" at 12:00:30 then "2" at 12:00:45 and then "3" at 12:00:52.5.
What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in ascending order.
Even if the conclusion follows from the premise I do not accept that the premise can possibly be true. Like with the previous argument, I think that it's impossible, even in principle, for me to have recited the natural numbers in the manner described.
I have attempted at least to explain why this is impossible (e.g. with reference to recording us doing so and then replaying this recording in reverse), but as it stands you haven't yet explained why this is possible. If you're not trying to argue that it's possible only that I haven't proved that it's impossible then that's fine, but if you are trying to argue that it's possible then you have yet to actually do so.
Can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum? It's not something that we can just assume unless proven otherwise. Even Benacerraf in his criticism of Thomson accepted this.
Would you then argue that infinity is like exposed electric or wiring and it cannot conclude?
That's almost right, the logic is valid, but not necessarily sound. Soundness requires true premises. Generally though, judgement of the premises is dependent on empirical knowledge, which all good philosophers know is unreliable. Therefore we have valid logic and if the premises are disproven they would be disproven by competing premises, and the judgement ought not be based on empirical knowledge..
The premise of infinite divisibility is provided from mathematical axioms. The conflicting premise which would be used to disprove this, the limitations of divisibility, cannot be derived from the unreliable empirical knowledge, and it has not yet been provided. So we need to defer judgement, until we can disprove the mathematical axioms in a more reliable way.
The fallacy in his reasoning is that it does not acknowledge that for all t[sub]n[/sub] >= t[sub]1/2[/sub] the lamp is on iff the button was pushed when the lamp was off to turn it on and the lamp is off iff the button was pushed when the lamp was on to turn it off. The lamp "arbitrarily" being on or off at t[sub]1[/sub] makes no sense.
This logical consequence can be shown when the experiment is explained more clearly:
A. At t[sub]0[/sub] the lamp is off, at t[sub]1/2[/sub] I press the button
B. At t[sub]0[/sub] the lamp is off, at t[sub]1/2[/sub] I press the button, at t[sub]3/4[/sub] I press the button, at t[sub]7/8[/sub] I press the button, and so on ad infinitum
The lamp being on or off at t[sub]1[/sub] must be a logical consequence of the lamp being off at t[sub]0[/sub] and the button-pressing procedure that occurs between t[sub]0[/sub] and t[sub]1[/sub] because nothing else controls the behaviour of the lamp.
With (A) we can deduce that the lamp is on at t[sub]1[/sub]. But what of (B)? If no consistent conclusion can be deduced then its button-pressing procedure is proven metaphysically impossible.
Not sure what you mean by potential cardinality.
Quoting noAxioms
Point being that you get no increase in computational power from parallelization.
Quoting noAxioms
No function is computable by a parallel process that's not already computable by a linear process. Talking computability theory, not software engineering.
Quoting noAxioms
I proved it at the supermarket today, unless you think my vat programmers fooled me again.
Quoting noAxioms
Well maybe it's all an illusion.
Quoting noAxioms
LOL
Quoting noAxioms
Coloring the steps reduces to the lamp.
Quoting noAxioms
My Quora feed gives me a lot of cute cat pics lately. Makes me happy. Quora certainly used to be a lot better.
If you play the recording in reverse, the very first movement of the tape or recording, no matter how small, must necessarily jump over all but finitely many of the vocalizations. For the same reason I've explained earlier. Cute thought experiment though.
But it's just like stepping backward from the limit of a sequence of real numbers. The first step, no matter how small, jumps over all but finitely members of the sequence. It's the same fact as saying that any circle drawn around a limit point necessarily contains all but finitely many elements of the sequence.
Quoting Michael
I didn't see any flaw. I didn't go back to look up that post, but I do remember responding to it. I can only ask you to reread my earlier response.
Quoting Michael
I already responded to this. It's the sequence 1, 1/2, 1/4, 1/8, ..., accompanied by the vocalizations 1, 2, 3, ... Every member of the sequence gets traversed, every natural number gets vocalized.
Since the limit of the sequence is 0, if you start at zero and take even the smallest step forward, you necessarily leap over all but finitely many elements of the sequence.
Do you understand this point? Mathematically I mean, nevermind the element of time, which is a red herring. Do you understand that any interval around the limit point of a sequence must contain all but finitely elements of the sequence? That's the key insight to untangle your example.
Quoting Michael
I've shown several times exactly how to do it, and I've proven that every number gets vocalized.
Quoting Michael
I get that you think that. If you would attempt to engage with my argument you might have an insight and develop better intuitions about limits of sequences.
Quoting Michael
I can't repeat myself again. I have nothing new to say. If you'd read my posts and have yourself a serious think, then come back with a substantive reply, we might get somewhere. We are not making progress.
Quoting Michael
That's the premise of your own example. It's not my premise. That's hilarious. In the end, you are reduced to denying your own premise.
Quoting Michael
It's your example, not mine.
Quoting Michael
Feel free to give a reference, else I can't respond.
The fact that there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 does not prove that the following supertask is metaphysically possible:
I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.
How does one start such a supertask?
Quoting fishfry
From Tasks, Super-Tasks, and the Modern Eleatics:
Also see my comment here where I try to explain where his arguments fail to "destroy" Thomson's.
Quoting noAxioms
Quoting Michael
So there is a common understanding of what the issue is. Your disagreement is about different ways of responding to it. Don't you think?
Ryle might have called it a category mistake and talked of putting a physical harness on a mathematical horse or (better, perhaps) putting a mathematical harness on a physical horse, He and many others thought that nothing further needed to be said.
But this problem makes me think that they were wrong. One issue that comes to mind is the issue of making a 2-dimensional map of a 3-dimensional sphere. Euclid doesn't work (accurately). But the problem is resolved by developing a different geometry, which breaks some of Euclid's rules. (I realize I'm oversimplifying here, but I hope I'm not hopelessly mistaken.)
One point to take into account here. This is a thought experiment, so, while the mathematics is real, the horse is not physical, but imaginary, and the difficulty is to work out what rules apply to that in-between context.
I followed the link you gave me and found another link - Quoting Lionino. This is your exposition of Toulmin's argument about synthetic necessities. Toulmin (for whom I have a lot of time) clearly identifies a class of propositions which orthodox philosophy has not recognized. But he is right.
In order to provide a nutshell explanation, I would say that the point is that the rules of a game can rule out possibilities which are physically possible, but violate the rules of the game. So in a way, they seem to be ruled out a priori or analytically, yet they are physical possibilities. Hence he classifies them as synthetic necessities.
You are interested in exploiting that to define metaphysics. Perhaps that works, perhaps it doesn't. (It's not as if there is any interesting alternative.) But the problem at hand is whether this helps with our problem. I think it does, because it suggests that it is not a matter of discovering what the rules are or what they imply. There is no truth of the matter, because it is a matter of deciding how to apply the rules to a situation which they were not designed to cater for. That's not the same as saying that it is an arbitrary decision, since decisions here may well have consequences elsewhere.
I beg to differ. A 16 processor machine can sustain a far greater work load than a single-processor machine. The Cray machines were highly parallelized (SIMD architecture) in which thousands of floating point operations were performed by every instruction. These machines were great for stuff like weather simulation.
With that I agree. But that same function can also be done by paper & pencil. You said 'powerful', a reference to how fast the work is completed, and more processors helps with that.
I notice that any scenario with a contradiction involves invoking magic. Suppose this physically impossible thing (infinite gods, stairs requiring faster-than-light speed, lamp switches that operate without delay. No magical measurement of something nonexistent. Zeno doesn't do that. No magic invoked, and the first premise thus produces no paradox.
Oh it serves its purpose, but correct answers are not promoted above the others, and apparently a great deal of their posters don't know what they're talking about when it comes to stuff like this.
Quoting Ludwig V
It is very valid to apply mathematics to physics, but it really helps then if that to which it is being applied is actual physics. Creation of a device to measure a nonexisting thing is not actual physics.
Zeno's horse is quite real. Almost none of the others are.
Quoting Metaphysician UndercoverExactly so. I have correct my post. I meant valid and wrote 'sound' in haste. A simple application of modus ponens shows the lack of soundness of Zeno's conclusion iff empirical knowledge is given any weight.
The conflicting premise seemed to be a denial of the completability of a supertask. He never suggests a limit to divisibility.
Quoting Barkon
I have no idea what that collection of words means, so while it may seem to you that I think it, I quite assure you that I don't.
Very true. I'm afraid what I wrote is a rather embarrassing case of tunnel vision. But it rather matters what mathematics you are trying to apply to what physics. Sometimes it's a case of finding the right mathematics to apply. Which means that it is the physics that's in charge, so to speak.
But that doesn't apply here. Indeed, there's a question whether this branch of mathematics applies to any physics - not that that's an objection to the mathematics itself - just that this isn't the right application of it. That would be a solution, though. (I won't mention the issue of possible future physics'. I don't say it's just arm-waving to discuss it, but it is pretty close.)
Quoting noAxioms
In that case, it is clear what the right mathematics is. (IMO) One of the ways in which Zeno is a better paradox-maker than the others.
Quoting noAxioms
If you mean Thompson's lamp, quite so. (Do I understand correctly that Thompson actually argued that supertasks are impossible?) It is a fairy tale which seduces us to look at it wrongly.
The lamp, and almost all the other examples that are not Zeno. They all seem to argue along the lines of
I don't see that. At best he showed that one example is undefined. To prove something impossible it must be shown that there is not a single valid one. To prove them physically possible, one must show only a single case (the proverbial black swan). Nobody has done either of those (not even Zeno), so we are allowed our opinions.
The physical premise of "for something to go from A to B, it must first go halfway there" is very questionable. A great example is a photon going from emitter to detector. Nothing in quantum theory says that the photon is at the halfway point at the time halfway between the emission and absorption events. The principle of counterfactual definiteness (PCD) says it is, and that principle has never been demonstrated to be the case. It is in fact not the case in mosl interpretations of quantum mechanics, including any of the ones that deny faster than light causality.
Yes. I realized soon after I had logged off what you were talking about, went back in and edited my response. Too late to avoid revealing how dumb I had been. Never mind, it happens.
Quoting noAxioms
That seems to me a good response, though not quite the knock-out blow one would hope for. But it seems to me also a perfectly good reply to a purely mathematical version whether last number is odd or even.
So supertasks are like Gettier problems. Whack one on the head, and another pops up, specifically designed to avoid your refutation. It gets really wearisome, but no-one seems able to find a general refutation. One just gets bored in the end.
Take my explanation of Thomson's lamp above:
Quoting Michael
The important part is in bold. If there is a problem with the button-pressing procedure, which there is in the case of A, then this problem remains even if the button is broken and doesn't actually turn the lamp on it turning the lamp on and off isn't the reason that the supertask is impossible but simply demonstrates that the supertask is impossible.
And this problem remains even if rather than press a broken button we recite the natural numbers or even recite a single digit on repeat.
The reason that the supertask in Thomson's lamp is impossible isn't because of what operations are performed but because of how the operations are performed: halving the time between each subsequent operation ad infinitum. This is proven impossible, and as such Thomson's lamp proves that all such supertasks are impossible.
A clock ticks 1 time per second.
You start with a cake.
Every second the clock ticks, cut the cake in half.
Make the clock variable, it ticks n times a second.
The limit clock as n tends to infinity applies an infinity of divisions to the cake in 1 second. There is no final operation.
There's nothing logically inconsistent in this, it's just not "physical".
The lamp starts off. Every time the clock ticks a lamp turns from off to on or from on to off as applicable. Thomson's lamp shows that this leads to a logical inconsistency.
If there was an easy knock-out blow to it, it wouldn't be a topic on philosophy/mathematical discussions.
Quoting Michael
I see that you have an opinion, and that you are attempting to rationalize this opinion. But you leave some pretty low hanging fruit in this post, and rather than have me point them out and you denying whatever it is I post, I invite you to step into my shoes and critique the above. If your opinion was the opposite, what portions of the above argument would you put in bold and say is wrong?
I want to see if you are aware of the issues against which you are arguing.
Your response to me never seems to be along such lines. Instead of pointing out faults in my assertions or whatever, you simply ignore the argument and post yet another rewording of a counterargument, making the same mistakes (from my point of view). So show me that you at least know where I think those mistakes are being made.
Quoting Michael
OK, the bold line is telling. There is something wrong with the procedure. I've pointed it out in several posts. The lamp isn't broken. That violates the mathematical definition of how the thing works. There is no physical lamp since physics cannot do what is described.
Quoting fdrakeThis is not a supertask, not even as the tick rate increases arbitrarily high, because the cake (if it is continuous, which a physical one isn't) is going to take forever to consume at any clock rate.
Why? The ticks per second is also going to infinity.
I don't really think it matters whether this is a supertask or not, though. It was an attempt to give an example that hits @Michael's argument.
Quoting Michael
By providing a standard mathematical object which is infinite, has no final element, tends to an end state, and has an infinite number of occurrences ("steps"), but occurs in finite time.
Ill repeat something I said above:
The fact that there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 does not prove that the following supertask is metaphysically possible:
I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.
Quoting Michael
Might show it's logically possible tho.
You think that the super task I described might be logically possible? How would you start it?
I have explained this repeatedly. If you have the sequence 1, 1/2, 1/4, 1/8, ... on the number line, the points go right to left. If you start at 0, the limit of the sequence, and move to the right by any nonzero amount, no matter how small, you necessarily jump over all but finitely members of the sequence. That's by virtue of the fact that 0 is the limit.
It's certainly true, and I've agreed to this many times, that an infinite sequence has a beginning but no end; so that you can not iterate through it in reverse. How you get from this utterly trivial fact to some kind of cosmic conclusion, I can not fathom.
Quoting Michael
Ok. I asked for a reference. Now I have no idea what I'm supposed to conclude from this. That one person thinks supertasks are impossible and another does.
Can you answer a specific question that I've asked you?
Do you understand that mathematically, if you take a step, no matter how small, from 0, you necessarily pass over all but finitely many elements of the sequence 1, 1/2, 1/4, 1/8, ...?
It's really important me to know if you at least understand this mathematical fact.
You tagged three people before that quote so I'm not sure if this is for me. But I did say I wasn't sure I understood the staircase problem so if I've got that wrong, so be it.
Quoting Ludwig V
I'm for that. The mathematics could not be more clear. Once you start talking about mythical lamps and staircases, the examples have all the moral force of Cinderella's coach. Fairy tales. Thanks for the quote.
Quoting Ludwig V
Yes, Gauss and Riemann et. al. Not sure how that helps us with the lamp, the staircase, or Cindarella's coach.
Quoting Ludwig V
The horse. Uh oh did I miss a story about a horse? Is that a horse of a different color?
Yes that's true. There's a set of cardinality 27. One such is the set {0, 1, 2, 3, ..., 26}. There are others, of course.
Quoting noAxioms
[math]\aleph_{26}[/math] is vastly larger than 27. It's infinite, for one thing, whereas 27 is finite. Not sure where you're going with this.
Quoting noAxioms
Of course many such sets exist, as shown by the von Neumann encoding of the natural numbers.
I am not understanding your point. Of course there are sets of all finite cardinalities. And since cardinal numbers are themselves defined as particular sets, there are sets of all cardinalities.
Quoting noAxioms
No. I'm talking about computability theory. A Cray supercomputer has no more computational power than I do using pencil and an arbitrarily large sheet of paper, which which I can implement a Turing machine.
You are thinking of complexity theory, in which the time and space resources of computations are important.
But in computability theory, a function is either computable or not. If it is, it can be computed by pencil and paper (taking a very long time, of course). If it's not, no supercomputer will help.
As an example, consider the Euclidean algorithm to compute the least common divisor of two integers. It's a simple algorithm that can be executed using pencil and paper. If I had two trillion-digit numbers, I could not feasibly do the computation with pencil and paper; but I could still do it in principle.
Complexity theory is about what can be done feasibly. Computability theory (Turing machines etc.) is about what can be done by algorithms, whether the computation is feasible or not.
A Cray supercomputer can not compute anything that I can't compute with pencil and paper. But it can do so much more feasibly once the inputs become large.
I found a SEP article on the subject.
https://plato.stanford.edu/entries/computability/
Quoting noAxioms
Computational power. With the complexity/computability distinction, I believe we're in agreement.
Quoting noAxioms
Ok. Not sure where we're going with this. I'm happy to do a Zeno supertask by walking across the room. Whether someone regards that as a supertask or tells me I forgot about the Planck limit and so forth are different issues.
Quoting noAxioms
Lot of troll accounts on the site and lots of people who don't know what they're talking about. Believe it's something to do with the Quora owners trying to make some money. Sadly I'm a bit addicted to the site.
Quoting noAxioms
The Zeno Wiki page doesn't mention a horse. Did I miss something? @Ludwig V mentioned a horse too.
In Zeno's Achilles and the tortoise, empirical knowledge shows that Achilles will pass the tortoise. But empirical knowledge has problems like what Hume showed with the problem of induction. Because of this, empirical knowledge does not prove the supertask to be impossible.
That the supertask is not completable is not denied, that it is not completable is what actually leads to the problem. In Zeno' paradox Achilles never catches the tortoise because the supertask is never completed. By the premises of the op, Icarus cannot reach the bottom of the staircase, just like Achilles cannot reach the tortoise. So "the supertask" by the nature of what it means to be a supertask, cannot be completed.
The problem is that empirical evidence shows us that tasks will be completed, Achilles will pass the tortoise, and in the op 60 seconds will pass. This shows that the supertask as a fiction. However, due to the problem of induction, empirical evidence does not provide a proper proof. That is why I suggested we look at the divisibility of time as the means for providing a better proof.
I don't really think it matters whether this is a supertask or not, though. It was an attempt to give an example that hits Michael's argument.
Quoting Michael
I was wondering about what is actually meant by 'metaphysically possible' or 'logically possible'. The latter is probably the same as 'mathematically possible', but I'm wondering how the former is distinct.
I notice you ignored my prior post. That itself indicates to me that you do not intent to actually consider points made against your stance. I was hoping for more confidence in it.
Quoting fishfryGotcha. No argument then. As I already pointed out, you had referenced power instead of computability: "there's no difference in computational power between parallel and serial processing." and I took it as a statement of work over time.
I brought this up in my simulation-theory topic. A simulation of Earth to a precision sufficient for consciousness can be done by pencil and paper, or by dominos falling, The latter is really interesting: set up dominos so that you get the function of a Turning machine. Not easy, but it seem that it can be done.
Plank length is not a physical limit, only a limit of significance. If I have it right, any pair of points separated by a distance smaller than that is not meaningfully/measurably distinguishable from just the two being the same point. It doesn't mean that the two points are necessarily the same point.
But I gave some QM examples that suggest a non-continuous model of reality.
Yes. Search for 'horse' in the last 20 posts or so.
Quoting Metaphysician UndercoverBecause of this, empirical knowledge doesn't prove pretty much anything to be possible or impossible. That's why science theories are supported by evidence and not by proofs. They'd be theorems, not theories, if they were provable.
I beg to differ. That simply does not follow from the description. Zeno describes a physical completable supertask, which is only as possible as the soundness of his first premise.
Again I differ. The supertask (if that premise is true) is not fiction. I mean, my opinion is that there isn't a physical supertask, but opinion isn't evidence, and I have no evidence (let alone proof) that it isn't a supertask.
Declaring something to be impossible is a strong claim and requires strong evidence.
Ok fair enough. I think of power as computability but perhaps I could have been more clear.No. I'm
Quoting noAxioms
I doubt that consciousness is computable, nor is the universe, and I utterly reject the notion that consciousness can be simulated by any computational device. Period. Consciousness is not a computable phenomenon.
We are not computers no matter how many TED talkers declare it so. Just as we aren't Newtonian machines, as we thought we were in the Newtonian machine age; nor are we flowing fluids as the Romans thought, in the age of their great waterworks. After all if we're computations, what are the odds we'd figure that out right when we're in the age of computation? It's historical relativism or whatever the phrase is.
But I do agree that if consciousness were computable, then the computation could be carried out by pencil and paper. Glad you made that point. Because if so, then where is the conscious mind? In the pencil? In the paper? In the air? In a neural network? I reject the idea.
Do people really think their web browsers or word processors are having subjective experiences? I know they think their neural networks are. The idea's absurd. Don't get me started :-) I would characterize myself as a mysterian.
Quoting noAxioms
Yes, I saw a domino logic gate on Youtube a while back. Any physical substrate will do. For computation, not consciousness. For consciousness you need some secret sauce not yet understood. It will turn out to be something other than a digital computation.
Perhaps it's some kind of analog computation, but that's not the same thing.
Quoting noAxioms
If reality is not continuous that goes a long way to solving Zeno. If it is continuous then walking across the room is a supertask.
Quoting noAxioms
I have the bad habit of only responding to my mentions, but I'll take a look. Thanks.
ps -- I checked out the Simulation thread and from there, saw your initial post in the "What is the Simulation Hypothesis" thread, and I agree with everything you said. I especially appreciated the distinction between simulation and VR, which is something a lot of the simulation discussions miss.
Aye that is true. I wrote wrong. I was imagining a clock that speeds up in its ticking to ape a convergent geometric series.
Maybe I'm not being clear, so I'll try one more time.
Here are two proposed supertasks:
a. I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum
b. I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum
Here is our premise:
P1. In both (a) and (b) there is a bijection between the series of time intervals and the series of natural numbers and the sum of the series of time intervals is 60.
However, the second supertask is metaphysically impossible. It cannot start because there is no largest natural number to start with. Therefore, P1 being true does not entail that the second supertask is metaphysically possible.
Therefore, P1 being true does not entail that the first supertask is metaphysically possible.
If you want to argue that the first supertask can end despite there being no largest natural number to end with, and so is metaphysically possible, then you need something other than P1 to prove it.
I said this:
Quoting Michael
You responded with this:
Quoting fishfry
I gave you this reference:
So I ask again: can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum?
I think you misunderstand Zeno's paradoxes. Zeno concluded that Achilles cannot overtake the tortoise. That is explicit. And therefore, it indicates that he is arguing that the supertask is not completed. The "supertask" of passing an infinite number of spatial divisions is never finished, therefore the faster runner never surpasses the slower.
The paradox is that physical evidence indicates that the faster runner always does overtake the slower, in reality, even though the logic proceeding from fundamental axioms proves that the faster overtaking the slower is a supertask which cannot be completed.
Due to the strength of the empirical evidence, we are led toward the conclusion that the fundamental axioms concerning the continuity of space and time, and the infinite divisibility of those continuums, must be faulty. Those axioms are the "unsound premises".
Quoting noAxioms
This is why we cannot simply accept the empirical evidence, and conclude that the supertask is a descriptive impossibility derived from faulty axioms. Empirical evidence is known to be unreliable. So, we need stronger principles to demonstrate the actual faults in the axioms.
Pretty much 1-1 odds. That's when the terminology became part of our language. You describe yourself in terms of the things you know.
We are water. The vast majority of mass would be lost (as would consciousness) if the water was taken away. Lots of pipes going here and there. It's a pretty good description for the Roman days.
In the process.
Gawd, I spelled it 'Turning' machine. More typos.
Anyway, yes, the discussion was inspired by that. Any moron can create a domino or gate, but creating a nor gate gets tricky. Any gate can only be used once, so it's impossible to create say a flip flop, normally a trivial thing created with a pair of nor gates.
I've not seen the video, but mention of it inspired me to design a Turing machine with the technology. Can dominos be used to run a physical simulation? I think it's possible since I found not obvious roadblocks. I'm tempted to start a topic on it, but not here since it isn't a philosophy topic at all.
I've also programmed analog computers in school, never on the job. It's a different sort of thing, I tell ya.
Your view of consciousness is modelled by a VR. One big distinction is that a VR cannot be implemented with paper and pencil (or dominos).
Quoting fdrakeOK, that would be pretty much what has been the topic of discussion this whole thread. If it completes in finite time, it's a supertask. Don't forget the inverse case where the clock starts fast and slows down to its final tick.
Quoting Metaphysician UndercoverCorrect, but a second unstated premise must be assumed in order to draw this conclusion, since without it, one can only say that the tortoise cannot be overtaken at any particular step. That second premise might well be that supertasks cannot be completed. That premise is indeed in contradiction with the first premise and empirical observation. At least one of the three is wrong.
Fundamental axioms? None of the premises are that. They're both easily doubted.
Or the premise of supertasks being uncompletable is wrong, or that empirical evidence isn't as strong as is asserted.
Asserting that your premise of choice must be the faulty one is a mistake.
I imagined you lot were talking about metaphysical rather than logical possibility. @Michael made a comment to the effect that such a construction was logically impossible. Which would be odd, seeing as such an object has a model in set theory. Even if it's not physically or metaphysically possible. All I wanted to add.
Does it? Consider these two supertasks:
a. I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum
b. I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum
The first is reciting every natural number in ascending order and the second is reciting every natural number in descending order.
Does the second have a model in set theory? Is the second logically possible?
That there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 says nothing about the possibility of (b) and so says nothing about the possibility of (a) either.
The sequence {1/2n} for n=1 to infinity has a finite sum, 1. That's numbers and an infinite set. So yes, geometric sequences have a model in set theory. They also have a finite sum, no last operation, and contain an infinity of operations. You can think of the partial sum, up to the nth term, as the total time elapsed on clock whose ticks last 1/2n. The limit construction exists without contradiction. The reason being that the sequence gets arbitrarily close to its greatest lower bound (0). And there is no smallest element, since the greatest lower bound isn't in the set.
Quoting Michael
So there is no final operation, there is an infinite sequence of operations, and it completes in a finite time. Since the completion time is just the limit of the partial sums.
There's your model. Logically possible. I leave now.
Nah. That's an appeal to metaphysical or physical impossibility. Not logical impossibility!
It is logically impossible to have recited every natural number in descending order because it is logically impossible to even start such a task.
I am so sorry. I started a hare by mistake. The horse first appeared in this comment
Quoting Ludwig V
So a horse here is shorthand for whatever physical object one is trying to put into mathematical harness. Zeno's horse is the tortoise, or Achilles, or both.
Quoting noAxioms
I asked about this earlier in this thread. You can find what I got from it here. I'm not at all clear what people who use the term metaphysics mean by it. For the time being I'm treating the "metaphysics" and "logic" as co-terminous, if not synonymous.
Quoting noAxioms
I have been wondering about exactly that point, and trying to work up the courage to articulate in this context. Thanks. If physics requires a non-continuous model of reality, then so be it, but then it would be empirical (physical) and wouldn't affect the geometrical concepts, would it? If what happened to the question whether matter was continuous or not is anything to go by, I think that a third alternative (not yet available) is most likely.
Well my quote above is not given from authority. Planck units are just a standard of natural units. A Plank length is a small distance, but the fact that they know that distance down to at least 7 significant digits means that far smaller space units are meaningful. Still, wiki says "Since the 1950s, it has been conjectured that quantum fluctuations of the spacetime metric might make the familiar notion of distance inapplicable below the Planck length", which is similar to what I was trying to convey.
There's lots of useful stuff on the wiki Planck units page that's better expressed than me trying to paraphrase it here.
Interestingly, a unit of Planck energy is said to be the equivalent of the chemical energy contained in the fuel tank of a fairly large car. Certainly not the minimum meaningful energy. One unit of Planck force is an even larger silly number. Such is the way with natural units.
A more complex model for the universe does not effect a simple geometric model at all, no. The simple model simply isn't fully applicable to the reality it is supposed to describe, just like Newtonian physics isn't fully applicable to the same reality, despite the fact that they'll continue to teach it in schools.
Somebody still suggests that matter is continuous? I mean, that sort of went out the window a couple centuries ago.
Quoting fdrake
Actually, I've been asking about the distinction between those two. Nobody has really answered. A nice example (not a supertask example if possible) of something that is one but not the other would be nice.
Quoting Michael
It may grind against your intuitions, but no logical argument against it has been presented. That you personally find it 'evidently absurd' carries no weight.
How does it start? That's easy. When the appropriate time comes, the number to be recited at that time is recited. That wasn't so hard, was it? It works for both scenarios, counting up or down.
Quoting Michael
I think that's actually a very difficult issue to resolve. It's basically the same question as whether an infinite regress is logically possible. I believe the infinite regress actually is logically possible, and it requires a contradictory premise to make it impossible.
Quoting noAxioms
If no particular step can overtake the tortoise, then the tortoise, by the described motion cannot be overtaken. Where's the need for another premise?
Quoting noAxioms
Following from the described premises, the supertask cannot be completed. It is logically implied that there is always further distance for Achilles to cover before overtaking the tortoise. Therefore the described task can never be completed. There is no further premise required, it is a logical conclusion, the described "supertask" cannot be completed.
Quoting noAxioms
The conclusion that Achilles cannot overtake the tortoise does contradict empirical evidence, that's the reason it's called a paradox. But I do not see how it contradicts "the first premise". Which premise would that be? The argument is valid, so how could it contradict a premise?
This is not the issue. It clearly does not have a start. The question is whether it is logically necessary for such a task to have a start. This is argued in "first cause" arguments.
@Michael @fdrakeThe problem is that what is described is an activity, and the way that we understand activities is that they have a beginning. Activities are all physical. But if you remove that requirement of "physical", then the activity might be proceeding without a beginning, just like it could proceed without an end. It's the way that we look at the difference between past and future, which makes it difficult. If there is no such difference, then the past supertask must be logically possible, just like the future one.
I don't have a particularly clear cut example to distinguish the two. And I don't know if the concept of logical possibility survives the existence of nonclassical logics. But what I envision for logical possibilities are things which don't by themselves, and with no other contextual information, entail a contradiction. Any possible fact is logically possible.
As for the merely logically possible - as in logically but not metaphysically possible - , I imagine procedures like Banach Tarski. Turning a sphere into two spheres using only the material in the first sphere. But that's just because I can't imagine a concept of space used in metaphysics (like extension) that makes central use of non-measurable sets (things with ill defined extension in principle).
Physically possible? That's getting hard. A universe that contains violations of the second law of thermodynamics is metaphysically possible. Like Lord of the Rings, Harry Potter. In the sense that there's a self consistent narrative going through those works of fiction whose behaviour is impossible to translate to our universe, those universes would be metaphysically but not physically possible.
So when I hear @Michael talking about the impossibility of a geometric series "completing" (so to speak) due to being unable to recite the terms in finite time, I hear a use of a metaphysical or physical context of events, speech and recitation which just isn't there in the definition of mathematical terms. And in that regard @Michael's response addresses physical or metaphysical possibility, rather than logical possibility.
The only reason I intervened is to pedantically point out the distinction, and that Michael let it slip for at least one post.
I don't think it impossible for a geometric series to complete. I think it impossible to have recited every natural number in descending order.
My issue is with supertasks, not with maths.
There is no first natural number to start with. It is logically impossible to have started reciting the natural numbers in descending order.
Good! Then it's logically possible for it to. An infinite number of things can complete without blowing up logic.
Obviously, the described process has no start, that is implied by the description. So your conclusion that it is logically impossible to have started such a process is irrelevant, as what is already known. You need to show that such a process, one without a start, is logically impossible.
That's what "first cause" arguments attempt to do. They describe the temporal aspect of "a process", "a thing", or similar term, in such a way that it necessarily has a beginning and an end in time, then they produce a logical argument from that description. It's an attempt to bring the realm of material (physical, or temporal) reality to bear on the realm of logical possibility, by stating premises which are supposed to represent the essence of material (physical) reality, and restricting logic with them. Another example of a similar restriction is the law of identity, and the other two fundamental principles.
That logic has not been shown. It's a non sequitur until it is spelled out.
No such implication exists, and no such statement is made. Asserting this would be another premise, and one that makes no sense either. And yes, it would follow that the tortoise cannot be overtaken if this additional premise is added.
Quoting Metaphysician Undercover
Your usage of 'clearly' implies you are referencing a second premise based on perhaps your intuition. What you may find 'clear' seems to be in direct contradiction with the first premise, I am presuming your 'clear' assumption is something on the order that there must be a first step, equivalent to asserting a bound to something explicitly defined as not being bounded. Of course you're going to run into contradictions if you add a second premise that directly denies the first premise. It isn't a paradox then, it's just wrong.
If that isn't your 'clear' premise, then state what you find so clear.
Michael also makes this mistake despite it being pointed out so many times.
Quoting MichaelTotally predictable response. We're like over 400 posts into this topic and you're you're stuck on the same fundamental mistake. You (as well as Meta above) seem to insist on an additional premise of the necessity of a bound to something explicitly defined to be unbounded. My method for performing the task made no mention of doing a first step, but it can be mathematically shown that any given step is done, and that the steps are done in order.
An unbacked assertion, especially when I showed how to do it. Your presented 'logic' seems to be the argument above, declaring a second premise that happens to contradict the thing you want to find impossible. The logic to which you refer is only valid for finite sets, but you cannot learn this.
You can disprove it by naming a number that isn't covered by my procedure, or by naming a pair of numbers that are recited out of sequence, or some other such demonstration of a violation of the task as described. That's how you go about it when dealing with unbounded sets. Hilbert's hotel is a great educational exercise showing how mathematics deals with infinities.
Quoting fdrakeI don't think it is the extension that is ill defined with that case, but rather a leveraging of the fact that the pieces are made of infinite points each, and you don't need 'more natural numbers' to count each one of them twice. I don't understand the Banach Tarski thing enough to know why 5 is a lower limit of the number of pieces.
Anyway, I chalk it up to another illustration of why the logical rules of bounded sets don't necessarily apply to unbounded ones. The posters above clearly cannot accept this, and so we go in circles.
OK, here you seem to use 'metaphysically possible' to mean 'possible in a universe with different physical laws'. But I don't find that very distinct from logically possible.
I don't think he says that time is the issue. It is his insistence on the need to eventually recite the highest number, after which there are no more. That number doesn't exist, so the task cannot be done because it missed at least that one.
But we're talking about supertasks, not geometric series. That a geometric series is possible isn't that a supertask is possible.
Given that there is no largest natural number it is logically impossible to even start reciting all the natural numbers in descending order.
I don't know why you think the existence of a geometric series proves otherwise.
No, I'm saying that something with no start cannot start and something with no end cannot end.
Your argument is effectively "by definition it has no start therefore it can start without a start." You're trying to take the very thing that makes it impossible as proof that it's possible.
This seems to be playing language equivocation games. You introduce the word 'start' here, undefined twice, once as a noun and once as a verb. Given certain definitions of both usages, I may or may not accept this additional premise you state.
The noun definition I will call Sn.
Sn1: The "start" of a sequence is a first step. This is exactly a bound, but you say 'No, ..." above, denying my calling it a bound, so you must mean a different defintion Sn2 of 'start' that this sequence doesn't have. I can't think of one that is distinct from a bound, so you need to help me here.
The verb is also used (cannot 'start'). Here I can think of at least two definitions:
Sv1: To start is to initiate the first step. This again is a reference to the bound.
Sv2: To to start is to commence the steps. There is a duration during which none of the steps has been performed, even in part. There is a duration when steps are being executed, and a duration when all steps have been completed. To start is to transition from the first to the second duration.
I suspect that you are actually equivocating multiple definitions of the verb to make your point. I mean, if you go with Sn1 and Sv1, then I actually agree with your added premise. A sequence lacking a specific step cannot execute the nonexistent step. That holds water. But then you equivocate to Sv2 and conclude that the existing steps cannot commence.
You are clearly using Sn1 as your noun definition here, which is a direct reference to the bound that we both acknowledge doesn't exist. This usage of the noun contradicts your opening word "No" in your post where you imply that your argument is something other than "an additional premise of the necessity of a bound to something explicitly defined to be unbounded". You contradict yourself.
Given the Sv2 definition for the verb, my argument is pretty much that, yes. It isn't ridiculous because I gave a precise description of how to do it. Your expressed ridicule isn't valid logic.
I cannot start reciting the natural numbers in descending order because there is no first natural number for me to start with.
That a geometric series has a finite sum is irrelevant to this very simple self-evident fact.
I'm pretty sure that one comes down to being able to split the pieces up into pieces that aren't measurable - IE can't be assigned a size - in a clever way, then applying some cool transformation to them that blow them back up into the sphere. But that's by the by.
Quoting noAxioms
I think that's a species of metaphysical possibility - a different physics. What would distinguish that from logical possibility, in my book, is that there are simply more ways of being noncontradictory than being unable to exist in our universe. Like flibbertygibbets. And nonmeasurable sets. And, maybe, abstract categories.
Quoting noAxioms
All I was pointing to was the conceptual explosion that happened when we finally split the atom. (Which, you will remember, was by definition unsplittable).
Quoting fdrake
This is a fascinating issue, mostly swept under the carpet in philosophy. I don't say that you are wrong.
I think it was Aristotle who first articulated the idea that a fictional story must be at least plausible. (Does that mean "possible"? - possibly). The idea that it requires "suspension of disbelief" was, apparently first articulated by Coleridge in 1817. There's a distinct tension between these two requirements. Both high-light that the audience/reader needs to collaborate with the author. It seems to me that the collaboration is at least sometimes secured by "arm-waving" by the author at awkward moments to distract the audience's attention and the audience not pressing questions that would be irresistible in other circumstances. The concept of magic is a good example. Science fiction stories usually put up a better front than that, but nonetheless... The issue comes home to bite philosophers when we offer examples - thumbnail stories. (I won't give examples for fear of setting off a hare and distracting us all). The difficulty for us is to distinguish arm-waving from actual possibilities (!).
Quoting Metaphysician Undercover
Yes. Disagreements between logic and experience are not unfamiliar. Experience usually wins, because logic is more adaptable than it seems. (I realize that may seem like heresy in a philosophical concept, but doesn't experience support it?)
Quoting fdrake
Surely a different physics will have to be consistent and complete - when it is finished. That looks very like "logically possible", doesn't it?
As for the rest, you seem to understand existence as a single category. Perhaps you believe the same of reality. That is not how I understand either term. Existence has many different modes? categories? which are defined contextually. Ditto reality. I understood a flibbertygibbet to be a silly person who talks too much, so they very much do exist in my universe and I envy you yours. Things like non-measurable sets and abstract categories exist all right, but not in the same way/mode/category as tables and chairs. So do fictional things like Achilles and his tortoise and the Gorgon's mirror.
Quoting Metaphysician Undercover
I'm afraid that if you condescend to use ordinary arithmetic, one can predict exactly when Achilles will overtake the tortoise, given data about how fast each contestant moves and the size of the handicap.
Not being a fully trained mathematician, I'm not sure about it, but I suggested this earlier and no-one has contradicted me - yet. Perhaps it is just too boring.
Quoting Ludwig V
Quoting Metaphysician Undercover
That is a very interesting take on the argument, though I don't understand how this applies to the law or identity. But then, I don't understand the law of identity, either. What are the other two principles?
I don't see the need for any other premise. Achilles is moving, and described as doing this in a way in which he will always have to move further before he can overtake the tortoise. Since he will always have to move further before he will overtake the tortoise, we can conclude logically that he will never overtake the tortoise in that described activity. Why do you see the need for another premise?
Quoting Ludwig V
Sure, but those mathematical principles are not the premises described by Zeno.
Quoting Metaphysician Undercover
Case closed, then.
I think so, but we'll have to see what noAxioms is talking about with the reference to a requirement for further premises. I think noAxioms looks at Zeno in a different way.
You've been perfectly clear, and I've clearly responded to your points several times already.
Quoting Michael
I have never made any such statement. I've repeatedly challenged you to name the first number not verbalized when we count forward 1, 2, 3, ... at successively halved intervals of time.
I ask you once again to tell me whether you appreciate the point that any interval containing the limit of a sequence must necessarily contain all but finitely elements of the sequence.
If you understand that, it addressed your counting backward argument. If not, let's discuss it.
Quoting Michael
Halving the time is your own thought experiment. It's not mine. Once I accept your own premise, I then work out the logical consequences. Halving the time is [i]not my premise[i].
Consciousness could perfectly well be a physical process, but not computable. So, what kind of process is physical but not computable? A task for some future genius to elucidate. FWIW Penrose believes that consciousness is not computable. He may be wrong, but he's Sir Roger and the rest of us are not.
Quoting noAxioms
That's my point. The Romans thought mind was a flow, because they had great waterworks, and so forth. We live in the age of computation so we think we're computers. The historical contingency is an argument against the theory, not for it.
Quoting noAxioms
You're agreeing with my point.
Quoting noAxioms
I agree with you that IF consciousness is a computation, then it could be implemented with pencil and paper. I regard that as an argument against the premise.
Quoting noAxioms
I've seen Searle argue that consciousness is physical but not computational. Some kind of secret sauce found in living things and not in digital circuits. Don't know much about analog computation with respect to consciousness.
Quoting noAxioms
The illusion can. But my consciousness can't. As Descartes noted, I may be deceived, but there is an I who is being deceived. So the VR theory doesn't solve anything at all, it leaves the mystery of what my own consciousness is.
It's always been unclear to me which aspect of simulate/VR Bostrom is arguing.
Quoting noAxioms
There is never a final tick in an infinite sequence, even if the sequence has a limit.
No worries. Like a certain Supreme court justice, I am not a biologist.
Quoting Ludwig V
Ok. I figured that out, just couldn't remember anything about a horse. I agree that a hare or a tortoise or Achilles does just as well. Thanks.
Given your reluctance to clarify the definition of the verb 'to start', I cannot respond appropriately to this statement. I gave a pair of options, or you can supply your own, so long as it isn't open to equivocation.
Quoting fdrake
Your confidence in your own understanding is then stronger than my confidence in mind.
Quoting Ludwig VSome of both, I'm sure. The impetus thing had to go (survived until Newton, not bad...), but one could argue that it is a poor description of inertia.
The smallest thing still is. Unfortunately the word got applied to something that was a composite object, and they kept that instead of renaming the assembly and keeping 'atom' for anything fundamental.
Quoting Metaphysician UndercoverNot always. Just a minute. I know, Zeno doesn't give the time, but we've been using a minute. The way the scenario is described has no effect on the situation compared to a different way of describing it.
Anyway, I deny that Zeno in any way suggests that the overtaking will never take place. He just says that another step always follows any given step.
Michael has added the verbalizing of the natural number count, but that doesn't change it taking only a minute.
Quoting fishfryThey can't both be right?
I think I am, yes.
Anything analog can be approximated with digital. But anything digital can be perfectly implemented with analog. Searle is perhaps referencing property dualism? I don't know if I got that right. Can't seem to articulate the differences between the variants.
I guess I'm even more skeptical than Descartes. I win! I didn't pick my handle for no reason. I try not to leave anything unquestioned.
VR says that all you know is potentially lies. You are not of this universe, but rather you are experiencing it. All very dualistic. The 'brain' in the body (if there is one at all, have you ever checked?) is not what's making any of the decisions.
If you think about it, the view can be empirically tested. Not so much with the simulation hypothesis.
Definitely the former. But Elon musk is arguing for VR, and references Bostrom's paper to support it, so he has no idea what he's talking about.
The comment above (and my reply) belongs in the other topic. I see you posted more or less the same question there.
or not a first tick. Zeno's dichotomy very much has a final tick. I can make a scenario that has a first and last, and gets singular in the middle somewhere. Just illustrating the classical snippet: Never say never.
Ok. We are ALL of whatever our latest technology is. Well ... maybe so. Something to be said for that.
Quoting noAxioms
Ok. But I'm arguing that the simulation theory, or the computational theory of mind, is suspect because of its very timeliness. We invent computers and the philosophers all go, "Ooh we're computers." That's a point against the idea IMO.
Quoting noAxioms
I think Searle was arguing against dualism in this instance. He was saying that mind is not a computation; but it's not something non-physical. Rather, there's something physical about living things that implements consciousness, in a way that rocks and digital circuits can't. I only saw him mention this on video, so perhaps he's added more detail in his writings.
Quoting noAxioms
Without axioms it's difficult to get reasoning off the ground. You have to start somewhere, right?
Quoting noAxioms
Yes but everyone agrees with that. There's a world "out there," and we experience it through our senses. Not sure what you mean by empirical testing here.
Quoting noAxioms
Right. And I saw a TED talk where George Smoot, the guy who discovered the cosmic background radiation anisotropy, was enthusiastically advocating simulation theory. Neil deGrasse Tyson too. A lot of people who should know better say trendy things for no reason at all. More arguments against simulation IMO. The pronouncements of celebrity scientists outside their expertise are always suspect.
Quoting noAxioms
Yes. Let's talk about this over there.
Quoting noAxioms
Ok. Just talking about standard mathematical sequences. It's a common misunderstanding in this thread. The sequence 1/2, 3/4, 7/8, ... has a limit, namely 1, but no last element.
The sequence 1/2, 1/4, 1/8, ... also has a limit, namely 0, and no last element. But if you put the elements of the sequence on the number line, they appear to "come from" 0 via a process that could never have gotten started. This is my interpretation of @Michael's example of counting backwards.
I accept this:
P1. If we can recite forward 1, 2, 3, ... at successively halved intervals of time then we can recite all natural numbers in finite time
But I reject these:
P2. We can recite forward 1, 2, 3, ... at successively halved intervals of time
C1. We can recite all natural numbers in finite time
If you want to claim that C1 is true then you must prove that P2 is true. You haven't done so.
I think Thomson's lamp and similar examples prove that P2 is false. See here.
Just the ordinary meaning of "start", e.g. "begin".
You ask me, right now, to recite the natural numbers in descending order. How do I begin to perform this?
I think it's self-evident that I cannot begin because there is no first (largest) number for me to begin with.
It is Achilles' run but with time reversed: https://plato.stanford.edu/entries/spacetime-supertasks/#MissFinaInitStepZenoWalk
This is what I mean by reciting backwards:
If I recite the natural numbers <= 10 backwards then I recite 10, then 9, then 8, etc.
If I recite the natural numbers <= 100 backwards then I recite 100, then 99, then 98, etc.
If I recite all the natural numbers backwards then I recite ... ?
It's self-evidently impossible. There's no first (largest) natural number for me to start with.
[quote=Internet Encyclopedia of Philosophy]According to this reasoning, Achilles will never catch the tortoise, says Zeno.[/quote]
https://iep.utm.edu/zenos-paradoxes/
The paradox is like this. Both Achilles and the tortoise are moving, but the tortoise has a head start. So at t1 Achilles is at location A and the tortoise is at location B. At t2, Achilles reaches location B, but the tortoise has moved to location C. At t3, Achilles reaches location C, but the tortoise has moved to location D. As this procedure will carry on without end, Zeno concludes that the faster runner cannot overtake the slower.
https://personal.lse.ac.uk/robert49/ebooks/philsciadventures/lecture24.html
Well, one could argue that it isn't a description of inertia, but of certain phenomena which are better described by inertia. Either way, impetus proved unhelpful and alternative conceptualizations proved more helpful.
Quoting fishfry
You are right that the historical contingency should make us suspicious. (Descartes, by the way, has a description of statues "animated" by a hidden hydraulic system - I think in Versailles). But I don't think the process is simply over-enthusiastic. It seems reasonable to try to apply a new discovery as widely as possible. That way, one discovers its limitations.
Quoting noAxioms
Oh, I don't know. Given the conceptual revolution that happened when sub-atomic physics arrived, it's not a bad idea to signal the change by leaving atoms where they were.
Quoting fishfry
That's more or less one Ryle's favourite arguments against dualism.
Quoting fishfry
Clearly "
Quoting noAxioms
That's true. But different descriptions of the same situation can affect how we think about that situation. An additional difficulty, I suspect, is that our descriptions are fictional (sorry, thought-experiments), which means that the context is limited and evaluations of descriptions much more difficult. They need to be assessed in a different way - as useful or not.
Quoting Metaphysician Undercover
Yes. You cannot necessarily decide that just one way of looking at things is true and all others false. They are better evaluated as useful or not. I think that applies here.
Quoting fishfry
Yes. The difficulty is how to evaluate a starting-point. True or false isn't always relevant. Which means that it can be difficult to decide between lines of reasoning that have different starting-points.
Quoting Metaphysician Undercover
So are you going to conclude, with Zeno, that motion is impossible? or that Zeno is analyzing the situation in a misleading way?
Yes, Zeno is analyzing in a misleading way, but only because the axioms of continuity and infinite divisibility are themselves misleading. So Zeno simply demonstrates how standard conventions are actually misleading us.
And here we are. a couple of millennia later, still being misled by the same conventions. This is because we have not yet determined the natural points of divisibility. And so, fundamental particles take every possible path when they move from A to B, because the direct spatial route does not allow them to get ahead of the tortoise.
Quoting Metaphysician Undercover
You mean because they allow the convergent infinite series?
Mathematically? Physically? (I'm inclined to think you mean physically, because of your reference to fundamental particles.)
Quoting Metaphysician Undercover
Well, we've caught them out misleading us before, so I suppose they may be doing it again.
Quoting Metaphysician Undercover
Is the direct spatial route not available because it contains a convergent regress?
What path does Achilles take? (I assume he is not a fundamental particle.)
I replied to much of your post, but all over there.
Quoting Michael
In that case I reject your premise. The lack of a first step does not prevent the beginning of the task, which is simply the transition from the time prior to any of the steps being taken, to the time during which steps are being taken.
Quoting Michael
I described exactly how to do that, and you found no fault with it, choosing instead to try a different wording of your additional premise. Why does my description fail? What step is missed? None, and it's done in finite time, so you apparently cannot find fault except by asserting additional premises, all of which take the form of asserting a need to perform a step that by definition doesn't exist.
Quoting Metaphysician UndercoverI know the story. You seem to have reworded it for your purposes, since the quote you give does not come from that site, but the site also seems to be conveying the story in its own words, not as reported by Aristotle.
Yes, and without justification, or at least without explicitly stating the additional premise that makes the conclusion valid.
Same thing. Does not follow.
Funny that the animation on that site below that quote shows 'lightning fast' Achilles moving at only twice the speed of the tortoise. It also shows Achilles slowing down to pretty much a halt, which is why he never passes the tortoise in the animation.
Are we resorting back to the beginning of the discussion here? If you cannot counter my posts, you just start over with the original story? Yes, you can keep the topic going a long time this way, but you're not helping your case.
Quoting Ludwig VYes, it affects how we think of them. It doesn't effect the situation, despite all the assertions to the contrary by several members.
A thought experiment is a valid method of deriving conclusions from premises. They only get fictional if the premises are faulty, such as the lamp, a device which cannot physically operate as described.[/quote]
It literally does.
Quoting noAxioms
No you didn't. You ignored it and just said "when the time comes I say the next number". That doesn't explain how the recitation can begin without a first number to say.
I am right now trying to recite the natural numbers in descending order but am silent because I cannot begin. It's been 60 seconds. Not only have I failed to recite them all, but I have failed to recite even one. Help me out here.
Now consider reciting its terms in reverse.
To recite its terms in reverse I am only allowed to say "0" or "1" but I cannot start by saying "0" and I cannot start by saying "1". Therefore I cannot start.
No appeal to a geometric series of time intervals can save you from this.
Yes - unless it is a fictional situation - whether in the philosophical or the literary sense.
Quoting noAxioms
That may explain why I have been confusing them. Thanks for that.
I have wondered whether one could replace the Thompson lamp with a question, such as whether the final number was odd or even. That would work if you start with an odd divisor and don't express everything in decimals. Perhaps it would work for all examples if you ask whether the number of steps taken is odd or even when the minute is up. I think.
P1. When the letter X is given a definition it retains this definition until it is redefined.
A1. At t[sub]0[/sub] X = 0
A2. Therefore, at t[sub]1[/sub] X = 0
B1. At t[sub]0[/sub] X = 0 and then at t[sub]1/2[/sub] X = 1
B2. Therefore, at t[sub]1[/sub] X = 1
C1. At t[sub]0[/sub] X = 0 and then at t[sub]1/2[/sub] X = 1 and then at t[sub]3/4[/sub] X = 0, and so on ad infinitum
C2. Therefore, at t[sub]1[/sub] X = ?
In all cases the definition of X at t[sub]1[/sub] must be a logical consequence of what occurs between t[sub]0[/sub] and t[sub]1[/sub].
Given that in C2 X cannot be defined as either "0" or "1" but must be defined as either "0" or "1" then C1 is necessarily false. The supertask described in C1 is impossible.
This addresses the very logic of a supertask without some dependency on a physical performance.
I meant, that they can mislead us when we apply the principles to the activities of the physical world. That's what Zeno's paradoxes show.
Quoting Ludwig V
What is evident, is that we do not know how things move, and the exact "path" through space, that things take, whether they are big planets, stars and galaxies, small fundamental particles, or anything in between.
Quoting noAxioms
Here's what Aristotle reported:
[quote=Aristotle Physics 239b 14-17]The second is the so-called 'Achilles', and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. [/quote]
How is this different from what I said? I gave a full explanation, as did the site I quoted. Aristotle just said "it amounts to this...", providing a shortened version, probably because the specifics were well known at that time.
Quoting noAxioms
I'm still waiting for you to explain how the conclusion is not justified, and why you think there is a requirement of an additional premise.
That's clear as crystal. Your conclusion coincides with mine, so I'm perfectly happy with the argument.
Quoting Michael
This puzzles me. Is this t(1) the same t as the t(1) in C3? It can't be. There must be a typo there somewhere.
One question, then - The state of X at any t(n), depends on its predecessor state at t(n-1), doesn't it? Isn't that a definition? Why is it inapplicable to t(1)?
Quoting Metaphysician Undercover
I think that's perfect. It's the conjunction of mathematics and - what can I say? - the everyday world.
What's difficult is the decision which is to give way - mathematics or the everyday world. Zeno was perfectly clear, but some people seem to disagree with him.
Quoting Metaphysician Undercover
That suggests that we do know roughly how things move. I don't think that's what at stake in Zeno's thinking. His conclusion was that all motion is illusory. The only alternative for him was stasis. But I guess we can do better now.
No, it was three separate situations. Sorry if that wasnt clear.
Quoting Ludwig V
It is applicable to t[sub]1[/sub], but given the supertask described in P3 theres no coherent answer to the definition of X at t[sub]1[/sub] (no final redefinition before t[sub]1[/sub]) proving P3 to be impossible.
Oh, I see now. You did explain, but I didn't pay enough attention.
Though I don't quite see how your B2 follows from your B1. But I don't think it is important.
Quoting Michael
You mean that we don't know the state of X at the last step before t(1), even though X must have been in one state or the other? (We don't have to work laboriously through each step. We just have to know how many there are steps there are between t(1/2) and the last step - we could work it out from that.)
It seems to me that we can work out the value of X for each and every step between t(0) and 1 if we work forward from t(0) but not if we try to work backward from t(1). In other words, whether X has a value at any stage depends on whether we define that stage in relation to the beginning or the end of the series. That seems very odd to me. But perhaps I've misunderstood. But I would be inclined to call a definition like that somewhat ill-formed.
It was redefined as 1 at t[sub]1/2[/sub] and never changed again, so is still defined as 1 at t[sub]1[/sub].
Quoting Ludwig V
There is no last step before t[sub]1[/sub], hence no coherent definition of X at t[sub]1[/sub]. But also at no point between t[sub]0[/sub] and t[sub]1[/sub] is there a step where X goes from being defined (as either "0" or "1") to being undefined, and the definition of X is always retained until redefined to something else. It's a simple contradiction.
If you're trying to find a "solution" you won't find one. We just have to accept that supertasks are illogical. It's that easy.
I'm not trying to find a solution, just to understand what's going on. Not so much why it's wrong, but why anyone would think it was right. Where does the illusion come from?
Quoting Michael
I think I've just understood the significance of your A and B propositions. They are what justifies your formulation of the problem as a contradiction.
Quoting Michael
If there is no last step before t1, there is no last-but-one step before the last step and no last-but-two step before that. And so on. The entire sequence unravels.
If you look at the series one way, it looks perfectly in order. If you look at it another way, it collapses entirely - it's not just a problem about defining the state of X at t1, but about defining the entire sequence.
Quoting Michael
Going back to your propositions A, B, C, it seems a fair guess that the problem is the insertion of "ad infinitum". That's the difference that causes X to become undefined. Our instinct that it should work derives from the fact that the series works perfectly well even if we do not insert any definite number of steps:-
D1. At t0 X = 0 and then at t1/2 X = 1 and then at t3/4 X = 0, and so on for n further steps where n is an even number.
D2. Therefore, at t1 X = 0
E1. At t0 X = 0 and then at t1/2 X = 1 and then at t3/4 X = 0, and so on for n further steps where n is an odd number.
E2. Therefore, at t1 X = 1
I think that's more or less what I was looking for.
The tortoise being overtaken is fiction, but mirrors real physical situations, unlike almost all the other examples in this topic. Describing the motion of Achilles as normal or as a supertask has zero effect on the ability of Achilles to overtake the tortoise.
I must clarify that the lamp itself is physically impossible, making it fiction. I said 'faulty', which it is not. It measures something undefined, so it isn't a contradiction (a fault) that the final state isn't defined.
Exact same scenario. But it's like asking if the smell of lavender is odd or even. There isn't a number that corresponds to the quantity of steps taken.
Quoting Michael
This is exactly why I asked for your definition of 'start' since you seemed to be committing an equivocation fallacy between two definitions. You copped out and gave a synonym (begin) that has the same two definitions.
Is it Sv1 or Sv2? Because you are using both here, playing a language game.
I am saying that the lack of a first step does not prevent the beginning (Sv2 definition, 'transitions from not doing the task to doing it') of the task. You reply that it literally does, but Sv1 is the literal definition (the finite, 'has a first step').
I would not state that the lack of a first step does not prevent the first step from occurring (Sv1). That would be a contradiction indeed.
You cannot show how that description doesn't work. Your only argument is that it doesn't perform a first step, but the description doesn't mention the need to do so, so the criticism is inapplicable.
You are not. It isn't a physically possible task. If you want to do a physically possible one, do Zeno's dichotomy. It's easy. You do it every day. The task is started despite the lack of a first step.
If you take the physical impossibliity away, then you failed because you didn't recite each number at the prescribed time. Your choice not to do so. You didn't follow my procedure. If it's been a minute, you admit to not following it.
Quoting Michael
Undefined. You give no indication of when each number is to be recited. When do I say the 71st to the last zero for instance? I can answer that with a scenario that is properly described. It isn't a supertask as described.
Quoting Metaphysician Undercover
That's apparently what somebody else reported about what Aristotle reported. I've seen it conveyed about 20 different ways. This particular wording says 'never' and 'always', temporal terms implying that even when more than a minute has passed, (we're assuming a minute here), Achilles will still lag the tortoise. The logic as worded here is invalid for that reason since the argument doesn't demonstrate any such thing. I've seen more valid ways of wording it (from Aristotle himself), in which case it simply becomes unsound.
It isn't much. I just didn't like the fact that the quote didn't match the site linked. Too bad Zeno's original argument is gone. Maybe he covered his ass better than the summary provided by somebody paraphrasing Mr. A.
The difficult thing is that many human beings are like naive realists, and they think that our sense perceptions of "the everyday world" are a direct copy of the way an independent world would be. From this perspective, we cannot look toward the everyday world to be what needs to give way. But from a more philosophical perspective, we know that sense perception doesn't really show us the way the world is.
So to be prudent, I'd say both sides need to allow give and take. This may be like the ancient division between Parmenides with being and not being, and Heraclitus with becoming. Plato described how the two seemed to be fundamentally incompatible, and Aristotle provided principles whereby they both could coexist as different aspects of reality.
Quoting Ludwig V
That was Zeno's conclusion, from his paradoxes, that motion is impossible. But I do not think that this was what he was sincerely trying to prove. Clearly he could observe motion, and he would know that this would be considered a ridiculous proof. So I think his arguments were designed to show that there is incompatibility between motion as observed, and motion according to the principles of logic applied to it. Zeno came from the Eleatic school, so the first principle was "being", stasis, but what he was demonstrating was that this principle was insufficient to understand reality. That's why Socrates and Plato took interest in the sophistry of the Eleatics. The Eleatics could employ logic to prove absurd things, and this showed the gap between the "becoming" of the physical world, and the "being" of the Eleatics and Pythagorean idealism. So I think that Zeno, even though he came from the Eleatic school, was apprehending the faults in that ontology, and was sort of poking fun at it.
Quoting noAxioms
I quoted that directly from Aristotle's Physics. I gave the page and lines, 239b, 14-17. Further, Aristotle compares it to the arrow paradox, and says "the 'Achilles' goes further in that it affirms that even the quickest runner in legendary tradition must fail in his pursuit of the slower"
Quoting noAxioms
The time length is irrelevant. The pursuer will "always" lag the pursued, for the reasons indicated. The pursuer must reach the point where the pursued was, and in the time that it takes to do that, the pursued will move further ahead.
Quoting noAxioms
The logic is invalid for what reason? There is no specic time periods mentioned.
Quoting noAxioms
I gave you Aristotle's wording. He rejects the arrow argument which demonstrates that motion is impossible, by saying that time does not consist of instants. So that is an attack on the premises of that paradox. He then says that the solution to the 'Achilles' "must be the same". But he doesn't show how time not consisting of instants would solve the Achilles paradox. The matter of instants appears irrelevant here, and the problem seems to be with the assumed nature of space, rather than time.
Quoting noAxioms
I can assure you, the quotes are taken directly from the referenced sites. I just went back to check. Click the links and you will see.
It's not just physically impossible, it's logically impossible. No physics can allow me to begin reciting the natural numbers in reverse. I can't even say one number, let alone all of them. And this is true even if we're not reciting the natural numbers in reverse but the sequence {0, 1, 0, 1, ...} in reverse, i.e. where each term, individually, can be recited in less than a second.
That there is no first number to recite is the very reason that it is logically impossible to begin reciting them in reverse and it astonishes me that not only can't you accept this but you twist it around and claim that it not having a first number is the reason that it can begin without a first number.
They're clearly being confused by maths. They think that because a geometric series of time intervals can have a finite sum and because this geometric series has the same cardinality as the natural numbers then it is possible to recite the natural numbers in finite time. Their conclusion is a non sequitur, and this is obvious when we consider the case of reciting the natural numbers (or any infinite sequence) in reverse.
There is a far more fundamental problem, and they're just ignoring it. I have no idea why. Perhaps because they can't look beyond the maths to what it would mean for us to actually carry out the tasks. This seems to be the mistake that Benacerraf made in his response to Thomson and which I addressed here.
.... and, as I think you must know, they think you are being wilfully dogmatic. That disagreement is what needs to be understood.
No, I don't think that they think that. It is a philosophical thesis. I'm not sure it is possible to articulate what people who have not thought about the question think the answer to it is.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
I don't think we have anything near the evidence required to divine Zeno's motives. We don't even have his articulation of the argument.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
But you don't know that he recognised what is so very clear to you, that the argument was ridiculous, or that he had "apprehended the faults in that ontology", though I admit that if he had understood what you understand, he might well have been poking fun at it. Still, other people since then have poked plenty of fun at it. But that's not a substitute for understanding the argument.
Quoting Metaphysician Undercover
I agree that it is very likely that Plato/Socrates was addressing the apparent incompatibility of the perceived reality of change and the Eleatic rejection of that perception as illusory. The "two worlds" solution has its problems and, for my money, Aristotle's solution was much better.
Quoting Metaphysician Undercover
The exact length is indeed irrelevant. But the dimension of time is not. On the contrary, it is embedded in the argument.
I imagined you were arguing toward the claim that anything which has no first event is not physically possible. Which is a bit different from the geometric series angle. Which shows that there are logical possibles which have no first event. The sticking point between you and us series heads seems to me that you've been arguing on the basis of "no first event" blocking logical possibility, which would then block physical or metaphysical possibility. Which is why I intervened, because while I agree with your argument strategy, the means you were arguing for it were imprecise enough to admit the geometric series counterexample. I do think your points can be steelmanned though.
You might come up with a mathematical model for "recitation of first element", which maps a series to its minimum or maximum. Since the series has a minimum, you can do that. It has no maximum, so you can't. It's the same principle as asking someone to count backwards from infinity. And then it's true, it's not logically possible to "recite" backwards from infinity, or "recite" the maximum of an increasing geometric series.
Going back to the first page of the thread, such a "recitation" for the state of Thompson's lamp, or just isolating the "state", could be construed by taking a time period and associating it with the states the lamp takes in that time period in order. If Thompson's lamp has states in a time period, they'll be picked out by that. However, the function which generates the values of Thompson's lamp has the property that for every time period X, there exists a time period Y such that if max( Y )>max( X ) then Y contains at least two states (on or off). You get those by going further toward the completion time. That property implies there is simply no "state" of the lamp at limit of 2 minutes. So it having a state is logically impossible.
What makes Thompson's lamp a paradox, then, is a physical or metaphysical intuition about the concept of the state of the lamp. There needs to be a beginning to the process, and it needs a unique isolable end state. Both the geometric series and Thompson's lamp have no unique isolable end state.
That will then bottom out in an inquiry about whether there are physical or metaphysical possibles which have non-unique and non-isolable end states.
NoAxioms has a habit of making astonishing claims, then instead of recognizing the incorrectness, arguing some twisted principles. Like above, noAxioms insisted Zeno did not conclude that the faster runner could not overtake the slower, then refused to recognize my references, insisting they were in some way improper.
Quoting Michael
This is the problem with mathematical axioms in general. As fishfry said, I can't really count the natural numbers, but I can state an axiom that the natural numbers are countable, and this counts as me having counted the natural numbers. So mathematicians really need to be careful to distinguish between the fantasy world they create with their axioms, and the true nature of what is "logically possible". Just because it can be stated as an axiom does not mean that it is logically possible. And when the axiom claims that something which is by definition impossible (to count all the natural numbers), is possible, then there is contradiction, therefore incoherency, inherent within that axiom. But would a mathematician accept the reality of an axiom which is self-contradicting?
Quoting Michael
The problem is the age-old incompatibility between being and becoming. Logic, and this includes mathematics, applies naturally to "what is", "being". But "becoming" has aspects which appear to escape logic, what lies between this and that, one and two, etc., and therefore it seems to be illogical. If we apply the logic of being, to the reality of becoming, we find paradoxes as Zeno demonstrated.
This implies that "becoming" requires a different form of logic. That's what Aristotle laid out with his definitions of "potential", and "matter", as the aspects of reality which violate the law of excluded middle. In modern times, much progress has been made with modal logic, and probabilities. But the truth is that these aspects of reality, those which are understood through probability, remain fundamentally illogical, and the so-called "knowledge" which is derived creates an illusion of understanding.
Quoting Ludwig V
It's simple, talk to people, ask them. Then you'll see that it's more than just a matter of thinking about the question, it is a matter of making the effort to educate oneself. Metaphysics is not apprehended as a valuable subject.
Quoting Ludwig V
Well, there is a lot of information available from Plato. In works like "The Sophist" and "The Parmenides", he takes a very critical look at the motives of some of the Eleatics, Zeno in particular. Specifically, in "The Sophist" he attempts the very difficult task of developing a distinction between philosophy and sophistry, even the sophist is engaged in philosophy.
Quoting Ludwig V
It's very clear from the discussion at the time, Plato and Aristotle, that Zeno knew he was using logic to produce absurd conclusions. There should be no doubt in your mind about that. He did not pretend to believe what he had proven, that motion is impossible, that the faster runner could never overtake the slower, etc..
I see your point. But you must know that there is a great deal of philosophy around your view of this. But I won't try to drag you through it, is because I'm not sure how relevant it is. Yet.
Quoting noAxioms
I agree with that. So when someone describes the situation in a way that seems to make that fact impossible, why don't we just reject it as inapplicable?
Quoting noAxioms
But we allow physical impossibilities into fiction all the time. They even crop up in philosophical examples. "The sun might not rise tomorrow morning". "Twin Earth has water that is not H2O". I won't even mention philosophical zombies, brains in vats or simulations.
Your point about the final state not being defined is about logic, not physics (despite some people thinking that it is about physics).
In any case, the final state is defined. It must (on or off) or (0 or 1). Wouldn't it be more accurate to say that it is undetermined? Or is the final state the one immediately preceding the limit; in any case, it is not determined. So is the one before that.... But it would be absurd to say that every state in the series is indeterminate. It seems that whether anything here is determined is a question of how you look at it - from the beginning or from the end.
True. And that implies time is discrete how?
Quoting fdrake
Good post
If time is continuous then supertasks are logically possible. Supertasks are logically impossible. Therefore, time is discrete.
Time being continuous is necessary but not sufficient for any given supertask being possible. Supertasks being impossible (especially the specific one you brought up) does not imply time is not continuous.
Any given example does not prove that supertasks in general are necessarily impossible. If the common necessity among all supertasks is time being continuous, the only way to prove the impossibility of supertasks is to prove time is not continuous.
(a and b and c and d) ? supertask
not supertask does not imply not a
I addressed this here and here.
Me and fishfry have insisted that this is a case of missing limit. I saw your post here, the only reply to it is noAxiom's, to which then there was no reply, only a mention by fdrake. I haven't checked your post with the "refutation" of Benecerraf yet, neither do I think noAxiom's post addresses it fully.
When it comes to this post, Quoting Michael
Is the failure of C2 really the consequence of the impossibility of C1, or is there an unstated premise?
Quoting Michael
What so many people disagree on is that. You think that the end of the sequence at t=1 is a temporal/logical consequence of what happens before. Others don't think so, because the lamp is being infinitely redefined until t=1.
That's why it's impossible to complete.
Quoting Lionino
Yes, I address that here.
It was more of taking the phrase "metaphysically (im)possible" to mean "there is (not) a possible world where" and seeing where that leads. And if it leads anywhere is that maybe the definition of metaphysically possible is «that which follows the rules of the game». That seems abusive of the meaning of the words, or the words are not well-defined (many would say so for "metaphysics").
One of the first replies of the thread is this:
Quoting jorndoe
If jorndoe is representing the view well, I am confident both have good reasons to make such equations; I was exploring ways to make the semantics of "metaphysical" not fully overlap with "logical" or "physical".
I understand the intuition you use to affirm that argument, I imagine others do too. At t=1 the sequence has ended, and the lamp must be either on or off. You use the same premise on it:
People disagree with the premise because we are not confident we can use such intuitions when the unintuitive concept of infinity is involved.
The important part from that post is this:
If the lamp is on at t[sub]1[/sub] then it must have been either turned on at t[sub]1[/sub] or turned and left on before t[sub]1[/sub], neither of which are allowed given the supertask, hence the contradiction.
It would be a mistake to apply (((P = Q) & (Q = R)) implies (P = R)) without checking very carefully whether "Q" means the same for both of them. It is not something one could take for granted. I wouldn't take that thesis seriously without cross-questioning the author very carefully.
Quoting Lionino
I doubt if it is possible to abuse the word "metaphysics". It has been abused so often in the past. In fact, it is so ill defined that I'm not sure what would count as abuse.
Three points:-
I have problems with the term "synthetic necessity" because I don't understand what that does to the meaning of "contingent". (I'm taking the Kripke-style interpretation that it means "In any world in which ...., this must be the case." - and in "in any world in which knock-out tournaments are played, it cannot be the case that two opponents in round 1 can meet each other again in round 2".) Tempting as it is, since logic is also a (language) game, or at least has rules, if metaphysics is that which follows the rules of the game", it aligns metaphysics with logic. But I do admire Toulmin's argument and recognize that he identifies an important class of propositions that have not figured much in philosophy.
I'm afraid I understand the possible worlds interpretation of possibility as simply possibility. Either way, of course, that aligns metaphysics with logic.
Many of the uses of apparently metaphysical language seem to me to be a simple matter of using what logic describes as "de dicto" as "de re" - possibly without being aware of what they are doing.
If I knew how to ask without leading them into philosophy, I would.
Quoting Metaphysician Undercover
The Stanford Encyclopedia is the best quick reference that I know of for something like this.
SEP - Zeno's paradoxes
From what I could find, Aristotle has very little about Zeno and nothing about his motives. But what he does summarize (some of) the arguments, which Plato doesn't.
I hadnt realized quite how close in time they were. It seems that the scenario in the Parmenides, which seems to be far and away the best source we have, could have taken place. Not that we know that it did. On the face of it Plato is not an implausible source if only separating out the history in Plato was not so complicated.
No evidence of your interpretation here.
Or here.
Now, this is another example of what I was talking about. Plato (and others) were confident that Zenos case was weak. Fair enough, but to go on, as Plato does, to accuse the sophists of deliberate deception or wilful blindness is completely unjustified (except when, as in the Protagoras,(?) Gorgias (?) someone boasts about doing so though it doesnt follow that everyone that Plato accuses of rhetoric and sophistry did so boast.). I have seen it often before, particularly in the last year on these forums. But it is most disheartening.
But accepting that connection is a long way from accepting that he had any doubts about the validity of his conclusions.
I was wrong about that. I elided Parmenides with the Eleatics, though the difference is, perhaps, somewhat metaphysical (!). However, the difference matters when it comes to Zeno, so now I can get it right. It does not follow that Zeno did not believe that his conclusions were not true.
All the quotations above are from SEP - Zeno of Elea
Well, whatever prompted you, the project makes sense to me and I agree with Toulmin. I'm not convinced about the relationship of those propositions with metaphysics or their classification in the analytic/necessary/a priori constellation. However, preserving those concepts doesn't seem to me particularly important. I would be quite happy to abandon all of them.
Given P2, what is the first natural number not recited? I seem to remember having asked you this several times already.
Quoting Echarmion
What is the first number not recited?
Any initial step necessarily leaps over all but finitely elements of the sequence. Same reason that any neighborhood of the limit of a sequence contains all but finitely many elements of a sequence.
I don't think you and I are making progress.
I have agreed repeatedly that we can't "count all the natural numbers backwards" since an infinite sequence has no last element.
Yes ok, but that supports the possibility that in the future, our current preoccupation with "mind as computer" will look as dated as "mind as waterworks" of the Romans.
Quoting Ludwig V
That it explains nothing? I agree. Like saying "God did it." Or saying the Great Sky Computer (GSC) did it. Except that God is not restricted to being a computation, whereas the GSC is, making God a less unreasonable hypothesis.
Quoting Ludwig V
Well you never "reach" 0, but 0 is the limit.
Quoting Ludwig V
If you allow the transfinite ordinals, the sequence 1, 2, 3, ... has the limit [math]\omega[/math]. And even if this seems unfamiliar, it's structurally identical to the sequence 1/2, 3/4, 7/8, ... having the limit 1, which is much more familiar.
Quoting Ludwig V
What is the starting point of no axioms? It's like playing chess with no rules.
Quoting Lionino
Quoting Lionino
There's something going on here about ends and limits. I understood that the issue here is that although the series does have a limit, it doesn't have an end. As an abstract concept, one need not be particularly puzzled by this. But when you locate the series in time, it gets difficult.
It's a dilemma. The definition of an infinite series defines all the members of the series. That takes no time at all - not even an instant. So the time factor is actually irrelevant. But in another sense, each term of the series needs to be worked out, by us, and that is a process. That process must take time; actually, it would take infinite time - i.e. can never be completed.
Quoting fishfry
Of course it does. I can't wait to see how it all plays out.
Though there is at least one case where the idea got transformed and returned with a vengeance. I mean the Pythagoras' and Plato's idea that ultimate reality is mathematical, meaning the only reality is the mathematical as opposed to the physical, world, returns as the idea that the physical world is mathematical. Weird.
Quoting fishfry
My word, there's a discovery! A hypothesis that is more unreasonable than God! This should get a Nobel prize of some sort.
Quoting fishfry
Yes, I do remember our earlier discussion of this. I don't pretend I understand them, but I do admit they exist - my allowing them or not is irrelevant.
Quoting fishfry
Did someone mention a starting-point of no axioms? It would be indeed be like playing chess with no rules - or discussing infinity.
Tegmark's trolling. And the world is mathematical to us just as it's sound to a bat. The world does whatever it's doing. We do the math. The world is described by the math to a good degree of approximation. It's a metaphysical hypothesis that the world "follows" the math. Clearly the world did not stop following Newtonian gravitation when Einstein came along. Both theories are just approximations to something deeper ... or perhaps nothing at all. Nobody knows.
Quoting Ludwig V
The computational theory of the world requires that the world is a computation. That is indeed more restrictive than the hypothesis that God did it. Computable functions are a tiny subset of all possible functions. There is no reason at all for the world to be computable. I find it unlikely.
That is (to repeat myself): The street corner preacher says that all of us are created in the image of God. The TED talker says that we are all created by the Great Simulator, who operates as a Turing machine. That is a most restrictive stipulation. Far less likely than God. It's ironic that the intellectual hipsters mock God and flock to simulation theory, which is a far less likely hypothesis.
In your opinion, how is simulation theory any less magical and unrealistic than God? And why should God be restricted to be a Turing machine? I never understand this point.
Quoting Ludwig V
That's why I prefer the 1/2, 3/4, 7/8, ... example. Same structure in more familiar clothing.
Quoting Ludwig V
Mathematicians have incredibly precise rules for infinity. The rules are the axioms of ZF or ZFC set theory.
A few quotes with no real context, does little. Anyway, it's off topic, and really sort of pointless to argue a subject like this. You have your opinion based on how you understand Plato, and I have mine. Due to the reality of ambiguity, i don't think there is a correct opinion here.
Quoting Ludwig V
The problem is that "sophist" was a word with a very wide range of application at that time. In the most general sense, you'll see Aristotle use it to refer to someone who uses logic to prove the absurd. Zeno might be a sophist in this sense. But also "sophist" referred to people like Protagoras and Gorgias, for their use of rhetoric. And "sophist" also referred to those who had schools and charged money to teach virtue. So there was a range of meaning, but "rhetoric" seems to be the essential aspect, and this is a mode of persuasion which is not necessarily logical. Accordingly, "sophist" has bad connotations, but as Plato demonstrates in "The Sophist", it's very difficult to distinguish a philosopher from a sophist. It appears like either the sophist is a type of philosopher, or a philosopher is a type of sophist.
Quoting Ludwig V
The issue is not the validity of the conclusions, it's the soundness. Take the Achilles for example, with two principle premises. First, to overtake the slower, the faster has to get to where the slower was. Second, in that time, the slower will move further ahead. So the faster does not overtake the slower, and this may repeat if the faster is still trying. It appears valid to me, so if we want to refute it we need to look at the premises, as Aristotle did. But when we try to understand how the premises are wrong, then there is disagreement amongst us, because we really can't demonstrate exactly what the premises ought to be replaced with.
,
Believe it or not, that's an incredibly helpful remark. Not only do I understand and agree with it, but it also enables me to get a handle on what metaphysics is. Sorry, clarification - I am referring to the whole sentence, not just the last five words.
Quoting fishfry
I had to look Tegmark up. No disrespect, but he does illustrate the observation that intellectuals are not exempt from normal human desires for fame and fortune, no matter how much they protest the contrary. There's also a normal human pleasure in astonishing and shocking the tediously orthodox Establishment.
Quoting fishfry
Yes, we had that discussion as well. You may remember that I had reservations. Same, but not identical, structures, I would say. But I don't expect you to like it. It doesn't matter until it becomes relevant to something.
Quoting fishfry
My apologies. I should have restricted my remark to those who dream up paradoxes. Though perhaps even that is wrong. They may be exploiting the rules themselves, rather than merely breaking them. The mathematical rules for infinity don't seem particularly helpful in resolving these problems.
Quoting Metaphysician Undercover
Well, I'm almost certain there isn't. But my disagreement with you prompted me to look more closely and acknowledge something that feels like error in one or two respects.
Quoting Metaphysician Undercover
Yes, I understand your account of this. It's important to add that Plato thinks that the sophist mimics the philosopher and what he says is accounted rhetoric because it mimics the speech of the philosopher. (He didn't have a concept of logic as we think of it.) The mimicry is the reason why he condemns both the man and what he says. How does he distinguish mimicry from the real thing? Mimicry seems to be true, but is not. So, in the end, the distinction between the two in his writings is the distinction between those who agree with him and those who do not. I'm not trivializing Plato. It is a universal problem.
Socrates (as presented by Plato) considered himself wiser than anyone else because he knew he didn't know anything, which doesn't seem to leave much room for anyone else (at least in Athens) to be a philosopher. However, his dialogues with sophists do not show Socrates treating them disrespectfully and this is something of a puzzle. The orthodox interpretation regards Socrates' respect as ironic. Maybe it is. But maybe Plato's practice was a bit less dismissive than all this implies.
It is very difficult. If you believe that you have got hold of an absolute guarantee of truth and someone else disagrees with you, the temptation to dismiss them, rather than just their view, is very great. If P implies Q and P is true, but someone rejects your conclusion, what are you to do with them? Have you ever read C. L. Dodgson's article on Achilles and the Tortoise? It faces the problem head-on. I won't spoil the plot. You should be able to get hold of it somewhere on the web. Wittgenstein faces this issue in his discussion of rule-following. I don't know of anyone else who takes the issue seriously.
Quoting Metaphysician Undercover
Yes, you are right. I was not accurate. Sorry.
Quoting Metaphysician Undercover
I had noticed. Which is why I keep trying to suggest other approaches. With little success, I admit.
In my reply to this quotation, I said Quoting Ludwig V
This was a mistake. I intended to spare you unnecessary verbiage in my reply. But what I said was annoying and unnecessary. I'm sorry.
The point of the article is very simple. Achilles and the tortoise are chatting after Zeno's race. Achilles observes:- "I was first past the post, so I won". The tortoise replies:- "I don't accept that." Achilles:- "What do you mean? The first competitor to pass the post is the winner of the race, and I passed the post first, so I won". Tortoise:- "I don't accept that". It continues for some time. There's no resolution - not even Achilles killing the tortoise - not that that would count as a resolution. But we all know what happens in real life when such situations arise.
There isn't one. I've answered this several times already. That's what it means for me to accept P1.
But you need to prove P2. You haven't done so.
So we're back to my post here:
You accept that (b) is impossible but you claim that (a) is possible. You have to prove this. P1 doesn't prove it.
That is the view that mathematical is somewhat of an empirical endeavor. Many disagree however, and think that mathematics is something fixed and representative of the world.
Quoting Lionino
Certainly mathematics is, in a sense, fixed. But what we are talking about it is applied mathematics. It seems pretty clear that arithmetic and geometry originated in severely practical needs of large empires. But it does seem to have taken off on its own, as it were, as a theoretical enterprise. Here, we are talking about applied mathematics.
I think what @fishfry means to say is that mathematics is the way the world is represented to us. That's the point of the comparison with what sound is to a bat. I would rather say that mathematics is the way we represent our world to ourselves.
It's true that the mathematical techniques we use are fixed - though we also develop new techniques, as in 17th century calculus or non-Euclidean geometries. But we have to work out how they can be applied to specific phenomena.
Also keep in mind that physics was absolute back then, and calculus was unheard of.
So Zeno's premise that in any task of changing location, one must first go to the halfway point. Zeno seems to have put that up there as a ridiculous premise, one in which he didn't believe. It was an attack on Aristotle's position which would find no fault with the statement. Given Zeno's beliefs, the premise above is false, and he attempts to demonstrate why, but of course he can only do so by begging his own opinion, which is the second premise that I've been going on about.
Not an example of a physical impossibility. Yes, i agree that physical impossibilities can be turned into fiction. Did I say otherwise?
The state of Achilles is that he is even with the tortoise. It's admittedly not final because he continues on after the task of overtaking it is complete and takes the lead. There's nothing about that where physics stops being relevant.
The lamp example? That isn't physics. Never was.
A list of valid options is not a definition of a state.
Synonym?
They are, or at least the existing ones are. None of the ones you listed was an existing step.
Quoting Metaphysician UndercoverSays the proponent that time stops.
See my comment above. I suspect Zeno believed his premise to be false, that one must first get halfway before getting to the goal. He was trying to illustrate this belief by driving Aristotle's assertions to absurdity, but he must beg his own beliefs to do this.
If there was a better worded argument provided by Zeno himself, perhaps Aristotle didn't convey the full argument, in the interest of waving away a suggestion that he is wrong. So many modern mathematical tools were not available to them back then.
To me it was just another wording, but apparently so since I see it referenced verbatim on so many discussions. Interesting is the total lack of mention of the tortoise.
He rejects most of the arguments because they contradict his own assertions.
The argument is the same with space. He says "time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles". Space qualifies as an 'other magnitude'.
Quoting Metaphysician UndercoverI said no such thing. Zeno very much is reported to have concluded such things.
Yes, but the meaning of limit here isn't the same as it is used in Calculus. It is in the sense used here.
His analogy/metaphor implies that mathematics is something that we impose onto the world instead of something that we derive from the world. His position is anti-realist therefore. If he was right, platonism about mathematics wouldn't be such a strong position today.
I don't think the calculus is relevant. In any case, I understood that it stated the problem rather than solving it - calculating the result to as close an approximation you aspire to, but never absolutely. I wouldn't be surprised if I got that wrong.
Quoting noAxioms
If you accept that Twin Earth is not physically possible, there's no need to argue about the sun example. Maybe your imagination is richer than mine.
Quoting noAxioms
Monochrome = (black, white or grey all over)? Red = (indefinite number of shades of red)? Sibling = (brother or sister)? Parent = (Mother or father).
Quoting noAxioms
I don't know, what do you think? I had in mind that every step is defined by the formula, which cannot be applied to any step unless it's predecessor is determined (except for the first step.) I wouldn't go to the stake for one or the other.
Quoting noAxioms
Yes. The first step exists if you are looking forward, but if you are looking backward, it doesn't. But in the normal world, the first step is the last step - i.e. exists whichever direction you are looking or even if you are not looking at all. This is Berkeley's world.
Quoting Lionino
I don't quite get what "anti-realist" means here. But you are right. I was trying to articulate the idea that counting is not a determinate description, but a system for generating determinate descriptions; we have to apply the system and discover what pieces of the number system apply in each case. Actually, one could see some sense in saying both that the mathematics is derived from the world and that it is imposed on the world.
The idea that mathematical entities aren't real, especially that they aren't abstract objects.
I believe Socrates (as portrayed by Plato) had great respect for the sophists. They displayed power and influence, and this piqued his interest. In Plato's dialogues, Socrates holds lengthy discussions with some sophists, and this would not be possible without the appropriate respect. On the other hand, I also believe that since the sophists presented themselves in a conceited way, as filled with a sort of complete or perfect knowledge, this produced a challenge in Socrates, to demonstrate their faults and weaknesses. Because Socrates had some degree of success in this personal challenge, Plato developed a level of disdain for them.
Prior to Socrates I believe that sophists were generally well respected, and this is evidenced by the power of their rhetoric. Socrates revealed the subjectivity of rhetoric, leaving the character of the sophists who employed it, exposed. The principal sophists who were exposed in this way, were the the politicians of Athens. But Socrates carried on toward exposing those in the even higher level, more exclusive schools of logic (I don't agree with you that there was no concept of "logic" at this time) like the Pythagoreans and Eleatics, and this allowed Plato to class them as sophists. This is where Zeno fits in. And Socrates is portrayed by Plato as having great respect and curiosity for the lofty principles held by these prestigious schools. Nevertheless, despite great respect for the individuals, he sees that there must be flaws in the principles, and therefore proceeds with his personal challenge of engaging the individuals to defend, and ultimately reveal those faults.
I think that the important point is the use of valid reasoning with unsound premises. This is how Aristotle attacked Zeno's paradoxes. But Aristotle didn't have a good understanding of the nature of knowledge, and the effects of faulty premises. He claimed that logic leads us from premises of greater certainty, to conclusions of lesser certainty, when in reality the opposite is true. Uncertainty in the premises is what introduces uncertainty into the conclusions. And the problem is that many premises are intuitive notions simply taken for granted, such as in the Achilles, the premise that the faster must first reach the place where the slower is, prior to passing. In reality, the faster passes the slower without ever sharing the same place.
The Aristotelian view of knowledge is still common place. You'll see that many here at TPF argue that there are fundamental principles, 'bedrock propositions' or something like that, which are beyond doubt, and support the whole structure of knowledge. In reality, those fundamental principles are the least certain because they are taken for granted, lying at the base of conscious thinking, bordering on subconscious knowledge. Those highly fallible intuitions are the ones most needing the skeptic's doubt, but it takes someone like Zeno to demonstrate this.
Quoting noAxioms
Huh? I said that time stops? I don't think so. I said that in the scenario of the op, 60 seconds will never pass. But clearly time does not stop. In that scenario, time keeps passing in smaller and smaller increments, such that there is never enough to reach 60 seconds, but time never stops. The claim that 60 seconds must pass or else time will stop, is derived from different premises which are inconsistent with the described scenario.
Quoting noAxioms
That's what I was arguing as well, but Ludwig produced references to show that this might not be the case.
Thanks.
Quoting Ludwig V
Well metaphysics is just "What is reality?" And it can't exactly be our math, because we can see that it wasn't quite what Newton wrote down, and in the end it won't quite be what Einstein wrote down. It's actually kind of strange that math doesn't exactly describe reality, but so well approximates it. Our theories get better and better but never get there. As if reality is the limit of our theories.
Or worse. Our math is like the bat's echoes. Just the only tool we have to understand the world, but greatly limited. And we think we know everything.
Quoting Ludwig V
Goes a step farther. The universe isn't just described by math, it "is" math. Which is a category error so massive that Tegmark must be trolling. The equations of motion describe the planets, they aren't the planets themselves. The map is not the territory. Just as the source code for a program must be executed on hardware in order to do anything.
Tegmark must be trolling. There is no other explanation. That so many take him seriously is a good reason to be skeptical of experts, celebrity scientists, and "public intellectuals."
Quoting Ludwig V
We're in agreement. Bostrom (we're all sims) and Tegmark (we're all mathematical structures) must be enjoying themselves tremendously. Most likely when they write serious stuff, nobody pays attention.
Quoting Ludwig V
Well it's relevant to the Thompson lamp. It's a mathematical model of a sequence with its limit point adjoined. The example is so familiar to me that I thought it would add clarity. To the extent it got in the way, perhaps I should rethink how I present the idea.
Quoting Ludwig V
Mostly philosophers who prefer to indulge in the vagueness of word games rather than the precision of math. But I concede that many smart people take these puzzles seriously. I respect that, but for some reason the fascination eludes me.
The lamp's defined at each point of the sequence, but it's not defined at the limit. There's no way to make the sequence continuous, se we are free to make the terminating state anything we like. There is no natural continuation. That seems perfectly clear to me. I don't know why it's not perfectly clear to everyone else. I actually have a difficult time seeing the other points of view.
Quoting Ludwig V
Maybe we'll get some new infinitary physics some day.
Let's focus on one thing at a time. Regarding your example of counting the natural numbers backward, or letting the sequence get smaller when time goes forward, the 1, 1/2, 1/4, ... idea; I have repeatedly asked you if you understand and agree that any interval of real numbers containing the limit of a sequence, necessarily contains all but finitely members of the sequence.
I need you to understand that in order for me to explain to you how the backwards counting puzzle is resolved.
Since I've asked you several times to just tell me, yes or no, do you understand what I said, and you have repeatedly ignored me, conversational progress can not be made on this point.
So let's stick to the inductive proof, in which you yourself proved P2 is true. Let's get back to the backwards counting example after you tell me, yes or no, do you understand the property of limit points of sequences that I keep asking you about and that you keep not answering.
Surely few if any people believe math is "fixed." Math is historically contingent and changes all time time, with a massive volume of new papers published every day.
If you are referring to some kind of Platonic math that's already known by God, that we are just discovering, that's an entirely different discussion.
Am I understanding you correctly?
Besides, math can't "represent the world," simply because there are Euclidean and non-Euclidean geometry. They can be used to represent the world; but they can't both be true, hence they can't both "represent the world." They can only be used to represent the world.
Math can not tell you what's true about the world. It can only be used to model various aspects of the world. That's different.
I'm asking, in what sense? Surely math has never been fixed. It's always changing. It's a human activity.
Do you believe that God's math book is already written? Is that how you are defining math, as a Platonic ideal "out there" that exists even before we've discovered it?
Quoting Ludwig V
Originated as, yes. But that doesn't restrict how math is seen today.
Quoting Ludwig V
I don't recall stipulating to any such restriction.
In any event, what is the difference between abstract and applied math? Only time and historical contingency. Non-Euclidean geometry was abstract and useless in 1840, and it became applied (to general relativity) in 1915. [Someone complained a while back about my phrasing. I could say Riemannian geometry, since there are other flavors of non-Euclidean geometry. I'm making a different point which I hope is clear].
Likewise number theory, regarded as supremely useless since the time of Diophantus; and now the basis of public key cryptography, the basis of modern digital commerce, since as recently as the 1980s. That's 2200 years of uselessness, only to become supremely useful.
The difference between abstract and applied math is time and history.
Quoting Ludwig V
I could have said that. I could have said your formulation too. My only point is that how humans model the world and the world itself are two different things. Aren't those Kant's phenomena and noumena? I'm not a philosopher but that's what he meant, right? Humans try to explain the phenomena. We can't know the noumena. That's Plato's cave analogy again.
Quoting Ludwig V
I think you are agreeing with me. Abstract today, applied tomorrow. Or often the reverse. We invent new abstract math to help us understand some real world application. It goes back and forth.
I agreed that if P2 is true then C1 is true, as I have agreed from the beginning.
This doesn't prove that P2 is true.
Since I don't know what "reality" means in its philosophical sense (which I designate by "Reality", but I do know, roughly, what you mean by "the limit of our theories", I would prefer to say "The limit of our theories is Reality". I'm of the school that teaches that the philosophical sense is metaphysics, and nonsense. But, since I arrived on these forums, I've had to recognize that, in philosophical discourse, "Reality" is a term in regular use and with some level of common understanding.
It's still a bit broad brush. I can understand it in the context of the inescapable inaccuracy of measurement in physics, etc, contrasted with the preternatural accuracy of (many, but not all) mathematical calculations. It's a version of Kant's regulative ideals and gives some content to phenomena/noumena and an explanation how they might be related.
Quoting fishfry
Not the only tool. We have sound as well. Not that we know everything, thank God.
Quoting fishfry
Well, I would certainly want to get him to explain what he means by "is". That might slow him down a bit.
Intellectuals have human motivations and follies just like everyone else - and some of them would do well to acknowledge that. I understand also that it is irresistibly tempting to explain people's failures to recognize conclusive rational arguments in ways that they will not like. But one needs also to understand that can be a trap. Hence Plato turned a classification of the philosophers he disagreed with into a term of abuse - "sophist", "rhetoric". You may have noticed that I'm engaged in some discussion with @Metaphysician Undercover about this issue in relation to Zeno. They, and, apparently @noAxioms cannot believe that Zeno believed his own arguments - and that's not an irrational response because they are incredible. Nevertheless, I can't believe that they believe that. It's not easy. But I think it is important not to follow Plato's example in this respect.
Quoting fishfry
I don't think there was anything wrong with your explanation. There's no such thing as the bullet-proof, instantly comprehensible, explanation. On the contrary, it helps to allow people space to turn what you say round and poke it and prod it. It's part of the process of coming to understand a new idea.
Quoting fishfry
Quite so. It's a sequence, but also a chain, because each point of the sequence depends on its predecessor. The reason it's not defined at the limit is that we can never follow the chain to its' conclusion - even thought the conclusion, the end, the limit, is defined.
It seems paradoxical, because the limit is established before the chain can begin. The first step is to define the limit and the origin; that gives us something we can divide by 2 - and off we go.
This may not be mathematics. But I do maintain it is philosophy.
The consequence is that the series "vanishes" if we try to look back from the "end". It's existence depends on our point of view. I don't suppose that any mathematician would be comfortable with that, but I plead that we are talking about infinity and standard rules don't apply.
Quoting fishfry
Quoting fishfry
Quoting fishfry
I agree with all of that. But I think it is very, even hideously, complicated.
It seems to me that we should always be specific about what is fixed and what is not. There may be disagreement about what goes in to which classification or what "fixed" means. But to say "math" without specifying further leads to confusion.
Arithmetic, for example, is (relatively) fixed, though it may be modified from time to time. The inclusion of 0 and 1 as numbers is an example. Number theory might count as another example - I'm not sure about that. But once the methods of calculation are defined, they are fixed and the results from them are fixed as well. One could say, however, that both methods and results are discovered rather than defined, because there are ways of demonstrating whether a particular procedure gives the right result or not - through the application of the results or through the application of criteria like the consistency and completeness of the system. Euclidean geometry is similar, so far as I'm aware.
Algebra, calculus, non-Euclidean geometry, infinity theory are all additions to mathematics, rather than replacements of anything. It is almost irresistible to speak of them as developed or created rather than discovered, but since they share something with arithmetic and geometry, there are some grounds for speaking of them as discovered, because they were always possibilities, in some sense. What is it that is shared? The best I can do is to say something like logic - a sense of what is possible, or permitted.
This is not irrelevant to this thread. Once we have realized that "+1" can be applied to the result, it would not be wrong to say that the result of every step is fixed, whether or not we actually do add 1 to the 3,056th step. The result of each step is "always already" whatever it is. (I think it derives from Heidegger, but that doesn't prevent it from being helpful.) It captures the ambiguity between "+1" as something that we do and something that is done as soon as it is defined, or even before that.
As a result of the simple recognition of a possibility, we find ourselves plunged into a new and paradoxical world. I mean that it is simply not clear how the familiar rules are to be applied. Which makes it clear that we have to invent new ones - or are we discovering how the familiar rules apply or don't? I don't think there is a determinate answer and "always already" recognizes the ambiguity without resolving it.
When we refer to a step in the series, are we talking about something that we do (and may not do) and which actually takes time or something that is "always already" done, whether we actually ever do it or not?
Yes.
Quoting fishfry
You are assuming a non-realist view of mathematical entities again. You can still have Euclidean and non-Euclidean facts in the world as different facts just like algebra and calculus are different facts. Many philosophers think mathematical objects are real objects that exist outside of space and time.
Quoting Lionino
So many questions about this.
It would be merely picky to ask whether "+" and "-" are objects, because it is obvious that they are operations to be carried out on objects. Still, there is a question what that means. But it would a distraction from the main event.
Take numbers. Does anyone deny that numbers exist? Does anyone claim that they are concrete in the way that bricks and timbers are? They are like objects in some ways, in that they can be distinguished one from another and counted. But what kind of objects are they?
Geometrical shapes like triangle and circles seem to be different. In one way, physical objects are called triangle and circles, but are acknowledged to be approximations to the ideals of geometry. Ideals are not physical objects and do not exist "in space and time". They certainly exist and are real in that sense.
But what does "outside space and time - beyond the fact that there is no possible answer to the question "Where are they now?" What does that mean - I mean what do "exist" and "real" mean here? True, we can say that they are abstract, but what does that mean - apart from "not physical" or "not concrete".
I don't know what platonists say about mathematical operations. Perhaps they would say they are relations among numbers (which for them are real things). Most kinds of dualism for me are overly problematic and I think only nominalism is really sensible when we get to the bottom of things.
Quoting Ludwig V
Putting it bluntly, nominalists and conceptualists and every kind of anti-realist strictly defined.
Quoting Ludwig V
In a way, immanent realists do.
Oh, well. That changes everything. I thought I was talking to a platonist and trying to get him to face up to some of his problems. But that's a bit futile.
Quoting Lionino
So you deny that numbers exist? Really?
Why do you keep reminding us that platonists exist? Is it perhaps because you think people should not say mathematics is thus and so, but be more specific? Or because people so often say that mathematicians think this and that when it is plain that only some mathematicians think those things? Those are tendencies that annoy me.
There are people today that say that there are no real infinities, whatever that means. I think this might be one example of such an assertion.
Where there's not-water? I accept that as a physical impossibility, yes, but overtaking a tortoise is not.
there's no need to argue about the sun example. Maybe your imagination is richer than mine.
Well illustrate. A list is not a parent, so I disagree with the '=' you put there. I'm sure there is a correct symbol to express that any member of that list satisfies the definition of parent.
Quoting fishfry
I replied to this in the simulation topic since discussion of it seems to be of little relevance to this topic.
Quoting fishfryThe two of us also seem to be on the same page.
Quoting Metaphysician Undercover
Not in those words. "Does not allow for a minute to pass", like somehow the way a thing is described has any effect at all on the actual thing.
Quoting Metaphysician Undercover
Anyway, I see nothing in any of the supertask descriptions that in any way inhibits the passage of time (all assuming that time is something that passes of course).
The OP scenario is pure abstract, and it directly describes a state beyond the passage of a minute.
Ah, it slows, but never to zero. That's the difference between my wording and yours. Equally bunk of course. It isn't even meaningful to talk about the rate of time flow since there are no units for it. The OP makes zero mention of any alteration of the rate of flow of time.
Quoting Ludwig V
Not to put words in anybody's mouth, but such a statement depends heavily on the definition of 'exists'. For instance, does the number 37 have a location somewhere in our universe? When was it created? That references a definition of "is a object in our universe". If you define 'exists' as 'is an abstract concept in some mind somewhere', then 37 exists as long as somebody is thus abstracting. It's still a version of 'is part of the universe'.
I am not really sure of the definition Lionino is using. I didn't get it from the brief context.
Look at Tegmark's view mentioned above. He is definitely using a definition of 'exists' that doesn't supervene on our universe, and suggests that the reverse is the case.
I don't have a very strong stance on this debate exactly, I am aversed to dualism in objects, so positions like platonism irk me immediately. For me, numbers exist more like Superman exists or an equation exists rather than how my hand exists.
Quoting Ludwig V
Because I think people should not claim X when whether X is far from being settled by specialists. Not exactly the same but close to how you put it:
Quoting Ludwig V
Tones, who knows about the topic much more than me, had a few corrections to make about it, some of which I have implemented already.
Yes. That annoys me as well. Though there has to be a little wriggle room, doesn't there? Philosophers, in particular, would be very constricted if such a rule were strictly enforced. Though I do agree that some philosophers would do well to be much more cautious than they are. For example, it is clearly wrong to treat the latest speculations from speculative cosmology as established fact.
Quoting noAxioms
Yes. And as you say, they were beyond the state of the art at the time, so what he was doing needs to be rather carefully described (unless you are going to propose time travel.) It is very difficult to handle anticipations of later developments in historical texts. Some people have seen anticipations of Einstein in Berkeley. In a sense, they may be there. But I think that's merely a similarity rather than an anticipation. I don't know how to represent this case properly.
Quoting noAxioms
Yes, and I think that @Lionino may have been protesting at such ways of talking. If one is not a platonist, the way to say what you want to say is to conceptualise "real" in a non-platonic way. To outright deny that infinities exist is just attention-seeking. Though perhaps philosophers are not exempt from such a very human temptations.
Quoting noAxioms
I've noticed a variety of extensions of the use of "=" lately, so I'm sorry if I misused it. I'm glad you recognized what I was trying to say.
Yes, for me, that is the most helpful approach. Different kinds of object - different modes of existence. If you haven't come across it before, you might find this reference useful.
[quote=Stanford Encyclopedia - Modal Fictions;https://plato.stanford.edu/entries/fictionalism-modal/]Fictionalism is an approach to theoretical matters in a given area which treats the claims in that area as being in some sense analogous to fictional claims: claims we do not literally accept at face value, but which we nevertheless think serve some useful function. [/quote]
The only downside I can think of is that it might lead to us conceding that God exists just because so many people believe that he/it does. But then, the same would apply to Zeus, Apollo, Thor, Loki, Horus, Ptah etc. So no-one could draw the conclusion that one is a believer in any of them.
Thanks, bookmarked. One of the schools of mathematical nominalism is fictionalism.
That goes down a rabbit hole of info and posts to even more topics. Good reading.
I was looking at Steffan's slideshow, and it goes into how Cantor's axioms are paradoxical because the set of all sets has smaller cardinality that the set of all subsets of that set. But for similar reasons as have been discussed in this thread, I'm not convinced by it since the axioms only seem relevant to finite sets (similar to a sequence having a first and last step only being relevant to a finite set of steps), and none of the sets in the paradox is finite. So it's a bit like saying infinity squared is larger than infinity, which it isn't.
Far be it for humble me to not be as distressed by this as the hardened mathematicians. I take their word that this has more teeth than I see.
Also of interest is the mention of Godel demonstrating that a goal to find a complete and consistent foundation of mathematics cannot be reached. Does this mean that there cannot be one, or just that we cannot know it to be complete and consistent?
Quoting Ludwig VI have issues with what most people label 'realism', so I'm probably further from platonism than are most. Real is a relation to me, and I use the word that way.
OK, there can be more than one use of the symbol. We seem to not be in disagreement.
The modal fictionalism link is appreciated.
Mind you, an in-house mathematician has criticised some of the content in the slideshow (and in the article). Here is our exchange (I don't think he will mind me leaking DMs in this case):
Let me remind you, the "thing" being described here, in the op is a fictitious scenario. It is one hundred percent dependent on the description, just like a counterfactual. We might say that "the factual situation" is that a minute will pass, but the counterfactual described by the op does not allow for a minute to pass. You seem to be unable to provide the required separation between these two, thinking that the factual and the counterfactual may coexist in the same possible world.
Quoting noAxioms
Right, as I said there is nothing in the op to inhibit the passing of time, in fact the passing of time is an essential part, it is a constant. However, the premises of the op restrict the passing of time such that 60 seconds will not pass.
Quoting noAxioms
There is nothing in the op to indicate that the passing of time slows. That is an incorrect interpretation. As you say, it isn't meaningful to talk about the rate of time in this scenario. What happens is that the speed of the person descending the staircase increases. And, as the speed increases, there is no limit to the acceleration indicated. The velocity is allowed to increase without limit. Even if we considered "infinite velocity" is a limited (which is of course contradictory), and assume that limit could be reached, this would still not imply "no time is passing". It would only make the spatial-temporal relationship unintelligible due to that contradiction.
This is actually very similar to the perspective of special relativity theory, which uses the speed of light as the limit, rather than infinite speed. This avoids contradiction but ti still renders the spatial-temporal relationship as unintelligible at the speed of light. From the perspective of the thing moving that fast, it appears like no time is passing, yet time is still passing. It's just a twisted way of making the passage of time relative to the moving thing for the sake of the theory. But there is no relativity theory stated in the op, nor any other frame of reference, so there is nothing to indicate a stopping, or even a slowing of time. The frame of reference which you keep referring to, in which 60 seconds passes, is excluded as incompatible with the described acceleration. The described acceleration is purely fictional though, like a counterfactual.
You yourself proved P2 true, and I don't understand why you aren't even engaging with my argument supporting that claim.
Reality is what's really going on in the world. Not sure why you regard that as problematic.
A bat has a particular view of the world, as does an ant, as does a sea slug. None of them, and that includes us, know what ultimate reality is. Not sure what your objection or concern is with this idea.
Quoting Ludwig V
Physics is inaccurate, but what if it's wildly inaccurate, as inaccurate as an ant's view of the world relative to the real world? We like to think that we're "close" to knowing reality because our physics works so well, but that's arguable.
Quoting Ludwig V
As I understand it, Tegmark believes the world is a mathematical structure, like a group o a topological space.
Quoting Ludwig V
I have the worst habit lately of only responding to my mentions and not reading the rest of these threads.
Quoting Ludwig V
Zeno's arguments are far more sensible than Tegmark's.
Quoting Ludwig V
I ended up spending all my time explaining the ordinals and that detracted from my resolution of the lamp.
Quoting Ludwig V
But the limit isn't defined in the lamp problem. There is no limit to 0, 1, 0, 1, ..., and therefore no terminal state that's more natural than any other.
Quoting Ludwig V
There need not be any relationship or rule that defines the elements of a sequence, but the sequence can have a limit. But in the case of the lamp, there's no natural conclusion because 0, 1, 0, 1, ... has no limit. I'm repeating myself but that's the point of the lamp. There's no natural terminal state.
Quoting Ludwig V
I'd say that the standard mathematical rules for dealing with infinity are perfectly clear, and do apply. It's vague, handwavy imaginings that don't apply.
Quoting Ludwig V
Not following at all. Math is constantly changing, has been for thousands of years, and is changing even as we speak. New math papers are published every day.
Are you referring to some kind of Platonic math, God's math textbook? Is that what you mean by fixed?
I am not understanding your point, I don't see what's complicated about this at all. Can you explain what you mean?
Quoting Ludwig V
I don't feel confused, so I must not be understanding you.
Quoting Ludwig V
Yes, sure, a fixed body of knowledge evolves. But that body of knowledge is added to every day by every math journal and university colloquium.
Quoting Ludwig V
Not only new results, but new ways of thinking are constantly introduced.
Quoting Ludwig V
So are you thinking of God's math book that humans slowly learn about? Well maybe that is fixed. So you're just regarding math as a Platonic body of knowledge that is "out there" somewhere. Is that what you mean? I'm not saying it's not true, I'm just asking if you're taking a Platonic viewpoint.
Quoting Ludwig V
I believe I lost track of what this paragraph referred to, sorry.
The world can not be simultaneously Euclidean and non-Euclidean.
Algebra and calculus can both be true of the world. They don't contradict each other.
Not sure I'm following your analogy.
You can't have facts about the world that are in conflict with each other.
What does the realist say about that? I did a quick lookup of mathematical realism and it has nothing to do with the relationship of math to the world; rather, Google says, "Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is." The Wiki article on the philosophy of math takes a similarly abstract view of mathematical realism. Nothing to do with the physical world.
That's a lot more subtle than saying that realism believes that math is literally true in the world.
I make no claim to expertise on these matters, actually I admit to ignorance. But I don't think you are using mathematical realism in the same sense as Google and Wikipedia.
No I didn't.
Your argument is analogous to this:
If I am immortal then when will I die of old age? I won't. Therefore, I have proved that I am immortal.
Agreeing with what follows if we can recite the natural numbers at successively halved intervals of time doesn't prove that we can recite the natural numbers at successively halved intervals of time.
I read your post. It is really helpful. I don't know enough to respond meaningfully, but I have a feeling I shall find my way back to it from time to time.
It did provoke the heretical speculation that it is an assumption that just one of these accounts applies to the whole of mathematics. Perhaps mathematics is not just one language-game, but a family of them.
I agree with "a bat has.... what ultimate reality is" But then, I wonder what the status of "what's really going on in the world". Is that ultimate reality? From what you say, the answer is not clear. My concern is that both "ultimate reality" and "what's really going on in the world" are not defined in a way that reminds me of the way that the last step in a converging series is not defined - and cannot be defined. Yet, the sun is really shining at the moment and there really is a war in Ukraine - in short, we all (including bats and ants and slugs) live in the same world and interact in it.
Quoting fishfry
But how can you say that an ant's view of the world is inaccurate? I think I can grasp what you are getting at when you say that physics is inaccurate. It reflects the fact that physics is an on-going enterprise. "What if it's wildly inaccurate.." is a style of question that I'm very sceptical of. It reminds me of "what if everything's a simulation?" I classify it as a speculation and not capable of a meaningful answer.
Quoting fishfry
One might interpret that belief as a dramatic way of putting the point that we can find a mathematical structure that applies to the world. If he doesn't mean that, I want to know what he means by "is".
Quoting fishfry
A very sensible policy. It is easy to drive oneself crazy by trying to respond to everything. But sometimes I can't resist intervening in discussions that haven't mentioned me. It doesn't always work, in the sense of developing into something interesting, but some times it does.
Quoting fishfry
That's my fault. Sorry. I did benefit very much.
Quoting fishfry
Yes, I understand that now. I was talking about the limit of the convergent series. The series "0,1,..." has no inherent limit. If it ever is limited, it is by some event "outside" the series. That's badly put. I just mean that I can stop following the instruction for any reason that seems good to me or even none at all. The series as defined is infinite.
Quoting fishfry
I didn't mean to suggest that wasn't the case. Thinking of the series backwards is a vague handwavy imagining. That's all. I intended to contrast that with a series that can be defined forwards or backwards. It's odd, that's all.
Quoting fishfry
Both sentences are true - the first sentence does not imply anything platonic, in my view. I think the difference between us is a question of emphasis rather than an actual disagreement.
Quoting fishfry
Yes, that was a step too far, and it is very speculative, more a musing than a thought. I should not have pursued it. Let's just let it go.
I am not talking about the fabric of space-time.
Quoting fishfry
Right, except for the kinds of realism that make it about the physical world, but that is one type among many.
Maybe you are misunderstanding what "abstract" means in those quotations. It doesn't mean something that we conceive in our minds, but a real object that exists independently of any conscious being, but that is outside space and time.
Quoting fishfry
Of course a single sentence doesn't represent a family of views. But one of the minimal characteristics of mathematical realism is that things such as "2+2=4" are true and they are true even if we are all dead in other words, it is about the world.
Quoting fishfry
I hope not, my sources are academic.
Quoting Lionino
If both of these are true, then we need to be very careful about what we mean by "the world". There is an application that takes "the world" to exist in space and time. Note, however, that the space-time world continues to exist even if we are all dead, even if we never existed at all. If "the world" includes everything that exists, then it can, of course, include things that exist "outside" of space and time - provided that we understand how anything can exist "outside" space, which seems to indicate a location, but does not.
Is not existing at any particular location in space the same as existing outside space? Where does platonism or Darwinism exist? or football or judo? Or the possibility of rain where I live tomorrow? Or the English language? Or the recipe for doughnuts?
Sorry - rhetorical questions. I realize that you are reporting the things that platonists might say.
Of course. Fully agreed. But that's YOUR hypothesis, not mine. Am I wrong about that? What am I missing here? Isn't that your example?
You have repeatedly asked me what happens if we go backwards, saying "1" at 60 seconds, "2" at 30 seconds, and so forth. That also is a purely hypothetical thought experiment. Why on earth are you proposing hypothetical non-physical thought experiments, then saying, "Oh that's impossible!" when I attempt to engage?
Don't know how to answer that. If there is an ultimate reality, whether we have access to it or not, that's what it is. If there isn't, then maybe it's all random. But then that would be the ultimate reality. God's point of view, as it were.
Quoting Ludwig V
No, those are only the things we perceive. Ants don't see the war but they do experience the sun. Creatures at the bottom of the sea don't see much of the sun. Ultimate reality is whatever is really out there, if there even is such a thing.
Quoting Ludwig V
Limited is a better world. Our view is limited too.
Quoting Ludwig V
Ultimate reality is the thing that physicists believe (or used to believe) they are trying to understand and learn about. That is no longer a core objective in physics, since the "shut up and calculate" school of quantum physics won the day.
Quoting Ludwig V
He says that the world literally is a mathematical structure. Not "is described by," but literally is.
https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis
Quoting Ludwig V
I'm probably missing a lot.
Quoting Ludwig V
I love talking about the ordinals so if you enjoyed that I'm happy.
Quoting Ludwig V
It's infinite in the sense of being defined at each of 1, 2, 3, 4, ..., but it is NOT defined at [math]\omega[/math], the "point at infinity." Or for a more concrete representation, it's defined at each of 1/2, 3/4, 7/8, ..., but it is not defined at 1. That's the point. The 0, 1, 0, 1, ... sequence only covers the infinite sequence. But there is no definition of the state at the limit point.
Quoting Ludwig V
That's @Micheal's example. But the sequence 1, 1/2, 1/4, 1/8, ... is a familiar sequence with limit 0. If you plot the points on the real line, it does indeed go backward, from right to left.
Quoting Ludwig V
Not sure what a sequence (not series, a series is an infinite sum) that can be defined forwards or backwards means. By definition, an infinite sequence is [math]a_1, a_2, a_2, \dots[/math]. It only goes forward. Though if the elements are decreasing (as 1, 1/2, 1/4, ...) the points go from right to left.
Quoting Ludwig V
Well, human endeavors are never fixed, they always evolve. In the case of math, it's a philosophical question as to whether there's something "out there" that it's evolving to.
Quoting Ludwig V
It looked interesting, I should take another look.
Perhaps I misunderstood. What then?
Quoting Ludwig V
Not conversant with the technical meaning of realism in this context, probably can't hold up my end of this.
Quoting Ludwig V
Is this a dualist point of view? What is outside of space and time?
I think the question, "Was 5 a prime number, before there was life in the universe?" is meaningless. Are you saying that a realist would say the answer is Yes? In that case, what is the difference between a realist and a Platonist?
I freely admit to my philosophical ignorance, so I am out of my depth in these matters.
Quoting Ludwig V
But no, that is not about the world. The world is what's real, what's physical.
So you (or the realists) are arguing a Platonic position then. I don't happen to think 2 + 2 = 4 is true before there was intelligence in the universe. Or, say, before the big bang. Or before the endless cycle of big bangs, "big bounce" speculative cosmology. I think it takes a human, or at least an intelligence, to give meaning to the proposition.
If we are all dead, if there is no life and no intelligence in the world, who can pass judgment on whether 2 + 2 = 4? How could it be assigned a truth value? Who would do the assigning, the agreeing and the disagreeing?
Again, I admit my thoughts in this area are naive and not the product of any directed study of what the smart folks think about these matters. So I don't put a lot of stock in my own opinions.
Let me ask you a different question. Before chess was invented, did all the games of the grandmasters exist "out there" in Platonic space? Did the collected games of Bobby Fischer exist before he played them? After all, each game could be encoded as a number, and the Platonists believe numbers exist independently of minds. I find that difficult to believe, that all the symbolic works of humanity exited before they were created.
Humans create. That's what we do. Humans are, if you like, the very mechanism by which the universe figures out if 2 + 2 = 4.
Quoting Ludwig V
I have no doubt, and I hope I am sufficiently conveying the humble limits of my knowledge in this area.
But these academic sources, are they Platonists? Dualists? Surely there are those who strenuously disagree and take something approximating my own point of view. That 2 + 2 = 4 has no truth value before there is an intelligence to assign it one; and that the chess games of the grandmasters did not exist before they were played, even if the list of moves is an abstract sequence of symbols that a Platonist must believe existed before time itself.
It's not mine. It's the hypothesis of those who claim that supertasks are possible. They try to use such things as the finite sum of a geometric series to resolve Zeno's paradox. They claim that because time is infinitely divisible it's possible for us to perform a succession of operations that correspond to a geometric series, and so it's possible to complete an infinite succession of operations in finite time.
I have been arguing firstly that it hasn't been proven that time is infinitely divisible and secondly that if we assume such a possibility then contradictions such as Thomson's lamp follow.
I was very clear on this in my reply to you on page 4, 22 days ago:
Quoting Michael
Most of the last few pages has been me trying to re-explain this to you, e.g. 10 days ago:
Quoting Michael
---
Quoting fishfry
It was brought up for two reasons. The first was to address the flaw in your reasoning. That same post 10 days ago was very clear on this:
Quoting Michael
If argument 1 is proof that it is possible to have recited the natural numbers in ascending order then argument 2 is proof that it is possible to have recited the natural numbers in descending order.
It is impossible to have recited the natural numbers in descending order.
Therefore, argument 2 is not proof that it is possible to have recited the natural numbers in descending order.
Therefore, argument 1 is not proof that it is possible to have recited the natural numbers in ascending order.
The second reason I brought it up was a proof that it is impossible to have recited the natural numbers in ascending order.
If it is possible to have recited the natural numbers in ascending order then it is possible to have recorded this and then replay it in reverse. Replaying it in reverse is the same as reciting the natural numbers in descending order. Reciting the natural numbers in descending order is impossible. Therefore, it is impossible to have recited the natural numbers in ascending order.
Or if you don't like the specific example of a recording, then the metaphysical possibility of T-symmetry might suffice.
Either way, the point is that it's special pleading to argue that it's possible to have recited the natural numbers in ascending order but not possible to have recited them in descending order. It's either both or neither, and it can't be both, therefore it's neither.
There's a confusion here. The remark you quoted, which the system attributed to me, is actually @Lionino. I could claim academic sources from what I'm saying, but I read them a long time ago, and if you asked my for attributions, I would have to spend a long time looking them up.
Quoting fishfry
I take your point. So the dots reflect the lack of definition and trying to run it backward finds the dots at the "beginning", so the "beginning" is not defined. But one could define a similar sequence that runs (0, 1/2,1/4.... 1), couldn't one? That would not be the same sequence backwards, of course.
Quoting fishfry
Quoting fishfry
Welcome to my world. Being out of one's depth in it is almost a prerequisite of inhabiting it, so that's not a problem. It would probably unfair to say that people who think they are not out of their depth are always wrong (compare relativity and QM). But it is certainly true that you need to be a bit out of your depth to be doing any serious work. If you have everything sorted out and pinned down, you've lost your grip on the problem. (Wittgenstein again)
Unfortunately "The world is what's real, what's physical" is a metaphysical remark (at least, it is if there are any philosophers around), so you've jumped into the water without, perhaps, intending to. The question is whether numbers, etc. are real things that are not physical; platonist-type theories see numbers as real things that "transcend" the physical world. Don't ask me what "transcend" means - or "thing", "entity", "object". They would probably prefer to tell you what transcendence etc. are. But that's the same question in a different mode. Their mode is metaphysics. Mine is linguistic.
What I was doing, in response to what Lionino was saying, was putting realism and anti-realism together - since they are defined in opposition to each other - and then asking what they disagree about. (There are many varieties of both sides of this coin, so I'm simplifying, and arguably distorting.) In particular, I'm trying to show that "real" is not 'really' in contention, since no-one could deny that numbers are real - what is at stake is different conceptions of reality. And you see how slippery this is because in mathematics, not only are some numbers real and some imaginary, other numbers (like transfinite ones) are neither. Worse still, the imaginary numbers are numbers and exist, so must be real - in the philosophical sense. (At least, you can put me right if I'm wrong here.)
What "real" means depends on the context in which you are using it. Some philosophers want to use "real" in a context-free sense. But that generates huge complications and confusion. Better to stick to contexts. (The same applies to "exists") That's why I try to avoid metaphysics and metaphysicians will classify me as a linguistic philosopher - and that is indeed where I learned philosophy.
Quoting fishfry
All right. Those are good questions. They lead one in a certain direction. I am very sympathetic, so it would be better to let a platonist answer them directly. But I don't think that platonism needs to rule out the possibility that humans might be able to create some things, such as fictional stories - (although Plato was very scornful about such things on moral grounds, though he made liberal use of them himself.) - and games.
But in this field, it is as well to understand your opponent's (colleague, hopefully, in a joint attempt to discover truth) position. So consider. Games like chess are unlike games like football. Once they are defined, all the possible games are defined (so long as you limit the number of moves). So you could argue that the Sicilian defence, for example, was not created, but discovered. That's the germ of platonism.
In the end, I think, one has to see these arguments, not as simple question of truth and falsity, but of how you think about things. The answers, then, are quite likely to be pragmatic or even moral.
Quoting fishfry
Yes, that's fine. There is an approach that sees humans (and perhaps some animals) as the means by which the universe becomes self-conscious. I think that's going a bit too far, but I can see the attraction.
My enemy in this field is dogmatism.
1. If we start reciting the natural numbers then either we stop on some finite number or we never stop
2. It is metaphysically possible to recite the natural numbers at successively halved intervals of time
If (2) is true then we can stop without stopping on some finite number.
Some take this as proof that (1) is false. I take this as proof that (2) is false.
I think that (1) is a tautology whereas no evidence has been offered in support of (2).
I agree. By "we" do you mean us human beings? You and I? If so, we will necessarily stop, if only when we die.
Quoting Michael
Assuming that there are people who believe this, it is reasonable to assume that they can offer what they think is evidence. So it's truth depends on what you mean by "evidence".
By "recite", do you mean some event that occupies a finite amount of time (larger than 0)? In that case, assuming you mean "all the natural numbers", 2 is false, or at least logically impossible.
What is "evidence" in a metaphysical realm?
How do you make this conclusion?
Its the conclusion of those who use the finite sum of a geometric series as proof that a supertask can be completed. If I recite the first number after 30 seconds, the second after 15 seconds, and so on, then I have recited them all and so stopped after 60 seconds, even though there is no largest number for me to stop on.
I think its nonsense as it contradicts the tautology given in (1), and so I reject (2).
Thats for those who assert the truth of (2) to answer. As it stands its just an assertion, and contradicts the tautology given in (1), so I have every reason to reject it.
I have to disagree. What you describe is a rate of acceleration which would produce an infinite speed. The rate at which you recite the numbers becomes infinite before 60 seconds passes. And, despite the fact that infinite speed is in some sense unintelligible, it is clearly not at all the same as being stopped.
No, it isn't the same as being stopped. Being stopped is an everyday occurrence. Infinite speed, is, as you say, unintelligible. If that's what underpins the supertasks, it makes sense of the narratives - apart from the fact that it doesn't answer the question whether the lamp is on or off.
The rules of this (language-game) still make no sense to me.
If one watches the lamp in a dark room, at some point it will appear to be on continuously.
I think that if the lamp is going on and off at an infinite rate, then it's not correct to say that it would be on at any particular time, or off at any particular time, because it is going on and off at a rate faster than our ability to determine a particular time.
Quoting jgill
We're talking about a time duration which is far beyond the distinctions which could be made by the human eye. This would be what is occurring in a tiny fraction of a second. It doesn't make any sense to talk about how the lamp would appear in this time.
Ok. Well I have no strong opinion about supertasks one way or the other, and can defend either position. Perhaps I'm misunderstanding what you're trying to say to me, and vice versa.
Quoting Michael
The swine!
Quoting Michael
Incredible. But why do you conflate me with them, whoever they are?
Quoting Michael
I agree.
Quoting Michael
I see no contradiction in Thompson's lamp, only a failure to define the terminal state.
Quoting Michael
Due to my misspent youth, my short-term memory only goes back 21 days.
Quoting Michael
Is this argument supposedly in response to anything I've ever said? I no longer believe you and I are even having the same conversation.
Quoting Michael
Shame on my for not remembering.
Quoting Michael
Well it's not my antecedent. And now you claim it's not your antecedent. So why are you and I arguing about it?
Quoting Michael
I may be mistaken on facts; but I doubt I've made any flaws in reasoning. On the contrary, one of my flaws in life is excessively precise reasoning.
Quoting Michael
You recited them all. Are we agreed or disagreed on that?
Quoting Michael
You recited them all. Which number didn't you recite? Are we agreed or disagreed on that?
Quoting Michael
Yes but it's an entirely hypothetical thought experiment. And not mine, either. It's Thompson's. Take it up with him.
Quoting Michael
You are equivocating "possible." If by possible you mean relative to the entirely hypothetical, abstract premises, then it's possible. If you mean physically possible, then it's doubtful, and false under current physics.
Quoting Michael
1, 1/2, 1/4, 1/8, ...
That's a perfectly sensible and well-known infinite sequence. If at each step you recite "1", "2", "3", and so forth, you'll get it done. But what of it?
Quoting Michael
Well the natural numbers have no last element, so I have agreed -- REPEATEDLY -- that you can't recite the natural numbers backwards. I have stipulated to this at least half a dozen times.
Quoting Michael
Under the successively halved time interval hypothesis, it's clearly possible to recite the natural numbers in ascending order. Proof: You can recite 1. And, if you can recite n, you can recite n + 1. Therefore, by mathematical induction, you can recite them all. I've asked you many times now to engage with this argument and you won't.
Quoting Michael
I just showed how it is. And rather than engage with the argument I've given several times, you just repeat your false claim.
Quoting Michael
If you play the recording in reverse, your first step, no matter how small, must necessarily jump over all but finitely many numbers. I have also given THIS argument many times now, and you pointedly refuse to engage.
Quoting Michael
No it's not, as I've explained several times now.
Quoting Michael
I have agreed with this several times.
Quoting Michael
I already showed that it's possible, under the hypothesis of successively halved time intervals.
Quoting Michael
Lost me there.
Quoting Michael
It's tedious at my end to dialog with someone who refuses to engage with any of my arguments, but nly keeps repeating their own talking points.
You seem sincere in your own confusions, but this is not productive and has not been for quite some time.
Sorry about that. I typically select the entire post and hit Quote, and it seems to lose a lot of the attribution.
Quoting Ludwig V
The limit is not part of the sequence. so that doesn't run the sequence backward. I am not sure what point you are making about the sequence. The dots merely indicate that the sequence progresses indefinitely.
Quoting Ludwig V
"If you know what you're doing you're not learning anything." Think I read that somewhere.
Quoting Ludwig V
Ok.
Quoting Ludwig V
I'm out of my depth on that. Don't understand what's meant by realism or anti-realism. Simply don't believe that 2 + 2 = 4 has a truth value before some intelligent entity shows up to pass judgment.
Quoting Ludwig V
Ok.
Quoting Ludwig V
Agreed.
Quoting Ludwig V
Ok. Don't think I disagreed with anything you said.
Quite so. I've been thinking that the rules of the game require one to classify that as a purely physical phenomenon. But I prefer versions of this problem that define a sequence (0,1,0,1,...) and align that with the lamp. Even better, I think, we can count the steps in the convergent series and not that odd and even numbers alternate and ask whether the last count when 60 seconds are up is odd or even. Nothing physical intervenes.
Quoting Metaphysician Undercover
That's true, but seems to be a purely physical limitation. It raises the question whether that means it is really on or off, or a some sort of in-between state. Fluorescent lights flicker on and off all the time (at least if they are running on AC, and we just say they are on. And it is true that for practical purposes there is no relevant difference between that light and sunlight or candle-light.
Quoting Michael
I'm not clear whether you are thinking of reciting as a human action that takes time. In which case, there will come a point in your recitation when you physically have to stop, but have not run out of natural numbers. (If we are talking about a series that is convergent in time, it will take longer to utter the word(s) than the time available.)
To put it another way, to say that the series of natural numbers is countable, doesn't mean that there is anyone (apart, perhaps, from God) who could actually, physically, count them. In my book, it just means that they are defined in such a way that they could be counted and, indeed can be counted, so long as one chooses to count only a part of the series. It doesn't follow from the definition of the natural numbers that anyone could count all of them.
Yes, I find that as well. I work round it by selecting only the quoted text, not including the link that gives the attribution. Then, you can hit "quote" and the system does pick up the attribution. Then, if you separately select the response, it is copied and attributed in the normal way.
Quoting fishfry
Neither am I, on reflection. I was trying to articulate the point that one can count forward, but not backward, so I don't think anything is at stake.
Quoting fishfry
Yes, I like that. I'm a bit of a contrarian, so I'm tempted to reply that I don't need my surgeon to learn anything while he's cutting me open. Indeed, I would be rather concerned if I thought he was. It applies better to artistic, experimental, open-ended activities - like philosophy and maybe mathematics, at least sometimes.
Quoting fishfry
If you don't understand what realism vs anti-realism means, you have understood correctly - as I see it. Some people would argue that the proposition that "2+2 = 4" does indeed only have a truth-value only when someone passes judgement on it but that 2+2 = 4 independently of anyone doing that i.e. is objectively true. There's a temptation to think that mathematical truth is eternal, i.e. always has been true, always will be true, whatever happens. But that's a mistake. It makes no sense to assign a place in the time series to 2+2 = 4; there is no meaningful way of doing that. (Grammarians recognize a tense that is called the timeless present which is exemplified in propositions like this.)
Quoting fishfry
I'm glad it made sense.
P1. If we can recite the natural numbers at successively halved intervals of time then we can recite every natural number in finite time
P2. We cannot recite every natural number in finite time
C1. Therefore, we cannot recite the natural numbers at successively halved intervals of time
I justify P2 with this tautology:
P3. If we start reciting the natural numbers then either we stop on some finite number or we never stop
See also here for my defence of Thomson's lamp and here where I explain that this reasoning applies to all supertasks, further justifying C1.
Though I would disagree with 'or we never stop'.
Since it should be P4, and it introduces the proposition that we could complete the supertask. In this case, there must be infinite time. In infinite time, it's no longer a supertask, but a measure of a greater and lesser infinity. If we imply an infinity and then imply a greater infinity, there is the middle of the two that we can call X, and for X, we can assume all natural numbers were recited, just like we assume natural numbers go on forever.
I followed your link and found this quotation from Benacerraf's Tasks, Super-Tasks, and the Modern Eleatics. I've put the passages of interest in bold and italicized the passage quoted from Thompson for clarity.
Pause here. I think Thompson means that contradicts That seems to be true.
But the passage continues (Benacerraf's words): -
That also seems to be true. The three sentences in bold in the first passage are not individually self-contradictory, but the conjunction of the three (the concept of a supertask) could be described as self-contradictory. Nor are Benacerraf's A or B self-contradictory. They could be both true, if a third state that is neither on nor off were possible. Perhaps Benacerraf was assuming that there isn't.
But @fishfry says that the final state is not defined. That would indeed be a third state which is neither on nor off. The idea that this is the case, is supported by the fact that both Thompson and Benacerraf feel the need to consider both alternatives. The point is simple enough - the definition of the infinite set is such that there can be no last step, in virtue of the definition, every step has a successor. So the last step is not defined. You may be thinking that there must be a last step in a convergent series in the series that we've been considering here, it is 1, or 0. But those are limits, not last steps. The series itself by definition cannot reach that limit, so 1 (or 0) cannot be a step in the series. One might say that one cannot complete such a series. I'm not sure of my ground here, but I think you will find that everything depends on what is meant by "complete" and it won't mean completing a recitation of all the steps in the series.
I think I'll leave it there.
PS Since I started writing this, the link to the post that I copied this quotation from seems to have become non-functional. Very odd.
You should read beyond the quote to where I respond to explain the error in Benacerrafs reasoning.
Jgill talked about how the lamp would "appear", and this implies a sense observation, and empirical judgement. The point I made is that the description describes something far beyond our capacity to sense, so it is incoherent to talk about how this described thing would "appear".
Something flashing on and off at a constant rate is not comparable, because the description is of a rapidly increasing rate. And the rate increases so rapidly that the prescribed rate becomes incoherent even to the mind, as well as the senses. This is just an example of how easy it is to say something, or even describe a fictional scenario, which appears to make sense, but is actually incoherent.
OK. You and @fishfry both believe that the supertask is impossible. But you believe that is because it is contradictory and fishfry believes that it is because the last step is not defined. Am I right about that?
There is no doubt that it is easy to do that. But it seems that people disagree about whether the scenario makes sense or is incoherent and even if they do agree, they still disagree about why.
Quoting Metaphysician Undercover
I agree that this isn't really about anything empirical, but it sort of seems to be.
In fact, one could simulate the on/off lamp so that at a certain rate you would see what appears to be a constant light. Flashing 0 and 1 cards would seem to be a zero with a one through it. These are models of the supertasks in a rough sense.
Admittedly not the real deals which are metaphysical fantasies.
It's easy to see why it's incoherent. Start out with the concept of infinite. We can easily see why it is beneficial to allow for numbers to be infinite. This allows that there is no limit to our capacity to count any quantity, or to measure any size of thing, because we can always have a large number. In the case of division though, we may assume that infinite divisibility would allow us to divide anything anyway, but this is really incoherent. That is because division implies, or requires logically, that there is something, an object of some sort, to be divided, and its divisibility will always be dependent on the sort of thing that it is. An object, or thing is a unity of some type, and as such there is always limits to its divisibility, whatever unifies also determines divisibility.
To propose a thing which is infinitely divisible is an incoherent proposition, because as a thing, it is already necessarily limited in its divisibility. This is the issue of the proposition of a finite object being infinitely divisible. That is incoherent because the finiteness of the object limits its divisibility. In whatever way it is finite, its divisibility is limited accordingly.
Quoting Ludwig V
This is the trick of the whole thing. It really is about empirical things. These empirical things are space and time, each of these is known through experience. Then we take these empirical things and pretend that they are absolutely abstract, purely ideal, and stipulate ideal principles like infinite divisibility. Then, someone creates a scenario, like the lamp or the op, which utilizes this purely ideal feature of infinite divisibility. Now we do not properly separate the purely ideal from the empirical, in our minds, so that "empirical time" interferes, and we say that 60 seconds must pass, it has to because experience tells us that it will. But that is allowing "time" to be an empirical thing.
Quoting jgill
The problem though, is that in the prescribed scenario there is no such thing as "a certain rate". The rate is not constant, but rapidly increasing. The only constant is the rate of increase. That rate of increase is what I say is incomprehensible and incoherent.
It's a bit more complicated than that. Bulbs like fluorescent ones flicker, but the light really is constant. It's like what is called "motion illusion" or the ? phenomenon. Film and television both rely on it. In a sense, the motion is an illusion, but in another sense, it isn't. The illusion of constant light, paradoxically, is real.
Quoting Metaphysician Undercover
Yes, but it is not difficult to abandon the (pseudo-physical) lamp for a purely abstract version, which does not have the same problems.
One is to align the infinite sequence "0,1,0,1,0,1...." with the steps in the regress and to ask whether 0 or 1 can be assigned to the last step. An even simpler version is to align the natural numbers with the steps in the regress and ask whether the final step is an odd or an even number.
Those are clearly abstract sequences, and don't have the complications of trying to align the steps of a physical sequence to the steps of the regress. I agree that trying to do that is very problematic and almost certainly incoherent.
Quoting Metaphysician Undercover
I don't have a problem with ideal principles. They are very useful. We need infinite divisibility for the same sort of reason that we need infinite numbers. The infinite numbers guarantee that we can count anything. Infinite divisibility guarantees that we can measure anything (that is measurable at all). Limitations on either are physical.
Quoting Metaphysician Undercover
I think you are being misled by the temptation to take the divisibility of "medium-sized dry goods" as the paradigm of divisibility. But even that depends on the level of description you are applying, or, if you prefer, the level of analysis you are using.
The weight of a medium sized dry good is infinitely divisible. We couldn't measure its weight or calculate its centre of gravity it if it wasn't.
The colour of something isn't divisible at all. (That's why colour is disregarded in physics and treated as a mental phenomenon, whatever that is.)
You may say that these are not "objects", but that reinforces my impression that you are unduly focused on just one phenomenon in the physical world.
Quoting Metaphysician Undercover
You may be right. I'm afraid that I'm like Augustine. I don't know what time is, though I do know, of course, what time it is right now and what time I woke up.
The great trap with infinity is the temptation to try to get round the endlessness of the sequence by hacking through each step in it.
The point I've been arguing since the beginning of the thread, is that if we abandon the empirical, and adhere strictly to the prescribed, purely abstract version, then nothing indicates that 60 seconds will pass. You see in both the op, and the lamp example, 'there will be a condition at 60 seconds' is an unwarranted conclusion simply added on, and not derived from the initial premises. The initial premises, being the described activity do not allow that 60 seconds will pass. This conclusion, 'there will be a condition at 60 seconds' is pulled from empirical evidence, and is completely inconsistent with the prescribed purely abstract version.
Quoting Ludwig V
No we don't need infinite divisibility, for the same sort of reason that we need infinite numbers, for the reasons I described. Any thing which is to be divided has its divisibility determined by the sort of thing that it is. And each type of thing is divisible in different ways. And, the way that the thing is divisible must be determined prior to division, or else we attempt to do the impossible. There is nothing that is divisible infinitely, therefore this ideal needs to be excluded as necessarily an attempt to do the impossible.
This is not the same as infinite numbers. The countability of a multitude cannot be determined beforehand, as the divisibility of a thing must be determined beforehand. So, we need infinite numbers because we do not know how large the multitude will be until we count, but we do not need infinite divisibility because it is impossible to divide something prior to knowing its divisibility, and the possibility of infinite divisibility is already excluded as actually impossible.
That is a fundamental feature of the difference between a unity and a multitude. If a unity is composed of a number of parts, then the number of parts is necessarily finite. To have infinite parts would violate the finitude of the boundaries implied by the concept "unity". But the concept of "multitude" has no such implied boundaries, principles of limitations. Therefore a "unity" is always limited in its divisibility, limited by the principles which make it a unity rather than just a multitude (I cannot say "multitude of parts" because "part" implies a whole), and a multitude is not necessarily a whole, so its countability is not necessarily limited.
Quoting Ludwig V
No, I am talking about the divisibility of anything, in an absolute sense. There is no such thing as a unity which has absolutely zero limitations on its divisibility. That is a fundamental feature of what it mean to be a unity.
Quoting Ludwig V
Colour is very much divisible. It is a collection of distinct wavelengths, and I believe it is divided by the harmonic principles of the Fourier transform.
.... apart from a geometrical straight or curved line. I grant you that that is a concept of an abstract, ideal object. I grant you also that such division does not necessarily affect the unity of the object in any way.
Quoting Metaphysician Undercover
That is the best representation of colour that physics can manage. But most people do not know about wave-lengths or Fourier transforms. So when I choose a red coat to wear to-day, how do I manage that? The colour that I am aware of is divisible in the sense that there are many colours and shades of colours. These correspond only roughly to the wavelengths of light.
Quoting Metaphysician Undercover
So how can we be sure that anything can be measured in terms of metres, if metres cannot be divided so that they exactly measure the length we are measuring?
I need to think about how to tackle your metaphysics.
An afterthought. Do I understand rightly that your analysis of wholes and parts applies to physical objects, and not to mathematical ones?
Thanks, guess I can no longer just copy/quote the whole response any more. Have to do it one paragraph at a time.
Quoting Ludwig V
Nothing is at stake for me or you, but our friend @Michael seems to regard it as important. It's obvilus that you can't count the natural numbers backwards.
Quoting Ludwig V
Agreed! That's why job applicants should not say they're looking for a challenging position. Employers want someone who can do the job!
Quoting Ludwig V
LOL. I feel better then.
Quoting Ludwig V
We must be in agreement on that then.
Quoting Ludwig V
Ok!!
I believe you're equivocating in the sense that you don't distinguish between the successively halved time intervals as:
a) An abstract, hypothetical, mathematical exercise; and
b) A physical situation that is incompatible with known physics and that may or may not be incompatible with the true nature of the world.
Can you clarify which sense you mean?
If you mean (a), then of course we CAN recite every natural number, so P2 is wrong; if (b), we can't currently and likely will never be able to.
When a mathematician says that 1/2 + 1/4 + 1/8 + ... = 1, they don't mean that you can perform this calculation with pencil and paper before lunchtime. They mean that the two expressions on either side of the equal sign denote the same real number.
Was this from you to me? That post of @Michael disappeared for me as well.
Metaphysical impossibility. Supertasks cannot be performed in any possible world. P3 is a tautology, P2 follows from P3, and so C1 is necessarily true.
Quoting fishfry
Here are three distinct propositions:
a) 1/2 + 1/4 + 1/8 + ... = 1
b) there is a bijection between this geometric series and the natural numbers
c) it is metaphysically possible to recite the natural numbers at successively halved intervals of time
(a) and (b) are true and (c) is false. Your argument rests on the assumption that (c) follows from (a) and (b), but it doesn't. (c) is proven false by P3, as well as arguments like Thomson's lamp.
You can continually assert that (a) and (b) are true, and I will continually agree, but until you can present actual evidence or reasoning to support (c), I will always reject it as per the above.
I still can't find it. I copied the quoted passage into my message, but not the commentary. Which is a pity.
Quoting Michael
There's another strictly philosophical issue. I know that metaphysics overlaps with logic. I'm still trying to work out whether it is identical with logic.
Quoting Michael suggests to me that it is a question of logic.
Subject to that, I do agree that c is false. I think that those who disagree with you think that they can stipulate by definition some sense in which it is true.
I have not said that. I have said that I have no strong opinion about supertasks and am entirely comfortable arguing either side.
Assuming lines to be infinitely divisible is problematic, just like assuming time or space to be infinitely divisible is problematic. As much as we like to claim that such a divisibility is "logically possible", it's really not. Such an assumption produces an unintelligibility similar to infinite regress, which is an incoherency, therefore illogical.
Quoting Ludwig V
I don't follow you. Why would you need to know something about divisibility to choose a colour?
Quoting Ludwig V
Again, I do not follow. Metres can be divided. We have centimetres and millimetres. But when we measure, at some point an approximation is made, a rounding off.
Quoting Ludwig V
I don't think that this is relevant. I believe the analysis applies to all objects. But there is a problem with supposed "mathematical objects", and this is that we assume them to be infinitely divisible. And this assumption creates incoherency. This incoherency renders the supposed objects as not true objects.
Ok. Possible worlds. I actually took a class where we talked about that, but I have a hard time understanding the concept. There are people who think possible worlds are real. I'm not one of them. And the whole metaphor is lost on me.
But there's a problem. You are assuming that which you wish to prove. You say, "Supertasks cannot be performed in any possible world." But that's the thing you're trying to convince me of. Nevermind that I don't necessarily believe in supertasks myself, but regardless, you are trying to convince me that there are no supertasks. So you can't just state that there are no supertasks. Right? Just a basic point of logic.
And even then I reject the claim on its own merits. I could argue (not that I do, but that I could -- hope that's clear) that if time is modeled by the real numbers (agreed, that is a dubious assumption) then I perform a supertask every time I get up to go to the kitchen for a snack. I named my refrigerator Zeno.
If I can make that argument, then there is some possible world in which it's true. Namely, in any possible world in which time is accurately modeled by the standard real numbers, supertasks are commonplace, every day occurrences.
I argue -- in fact I believe I've made this argument several times already -- that because supertasks are abstractly conceivable, they are NOT metaphysically impossible.
A
In short, (1) You assumed what you're trying to prove; and (2) Your assumption is wrong. Supertasks are conceivable and as far as anyone knows, even physically possible. It's for the physics of the future to know.
Also, it's conceivable that future physics will incorporate physically instantiable infinities, and that supertasks will be possible. I just don't see how you can ignore that possibility.
Quoting Michael
(a) and (b) are mathematical truths we all agree to, I hope. But "metaphysically impossible," I don't even know what that means.
If I make the assumption of successive halving, I can recite all the numbers in finite time. I agree this is a purely fictitious, hypothetical game.
We agree that it is physically impossible, relative to currently known physics.
I simply do not understand why you jump to saying that means it's metaphysically impossible.
By analogy with previous scientific revolutions, we must allow for the possibility that actualized infinities could become part of physics; and that physical supertasks might be witnessed in the Superconducting Supertask Collider of the future. This is a grant proposal waiting to happen.
I just don't see how you can call this metaphysically impossible. So was heavier-than-air controlled flight, once.
Quoting Michael
I don't even know what you are talking about. No argument I make assumes a and b and concludes c. (a) and (b) are trivialities. (a) is freshman calculus and (b) is the very definition of an infinite sequence. Together they don't imply anything. You're imagining some argument I didn't make and wouldn't make.
Quoting Michael
I wish we were having the same conversation. "My argument" assumes (a) and (b) and concludes (c)? I can't even relate that to anything I can think of. What do (a) and (b) have to do with what's metaphysically possible? And what does metaphysically possible even mean? I should have asked that earlier. I don't know what the phrase means to you. And it's not one I use in my own mind, myself. I think just about anything is metaphysically possible, if I were pressed to give an opinion on the subject. If not pressed, the thought of metaphysical possibility never enters my mind. So probably I have no idea what you mean.
Quoting Michael
I have surely presented my reasoning, which I will repeat here.
P1) It is metaphysically possible that time is accurately modeled by the standard real numbers;
P2) 1/2 + 1/4 + 1/8 + ... = 1 in the real numbers;
C1) A Zeno supertask is possible, by moving in any direction for one second, and thereby traversing a countably infinite sequence of finite intervals of time [0, 1/2], [1/2, 3/4], [3/4, 7/8], ...
C2) Therefore supertasks are metaphysically conceivable.
Secondly:
P3) It is conceivable that future physics will allow for supertask; and therefore
C2) Supertasks are metaphysically conceivable.
You could probably help me out by clearly defining metaphysically impossible.
And please reframe your argument about my claiming (a) and (b) imply (c), when in fact I make no such argument.
Lost in the ether, forever.
I expect we'll survive.
Quoting fishfry
Thanks for clarifying that. I find it quite hard to remember what everyone's position actually is. It gets lost in all the detail.
Quoting Ludwig V
I would be very grateful if you could help me clarify this. When you say:-
Quoting fishfry
That's not quite as simple as it looks. The left-hand side will never equal the right-hand side as long as I try to make them equal by adding further steps in accordance with the same rule (...1/16, 1/32...). That's what it means to say that 1 is the limit, not the last step. But if I add 1/8 again, the two sides will be equal. Does that count as completing the sequence?
Quoting fishfry
Whether possible worlds count as real depends entirely on what you mean by "real". For some people, "real" comes down to true. If it is possible that it will rain tomorrow then possible worlds are real because it is true that it will rain tomorrow. For others, a possibility is not actual, so cannot be real.
Quoting fishfry
Quite so. But I think there is a confusion going on here. If you'll allow a temporary and artificial distinction... Roughly, it's the difference between an analysis, which doesn't change or affect its object, and a division or separation which does. That's the difference between measuring a plank of wood as 10 cm long and cutting it into 1cm lengths. The first is an analysis, the second is a division.
There are infinite ways in which I can mark out the plank, and they are all true at the same time and the physical object that is the plank is unaffected by any of them. True, the marks will be physical objects, so there will be limits to what I can do. But the system allows me infinite possibilities, including a convergent series. None of these makes the slightest difference to the plank. So when you visit Zeno for a beer, the fact that there are infinitely many analyses of your journey does not make the slightest difference. It's all in your head.
(Here's a thought. When you drink your beer, you have to drink 1/2 of it and then 1/4 of it and then... Your beer will never be finished. :smile: But then, a similar argument would show that you can't even start drinking it. :sad: )
Quoting fishfry
It simply isn't clear. "Metaphysics" is a word looking for a meaning. There is some connection with logic, but what differentiates the two is a mystery.
Because it leads to contradictions as shown by Thomson's lamp, defended here and expanded on here.
Also because it's the conclusion of this sound argument:
P1. If we can recite the natural numbers at successively halved intervals of time then we can recite every natural number in finite time
P2. It is metaphysically impossible to recite every natural number in finite time
C1. Therefore, it is metaphysically impossible to recite the natural numbers at successively halved intervals of time
I justify P2 with this tautology:
P3. If we start reciting the natural numbers then either we stop on some finite number or we never stop
Quoting fishfry
Metaphysical impossibilities are things which are necessarily false; e.g. see Kripke's Naming and Necessity in which he argues that "water is H[sub]2[/sub]O" is necessarily true even though not a priori (i.e. logically necessary).
But I would even go so far as to say that supertasks are logically impossible (as shown by the above argument and Thomson's lamp). I simply went for the phrase "metaphysical impossibility" because it's the weaker claim and so easier to defend.
Quite so. And we know that it is an approximation because we know what more and less accurate or precise measurement is. The exact measure, in the physical world, is the limit that empirical measurements can approach and never reach. That's mathematics and logic.
Quoting Metaphysician Undercover
I'll take that. I wouldn't put it the same way, but it's near enough. I think, by the way, that you would have a tough job to convince mathematicians that there is an incoherency in the concept of the infinite. But that's not my problem.
Irrational numbers are built in to the system, so are recurring numbers, and so is infinity. You can't have one without the other. Sad, but true.
Where do i get one of these metaphysical beers?
Quoting Ludwig V
I clearly explained though, it isn't "infinite" which is incoherent, it is "infinite divisibility" which is. "Infinite divisibility" is a specific application of the term "infinite" which is incoherent. It is incoherent because the concept of "infinite" is incompatible with, inconsistent with, or contradicts, what is implied by the concept "divisible". Therefore the two together as "infinite divisibility" is self-contradicting.
Mathematicians have made "infinite" into a new term, which really has very little resemblance to its metaphysical roots. This is very evident in set theory. There is no consistency between "infinite" in mathematics and "infinite" in metaphysics. Because of this, mathematicians are incapable of understanding metaphysical "infinite". There is no word for it, as that word is used for something else in mathematics, so the concept escapes their grasp. This leaves mathematics, and mathematicians in general, as fundamentally incapable of dealing with the metaphysical problems involved with the concept "infinite".
At your local metaphysical beer shop, of course. I'm sure Google knows its address and will give you directions. (Shops never stock both metaphysical and mathematical beers at the same time. They fight, you know - very messy!)
Quoting Metaphysician Undercover
Of course you did. I'm sorry. But in any case you've just accepted that mathematical objects aren't true objects. So what's the problem?
Quoting Metaphysician Undercover
I would be happy to accept that there are two concepts of infinity here. I think that may be because their concept has its roots in mathematics, whereas the metaphysical concept has roots elsewhere..
Quoting Metaphysician Undercover
So we just have a case of Domains of Magisterial Authority, and no need to fight about it.
Our only remaining issue is whether the problem of Achilles and the tortoise and Thompson's lamp is a mathematical problem or a metaphysical problem. Or maybe it's just a question of understanding two solutions to the same problem. They clearly won't be incompatible.
I think it would be better to stick with the strong claim. At least it is more comprehensible.
Quoting Michael
Yes Kripke does claim that. But he waters down the meaning of "necessarily". For him, it no longer means "in all possible worlds", but "in all possible worlds in which certain conditions hold". But contingent means, or used to mean, "true or false depending on certain conditions". So, on this account "necessarily" means what "contingent" used to mean. Talk about having your cake and eating it!
The problem is the incoherency. Maybe, if there was no incoherency inherent within mathematical objects, they could be true objects.
Quoting Ludwig V
There's still the issue of incoherency, and the work of the metaphysician is to seek and destroy that sort of evil. Therefore, the fight is on.
Quoting Ludwig V
It's both, and that's the reason for the fight.
Quoting Ludwig V
When one uses incoherent principles, and the other does not allow incoherency, then they will not be compatible.
Yes, I am experiencing that myself. No idea what's being argued in some of these threads or what claims anyone is defending.
I try to take a useful role around here clarifying some of the vague and imprecise mathematical ideas people have. Misunderstandings around mathematical convergence and limiting processes abound in supertask and Zeno discussions. My main goal is to try to rigorize and clarify the mathematical discussions, without necessarily taking strong positions on the paradoxes themselves. I don't care if there are supertasks or not, but I am driven to straighten out the bad thinking around limits (or die trying, is more like it).
Quoting Ludwig V
Don't remember what that was in reference to, but based on the next para perhaps we're talking about mathematical infinite sums?
Quoting Ludwig V
The two sides are equal. Unlike a For loop in programming, the left side is not a process or a sequence of steps. The left side happens "all at once" in exactly the same way that 1 + 1 = 2. It's true right now and it's true for all time. 1 + 1 and 2 are different text string expressions for the same abstract object, the number we call 2.
Likewise 1/2 + 1/4 + 1/8 + ... and 1 are two text string expressions for the same abstract object, namely the number we call 1.
In programming with think of loops as taking place in time. And they do. A loop in a programming language is a notation for a physical process that inputs energy and outputs heat and takes a certain amount of time. And of course the computer process never finishes, it just hits the computational limits of the computer and doesn't get any larger after a certain point.
But in math, 1/2 + 1/4 + ... is added together all at once. And the sum is exactly 1, right now, right this moment. And the reason the sum is 1 is because we cleverly define it to be that way. We define the sum of that series to be the limit of the sequence of partial sums: 1/2, 3/4, 7/8, ... And that sequence has the limit 1, because we have carefully set up the definition of a limit in such a way that it's true.
Am I addressing or at least understanding your concern? It's a common one when it comes to convergence.
In math, the notation 1/2 + 1/4 + 1/8 + ... does NOT denote a process or a sequence of discrete steps.
Rather, it's the sum of infinitely many numbers, taken all at once. And it evaluates exactly to the same abstract number that the notation "1" evaluates to. It is no difference than 1 + 1. It's 2 right now, not later one when the process completes. It's not a process. It's a sum. A binary operation. Two numbers go in, one comes out. Or infinitely many numbers go in, and their sum comes out. Immediately. No resources such as time or energy are consumed. The sum just is.
Quoting Ludwig V
Some people regard all possible worlds as equally true. That viewpoint doesn't resonate with me.
Quoting Ludwig V
You lost me here. I believe I was arguing to @Michael that it's at least conceivable that we execute a Zeno walk on the way to the kitchen for a snack; and that therefore, the idea is at least metaphysically possible. That's all I'm saying.
Oh maybe I understand ... you're saying that just because the path can be infinitely subdivided, does not mean that I'm actually executing that sequence. I think I disagree. I have to traverse each of the segments to get to the kitchen.
But all I'm saying is that it's at least conceivable; in which case it's not metaphysically impossible. I don't have to argue strongly that it's true; only that it's at least barely conceivable.
Quoting Ludwig V
Right. Aleph-null bottles of beer on the wall, aleph-null bottles of beer. You take one down, pass it around, aleph-null bottles of beer on the wall ... :-)
Quoting Ludwig V
I think this was @Michael I was asking. Did my quoting get messed up? Michael keeps saying supertasks are metaphysically impossible, and I want to make sure I understand what he means by that.
Thompson's lamp does not lead to a contradiction showing that supertasks are impossible. That's your interpretation, which you are failing to explain or defend to my satisfaction.
Quoting Michael
P2 is false. As shown by P1. But look, Michael. We have not said anything new to each other in about a week. May we not just agree to disagree? I already pointed out that you claim P2 is true essentially by claiming it's true. Circular.
Quoting Michael
How in this great vast wonderful world of ours, does P3 justify P2? They're not even related. You are just confused about what it means to iterate an infinite sequence in math. It doesn't stop, because there's no last element. But under the successive halving hypothesis, all numbers are recited. One day you are going to wake up and this is going to be as clear to you as it is to me. Till then, I can't really repeat it any more times than I already have.
Again:
[b]
(1) The sequence 1, 2, 3, 4, ... never stops. It has no last element. You can always find the next one.
(2) Under the successive halving hypothesis, all numbers are counted. Because as can be plainly seen, there is no number that isn't.
[/b]
When you understand that both these are true, you will be enlightened. I implore the math gods to bring you this insight in a dream. Else my fingers are going to fall off repeatedly explaining it to you.
Quoting Michael
Ok thanks for that. So something is metaphysically impossible if it's necessarily false.
But I do believe that under that definition, I have shown that a Zeno-like supertask is NOT metaphysically impossible, because it is NOT necessarily false. A Zeno-like supertask is indeed POSSIBLE under the assumption that time is like the real numbers, and that an interval of 1 second is equal to an interval of 1/2 second plus 1/4 second, dot dot dot, exactly as it works for the mathematical real numbers.
I would be the first to admit that such a think is highly unlikely. But it is not inconceivable, and therefore is is not necessarily false, and therefore it is not metaphysically impossible.
That is my rejoinder to your claim that supertasks are metaphysically impossible. I can conceive of a circumstance in which they're possible.
But I would even go so far as to say that supertasks are logically impossible (as shown by the above argument and Thomson's lamp). I simply went for the phrase "metaphysical impossibility" because it's the weaker claim and so easier to defend.[/quote]
Quoting Michael
Thompson's lamp shows absolutely no such thing. And of all the paradoxes you could think of, Thompson's lamp is the least relevant thought experiment possible. It just shows that there's no natural way to define the limit of a sequence of alternating 0's and 1's. A trivial observation. Nothing to do with the metaphysical impossibility of supertasks.
Quoting Michael
I'm going to skip engaging with the rest of this. I have articulated my objections to your claims as clearly as I humanly can; and many times over already. I haven't said anything new to you in my last half dozen posts. You just prefer not to engage with my arguments. I wish you would.
The argument form I am using is called modus tollens and is valid:
If P then Q. Not Q. Therefore, not P.
P = we can recite the natural numbers at successively halved intervals of time
Q = we can recite every natural number in finite time
If P then Q does not prove P and does not prove Q. So you are wrong to claim that P1 shows that P2 is false.
Quoting fishfry
If we stop on some finite number then we don't recite every natural number. If we never stop then at no time have we recited every natural number. Therefore P3 entails P2.
Quoting fishfry
The "successive halving hypothesis" leads to contradictions namely Thomson's lamp and reciting every natural number in finite time and so it is necessarily false.
Quoting fishfry
It entails contradictions. Therefore either it isn't conceivable or contradictions are conceivable. If the latter then being conceivable does not entail metaphysical possibility. If the former then you are simply mistaken in claiming it conceivable; you are failing to fully understand what it means to perform a supertask.
Quoting fishfry
I explained it here. I think it very clearly shows that having pushed a button an infinite number of times leads to a contradiction. And here I explain that this reasoning extends to all supertasks.
A formalist is free to use the name "1/2 + 1/4 + ..." to denote 1, but the formalist cannot interpret "1/2 + 1/4 + ..." as an expression implicitly representing part of an infinite summation, because the formalist considers expressions to have no meanings other than being finite states of a syntactical parser when proving a theory in a finite number of steps.
Frege fell into a similar trap as the supertaskers in the Grundgesetze when he proposed his law V. He wanted there to be a one-to-one correspondence between every function and it's representation as a table of values, even in the case of functions with infinite domans. So he proposed Basic Law Vb with disastrous consequences:
{x??x} = {x??x} ? ?x(?x ? ?x).
To a finitist or potentialist, Law Vb can be interpreted as introducing fallacies of induction into Set Theory, since they will likely interpret the sets-as-extensions on the left hand side as denoting a finite amount of observable information, and they will likely interpret the function on the right-hand side as denoting an unbounded amount of implicit information, meaning that they cannot regard Law Vb to be a reliable rule of induction. Furthermore, according to their reasoning Law Vb cannot be regarded as constituting a definition of the right hand side, unless one gives up the idea of functions having infinite domains).
This is the kind of mistake that Benacerraf makes in his response to Thomson, as explained here.
The lamp is not defined as being on or off at particular times; it is turned on or off at particular times by pushing a button.
It is an important distinction that some are failing to acknowledge.
I'm entirely in favour of the project, but, to be honest, I don't think it is worth dying for.
Quoting fishfry
I think that's the first time I've encountered anyone on these sites who understands the difference between "discrete" and "discreet". Not patronizing, just saying.
Quoting fishfry
Now you have me a bit puzzled. In my book, that means that the equation is about the complete series, which seems at odds with the idea that it can't be completed. What does "complete" mean? Or does it mean the sense in which it is "always already" complete? (see below)
Quoting fishfry
It might be easier to understand if you thought of them as regarding all possible worlds as equally possible. I could understand that. I hope they don't mean that all possible worlds are equally actual....
But some people tend to think only of one kind of possibility - logical possibility. But there many other sorts - physically possible, legally possible, practically possible, etc. etc. I say that possible means different things in different contexts, but it may be that I should be saying there is cloud of possible worlds for each kind. Or maybe physically possible worlds are a subset of logically possible worlds. It's all very confusing. But I shouldn't get too snooty. There is, apparently, a need to this concept in modal logic, but I don't understand what it is.
Quoting fishfry
Quoting fishfry
Well, in that case, you are also traversing the infinitely many possible points along the way, as well as the convergent series based on "
Quoting fishfry
Yes. Thanks for clarifying that for me. That's what I was trying to express when I started babbling on about "always already" in that post that you couldn't get your head around. The comparison with Loop program captures what I've been wrestling with trying to clarify. All that business about getting (or not getting) to the end... It's important though that it's a physical process which takes time. You can switch it off at the end of 60 seconds, and see how far it got, but it won't have completed anything, will it?
Quoting fishfry
There's no clear criterion for what is conceivable and what is not, in spite of generations of logicians. It seems pretty clear that some people have a much more generous concept of that than I do. There are famous philosophical issues around that many people seem able to conceive of, but I can't. I don't know what's wrong with me.
Quoting fishfry
Make sure you get one of the ones that you can't finish drinking. You would not be popular if you passed round one of the ones that you can't start drinking.
Quoting fishfry
You won't have bothered with this exchange - his comment, my reply:-
Quoting Michael
Quoting Ludwig V
My impression of Benacerraf is that he is defining Thomson's Lamp as a boolean valued function
[math]l : \overline {\mathbb {N}} \rightarrow \mathbb {B}[/math]
on the domain of the extended natural numbers [math]\overline {\mathbb {N}}[/math] which introduces an additional point [math]\omega[/math] at "infinity", and then arguing that the value at [math]\omega[/math] can be chosen arbitrarily and independently of the function's limiting value, if any. But if this the case, then he isn't engaging with Thomson's argument and has merely shifted the goal posts to declare victory in an incomparable axiomatisation.
But the point about Frege's Law Vb also applies to the extended natural numbers; Thompson's lamp when defined as the function [math] l [/math] has a domain consisting of two definite and maximally separated points 0 and [math]\omega[/math] and a number of points between 0 and [math]\omega[/math] that is intensionally described as being countably infinite. However, if Frege's Law Vb is rejected for reasons mentioned previously, then although [math] l [/math] still has the aforementioned intensional properties, it does not possess an extensionally well-defined number of points, in which case it cannot be considered to represent the metaphysical notion of a supertask.
Essentially, mathematical analysis will fail to persuade unless one is already a true believer of supertasks.
That about sums it up.
That looks to me like a valid argument to the conclusion that the Thompson lamp is a physical impossibility, because switches and electrical currents are not infinitely fast. What's wrong with that?
My problem is that I don't understand what metaphysics is, if it is not logic.
I may be wrong, but it seems to me that those who believe in supertasks think that by stipulating that the lamp can be switched infinitely many times in a limited time, they can make it so. But they can't. Supertasks play on the difference between the physically possible and the logically possible to create an illusion.
After completing the supertask the lamp must be either on or off, but as I explain here, Thomson's lamp shows that if we have pushed the button an infinite number of times then it is logically impossible for the lamp to be either on or off after the supertask is completed. This is a contradiction, therefore Thomson's lamp shows that it is logically impossible to have pushed a button an infinite number of times.
Quoting Michael
That's right. But there's nothing special about the lamp. It is impossible to complete any action an infinite number of times.
I wasn't accurate enough in my last post. The representation of the series is misleading. (1/2,1/4,1/8...) invites one to think of the dots as an abbreviation for a longer sequence, which could be written out in full. But there is no possibility that it could be written out in full. The notation does not define an end,
In the message you linked to, you concluded:- Quoting Michael
I think that this is what @fishfry was saying. (Substituting "logically impossible" for "metaphysically impossible".)
So what is the argument about?
Your post seems to add nothing new, and does not appear to engage with any of the points I've made. I have nothing to add till I see a need to write something I haven't already said.
I've enjoyed the chat but progress has not been made in some time.
I love to spread the gospel of Cauchy and Weierstrass.
Quoting Ludwig V
I'm sure I'm not the only one. I've always been a finicky speller.
Quoting Ludwig V
Well "completed" is a loaded term, but it has no mathematical meaning. Mathematicians don't use the word. Given a sequence, it may or may not have a limit. The word completed is not a technical term.
On the other hand, the real numbers are the completion of the rationals, meaning that the reals consist of all the limits of all the Cauchy sequences of rationals. But that's a technical definition, it doesn't have the overtones you're giving to it.
Quoting Ludwig V
"Complete" is not an applicable mathematical term. Unless you want to say that sqrt(2) completes the sequence 1, 1.4, 1.41, 1.414, ... That's an acceptable usage. But it doesn't mean there is any kind of magic jump at the end. It just means the terms of the sequence are arbitrarily close to the limit.
Quoting Ludwig V
I even disagree with that. But some of the possible worlds people (David Lewis, I believe) claim that the possible worlds are real.
Quoting Ludwig V
Most comprehensively in On the Plurality of Worlds, Lewis defended modal realism: the view that possible worlds exist as concrete entities in logical space, and that our world is one among many equally real possible ones. -- David Lewis
The day I read that is the day I gave up on caring about possible worlds.
Quoting Ludwig V
Well it all went over my head when I took a MOOC on the subject.
Quoting Ludwig V
Yes. If time is like the real numbers, then Zeno supertasks are a everyday occurrence. You execute one when you roll over in your sleep. This is the point I made to @Michael.
Quoting Ludwig V
The mathematical series sums immediately. The loop takes time and never achieves its limit, since computing resources are bounded. Under the thought process experiment of "adding the next term" at successively halved time intervals, I'd say it completes in finite time. But that confuses people because we're conflating math and physics.
Quoting Ludwig V
I agree. @Michael keeps saying supertasks are metaphysically impossible, and I think they're metaphysically possible.
No. I'm saying that there's no natural way to define the terminal state. There are lots of ways to defined it. I define it as a plate of spaghetti. That's entirely consistent with the rules of the lamp problem, which only defines the state of the lamp at the points of the sequence, and not at the limit; and it's not a real lamp, so turning into spaghetti is no more unrealistic than cycling at arbitrarily small time intervals.
No it's not, as explained here.
Quoting fishfry
As far as I can see I've addressed everything you've said.
Yes, as I further explained in this comment.
Now I'm confused. I thought you didn't know what "metaphysics" means - or what metaphysics is.
Quoting fishfry
I'm puzzled now about "natural". If the terminal state of the lamp is not defined, there is no way to define it - natural or otherwise. Or, possibly better, any arbitrary state will do. Hence the plate of spaghetti.
Quoting fishfry
Yes, of course - and since it is not defined, Michael can derive a contradiction - two equally possible or impossible states.
Quoting fishfry
H'm. I would be quite happy with that acceptable usage. But it suggests that 1,1.4,1.41, 1.414... is incomplete, and we are back with the temptation to think that series can somehow be completed. It's probably better to stick with "not applicable".
Quoting fishfry
I think that's the heart of the problem. My only hesitation is that the lamp is imaginary, so it sits on an ill-defined boundary between the two. I'm very suspicious of the idea that anything anyone can imagine is (logically) possible. Twin Earth is a good example. But there's a raft of others.
Quoting fishfry
I don't know what to say. Ryle would go on about category mistakes. In poetry (or politics) people sometimes talk of a "tin ear". That's exactly what this is - a rhetorical gesture that confuses "concrete" with "well defined" and with - well - concrete. It's protesting too much. There must be some repressed doubt going on there.
Quoting fishfry
You are lucky. It will spare you a world of grief and confusion. Modal logic can look after itself.
Quoting fishfry
The system is telling me that you mentioned me in the context of this comment in the thread on the Fall of Man paradox, but I can't find any mention of me. But the system is doing some weird things anyway, so I'm not going to worry about it. I do regret not having been aware of the thread sooner. I thought it had something to do with theology.
I can't contribute to the discussion you are involved in there, but this quotation: Quoting keystone
does make me think that the same problem, of the interface between mathematics and empirical reality, is at the heart of this paradox as well.
What I've explained though, is that infinite divisibility is really incoherent due to self-contradiction. So the supertask is not even logically possible. It just appears to be, when not subjected to critical analysis.
So the matter of doing something an infinite number of times never comes about, because the conclusion of an infinite number of times is only produced from the premise of infinite divisibility. If we remove that premise, and just start talking about doing something an infinite number of times, it's obvious that there is no end to such a task. It's only the contradictory notion, that a finite thing can be divided an infinite number of times, which produces the paradox.
From past experience I understand that @fishfry is very slow to accept the reality that some principles employed by mathematicians are incoherent.
.... and when one analyses it, it is a confused mixture of physical possibility and logical possibility, each of which are coherent on their own.
Quoting Metaphysician Undercover
A finite thing certainly cannot be divided an infinite number of times, if by "divided" you mean "physically divided", subject to clarification of what you mean by a finite thing.
At the same time, it is possible to divide it into halves, quarters, etc. (how many fractions are there?) and into feet, inches, etc. and into metres, etc, and according to an indefinite number of other units of measurement. To physically divide in one of those ways excludes dividing it in any other way, so you can't divide it by all those things at the same time. But those possibilities do all exist, all at the same time.
On the other hand, the number 2 is finite, in one sense. Yet it cannot be physically divided at all (because it is an abstract thing), yet it can be divided by a familiar mathematical operation, and that operation can be applied to it an infinite number of times. (No, I'm not talking about space or time.)
Quoting Metaphysician Undercover
I wonder if that's because the principles that you are applying to mathematics do not apply to mathematics? For example, numbers are abstract objects; they do not exist in space and time. Geometry is not about physical objects, but about ideal objects, which do not exist in space and time. Abstract entities that do not exist in space or time are not subject to the restrictions you wish to impose on space and time - obviously. You may or may not regard such entities as not true (or not real) objects, but that's neither here nor there.
I don't think infinite divisibility is a logical possibility, that's the point I'm making. Infinite division is logically impossible.
Quoting Ludwig V
I mean to "divide" in any sense of the word. The qualification of "physical" is irrelevant. Division is an action, an operation, and we are talking about the possibility of dividing something an infinite number of times. Do you think that this is logically possible? It's like counting, which is another activity. Do you think it's possible to count infinite numbers?
Quoting Ludwig V
You may say that it is possible to divided indefinitely, but that does not mean that infinite divisibility is possible. Take pi for example. You can get a computer to produce the decimal extension for pi, "indefinitely", but you never succeed in reaching an infinite extension. Divisibility is the very same principle. Some mathematical principles allow one to divide indefinitely, but you never reach infinite division. That is because infinite division, therefore infinite divisibility, is logically impossible.
Quoting Ludwig V
No, the mathematical operation of division cannot be applied to an infinite number of times, for the reason explained above. If the action is completed it is not infinite, and if it continues indefinitely, at any stage in its progression, it is not infinite either. Quite simply, there will always be more dividing to do before an infinite number of times is achieved, and if you stop it is not achieved either. Therefore it is very clear that such an activity (infinite division) is logically impossible. In general, an infinite activity, or operation, is logically impossible.
So for instance, physical determinism is a metaphysical issue. There are some valid interpretations that are deterministic, and some valid interpretations that are not. Therefore determinism is metaphysically possible, and therefore determinism is physically possible.
Everybody posting seems to treat 'metaphysically possible' as some weird sort of realm between physically and logically possible, resulting in confusion when nobody can come up with an example of something distinct.
So any argument against the lamp, or recitation of numbers, becomes a logical argument because these arguments have no application to physics. They are all metaphysically impossible for the very reason that they are physically impossible.
This seems to say it. It is a logical issue, but with applications to the physical when the scenario in question doesn't involve physical impossibilities.
Quoting MichaelI am willing to accept this statement, but you are not willing to engage with any of the faults identified with your logic. Hence I can only presume you have no counters to them, resorting only to changing the subject every time a fallacy is pointed out. I for the most part have dropped out due to this lack of engagement.
Quoting Ludwig VThis is Zeno's strategy. Just beg your conclusion.
Quoting Ludwig V
There it is. Not possible due to the asserted necessity of a bound of something which by definition has no bound. All the arguments against seem to take this form. Even Zeno avoided this fallacy, and his argument was made before the mathematics of infinite sets was formalized.
Quoting fishfry
Myself as well. I have dropped out some time ago, and not surprisingly, nothing new has been posted. But I did chime in to define 'metaphysically possible' since the term seemed to be used in a way in which it was somehow meaning something different than physically possible, which it cannot be.
I respectfully disagree. You've addressed none of my points.
I don't know what @Micheal means by metaphysically impossible. I know what metaphysics means. More or less. Not an expert.
Quoting Ludwig V
Contrast with the staircase. The walker is present at each step, and the terminal state is undefined. If we define the terminal state as "walker is present," that is natural, ie continuous. If the walker is defined to be absent at the terminal state, that's discontinuous: 1, 1, 1, 1, ... with terminal state 0. That's unnatural.
Likewise Cinderella's coach. Coach at midnight minus 1/2 second, coach at mid minus 1/4, etc. Pumpkin makes it discontinuous. The natural continuation would be for it to remain a pumpkin.
I'm defining natural as continuous.
Quoting Ludwig V
There is no natural continuation. No terminal state that makes 0, 1, 0, 1, ... continuous.
Quoting Ludwig V
Yes, the completion in that case is sqrt(2). That's how we define the irrationals as particular sequences of rationals.
Quoting Ludwig V
My point is that the lamp is fictitious and violates the laws of physics. So its terminal state need not be on or off. It could be a plate of spaghetti. That is no more fictitious than the lamp itself.
Quoting Ludwig V
I don't just oppose the modal realists. I think they're insane. Or trolling.
Quoting Ludwig V
My feelings exactly.
Quoting Ludwig V
I no longer know what any of these threads are about. Perhaps I never did.
That's funny, coming from someone who can't understand the axiom of extensionality because you don't understand material implication.
Logical equivalence does not imply "the same as". I have no problem with the axiom of extensionality. I have a problem with people who conflate the axiom of extensionality with the law of identity, to interpret that axiom as saying two equal things are the same thing.
Our conversations have proven and confirmed to me that you do have a problem with the axiom of extensionality. You are unable to engage with the mathematical formalism, and therefore you do not understand what it says, and how it is to be used. And the reason you can't engage with the formalism is that you don't seem to understand material implication.
That's what I determined. You've said nothing to convince me otherwise recently. If I have misconstrued your position, I'd be grateful for any correction. But I don't think I have, because before that you refused to even acknowledge my proof that 2 + 2 = 4 from the Peano axioms.
That just shows you won't/can't engage with the symbolism. Either way, if you are not willing to do that, then there's little else I can say.
[Note: These next two paras inserted a bit later]
For what it's worth: The axiom of extensionality is the definition of a symbol. Nobody is saying it means anything at all. If you denied that sets exist, I would agree with you. They're mathematical fictions.
But they are useful, because we can base almost all modern math on them; and two, they're interesting in their own right. You are the only one trying to give the axiom of extensionality metaphysical implications that are not really there.
Quoting Metaphysician Undercover
Are these people in the room with us right now?
I have no problem acknowledging that 2+2=4. I have a problem with people who claim that "2+2" symbolizes the same thing that "4" does. And so, I refused to accept your claim to have proven that "2+2" signifies the very same thing as "4" does. Simply put, if the right side of an equation does not signify something distinct from the left side, mathematics would be completely useless.
You can say that I have a problem with formalism, because I do. Like claiming that accepting certain axioms qualifies as having counted infinite numbers, formalism claims to do the impossible. That is, to remove all content from a logical application, to have a logical system which is purely formal. If such a thing was possible we'd have a logical system which is absolutely useless, applicable to nothing whatsoever. Attempts at formalism end up disguising content as form, producing a smoke and mirrors system of sophistry, which is riddled with errors, due to the inherent unintelligibility of the content, which then permeates through the entire system, undetected because its existence is denied.
Formalism as a philosophy considers mathematics to be reducible to a finite single-player sign game of perfect information in which proofs refer to deterministic winning strategies, and hence Formalism does not support the Platonic interpretation of abstract mathematics as denoting actually infinite objects, whatever the formal system concerned.
So I think your problem is actually with Platonic myths that have become psychologically wedded to innocent formal definitions, and in particular the formal definitions of limits and total functions that are ubiquitously misinterpreted in both popular and scientific culture as denoting a non-finite amount of information, E.g as when the physicist Lawrence Krauss misleads the public with nonsense about the physical implications of Hilbert Hotels.
See that phrase, "perfect information"? That's why I say formalism attempts to do the impossible. In other words, it assumes an ideal which cannot be obtained, therefore it's assumption is necessarily false.
Quoting sime
I view formalism as a form of Platonism. It's a Platonist game in which the participants deny their true character, that of being Platonist. Notice "perfect information" is the foundational feature of Platonist idealism. That perfection is the only thing which supports the eternality of Platonic ideals. So formalism and Platonism are really just the same thing, even though the formalists will claim otherwise.
Perfect information isn't an assumption of formal reasoning, rather it is regarded to be a necessary condition of the meaning of "formal" reasoning in that it is by definition finitely deducible and does not require appealing to unformalized intuitions about infinite and ideal objects. Most importantly, the condition of perfect information ensures that formal reasoning cannot interpret an expression such as {1,2,3,...} as representing an abbreviation of some ideal object; the former expression must either be formally treated as a finite object of some type, else the expression must be considered illegal.
It is actually by sticking to formal reasoning that the illusion of the ideal is never obtained. The opposite impression is due to Platonists disguising themselves as formalists, which might be said to even include Hilbert himself.
Formalism makes the reasonable demand that whatever informal intuitions originally motivated the construction of an axiomatic system, and whatever informal interpretations one might subsequently give to the signs of that system, the methodology of theorem-proving should be purely algorithmic and make no appeal to such intuitions, whether such intuitions be rooted in platonism or in Kantian intuition.
Quoting Metaphysician Undercover
The irony of Hilbert, is that his formalism ultimately led to the rebuttal of his own informal intuitions about infinity, namely his presumption that a closed axiomatic system must possess a finite representation of it's own consistency. Had Hilbert better understood the implications his formalism, and especially the finite formal meaning of The Law of Excluded Middle which he apparently accepted for instrumental purposes, then Godels incompleteness theorem might not have come as a shock to him. It is evident that Hilbert was a methodological formalist who didn't mean to insinuate that mathematics was a meaningless game void of semantics, but only that the terms used to denote sets, formula and constants shouldn't require interpretation for the purposes of theorem proving. Unfortunately, his intuitions misled him regards to the outcome of his formal program.
If we inspect the finite activity of theorem proving in a formal system, we see that every term that is informally interpreted as denoting an "infinite object" only possesses finite conditions under which the term is introduced into a theorem and under which the term is eliminated from a theorem.
Different formal systems can be regarded as differing only in regards to their ability to distinguish types of finite object. E.g Intuitionism that formalizes choice-sequences can distinguish uncompleted finite sets from ordinary finite sets, whereas ZFC as a theory of first-order logic can only distinguish finitely defined functions from finite sets - so whilst ZFC might be informally said to be a theory about "infinite sets", this isn't the proof-theoretic formal meaning of ZFC, and so a formalist is free to reject the platonic myths that surround ZFC.
[quote=Wikipeida;https://en.wikipedia.org/wiki/Recursion]Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition.[/quote]
I know it is only Wikipedia, but I'm sure that more authoritative references could be found.
What is wrong with that?
Thomson makes a similar point:
There's nothing wrong with defining, or performing, a recursive function. There is a problem with claiming that it is possible to have completed a recursive function.
Quite so.
Can you prove that it is impossible to complete a given recursive function?
According to those who claim that supertasks are possible, after two minutes the echo isLampOn command will run. This is the first contradiction; the while (true) command ensures that this cannot happen.
But even assuming that echo isLampOn does run, it cannot output true and cannot output false, but also cannot output anything other than true or false. This is the second contradiction.
fishfry's solution is to inject some additional code after while (true) { ... }, assigning some arbitrary value to isLampOn, but in doing so he is no longer addressing the problem as posed.
See also this.
Math (arithmetic etc) with infinity as a value or function argument is not valid.
An explanation why that's so is Hilbert's hotel explanation:
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
You wanted something new so see the above.
I see no contradictions because division by 2 results in even number of total divisions.
Therefore if initial state of the lamp is off then it follows that the lamp will be off after 2 minutes.
Under assumption that those 2 minutes must pass the lamp will therefore be off with
And if those 2 minutes never pass due the nature of infinitely small wait time then that doesn't matter because we already know what logically will happen otherwise.
objections?
https://en.wikipedia.org/wiki/Thomson%27s_lamp
I don't know what you mean by this.
The wait(i *= 0.5) simply means that pushButton is called at successively halved intervals of time, i.e. it is called for the first time after 60 seconds, for the second time after a further 30 seconds, for the third time after a further 15 seconds, and so on.
Quoting SpaceDweller
Given what while (true) { ... } means, it is logically impossible for console.log(isLampOn) to ever run.
2 minutes must pass, and while (true) { ... } must still be running after those 2 minutes.
In this case there is no wait time between operations; the next occurs "immediately".
Does this script end in an instant, or does it continue forever? If it ends in an instant, does echo isLampOn output true or false?
But given the meaning of while (true), it must continue forever; echo isLampOn never runs.
If you think that this changes by introducing a wait time within the loop then you've obviously made a mistake.
Successive Division by even number results in even number of divisions, no matter how many times you divide some number by 2 total count of divisions will be even, not odd.
Therefore lamp state would not change after that count of divisions.
Quoting Michael
Yes, but it will happen even count of times, that's my point.
Quoting Michael
Yes it's infinite loop, I misquoted that, regardless if the loop is finite or not count of divisions is even and so the result is known upfront.
This would not be the case if you divided by 3 or some other odd or real value.
You second example is completely different story though because there is no division involved.
Example with a finite number:
4 / 2 = 2
2 / 2 = 1
Count of divisions is even (2) because divided by even number (2) therefore lamp state does not change, it's off.
You continue this division to infinity and the result will be the same because you divide by 2.
ex. 1/2 = 0.5 / 2 = 0.25 etc... it doesn't matter how far you go.
I'm not sure how to try better express myself.
But let's try division by 3 with same sample before:
4 / 3 = 1.33
count of divisions is odd so the lamp is ON,
You can continue dividing and it's uncertain that the result will be even.
I stopped with 1.33 because the result is lower than 3, same way I stopped in my previous example because 1 was lower than 2.
2 + 2 and 4 symbolize the same set. You are the one strawmanning the claim that somebody says they're the same thing.
YOU are doing that. Not any mathematicians, unless they are speaking casually.
2 + 2 and 4 symbolize the same set. Not necessarily the same thing, whatever that might happen to mean in someone's lexicon. "Thing" is not a term of art in mathematics. Set is. I have explained this to you certainly more than ten or twenty times.
Quoting Metaphysician Undercover
I have never made that claim. In your refusal to engage with a single thing I've ever said to you, you BELIEVE, in your heart of hearts, that I have claimed that 2 + 2 and 4 are the same "thing." I will credit you with sincerity for once.
But I have not said that, and if you will go back through my posting history as long as you want, you will never find that I ever said that.
I have said, repeatedly, that 2 + 2 and 4 represent the same set. And that, by the rules of the formal system known as set theory, is as true a statement as a statement ever could be true. It's as true as saying that bishops travel only on diagonals. It's not true by virtue of any property or aspect of the world. It's true by virtue of the rules of chess.
Likewise, the fact that 2 + 2 and 4 symbolize the same set, is derivable from the rules of set theory.
Quoting Metaphysician Undercover
Ah, the old tautology argument. "Since everything in math follows from axioms, the theorems are already inherent in the axioms, hence they add no new information, hence they are useless."
I counter that argument by asking whether the sculptor adds value in finding the statue within the stone.
Of course he does. The theorems follow logically from the axioms; but they do not necessarily follow obviously. It often takes great ingenuity and insight to derive theorems that are interesting, useful, and beautiful. That's mathematics.
Quoting Metaphysician Undercover
I have observed with my own eyes your inability and unwillingness to engage with my Peano proof that 2 + 2 = 4, and the statement of the axiom of extensionality.
I don't "say" the sun rises in the east. I observe it to happen with great regularity. I don't "say" you have a problem with formalism; I observe it.
I will admit that you haven't failed to engage with formalism as many times as the sun has risen. But I'm willing to bet that's only because I haven't challenged you that many times.
Quoting Metaphysician Undercover
You are making a category error akin to complaining that bishops "make church laws, be a judge in church matters and to enforce observance of these laws," in response to my telling you that bishops move and capture along diagonals.
You are railing against a metaphysics that is not claimed by any practitioners of mathematics.
Quoting My vetaphysician Undercover
You're just making the unreasonable effectiveness argument. If math is a flagrant fiction, why's it so darn useful?
You are not the first person to notice this, and you are not special for having noticed it. You are "special" in a different sense, for thinking you've had a profound insight, when it's such a commonplace. Bertrand Russell said, "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." He was making the same point.
Quoting Metaphysician Undercover
I see the point you're making, but what of it? Do you object to chess because it redefines what bishops are? You're so against formalism that you reject it entirely? Why do you bother to read and write? Language is just an abstract formalism for notating thoughts, and we know words are just an approximation to thoughts. Why aren't you equally against language?
Quoting Metaphysician Undercover
Just as language is, as some philosophers have argued.
Quoting Metaphysician Undercover
Just like language.
"Man and His Symbols," Jung. Humans use symbols. You are standing four-square against mathematical symbolism but not other forms of symbolism. Presumably you're not picketing art museums and book publishers.
Why is that?
Was that the pseudocode I was supposed to look at?
I entirely agree, with one exception. Code is a notation for a physical process executing on a finite computer. The mathematical lamp problem is the same code, but interpreted in abstract mathematical space where the process never ends. With that caveat I accept your pseudocode.
BUT!!!!!! You have not defined the terminal state. So why do you think there should be a sensible answer for what it is?
After all, there is no number that can serve as the limit of the sequence 0, 1, 0, 1, ...
I don't see how your expressing the problem in pseudocode adds anything. We all have agreed to it long ago, even before you wrote it down. That's the premise of the problem. But the question is about the terminal state, which is not defined.
I don't see how recursion qualifies as an infinite activity.
Quoting fishfry
My point has always been that "same" in this context is not consistent with "same" in the context of the law of identity. So, to say " 2+ 2 and 4 symbolize the same set" is to use "same in a way which is in violation of the law of identity.
Whether we are talking about "same thing", "same set", "same number", or "same kick in the ass", is irrelevant. The point is that this specific use of "same" is very clearly in violation of the law of identity. If the law of identity indicates that only a thing can be said to be "the same", and you do not believe that a set is a thing, and you want to say that a set is the same, then I suggest that you do not agree with the way that "same' is used by the law of identity. Is this the case? Do you believe that mathematicians have a better definition of "same"?
Quoting fishfry
In case you have never noticed, fiction is extremely useful. I suggest you begin with a look at the obvious, deception. Deception demonstrates that fiction is very useful in convincing others, to help us get what we want from them. And, so is mathematics.
In this context, same means that we may write the symbol "=" between the sets. The axiom of extensionality tells us when we may write A = B. If you prefer to be formal, we could simply never say, even casually, that A and B are "the same." Only that they satisfy the axiom of extensionality and therefore we may write A = B. And if you don't like the equal sign, we could use some other symbol.
Quoting Metaphysician Undercover
We need not use the word same if it bothers you. It's sufficient that the statement A = B is true (within set theory) exactly when A and B satisfy the axiom of extensionality. And "=" doesn't mean anything, it's just a symbol.
Quoting Metaphysician Undercover
Same is only being used casually. When pressed, we always revert to the definition of = given by the axiom of extensionality.
Quoting Metaphysician Undercover
Then why are you disagreeing with me? If you say that mathematics is a useful fiction, that's exactly what I would say. And if you say it's a useful deception, that's fine.
May I ask, is chess similarly a useful deception? Language? You didn't respond to my point earlier that language is also a formal symbology that attempts to capture, however imperfectly. some aspect of abstract thought.
The terminal state isn't just undefined; any proposed terminal state is inconsistent. The lamp cannot be either on or off after two minutes even though it must be either on or off after two minutes. This is a contradiction, therefore it is impossible to have pushed the button an infinite number of times.
Quoting fishfry
We're discussing the consequence of having pushed a button an infinite number of times, not the limit of some infinite sequence of numbers. These are two different things.
As Thomson says, "the impossibility of a super-task does not depend at all on whether some vaguely-felt-to-be associated arithmetical sequence is convergent or divergent."
Great, I prefer the word "equal". It's better suited for that purpose. "Equal" generally allows that the two things which are said to be equal are not necessarily the same. "Same" is defined by the law of identity as indicating one thing only. That is the commonly expressed difference between "same" and "equal". "Equal" indicates a similarity of two things by both sharing an identifiable property, while "same" means that you are referring to one thing only.
Quoting fishfry
Generally I disagree with your wording, as indicated above. The axiom of extensionality indicates what is required for two sets to be equal, yet you state this as "the same". That I take as a mistaken use of words.
Quoting fishfry
I didn't say that though. I simply gave an example of how fiction is useful, one that was obvious. Many times fiction is used in ways not intended to deceive, like the use of counterfactuals in logic, for example. So, the issue is complex, because mathematics, like fiction in general has many uses.
Quoting fishfry
These are broad generalizations which I can't relate to because I do not accept them as valid generalizations, so I do not reply. For example, you say that language "attempts to...". But language doesn't attempt anything, individual people attempt to do things with the use of language. And, there is such an extremely broad range of things which people attempt to do with language, that it doesn't make sense to make the generalization that what people attempt to do with language is to capture "some aspect of abstract thought".
For instance, a supertaskist might appeal to the fact that one cannot say how many moments of time has passed during a minute, or how many physical operations took place in one's computer to sum 1 + 1, and they might appeal to this inability to measure, divide and count experience or events as grounds for being open minded to the idea that space-time is a literal continuum.
But in that case, how does the supertaskist propose identifying what isn't a super-task? If super-tasks are to have empirical meaning and inferential value, the supertaskist must delineate task from super-task, but how can they delineate them on a non-ad hoc basis?
I could say I was using the word same in a casual sense.
Or, I could say that when i comes to sets, two sets are the same if they have the same elements. Same set. Equal sets or same set. It's the same.
Quoting Metaphysician Undercover
In set theory, two sets that are equal are the same set. Two numbers that are equal are the same number. You're playing word games.
Quoting Metaphysician Undercover
Same set. That's what it means.
Quoting Metaphysician Undercover
Commonly meaning you've said it a couple hundred times, to little effect. You've had the extensional/intensional distinction pointed out to you many times to no avail.
Quoting Metaphysician Undercover
Well if we're talking about sets, equal and same are the same. And I'm talking about sets.
Quoting Metaphysician Undercover
Sets that are equal are the same set.
Quoting Metaphysician Undercover
They're the same set. If you can give me an example of two sets that are equal yet not the same set, well ... you can't.
Quoting Metaphysician Undercover
So math is a useful fiction.
Quoting Metaphysician Undercover
You are agreeing with me.
Quoting Metaphysician Undercover
You object to symbolic notation for math. Do you object to symbolic notation for music? For thought and speech? For chess?
Quoting Metaphysician Undercover
You are avoiding the issues I've raised.
You can't push a button an infinite number of times. This is not a physical experiment. It's purely an abstract mathematical exercise.
You can define the terminal state to be on, off, or a plate of spaghetti and be consistent with the rules of the game. It's not a real light bulb, it's not driven by a real circuit. You're confusing yourself on this point.
Quoting Michael
You can't push a button an infinite number of times. It's not a real button, it's an abstract thought experiment about an entirely fictional entity.
And since when can't we use math to model some real or fictional situation? Did you complain when Maxwell wrote down his equations of electromagnetism? Hey it's electricity and magnetism buddy, it's not math. Surely you are not trying to make this absurd claim, are you?
The sequence 0, 1, 0, 1, ... which clearly has no limit, is a perfect mathematical model of this problem. You can define the terminal state -- that is, the state at [math]\omega[/math], as 0, or 1, or a plate of spaghetti. It makes no difference to the original sequence.
Quoting Michael
That's funny, because according to SEP, he used the asymptotic density of the sequence 0, 1, 0, 1, ... to argue that the final state must be 1/2. He made that exact argument using that exact sequence. So you are mistaken, because Thompson has used the exact same reasoning I did. Please read this section:
https://plato.stanford.edu/entries/spacetime-supertasks/#MissLimiThomLamp
No you can't. I addressed this in my initial defence of Thomson here, and even more clearly below.
You're claiming that "a plate of spaghetti" is a coherent answer to the question "is the lamp on or off after two minutes?"
So I think the confusion is yours.
Quoting fishfry
He discusses the sequence and its sum, but only to show its irrelevancy, hence the earlier quote.
From his paper:
By contrast, Benacerraf changed the goal posts by giving Thompson's Lamp a formal treatment from the perspective of classical analysis, in which he interpreted Thompson's Lamp as being an incomplete description of an extended abstract function with a point at infinity, whose value can be chosen as being continuous with some property of the original sequence. However, Bencerraf's formal interpretation doesn't have a temporal interpretation in the sense demanded by the intuitionists. And his methodology runs into problems in situations where the function to be completed has conflicting notions of continuity, as in The Littlewood-Ross Paradox.
Conclusion: set theory is in violation of the law of identity. I've explained to you why this is the case. Do you agree with me?
I don't think it has anything to do with formalism or intuitionism or anything like that.
Our starting premises are:
P1. If the lamp is off and the button is pushed then the lamp is turned on
P2. If the lamp is on and the button is pushed then the lamp is turned off
P3. The lamp is off at t[sub]0[/sub]
P4. The lamp is either on or off at t[sub]1[/sub]
Thomson asks the following question:
Q1. If the button is pushed an infinite number of times between t[sub]0[/sub] and t[sub]1[/sub] then is the lamp on or off at t[sub]1[/sub]?
Compare with:
Q2. If the button is pushed an odd number of times between t[sub]0[/sub] and t[sub]1[/sub] then is the lamp on or off at t[sub]1[/sub]?
Q3. If the button is pushed an even number of times between t[sub]0[/sub] and t[sub]1[/sub] then is the lamp on or off at t[sub]1[/sub]?
Benacerraf claims that we can simply stipulate that the lamp is on at t[sub]1[/sub].
There are two problems with this.
The first is that we cannot simply stipulate the answer. I cannot stipulate that the answer to Q2 is "off" and I cannot stipulate that the answer to Q3 is "on". The answers must be deducible from the premises.
The second problem is that the following is deducible from the first three premises:
C1. If the button is pushed an infinite number of times between t[sub]0[/sub] and t[sub]1[/sub] then the lamp is neither on nor off at t[sub]1[/sub]
This is because if the button is pushed at least once then for the lamp to be on at t[sub]1[/sub] the button must have been pushed to turn and leave it on for t[sub]1[/sub], and for the lamp to be off at t[sub]1[/sub] the button must have been pushed to turn and leave it off for t[sub]1[/sub], neither of which are possible if the button is pushed an infinite number of times between t[sub]0[/sub] and t[sub]1[/sub].
C1 contradicts P4.
Given that P4 is necessarily true, it follows that the antecedent of C1 is necessarily false. So it is metaphysically impossible to have pushed the button an infinite number of times.
Yes, and that sounds identical to the philosophy of intuitionism :) It doesn't matter what type of object is associated with the lamp's output.
By contrast, Benecerraf et al argue along more classical lines, by defining an abstract completion of the sequence that doesn't contradict Thompson's premises, but which requires changing the original problem by adding an point at infinity to accommodate such a completion.
No, I'm saying that it isn't completable, even hypothetically. The fact that there is no definite end value is just one way to demonstrate that it isn't completable; the lamp must be either on or off after two minutes, but if the button has been pushed an infinite number of times before then then it cannot be either.
Quoting sime
I don't think it has anything to do with mathematics. This is perhaps clearer if we don't consider the button to turn the lamp on and off but instead consider it to alternate between two or more colours.
What number would you assign to the colour red, and why that? What number would you assign to the colour blue, and why that? Shall we use [math]e[/math] and [math]i[/math], because why not?
The logic of the lamp just has nothing to do with some sequence of additions and subtractions. The code here properly demonstrates the logic.
I think it does. We need to examine the process in reverse, and remember that the lamp is on iff the lamp was off and the button was pushed to turn it on. We're discussing a supertask after all, not simply the infinite sequence {off, on, off, ...}.
If the lamp is on at t[sub]1[/sub] then either:
a) the button was pushed to turn the lamp on before t[sub]1[/sub] and then it was left on until t[sub]1[/sub], or
b) the button was pushed to turn the lamp off before t[sub]1[/sub] and then it was left off until t[sub]1[/sub] when the button is pushed to turn it on
Neither (a) nor (b) are possible given the defined supertask the lamp is never left either on or off therefore the lamp cannot be on at t[sub]1[/sub]. And then the same reasoning shows that it cannot be off at t[sub]1[/sub] either. Yet it must be either on or off. This is a contradiction.
This fact has nothing to do with one's interpretation of mathematics (and nothing to do with the limit of some proposed infinite sequence of numbers).
It seems pretty clear that "same" does not have the same (!) meaning in the context of set theory that it has in the context of the law of identity. How could it? The definition that applies in the context of the law of identity is inapplicable to the context of set theory, and vice versa. So why don't you conclude that the use in the context of the law of identity violates the use in the context of set theory? It seems to be an arbitrary choice.
The meaning of "same" depends on its context. There are many other concepts of which that is true.
The contradiction is the result of the fact that there is no criterion set for the final step in your process - i.e., the end state is undefined.
Quoting fishfry
Surely, the contradiction is the result of the lack of any definition of the terminal state. If the terminal state could be a plate of spaghetti, why couldn't be a lamp that is neither on nor off?
I really cannot see what you two are arguing about. Why does the difference matter?
Quoting fishfry
The plate of spaghetti is a great dramatic way of making the point that there is no definition. But the series is defined on the basis that its limit is 1. You can't derive 1/2 from a plate of spaghetti.
That's precisely why supertasks are impossible.
The lamp is off at t[sub]0[/sub]. The lamp must be either on or off at t[sub]1[/sub]. But if the button has been pushed an infinite number of times between t[sub]0[/sub] and t[sub]1[/sub] then the lamp can neither be on nor off at t[sub]1[/sub].
It's strange that some are taking the very thing that proves that supertasks are impossible as proof that the proof isn't a proof.
My point is that I think that the disagreement between you and @fishfry is about different ways to make the same point.
Quoting Michael
Quite so. Wittgenstein made much of the endlessness of infinity and asked how it was possible. You may know what his answer is. If you don't, it is easy to look it up. (It would be far too long to try to outline it in this context and you likely know anyway.
My question is, how do we know that it is not possible to complete a recursive function (in the sense of writing or speaking each step that the function defines)? BTW, I don't think there is any particular problem about defining such a function, though you could argue that it is the result of misapplying an operation that is perfectly harmless in other contexts, like ?2 or a self-referential pronoun like "I" or "this".
For example, proving that ?2 is irrational doesn't rest on trying and failing to write it down, but on showing that the assumption that ?2 is rational leads to a contradiction - reductio ad absurdum.
In the case of infinity the argument rests on mathematical induction. I understand that not everyone is happy with that argument but it seems OK to me. A recursive function is defined, but in such a way that its end is not defined.
So even if a sequence cannot be completed empirically, so to speak, there are ways of proving that it is endless with attempting the impossible refutation of writing down each step.
The code here is effectively the same as a recursive function.
Quoting Ludwig V
I'm arguing that supertasks are metaphysically impossible. He's arguing that supertasks are metaphysically possible.
LOL.
No, I don't agree with you.
Me either.
Quoting Ludwig V
Don't follow. The limit of 0, 1, 0, 1, ... can not be 1. Nor can it be 0. It's a sequence that has no limit.
Quoting Ludwig V
I do not think @Michael and I are making the same point.
I'd agree except that the law of identity was first, set theory came along after. So set theory violated the law of identity, which was already established. If it was the other way around, then we'd want to look at the reasons why someone would be trying to enforce the law of identity, which seems like a useless, obvious, self-evident tautology, in violation of how set theorists were defining "same".
Well, if the law of identity is an obvious self-evident tautology, then it appears like there must be something wrong with set theory if it's in contradiction with what is obvious. We can see something like this in Aristotle's work to establish the law of identity. He claimed that the law of identity was necessary to battle against sophists who could logically demonstrate absurdities. If the base axioms of a logical system are contrary to what is self-evident tautology, then that system will be able to prove things which are contrary to what is obvious, one can prove the absurd.
Quoting Ludwig V
Very true, but we need to pay attention to subtleties to avoid deception. Suppose you and I both drive the same make and model of car, and same year and colour as well. Notice, that "same" is being used in a qualified sense, referring to different properties "same make", "same model", "same year", "same colour". Now, suppose I say that I drive the same car as you. This would be incorrect. But why is it incorrect, because everything about it seems to be the same? The thing is that everything about it is not the same, only those named qualities are the same, and that's why it's incorrect to say that it is "the same" in that unqualified sense.
Now, when we say that "two sets are the same", this is incorrect for the same reason. Not everything about the two is necessarily the same, only the stipulated required qualities. So it is incorrect to say that the two sets are the same, in the unqualified sense, because some features like the ordering of the elements may be different.
Temporal priority is not logical priority.
Whether a proposition is obvious (self-evident) or (self-evidently) absurd are subjective.
Quoting Metaphysician Undercover
There are plenty of ways to formulate that law without using the word "same". In any case, "same" in that context just means "same object", so it isn't absolute. moreover, If you drive my car, you don't drive it at the same time.
Quoting Metaphysician Undercover
There is no unqualified sense of "same".
Quoting Metaphysician Undercover
We agree!
Quoting Metaphysician Undercover
There is no unqualified sense of "same".
Quoting Metaphysician Undercover
I'm sure that Aristotle would not object to my regarding that as not a logical argument.
I'm sorry. I was talking about the convergent series. Didn't check
Quoting fishfry
Perhaps not. But if the last term in the series is not defined, contradictions are likely to follow from the attempt to identify it. Equally, if something gives rise to a contradiction, the definition will be faulty. So, if you are right, I need to ask why it matters.
Not quite. The code specifies a process which must take time. The function does not.
Quoting Michael
Thank you. I must have got confused.
My confusion.
Quoting Ludwig V
There is no last term in any infinite sequence. There may (or may not) be a limit. Big difference.
There is no last term of the sequence 1/2, 1/4, 1/8, ...
It has a limit of 0. But 0 is not an element of the sequence.
Also the terminology I'm using is that a sequence is a comma-separated infinite list, 1/2, 1/4, 1/8, ...
A series is an infinite sum: 1/2 + 1/4 + 1/8 + ...
The elements of a sequence are not "premises," as @Michael seems to think. They're just numbers. Some sequences, like 1/2, 1/4, 1/8, ... have a limit, in this case 0. Other sequences, like 0, 1, 0, 1, ... can't possibly have a limit. That doesn't stop you from defining a "terminal state," which I've formalized as a value at [math]\omega[/math]. But there's no need for there to be any logical relation between the sequence itself, and the arbitrarily-defined terminal state. That's why lamp on, lamp off, or lamp turns into a plate of spaghetti, are all equally valid terminal states to the sequence 0, 1, 0, 1 ...
Even if you insist that the terminal state must be either 0 or 1, there is no logical way to prefer one over the other.
Yes. The exact status of 1 or 0 in these cases is more complicated than I realized.
Quoting fishfry
So can you help me to describe the role of 1 in defining the series 1/2, 1/4, ... when the limit state is 0? (Or indeed when it's the other way round?)
What do you mean I can't? I already have, numerous times in this thread.
The point is that the sequence 0, 1, 0, 1, ... has no limit. We are free to define a terminal state that is not a limit, but is just a value assigned to a point beyond the sequence, traditionally notated as [math]\omega[/math]. This value could be 0, it could be 1, or it could be a plate of spaghetti.
Now I will agree with you that we COULD make a rule that the terminal state must be 0 or 1. That rules out the spaghetti. But there is no logical preference between 0 and 1. Neither value could be the limit of the sequence, since that sequence does not have a limit.
Quoting Michael
You can't have addressed it, since what I said is true.
Quoting Michael
It's consistent with the rules of the game. But if you add the rule that it must be 0 or 1, then either answer will do, and neither answer has any logical relationship to the sequence, since the sequence doesn't have a limit.
SEP is clear that Thompson was clear on this point, in full agreement with me.
Quoting Michael
You know, I lost track of whether I said this to you, or you said this to me. I'm sure we both feel that way. Can't we just agree to disagree? This poor deceased equine has been flogged well into the next life.
Quoting Michael
All possible terminal states are equally irrelevant, because the sequence has no limit.
May I ask, you do you understand that? Do you understand that the sequence 0, 1, 0, 1, ... has no limit, and can not be made to have a limit?
Quoting Michael
Completely irrelevant to anything. Nor do mathematicians say any such thing, except by using asymptotic density (which I believe SEP refers to as Cesaro summation).
You haven't proved your point. You haven't even made a point.
So, the terminal state not being defined does not prevent me defining one arbitrarily?
Isn't it the case that there is a requirement - that the terminal state not be defined by the function.
Have you even read Thomson's paper? This is the most relevant part:
The answer to his question isn't "a plate of spaghetti", it isn't "0", it isn't "1", and it isn't "1/2".
The only coherent answers are "on" and "off" except as he argues, it can't be either, and so therefore it is impossible, even in principle, to have succeeded in pressing the button an infinite number of times.
Quoting Ludwig V
This is the bit with the ordinals I was using earlier.
What is a sequence? A sequence is a function whose domain is the natural numbers 1, 2, 3, ...
For each natural number, there's a value, like 1/2, 1/4, 1/8, ...
We can conceptually adjoin a "point at infinity" to the natural numbers, traditionally called [math]\omega[/math] in this context, so that our extended natural numbers look like
1, 2, 3, 4, ... [math]\omega[/math].
This structure, the extended naturals, are called [math]\omega + 1[/math], because the natural numbers themselves are called [math]\omega[/math]. That's just another name for the more usual notation [math]\mathbb N[/math], except that [math]\omega[/math] implies the set of natural numbers in their usual order.
So if a sequence like 1/2, 1/4, 1/8, ... is a function on the domain [math]\omega[/math], an "extended sequence" (not the official terminology) is a function on [math]\omega + 1[/math]. Such a function might look like this:
1/2, 1/4, 1/8, 1/16, ..., 47
Here I just defined 47 as the terminal state.
Another possible completion is
1/2, 1/4, 1/8, 1/16, ..., plate of spaghetti
There is absolutely nothing wrong with that. I've just adjoined a plate of spaghetti to the rational numbers, and defined the terminal state of this particular sequence to be the plate of spaghetti.
Now in this case there is ONE completion that is "natural," namely:
1/2, 1/4, 1/8, 1/16, ..., 0
That completion is just as arbitrary as any other. But it has one supreme virtue: 0 happens to be the limit of the sequence. So that's why I call it natural.
The problem with the lamp is that the sequence 0, 1, 0, 1, ... has no natural completion. That's why we can define the terminal state of the lamp as on, or off, or a plate of spaghetti, or Cinderella's coach.
All completions are legal; and no completion is natural; because the lamp sequence does not have a limit.
Have a nice evening. I'll forego responding in kind.
I'm afraid it's not, because it doesn't address the issue of supertasks.
For supertasks, we have this:
P1. If the lamp is off and the button is pushed then the lamp is turned on
P2. If the lamp is on and the button is pushed then the lamp is turned off
P3. The lamp is off at t[sub]0[/sub]
P4. The lamp is either on or off at t[sub]1[/sub]
Q. If the button is pushed an infinite number of times between t[sub]0[/sub] and t[sub]1[/sub] then is the lamp on or off at t[sub]1[/sub]?
It makes no sense to answer this question with "a plate of spaghetti" or "[math]1\over2[/math]".
I think you'll find that's because it makes no sense to answer the question.
In other words, it also makes no sense to answer the question with "on" or "off".
I understand why those answers seem more natural, but that's an illusion.
I understand that. What seems important to me is that the convergent series is the result of a calculation which involves 0 and 1, while "0,1, 0, 1, ..." doesn't involve any calculation at all. You could also have a series "a, b, a, b, ..." or "fish, chips, fish, chips, ..." The calculation involves numbers, but "0, 1, 0, 1, ..." only involves numerals.
The lamp is either on or off at t[sub]1[/sub]. The fact that it makes no sense for it to be on and no sense for it to be off if the button has been pushed an infinite number of times before that is proof that it makes no sense for the button to have been pushed an infinite number of times.
Exactly. The contradiction follows from the fact that no final state is defined.
Well, it's not a calculation. The limit, or the terminal value, is not necessarily the result of any logical process. I keep trying to explain this to @Michael. The limit of 0 does not "come from" the sequence 1/2, 1/4, 1/8, ... in the sense that you could put those numbers into a machine and 0 would pop out.
Rather, you are given the sequence; and given the limit; and you can apply a formal definition to see that 0 is indeed the limit of the sequence. It's conceptually sort of the other way 'round from thinking that the limit is the result of some logical process applied to the sequence. Although "find the limit of this sequence" is a common calculus problem, so there are heuristics and strategies to find limits of sequences that have closed-form representations. But not all sequences do, so my remark stands.
That's the think with the lamp. You can define the terminal state as on or off. Neither makes sense, and neither is forbidden. The sequence still stands on its own terms. On, off, on, off ... All that happens, as you count through 1, 2, 3, ... Then you arbitrarily assign a terminal state. On, off, or a plate of spaghetti.
@Michael just doesn't get this or just doesn't want to get this.
If Thompson were here I'd explain this to him. Thompson and @Micheal are making the same conceptual error. There is no logical relationship at all between a sequence, and some arbitrary terminal value. If the terminal value happens to be the formal limit of the sequence, that's a lovely special case that happens a lot in calculus class, where all the problems are designed to work out nicely for diligent students.
But it can't happen with the lamp. The sequence on, off, on, off ... does not have a limit
End of story as far as I'm concerned.
I know, but if there is an existing law, and someone does something that is contrary to that law, then that person violates the law. That is a simple fact.
Quoting Ludwig V
I believe that what is attempted with the law of identity is to express an unqualified sense of "same". You seem to think it fails. Why?
The 1/2 was Thompson's idea. The plate of spaghetti is mine, inspired by Cinderella's coach, another omega sequence paradox that's not usually recognized as one.
The thing you and Thompson keep missing is that the lamp is not real. It violates the currently known laws of physics; and because it stipulates a circuit that can change state in arbitrarily small intervals of time as modeled by the mathematical real numbers, it's difficult to imagine, even speculatively, that the lamp could ever be physically realized even under far future physics.
So there is no "logical" way to connect the sequence, with its arbitrary terminal state, which you can define as on or off. Or as a plate of spaghetti, if you so desire. There's nothing about the lamp that's real. The lamp can turn into a plate of spaghetti exactly as Cinderella's coach turns into a pumpkin. By fiat in a fairy tale.
Please see my response to @Ludwig V above, where I expand on this theme.
https://thephilosophyforum.com/discussion/comment/907876
The rest periods at step 1, instead of being 30 seconds long, is that really just 2 times as long of rest than at step 2? Or is it 4 times as long as step 3, or is it 6 times as long as step 4? In which case, this story has as much to do with increasing speed between steps as it does shortening rests on each step. So rest and motion are needed to place steps in flight beneath you. Is it the increasing speed that shortens the rest, or the shorter rests that increase the speed? How can any rest cause motion to increase?
And why not a bottomless pit if you are to reach an impossible bottom?
This whole image is that of a square circle. I dont see what there is to resolve.
There can exist no infinite anything. Thingness, such as a step or a series of steps, is finitude. Infinity is not a thing to which you can add 1 or subtract from, certainly not when describing actual steps. Calling a thing or a series of things infinite, removes the thing or things from your sight, removes them from the chalkboard, and shows you the same infinity as imagined in the infinite series of fractions between step 1 and step 2. There is no such thing as a half step. Not is there a such thing as an infinite series of steps. There is only a whole distance later conceptually halved, as when you conceive of halving some existing whole step infinitely.
The infinite or infinity as a noun, is best used for dramatic effect. Its not a thing, like a noun is best employed. Infinitely as an adverb, sets out some activity that, by definition, cannot conclude. Thereby banishing all finitude, which marks conclusion, such as a step, or a series of steps, or a noun.
Halving as a verb, like stepping as a verb, can be conceived of as continuing infinitely. But you never find the infinite. There need be no infinitely small fraction. Saying the stars and the atoms in the multiverse are infinite in number means you dont know how to count them, so for dramatic effect, we invoke the infinite. But the infinite finds no home, no place in the physical world, in the form of the finite, save the mind that conceives of some activity that can continue infinitely.
We might as well start this by saying, there were three steps to the basement, but before he took the first one Icarus had to get off the couch, but as he did so, he realized he had to first sit up, and then realized he had to move his legs to the floor, and increasing his effort between each new realization, he realized he had to move his first leg, by first turning his foot, after ending his knee etc. infinitely, as he turned his eye and saw a corpse on the couch with him .
Quoting keystone
No it doesnt. There is no infinite it that could simultaneously do anything, such as exist.
A supertask is not simply an infinite sequence.
With a supertask we are given some activity to perform and we assume that it is physically possible to perform this activity at successively halved intervals of time. We are then asked about the causal consequence of having done so.
We do not get to introduce additional (and nonsensical) premises such as "and then the lamp magically turns into a plate of spaghetti, prior to which the lamp was neither on nor off."
The lamp must be either on or off after two minutes. If the lamp is on after two minutes then it is on only because the button was pushed to turn it on, prior to which the lamp was off. If the lamp is off after two minutes then it is off only because the button was pushed to turn it off, prior to which the lamp was on. The supertask doesnt allow for either of these scenarios and so is proven impossible in principle.
Quoting fishfry
I take the point. I may not have stated it accurately enough, but the crucial thing, it seemed to me, is to realize that the limit is part of the definition from the start - not, as I think you're saying, something that is worked out from the sequence itself.
Quoting Metaphysician Undercover
If I say that Hesperus is Phosphorus, I am saying that they are the same object.
Quoting Fire Ologist
You are right, Language is a great trap here. I would like to use "endless" or "endlessly" and even "endlessness" instead. That would make it more difficult to talk about conclusions. But we are lumbered with a world which uses "infinity". Natural language allows this, but has no guard rails to prevent us from talking nonsense.
Quoting Fire Ologist
The difficulty here is that it is possible to defined an infinite series in a finite frame, which leads people to think of apply the abstract idea to the physical world. Sometimes that works, as in physics, so we can't just say that such ideas have no place in the physical world.
Quoting Fire Ologist
Yes, we do. We don't find them by failing to count them, but through various arguments. The proofs that ? or sqrt(2) or that there is no largest natural number are all well established. So is the possibility of a convergent series.
Quoting Fire Ologist
True, if you are thinking of a staircase. But nobody would contest that. But if you think of the distance between my eyes, you can certainly divide that by 1/2 or 1/4 or...
The distance between your eyes is a whole. You need to grab that finite whole thing first from the physical world to then posit the concept of half of that whole. The half wasnt grabbed from the physical world. Otherwise it would have been a smaller whole distance to start again. The half-distance comes after a whole is firmly in hand. Lets say its two inches between eyes. You cant identify half that distance by referring to your face. You call some smaller distance half, numerically, in reference to the numerical value the refers to the whole distance. The whole distance refers to your eyes.
Imagine someone says in order to walk from the goal line to goal line in football, you first have to walk one-hundredth of a distance, but before that you have to walk one-third of the distance to that first one-hundredth mark, etc. etc. infinitely.
Calling these smaller distances fractions is semantics with reference to mathematical concepts. No fraction ever exists. What exists would be one whole distance from goal to goal, one whole yard, one whole foot we can rename the measures fractions by referring them to some greater whole, but then we need to have the greater whole first before we can measure a fraction. We must walk the entire football field first before we can conceive of a whole yard being 1/100th.
Infinity is like that. It cant refer to a physical, identifiable thing or be contained in an object. It can only refer to numbers, which are concepts. There are no infinite series of steps.
However, it is difficult to see how one could make much metaphysical hay out of Thompson's Lamp. Unlike Zeno's thought experiments, which deal with examples of ordinary motion, Thompson's Lamp is blatantly unrealistic. It is not like anything in this world. Its "motion" is not even piecewise-continuous. So what if it's indeterministic? Who said that it must be? Neither intuitions nor physics are of much help here.
The most one can do here is argue that while, of course, Thompson's Lamp is not physically possible, it is possible in a "metaphysical" sense. And at the same time, its indeterminate state is not possible in that same sense. But that argument would be hard to put across (and I haven't seen anyone seriously try). Just what kind of goofy sense of possibility slices the problem in just such a way? And why would we be obligated to take it seriously?
Continuous motion suffers from the same problem. We can imagine sensors at each successive half way point that when passed turn a lamp on or off. Is the lamp on or off when we finish our run?
The simple solution is to say that motion isnt continuous. Discrete motion at some scale is a metaphysical necessity.
Quantum gravity and quantum spacetime are viable theories so its not a hard pill to swallow.
If I say that Hesperus is Phosphorus, I am saying that they are the same object (heavenly body). If I say that Ringo Starr is Richard Starkey, I am saying that Ringo Starr is the same person as Richard Starkey.
If I say that Venus is Venus, I say nothing at all. But even then, it makes a big difference whether you are talking in a context in which Venus is a goddess or a planet.
Quoting Fire Ologist
I suppose you can. But then I can define as a whole anything I like. A spoonful of sugar. A rainbow. Six inches of two-by-four. The distance between my front door and the shop on the corner. What counts as a part is defined in relation to that. But each part is a whole in its own right. The leg of a chair. The branch of a tree. The handle of a door. Half of a penny. It's just a convenient trick of language.
MindQuoting Fire Ologist
"Grabbed" from the physical world is a completely inappropriate metaphor. Nothing is grabbed. Something was defined. In any case, if the whole thing was "grabbed from the physical world", it follows that both halves of it were "grabbed". If they weren't, nothing was "grabbed".
Quoting Michael
The simple solution is to recognize the difference between an analysis and a dissection. A dissection physically separates an object into separate parts (and the parts then become wholes in their own right). An analysis has no physical impact on the object at all. One can analyse a distance into metres, centimetres, millimietrs or yards, feet and inches or any other units you like. You can analyse it into any fractions you like. All at the same time. The object doesn't change.
If I ask you what the minimum unit of space is, I can analyse that distance into fractions, however small it it is. Whether I can physically divide an object into those fractions is another question.
I don't see how it's a problem for continuous motion that you can imagine something else taking place alongside of it.
Quoting Michael
Solution to what? I have yet to see a problem in need of a solution here.
The problem is that if motion is continuous and if the sensors are set up as stated then the lamp can neither be on nor off after the run is completed, which is a contradiction.
One or more of the premises is necessarily false. So either motion is not continuous or we cannot set up sensors at an infinite succession of halfway points. The latter would seem to suggest that there arent an infinite succession of halfway points and so would entail the former anyway.
That's not the case. "Object" is not implied. You are simply saying that whatever it is that the two names refer to, it is one and the same.
The issue with "=" in mathematics is that the meaning of, or what is referred to by, "2+2" is not the same as what is referred to by "4". Someone might stipulate by axiom, that these two do refer to the exact same thing, but that does not reflect the way that the symbols are commonly used in the application of mathematics. In this way the axiom would be false, in the sense of a false definition. That's why such an axiom is misleading. This is also the problem with formalism, what is stipulated by the formalist is not consistent with the way that logic is applied, therefore it is a false description of logic.
Quoting Ludwig V
Again, this is not true. When you say Ringo Starr is Richard Starkey, all you are saying is that these two names have the same referent. It is only upon analysis, if one seeks to determine whether it is true or not, or something like that, that one would determine that the two names both refer to a person.
You are just restating - reimagining - Thompson's Lamp thought experiment, which has nothing to do with continuous motion as such (and repeating once more your baseless conclusion).
Im using Thomsons lamp to show that continuous motion entails contradictions.
Quoting SophistiCat
Its not baseless. Ive explained it quite clearly here and here and in many other comments.
Well, as far as I can see, you haven't done either: you haven't demonstrated any contradictions in TL, nor linked it to continuous motion. But I am not going to wade into that trainwreck. I chimed in to comment on the TL. You can engage with that if you want, or leave it alone.
A single thing that can be grabbed is defined as you say as a unit. A single thing. Like one whole step.
So now we have conceived of the unit. Weve defined it as 1. As a whole.
Only now can we posit or define infinity. Only now can you keep the conception of the unitary whole and define half. Only once you have a single unit can you add to that unit more units infinitely. But at each step, if you refer to the physical thing, you have a finite number of units. And you cant posit or define or conceive of half without reference to half of some other defined, conceived thing, and that thing must be a whole unit.
There is no infinity apart from the mind that conceives it. There are things apart from the mind that conceives of the unit.
There is no infinite thing to begin with. Only unitary wholes. And infinite staircase is an infinite finite unit - a square circle. There is no infinite thing, so pondering the paradoxes that arise from traversing an infinite distance, or descending and infinite staircase misapplies infinity to unitary whole, single, definable things.
Infinity applies to numbers. Numbers arent physical things, like stairs.
Thomson does that himself in his paper. I am defending his paper and explaining why Benacerraf's response to it fails. See here where I first brought it up.
Perhaps you could explain which part of my (or Thomson's) reasoning you reject? You're a coder so perhaps you could even address the code here. Simply saying "it's wrong" is hardly a meaningful criticism. If that's all you have to say then I will simply reply with "it's right".
Quoting SophistiCat
I did so in the post yesterday. Just as if we push a button an infinite number of times within two minutes the lamp can neither be on nor off after two minutes, if we run through an infinite succession of sensors when running a mile the lamp can neither be on nor off after finishing the run.
Given that the lamp must be either on or off, this is a contradiction, and so therefore it is proved that one cannot have run through an infinite succession of sensors.
A sequence is a function whose domain is an ordinal. If the domain is finite, then the sequence is finite. If the domain is countable, then the sequence is countable. If the domain is w (read as 'omega' the set of natural numbers), then the sequence is denumerable. If the domain is uncountable then the sequence is uncountable. Sometimes we omit 0 from the domain so that there are sequences whose domain is the set of positive natural numbers.
Let t be the following sequence whose domain is the set of positive natural numbers:
t(n) = 0 if n is odd
t(n) = 1 if n is even
t is a mathematically defined sequence. It is not required to express its values per some other arithmetical formula.
t has no greatest member in its domain, there is no last value for t, and t does not converge.
Let x be any mathematical value. Let s be the following sequence whose domain is the set of positive natural numbers along with w itself (the domain of s is {n | n is a positive natural number or n = w}):
s(n) = t(n) if n is a natural number
s(n) = x if n = w
s is a mathematically defined sequence. It is not required to express its values per some other formula.
s has a greatest number in its domain, and the last value for s is x.
This is a way of saying that for any x, there is a sequence in which x is the last value.
Understood.
Quoting Michael
The lamp is not physically possible. Supertasks are not physically possible per currently known physics. Surely we agree on these two things. Yes? No?
Quoting Michael
Well this is exactly the point I'm making. Why do we get to introduce a nonsensical premise such as a lamp and a switching circuit that can change states in arbitrarily small intervals of time, in contravention of the principles of quantum physics and electrical engineering?
My point is that once we've entered the realm of speculative fantasy, where do we stop? I say the lamp switches in arbitrarily small intervals of time, AND turns into a plate of spaghetti.
And you say ... what? That it CAN violate some laws of physics, but CAN'T violate others? Why?
Quoting Michael
Suppose that I accept this additional stipulation to the problem. Then I say it's on at two minutes. Or I say it's off at two minutes. Both are consistent with the premises of the problem; and neither is to be preferred, since neither are the limit of the sequence. You can make it anything you like. I use the spaghetti example to illustrate the arbitrary nature of any stipulated terminal state. One answer is as good as any other.
Quoting Michael
Now you (and Thompson) are trying to reason logically about a hypothetical situation that is entirely fictional, namely a lamp that switches in arbitrarily small intervals of time. You accept that, and then try to flim-flam readers by appealing to readers' experience with actual lamps.
But the Thompson lamp is not an actual lamp! So you can not reason about it as if it were the lamp in your living room. There is no button, there is no bulb, there is no electric bill to be paid at the end of the month. There is only an entirely fictional situation that you are pretending to reason about as if it weren't entirely fictional.
Quoting Michael
No. Not so. The terminal state of the lamp has nothing at all to do with what has come before. It's just like the mathematical sequence 0, 1, 0, 1, ..., to which I arbitrarily assign a terminal state (a state at the point at infinity) of 47. It's arbitrary. It's legal. It violates no laws of God or man.
Quoting Michael
You are as wrong as can be. I have explained this to you muliple times.
It's not a real lamp. It is not constrained to buttons and it's not plugged into a wall and connected to the electric company. It's a fiction.
Tell me this. Did Cinderella's coach turn into a pumpkin or not?
Do you not see that Thompson's lamp has the exact same ontological status as Cinderella's coach? It's a fairy tale. It's silly to try to reason as if it were a real thing, subject to the laws of this world.
Yes, and now I hope I did not overstate my case. I'm making an abstract philosophical point. The main thing is that the terminal state (the state at the point at infinity) bears no necessary relationship to the sequence that precedes it. But if we are given a sequence that does happen to have a limit, we can generally determine what the limit is. I don't recall how we got started on this, but the lamp sequence doesn't have a limit so the terminal state has no natural answer.
Pretty much sums up this thread.
We stop at the single issue being discussed: performing some action at arbitrarily small intervals of time. So taking the code here we assume that each line is run in an instant with the exception of the wait i *= 0.5 line which waits for the specified time in seconds. The logic of the code still behaves exactly as we would expect. Thomson is asking us what is output when echo isLampOn runs.
If your only solution is to insert the line isLampOn = 'a plate of spaghetti' after while (true) { ... } and before echo isLampOn then you are not answering the question as posed.
If you cannot make sense of the echo isLampOn line without inserting some arbitrary code before it then you must accept that it doesn't make sense for while (true) { ... } to complete. The arbitrary code you are trying to insert is a smokescreen to disguise this impossibility, exactly like your magic turning the lamp into a plate of spaghetti.
Yes, but once you have defined your half, you can treat it as a unit and define a half of a half... and repeat indefinitely. What limits that process?
Quoting Fire Ologist
That's certainly where are problems are. But you need to state this carefully. For example, there are no infinite natural numbers and while numbers are not physical things like stars, they do apply to physical things. The tricky point is that the idea of infinity is embedded in the number system, not some accidental additional property.
Quoting Fire Ologist
Infinity is certainly not a concept and not a physical entity - I doubt that it should be called an entity at all. I would love to know what things apart from the mind "conceive of the unit".
Quoting Metaphysician Undercover
Ostensive definition can only work if you know, or can work out from the context, what kind of thing (category) is being defined. When you gesture at a red car and say this is red, you will misunderstand if you take red to mean a car or a wheel or a heavy object.
Quoting Metaphysician Undercover
It's not enough to know that the two names have the same referent. You need to know, in Wittgenstein's phrase, where the referent "is stationed in the language".
Quoting fishfry
I think that there some rules that apply in fiction (imaginary stories), because the story needs to have plausibility. But I don't know how to work out what they are. Coleridge, I think it was, said that there needs to be a "suspension of disbelief" for any fiction to work. The reader/audience needs to co-operate and not ask awkward questions. But there are limits. There needs to be some realism for the story to be recognizable at all.
Quoting TonesInDeepFreeze
I don't quite see why x is the last value, nor why you think that defining the set in that way gets round the point that w is not derived from the criterion from which all the other numbers in that set are derived.
Exactly! You have to take the thing you call a half as a single whole unit before you can take some measure again. A half is just a measure conceived of after there is a unit. Only unitary whole things can be touched or stepped on, like a step.
I'm missing something here. When I step on a step, do I step on the whole step, or just a part of it. When I sit on a chair do I sit on the whole of it. It depends how you interpret the words, that's all.
I merely defined two different functions with two different domains.
x is the last value in the sense that it is the value of the last member of the domain of s where the domain of s is ordered in the usual manner for an ordinal.
The function s is the union of the function t with {
If, as suggested, the thought experiment is a kind of fiction, then we see the fiction we're told doesn't mention anything about a last moment in the sequence of moments, so the fiction does not disallow us from extending to another fiction in which there is a last moment and such that the value of the action at that last moment is whatever x we want it to be. That is, at all the moments mentioned in the fiction, the lamp is on or it's off. That doesn't preclude another fiction in which there is a last moment in which any number of things can be the case: (1) The lamp is on, (2) The lamp is off, (3) The lamp explodes into bits and is off, or (4) The lamp expands to the size of the sun and is on, or (5) the lamp transforms into a pepperoni pizza.
[RETROACTIVE EDIT: The above paragraph is incorrect. Whether explicitly or implicitly it is understood that Thomson's lamp requires that the lamp is on/off at time t only if it at some before time before t it was off/on, then the button was pushed on/off and not off/on again before time t. Therefore, another fiction in which that is not the case is not a fiction that addresses Thomson's lamp. This edit also carries forward for any other similar comments I might have made.]
/
Quoting Ludwig V
At least in mathematics, we have the adjective 'is infinite' so that there is not a set called 'infinity' but rather many sets having the property of being infinite. (There are also such things as points of infinity in different number systems and things like that; but this is different from the notion that there is one particular object in mathematics called 'infinity'.)
Meanwhile, in other fields of study, such as philosophy, writers do speak of a concept of infinity. I am not opining here whether, despite philosophical discussions of infinity, infinity is or is not a legitimate concept.
/
Regarding another poster:
Ringo Starr is not a name. Richard Starkey is not a name.
'Ringo Starr' is a name. 'Richard Starkey' is a name.
The denotation of 'Ringo Starr' = the denotation of 'Richard Starkey'
Ringo Starr = Richard Starkey
Ringo Star is Richard Starkey
The sense of 'Ringo Star' not= the sense of 'Richard Starkey'
'Ringo Starr' not= 'Richard Starkey'
'Ringo Starr' is not 'Richard Starkey'
Im just saying the notion of an infinite staircase is impossible to conceive as steps and groups of steps are unitary wholes, and infinity never unifies or finishes multiplying. You cant apply infinity to finite things. There is no infinite number of steps between the 1 yard line and the 2 yards line. There is a single yard. You can mathematically take the single yard and mathematically divide it in half, and take one of the halves and divide it infinitely. But that has nothing to do at all whatsoever with taking a one yard physical step on a football field.
Can you answer this question that I asked earlier? What is the difference in the ontological status between Thompson's lamp and Cinderella's coach?
Quoting Michael
But it's not my only solution. I've said (several times) that "Lamp is on" and "Lamp is off" are also valid solutions. Just about anything in the universe, physical, abstract, or fictional, is a valid solution.
After all the times I've explained this to you, I don't mind if you say, "You're wrong." But how on earth can you say that MY ONLY SOLUTION is X? I have already said that ANYTHING AT ALL can be a solution; and that if you insist that on/off are the only legal solutions, then either one is valid.
I don't mind having a difference of opinion, but you seem to have not read a single thing I've said.
Quoting Michael
You accuse ME of magic! YOU are the one with the lamp that switches state in arbitrarily small intervals of time. I'm playing by YOUR rules, which clearly allow magic.
Of course I have already made that point to you many times as well.
You've solved my problem. I need the willing suspension of disbelief to converse with @Michael. Indeed, that's the question I asked him. What's the difference between Thompson's lamp and Cinderella's coach? Why am I supposed to treat the lamp as if it has a button, or is operated by a computer program (that likewise can wait for arbitrarily small time intervals), or is subject to some rules of rationality but not others?
I would gladly put these questions to Thompson, but Wiki says he died in 1984. I dare say that if there's an afterlife, he is now eating his plate of spaghetti.
Modeling is not physical, so the models built with infinity will never pose a problem when descending stairs. There is no paradox because the paradox seeks to mix actual stairs with modeling.
The lamp is on only if the button was pushed to turn it on, prior to which the lamp was off. Even if you want to introduce magic it is on only if magic turned it on, prior to which the lamp was off.
So if you want to say that the lamp is on after two minutes then you must accept that at some final time prior to two minutes the lamp was off. This is a simple logic fact.
Except the supertask doesn't allow for this. So, as Thomson argues, the lamp cannot be on after two minutes. And for the same reasoning cannot be off after two minutes. This is a contradiction and so the supertask is proven impossible in principle.
According to what principle of physics?
If the lamp can violate the known laws of physics, what are the limits of its magic?
Quoting Michael
But I didn't. YOU (and Thompson, who attained his terminal state in 1984 and is now enjoying his spaghetti) introduced magic. I'm just playing by your rules.
Quoting Michael
Not at all. I've given many counterexamples. The sequence 0, 1, 0, 1, ... has no limit. If I define its terminal state (that is, its value at the point at infinity) as 47, that's perfectly legal. And as sensible as any other terminal state.
Quoting Michael
Of course it does, and I have repeatedly explained how.
A supertask is not simply an infinite sequence of numbers.
In our hypothetical scenario with hypothetical physical laws we are still dealing with the ordinary logic of cause and effect.
It is implicit in the thought experiment that it is only by pushing the button that the lamp is caused to turn on and off, but strictly speaking this premise isn't necessary as the logic applies regardless of the cause even if it's magic.
If the lamp is on then something caused it to turn on, prior to which it was off. If it is turned on then it stays on until something causes it to turn off.
Given this, if the lamp is on at t[sub]1[/sub] then either:
a) it was turned and left on prior to t[sub]1[/sub], or
b) it was turned and left off prior to t[sub]1[/sub] and then turned on at t[sub]1[/sub]
But as Thomson says, "I did not ever turn it on without at once turning it off ... [and] I never turned it off without at once turning it on", and so both (a) and (b) are false. Therefore the lamp is not on at t[sub]1[/sub]. Similar reasoning shows that the lamp is not off at t[sub]1[/sub] either.
I'm afraid there was a typo in my last post. I posted "Infinity is certainly not a concept", which is rubbish. I meant to post "Infinity is certainly a concept". Apologies.
Quoting TonesInDeepFreeze
Quite so.
Quoting Michael
I'm not at all clear how the ordinary logic of cause and effect would apply in the context of hypothetical physical laws. But we are clearly not dealing with the ordinary physical world, and that leaves us free to imagine anything at all.
Quoting Fire Ologist
The first sentence is fine. I don't get the second sentence. You seem to be saying that the paradox is real. But mixing up actual stairs with models of stairs just produces a confusion, so the paradox is just an illusion - in my opinion.
Quoting fishfry
Yes, I realized that and was hoping to produce a formulation that would allow a more constructive discussion.
Quoting Heracloitus
I don't know about Bergson. I think it is clearer to distinguish between "physical division (separation)" and "mathematical division (analysis)".
Quoting fishfry
There are two cases in play at the moment - "0, 1, 0, 1, ..." and "1/2, 1/4, 1/8, 1/16.." Comments switch between them without always being clear. You are, however, quite right that the first sequence doesn't have a limit and the second one has what we could call a natural limit.
If it's on at t[sub]1[/sub] then either it was left on before t[sub]1[/sub] or it was left off before t[sub]1[/sub] and then turned on at t[sub]1[/sub].
This is a straightforward logical point that does not depend on what the physical laws are.
To make it very simple, Thomson's lamp proves that these premises are logically inconsistent:
P1. The lamp is turned on and off only by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t[sub]0[/sub]
P5. The button is pushed at successively halved intervals of time between t[sub]0[/sub] and t[sub]1[/sub]
P6. The lamp is either on or off at t[sub]1[/sub]
So Thompson's lamp is not merely physically or metaphysically impossible, but logically impossible. Quoting Michael
But which is not defined.
It's more than that; the lamp can't be on and can't be off, even though it must be one or the other. This is a contradiction, and so therefore the supertask is proven impossible in principle.
Does the difference matter?
Exactly. There is no paradox caused by an infinite staircase, because an infinite staircase is a square circle, barely conceivable if conceivable at all.
What if every time I bit an apple and ate it, in the time I chewed the bite, the apple grew in size bigger than the bite I just took? Do I need to figure out the math here to see how to prevent the apple from eating me?
Quoting keystone
Staircases are always, and only, actually, finite, as any object is. The endlessness of the staircase is brought to an end at the bottom, so it is not endless, so there is no sense to the word despite.
This is not a paradox, but a confusion of concepts (like the number 1 or infinitely) with actual things (like a one step down one stair, and never reaching the bottom or doing so in a minute).
:up:
I know I'm right :-) Now if I can just get @Michael to agree!
I have always agreed that the sequence "0, 1, 0, 1, ..." does not converge.
I disagree with your claim that with respect to Thomson's lamp we can simply stipulate that the lamp is on after two minutes. See my previous post and my initial defence of Thomson on page 13.
@fishfry will speak for himself. But I think the point is that, even a convergent sequence, which does have a limit, does not have a end or last step defined - indeed, is defined as not having one. That means that any answer whatever is equally valid and invalid.
That's not true, as I explained here, and as I alluded to above. It is not just the case that whether the lamp is on or off after two minutes is undefined but that the lamp cannot be either on or off after two minutes.
As Thomson says in his paper, "the impossibility of a super-task does not depend at all on whether some vaguely-felt-to-be associated arithmetical sequence is convergent or divergent."
I'll think about that.
I dont understand. How do you ever arrive at the two minute mark?
1 minute, half a minute later, quarter minute later than that, etc., infinitely you never arrive at the two minute mark.
Like the endless bottomless staircase that for some reason had a bottom with a dead guy.
??? Two minutes and a bottom step subvert the issue and banish infinity from the math of it.
These are manufactured conundrums mixing what is an actual occurrence (walking down a step, turning a light on or off) with what is not an actual occurrence but a conceptual/mathematical idea that has no weight or influence on walking or flicking light switches.
There no paradox because the lamp cant be on or off. There is just an endless motion (invented in a mind that can invent things and objects that cant actually be built and tested).
If you are going to ponder whether the light is actually, physically on, you have to ponder whether a switch can be switched as rapidly as you would have to switch it as you approached two minutes. The light might be off because you broke the switch. I know, to a mathematical/logical student I am missing the point, but then, if we can assume functioning switches to lay out the mathematical intrigue, why cant we assume the mathematical intrigue of never reaching two minutes and therefore never able to determine an on/off state at the two minute mark? And conclude that such a scenario builds a lamp that will endlessly turn on and off at ever increasing speeds with no limit or single state thereby enlightening or darkening said limit.
As you approached 2 minutes with really small fractions (but never got there if you are being consistent), would the speed between switching the light on and off eventually approach and then exceed the speed of light? (Thats one solid switch!). So how would we be able to tell whether the light we saw was the light from the latest switching motion or light from a few switches ago? Wed need a new measure besides our eyes, and could not watch the experiment in real time anymore.
We cant mix real things with infinity as a limiter. Real things always include finite limits. The positing of infinite physical steps, or infinite half steps, pose logical perplexities, but not actual paradoxes.
I don't know what you mean by "arrive" at the two minute mark. Two minutes just pass. That's how the world works.
Imagine I am facing a clock with my back to a lamp. The experiment starts at 12:00. Through some automated process the lamp turns on and off at successively halved intervals of time. When the clock shows 12:02 I turn around. Is the lamp on or off?
Haha! Ok, so the switch clicks at 1 minute, then it clicks again at 30 seconds more, then it clicks again at 15 seconds more, then again at 7.5 seconds, etc.
What will the switch and light be doing at 2 minutes?
Nothing in relation to the switch as described above. Something beyond the parameters the initial scenario set in motion.
It may be on. It may be off. It will probably be broken or melted. But in relation to the switch described above those possible outcomes have no bearing because the switch was designed to serve a function before two minutes time can elapse.
If the light clicks on at two minutes, or off, either the switch malfunctioned, or it can do more then switch itself at half intervals of the prior lapsed time.
When the clock shows 12:02 I turn around. Is the lamp on or off?
Whichever it is, it wouldnt be a function of a switch that operates by waiting half the time to switch and then half of that time to switch again, and then half of the half, etc, because such a switch would have no direction or programming to follow at two minutes, because it is designed to operate before two minutes could pass.
The lamp is off. After one minute the lamp turns on. Is the lamp on or off after two minutes? It's on because it was turned on after one minute and then never turned off again.
If you can't apply the same reasoning to the lamp having turned on and off an infinite number of times before the end of two minutes then you must accept that it makes no sense for the lamp to have turned on and off an infinite number of times before the end of two minutes.
No, thats a different equation. That switch is more easily predicted. The switch on Thompsons lamp requires some serious calculation to determine its state after say one minute fifty-nine seconds. It can be calculated though. But it cant be calculated at 2 minutes time, because it was designed to function within two minutes. Once two minutes lapse, the switchs function has no relationship to whatever would be the state of affairs (which Im telling you would likely be a fire, so in a sense, Im guessing the lamp would be on).
Quoting Michael
Id say we never had a paradox. There is no paradox at 2 minutes, because by the time we get to 2 minutes, the initial switch and algorithm it functioned by are no longer in play, unable to be held at odds with whatever might be the case at 2 minutes.
The lamp is off. It turns on and off only as described:
Scenario 1
The lamp turns on after 1 minute.
Is the lamp on or off after 2 minutes? It's on.
Scenario 2
The lamp turns on after 1 minute and off after a further 30 seconds.
Is the lamp on or off after 2 minutes? It's off.
Scenario 3
The lamp turns on after 1 minute, off after a further 30 seconds, and so on ad infinitum.
Is the lamp on or off after 2 minutes?
In all three scenarios the switch is "designed to function within two minutes."
Quoting Michael
Not if you want to answer the question in scenario three. Or more precisely, not designed to function at or after two minutes.
That's also true about the first two scenarios neither switches after 1 minute and 30 seconds and yet we can still answer the question about the lamp after 2 minutes, and even after 2 years.
So what are we disagreeing about?
Whether this is paradox, or whether there is an answer to question 3?
I dont think there is an answer to question 3. Because the switch is not designed to ever present the question.
I don't know what you mean by this.
Given that the lamp must be either on or off after two minutes we must ask the question. If you cannot provide a coherent answer then you must accept that your premise that the lamp has turned on and off an infinite number of times is necessarily false, and so that supertasks are metaphysically impossible.
How is that? How is it on or off at or after two minutes?
It cannot be a function of a switch that operates by switching every half of the prior interval. Some other function needs to be introduced into the picture to ask about the state of the lamp at 2 minutes or beyond.
Why is it given by this switches function that the lamp will be anything in particular at or after 2 minutes? You are just assuming something exterior to the premise about time and lamps.
In that case, if the lamp must be on, or off, at two minutes, is that state caused by the switch?
Because it's a lamp. If it exists at 12:02 then it's either on or off, and it exists at 12:02.
Quoting Fire Ologist
And that's precisely why supertasks are impossible.
We are imagining, for the sake of argument, that we are in some alternate universe with physical laws that allow us push a button to turn the lamp on and off at successively halved intervals of time. We want to know what happens to the lamp if we do this (and only this).
As you say, we don't get to "just [assume] something exterior to the premise about time and lamps."
So the only premises we are allowed to work with are:
P1. If the lamp is off and the button is pushed then the lamp is turned on
P2. If the lamp is on and the button is pushed then the lamp is turned off
P3. The lamp is off at t[sub]0[/sub]
P4. The lamp is either on or off at t[sub]1[/sub]
P5. The button is pushed at successively halved intervals of time between t[sub]0[/sub] and t[sub]1[/sub]
So we ask: is the lamp on or off at t[sub]1[/sub]?
If you cannot provide a coherent answer then you must accept that the premises are inconsistent. Given that there's nothing problematic about P1 - P4, we must accept that the problem lies with P5. It is necessarily false. Supertasks cannot be performed.
And thats precisely why the question of whether the lamp will be on or off at two minutes will never present itself.
If you want to talk about how lamps and intervals of time work in an alternate universe why are we using any premises from this universe about lamps and so on?
The three stooges were hungry and looking for ideas on what to eat. Larry says, if we had ham, we could have ham and eggs, if we had eggs. Mo hits him on the head.
If we want to ponder what a lamp will be doing at two minutes time, we could just hook it up to a switch that switches every half of the interval of time prior, if we had a universe where such a switch existed.
This conversation (and not your fault) is starting to feel like a descent on a bottomless staircase.
It does present itself because the lamp must be either on or off after two minutes as per the law of excluded middle.
We want to know what would happen to the lamp if we were to push the button to turn it on and off at successively halved intervals of time within two minutes. Nothing else happens to the lamp except what we cause to happen to it by pushing the button.
Is it left on or off after we stop pushing the button? If you cant answer this then you must accept that supertasks are metaphysically impossible.
Then stop talking about at two minutes or after two minutes. Thats some other scenario.
Dont you see that?
Two minutes or more is outside the universe of at successively halved intervals of time within two minutes.
The question is the lamp on or off at or after two minutes is talking about some other lamp.
Maybe you are really picturing a person pushing a button? Do you think you could push a button to work the premise of switching the lamp at successive half intervals? A lamp switch like this would have to be automated, and it would not need any instructions or programming at or beyond two minutes to function.
I feel like I just suggested ham and eggs again.
That's what I've been arguing since the beginning of the thread. And the reason why it can't be on or off is that in the described 'possible world' two minutes cannot pass. So the possible world in which "after two minutes" makes sense is incompatible with the possible world of that lamp.
Quoting Fire Ologist
You are exactly correct. In the described scenario, the stipulated possible world, the two minute mark cannot be reached. It's a Zeno-type argument. Michael insists that two minutes will pass, but that's a reference to a different, incompatible, possible world, the one designed around our empirical experience.
Quoting Michael
See, you are referring to a different possible world here, the one derived from empirical experience. But this possible world is inconsistent with the one that the lamp is in.
Quoting Fire Ologist
Believe me Fire Ologist, I tried my best. Two months later Michael is still stuck in the same infinite loop.
I posted a bit earlier in in the thread, then noticed recently it was up to 23 plus pages so I was curious if everyone had gotten to the bottom of the staircase.
But found out we are still counting the same steps over and over again!
I think Ill give it another two minutes then Im done.
Possibly done for an infinite number of minutes.
At that point I might post again apparently.
Ok. Agreement is good. I appreciate that.
I hope you agree that building a mathematical model of the problem is a valid way to approach it.
I do recognize that "a supertask is not just an infinite sequence," as you've often told me. I agree with that too.
But if you will accept that "0, 1, 0, 1, ..." does not converge," I can work with that for a start.
Quoting Michael
I get that. That's where we disagree. I'll try to clarify my argument the best I can.
Quoting Michael
Coming up ... at least just your previous post, not necessarily going back through the thread. This is page 24, that's a long time ago. Is there something I should see in there that you haven't said recently?
Okm to your previous post.
Quoting Michael
I agree with that.
Quoting Michael
I understand your point of view. But here is no cause and effect between the oscillating lamp sequence, and its arbitrarily defined terminal state.
Quoting Michael
Ok, fine. I'll work with you on that. Let me replace your earlier 60-second model with a 1-second model. That is, there's a final time -- let's call it midnight, in homage to Cinderella's coach. Let's say the light is on at midnight minus 1 second; off at midnight minus 1/2 second; on at midnight minus 1/4 second.
In general, it's on at [math]\text{midnight} - \frac{1}{2^{n}}[/math] when n is even, and off when n is odd, for n = 0, 1, 2, 3, ... That is, the lamp is on at the even-exponent inverse powers of 2, and off at the odd-exponent inverse powers of two.
So we have an easy way to know if the lamp is on or off. For example, what is the lamp state at midnight minus 1/128 seconds? Well, 128 is 2 to the 7th power, and 7 is odd, so the lamp is off. So far so good I hope.
Quoting Michael
Well no, this is exactly the spot where you're wrong. Or being more charitable, the spot at which we disagree.
The mathematical model that we've just agreed to (assuming you agree with my notation) defines the state of the lamp at every inverse power of two seconds before midnight.
Say I claim the lamp is on at midnight. If you pick some earlier point, we can determine if the lamp is on or off.
But so what?
Two things are true: The lamp state follows the rule we've given; and the lamp is on at midnight.
That's perfectly fine. There is no contradiction I think you should be much more clear that you think that's impossible.
Likewise, the state at midnight could be off. That too, is perfectly consistent with our lamp state algorithm that works for any point BEFORE midnight.
Quoting Michael
No it's never "left off" or "left on" prior to midnight. Whatever is its state at [math]\frac{1}{2^n}[/math] before midnight, it will have the opposite state at [math]\frac{1}{2^{n+1}}[/math] before midnight.
None of that has any bearing whatsoever on the terminal state, which can be anything we like, because, since the on/off sequence has no limit, there is no natural final state.
This "leaving on" or "leaving off" idea is not founded on anything. It doesn't even make sense. It's a consequence of failing to think clearly about the problem.
Quoting Michael
So what?
Quoting Michael
On the contrary. I just showed that it's perfectly sensible for it to be on OR off. Either way is consistent with what's come before; and neither is to be preferred.
I hope you can come even a little ways toward my view. Before midnight, it's an infinite sequence. It has no end. It's never left on or left off. Whatever state it's in, it flips state at the next inverse power of 2.
The basic confusion is not understanding that an infinite sequence has no end. The state at midnight is arbitrary and is not related to what's come before, even under your pretend physics.
In online discussions of infinity, someone is bound to come along and say, "Infinity is a concept." Which always makes me say, So What? 3 is a concept. 47 is a concept. Civilization is a concept. The rules of the road on the highway are a concept. Saying infinity is a concept doesn't actually tell us anything about it. I don't even know what it means to say infinity is a concept. Is there anything that isn't a concept?
Quoting Ludwig V
I gave it my best shot just now. cc @Michael
You totally get it. @Michael's turn :-)
I understand that it has no end. That is why I am arguing that it is metaphysically impossible for an infinite succession of button pushes to end after two minutes.
[math]0, 1, 0, ..., 1\\0, 1, 0, ..., 0[/math]
Such sequences may make sense in the context of abstract mathematics but they do not make sense in the context of a lamp being turned on and off.
As a comparison, even though imaginary numbers have a use in mathematics it is more than just physically impossible for me to have [math]\sqrt{\text{-}1}[/math] apples in my fridge; it is metaphysically impossible.
No pretend physics can allow for me to have an imaginary number of apples in my fridge and no pretend physics can allow for the above two mathematical sequences to model the state of a lamp over time.
With Thomson's lamp, these are our premises:
P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t[sub]0[/sub]
P5. The button is pushed at least once between t[sub]0[/sub] and t[sub]1[/sub]
P6. The lamp is either on or off at t[sub]1[/sub]
And these are our conclusions:
C1. If the lamp is on at t[sub]1[/sub] then the button was pushed to turn it on, prior to which it was off
C2. If the lamp is off at t[sub]1[/sub] then the button was pushed to turn it off, prior to which it was on
C3. The button is pushed [math]n \in \mathbb{N}_1[/math] times between t[sub]0[/sub] and t[sub]1[/sub]
These conclusions prove that a supertask is not performed.
Yes, of course you are right. It is no more help than saying that a table is an object. I was trying to re-instate the line between, let us say, a mathematical reality and a physical reality - or between mathematical possibility and physical possibility. (I think we are agreed that what creates the difficulties here is the confusion of the two in the definition of the supertask.)
Quoting Michael
I suggest that what creates the problem here is the idea that the mathematical induction is a process that takes time. Perhaps the temptation to do this derives from the analogy with a Hume's inductive process that creates so many issues about empirical laws or generalizations. But once you pose the challenge of actually executing the process to the bitter (and non-existent) end, or think that you can stipulate what happens at the end, you are enmeshed in contradictions. (Achilles' race with the tortoise has exactly the same issues, but in the medium of space rather than time.) This leads us to think that there is some sort of miracle involved in arriving at the fridge to get a beer. But the mathematical induction is one analysis of many that can be applied to either space or time, and does not in any way affect our walking about our kitchen or arriving on time at a party.
There's no miracle. Motion isn't continuous; it's discrete.
Your argument doesn't prove that.
Others do, like Zeno's and Bernadete's.
Yes there is a miracle. The miracle is the start, the initial act, or force, which breaks the previous inertia, causing the new motion toward its end (in this case the beer in the (fridge).
In the case of the op, and also the lamp, the defined start is a rate of acceleration which will continue without an end. This is why it confuses it is acceleration without end. That is not an unusual way to define the effect of a force though, gravity is defined in this way (9.8 metres per second squared). The universal law of gravity describes an acceleration without end.
The commonplace nature of such a description, acceleration without end, makes it appear like there is no miracle involved with acceleration. But this is only because, in these commonplace activities, there is always contrary forces which negate the acceleration before the speed becomes infinite. There is commonly an end to acceleration. So this only serves to confirm our belief that there is no miracle involved with acceleration. However, the unintelligibility of examples like the op and the lamp, where no contrary forces are invoked, demonstrate to us, that acceleration truly is miraculous.
Exactly. You dont have infinity any more anywhere in the whole scenario if you want to talk about 2 seconds and what happens at 2 seconds or after.
What is infinite is the number of switching that would occur before two seconds. Once you are looking at second number 2, you have to ignore all of the premises and activity of the scenario. Midnight is not part of the conversation. Its a limit that will never be reached so the state of the lamp at midnight is indeterminable because the function of the switching will never be operating at midnight.
I think there are abstract things and concrete things. But physics these days pushes hard on the nature of physical things. Is there a philosopher in the house?
Who is pushing this button, by the way? And how? Magic again?
It's a premise of the thought experiment that there is a sequence of steps at successively halved intervals of time. You are now arguing that since such a premise is physically impossible, it's metaphysically impossible. You are wrong on the facts and wrong on the logic.
And its your premise, not mine.
Quoting Michael
It's a mathematical model of a lamp being turned on and off. What if instead of a lightbulb, we have a big LED that displays 0 or 1? You have an objection to that? Based on what?
I already showed you how to model the process using the inverse powers of 2. You ignored that, since it refutes your argument.
Quoting Michael
Tell that to an electrical engineer or quantum physicist, both of whom use imaginary and complex numbers as essential tools of their trades.
Quoting Michael
Wow. I guess you don't believe in negative numbers, fractions, or irrational numbers either. You are really flailing with these weak arguments.
Quoting Michael
You accept "on" and "off" but not one and zero? Again: replace the bulb with an LED that alternately flashes 1 and 0.
Quoting Michael
I must have made a pretty good argument in my previous post, since in your response you went into wild irrelevancies and repeated arguments I've long lost interest in, and never engaged with a word I said.
You really reject the complex numbers, negative numbers, fractions, and irrationals? Is mathematical nihilism your last defense?
I didn't say that imaginary numbers don't have a use. I said that I cannot have an imaginary number of apples in my fridge.
Quoting fishfry
No.
Quoting fishfry
Yes, and this premise is proven false. See refutation by contradiction and modus tollens.
Quoting fishfry
It doesn't refute my argument because it doesn't address my argument.
My argument is that the lamp cannot be on unless it is turned on (whether by pushing a button or by magic), as argued above.
Therefore, off, on, off, ..., on makes no sense in context. The lamp being off must precede the lamp being on. Given that this is inconsistent with the premise that the button is pushed at successively halved intervals of time it is proven impossible in principle to push the button at successively halved intervals of time.
Eh. You have a collection of apples in your fridge whose count is referable to by the symbol of a natural number. You also have an apple in your fridge, which if rotated 90 degrees counterclockwise, could have its current orientation referred to as i multiplied by its previous one. It's not like either number is somehow more physical than the other one, people just intuit the naturals (and positive computable reals) to really "be there" whereas the other numbers aren't.
Whereas they live in rotations and waves and the columns and rows of accounting books. None of those concepts have, to my knowledge, a baked in proclivity in our wetware. Whereas natural numbers, counting, cardinality, size comparisons etc and their symbolic representations do.
At the very least you've got work to do in "dephysicalising" or "physicalising" the intuitions regarding number and processes you have.
Im not claiming otherwise. Im only claiming that I cannot have an imaginary number of apples in my fridge.
That some number is sensible isnt that its sensible to use that number in any and every context.
Similarly, the sequence 0, 1, 0, , 1 might make sense in some contexts, but the sequence off, on, off, , on doesnt make sense in the context of a lamp being turned on and off by pushing a button.
Well, you said that as if it makes a point. Which escapes my simple mind.
Quoting Michael
If after all this, your point is that Thompson's lamp is a silly thought experiment, at long last we are in full agreement. It doesn't tell you much about supertasks, though. Perhaps I should have tried to expand on that line of argument earlier. Thompson's lamp is not a good example of a supertask, because the sequence doesn't have a limit or any natural termination point.
Zeno getting up from the couch to get a snack in the kitchen, under the assumption of arbitrary divisibility of time and space, is a much better example. I have argued that that is a true supertask, one that is metaphysically possible.
But I was thinking about your button presses. I don't see why the sequence doesn't run at every time prior to midnight; and then at exactly midnight, the Supreme Button Pusher doesn't just flip a coin and turn the lamp on or off accordingly. Makes as much sense as the rest of the story.
Quoting Michael
It (my inverse powers of 2 model) shows that at any time prior to midnight, we know the state of the lamp. And that at midnight, it's undefined and entirely arbitrary.
Quoting Michael
Ok. At midnight the Supreme Button Pusher flips a coin and turns the lamp on or off, accordingly. I don't see any problem.
Quoting Michael
Not at the termination point. Only prior to midnight. Not AT midnight. As I've indicated. It's fundamental to the nature of an infinite sequence with a termination point adjoined at the end.
Quoting Michael
No no no no no. "the premise that the button is pushed at successively halved intervals of time" is only valid for the infinite sequence itself, and NOT for the termination point.
If the Supreme Button Pusher turns the lamp on at midnight then it must have been off before He turned it on, because that's what "turning on" means. If He turned it off at midnight then it must have been on before He turned it off, because that's what "turning off" means.
For the entire continuous time that the lamp exists not just the time during which the button is pushed it being off must precede it being on, regardless of magic or divine intervention. As such the sequence off, on, off, ..., on makes no sense, much like having an imaginary number of apples in the fridge makes no sense.
We must accept that the button can only be pushed [math]n \in \mathbb{N}_0[/math] times between t[sub]0[/sub] and t[sub]1[/sub].
But also note P1 in the argument above, implicit in Thomson's argument. You dont get to just introduce God to deflect from the incoherent causal consequence of having pushed a button an infinite number of times.
Quoting fishfry
A supertask is just performing an infinite succession of tasks in finite time. Pushing a button, which just so happens to turn a lamp on and off, is as good a supertask as any.
But lets assume that pushing the button displays the time that the button is pushed persisting until the button is pushed again and is pushed (only) at successively halved intervals of time starting two minutes to midnight. What time is displayed at midnight?
The sequence approaches midnight but because the button is never pushed at midnight the display can never show midnight, but must show something because its never turned off. It always only displays the time that the button was last pushed, but in this scenario there is no last button push, entailing a contradiction.
As Thomson says, "the impossibility of a super-task does not depend at all on whether some ... sequence is convergent or divergent."
I don't think that either of them suggests that space or time may be discrete. In any case, you seem to accept that that's a different topic.
Quoting Michael
That seems to me to be a good diagnosis of the issue with supertasks. All that is then needed to free people from the illusion that walking to the fridge can be mathematically analysed in many ways, none of which affect physical reality.
Quoting fishfry
The trouble is that many philosophers seem to be hypnotized by physics, and seem to forget that physicists develop their theories and conduct their experiments in ordinary human reality.
Quoting fdrake
Spot on. The difference between analysis (in the head) and dissection (on the bench) seems obvious, but turns out to be quite difficult to trace in certain situations.
Just a passing thought on a thread that behaves in a similar fashion.
Well, physics got strange in the 20th century, leading to a lot of philosophy, good and bad. There's a philosopher of physics named Tim Maudlin whose videos were showing up on my YouTube feel a lot a couple of years ago, but not at all lately. He knows a lot of physics. Seems to be one of the few philosophers who does.
I understand your point. I really do. But you are expecting a limit point to have an immediate predecessor, when it does not.
Consider the sequence of times at which we "sample" the state of the lamp. I accept your point that the lamp exists continuously, so that (for example) if we want to know the state of the lamp at, say, 1/3 second before midnight, we would just note that at 1/2 second before midnight, it was off (I think we set it up that way earlier. On at 1 sec, off at 1/2, on at 1/4, all times before midnight.
So it's off at 1/2 sec before midnight, and it will stay off till we turn it on again at 1/4 sec before midnight.
Consider the infinite sequence 1, 1/2, 1/4, 1/8, ... Mathematically, it has the limit 0; and in the lamp puzzle, the limit 0 represents midnight; that is, zero seconds before midnight.
Now two things are true:
For any of the times in the sequence -- 1, 1/2, 1/4, etc. -- it has an immediate predecessor (except for 1 of course).
But 0 does not have an immediate predecessor.
It makes no sense to ask what is the state of the light "immediately before" time 0. Such a thing is not defined.
If you want to interpret this as meaning that the premises or conditions of Thompson's lamp makes it impossible; then I'm fine with that. I'm fine either way. I think you can just arbitrarily define to be the state at midnight as on, off, or turning into a plate of spaghetti.
But perhaps that's just another way to say that the problem is impossible. If so, I can live with that.
But tha does NOT mean that supertasks are impossible. It only means that THIS PARTICULAR supertask is impossible. There might be others that ARE at least metaphysically possible. I've argued this several times. A Zeno walk from my couch to my fridge is one such, under the (perhaps unrealistic, but at least metaphysically feasible) condition that time is modeled by the mathematical real numbers, infinitely divisible and Cauchy-complete.
Surely you must take my point. Even if the lamp is impossible, some other supertask might still be possible. Thompson's lamp is a rather poor example of a supertask, because its underlying sequence can not be made to converge to a limit.
Quoting Michael
I take your point, and for sake of discussion I'll agree that Thompson's lamp is impossible. I also think you can just complete it arbitrarily. But if you don't like that, then I'll agree it's impossible.
I do understand the point you think you are making. But you are just unhappy that the limit of a sequence has no immediate predecessor in the sequence, and that's just a mathematical fact that's true and that you don't like.
Quoting Michael
Ok. I'll agree for sake of discussion that Thompson's lamp is impossible.
Quoting Michael
No, it's a terrible example of a supertask, because it can not possibly be made to have a limit.
Consider a Zeno walk from my couch to the kitchen. First I go 1/2 the distance. Then I go 1/2 the remaining distance, and so forth. So at each step I've covered 1/2, 3/4, 7/8, ... of the distance.
After one minute (say I go half the distance in half a minute, etc.) I am in the kitchen.
Now this is the same exact problem as with the lamp, namely that the limit of the kitchen has no immediate predecessor, just as the limit 1 of the sequence 1/2, 3/4, 7/8, ... has no immediate predecessor in the sequence.
But ending up in the kitchen has the virtue of at least being the limit of the sequence. So it's a natural solution. And, under the assumption that time is like the mathematical real numbers, it's a true supertask. I performed infinitely many steps in finite time.
Quoting Michael
Midnight. What time does YOUR clock say at midnight? Of course the clock says midnight at midnight.
You are just psychologically uncomfortable with the fact that the limit has no immediate predecessor in the sequence.
Quoting Michael
I'll concede your point that the lamp is impossible. That does not necessarily entail that supertasks in general are impossible. Only that this one is. But there is no time immediately before midnight, for the same reason there's no inverse integer power of 2 immediately before 1. It's just how infinitely divisible sets work.
Quoting Michael
Oh, that is interesting. So I will take from this the happy conclusion that Thompson and I are thinking along exactly the same lines.
I don't know enough about supertasks to know if every supertask involves demanding that there be an immediate predecessor to the limit of a sequence.
But since there isn't an immediate predecessor to the limit of a sequence, I have to conclude that whoever is defining supertasks such that a mathematically impossible thing must become possible, to that extent, supertasks are impossible.
But they're just making a mountain out of a mathematical molehill. The ordinal [math]\omega[/math] does not have an immediate predecessor, and that's a fact.
So you are making this argument:
P1: Every supertask involves finding the immediate predecessor of [math]\omega[/math].
P2: [math]\omega[/math] does not have an immediate predecessor.
Therefore:
C1: There can be no supertasks.
I believe this is the heart of your argument. I'll make another run at the SEP article because it's interesting that Thompson said that convergence doesn't matter.
ps -- Ok, SEP says this:
[quote=SEP]
Benacerraf (1962) pointed out a sense in which the answer is yes. The description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. It may still be possible to complete the description of Thomsons lamp in a way that leads it to be either on after 2 minutes or off after 2 minutes. The price is that the final state will not be reached from the previous states by a convergent sequence. But this by itself does not amount to a logical inconsistency.[/quote]
So I have been expressing Benacerraf's point all along. Good to know. I may be wrong, but I'm at least in the ballpark of professional philosophical thought.
pps -- I love the example of the bouncing ball electrically connected to the lamp. That actually gives what I would call a natural solution in which the lamp is on, based on the fact that a bouncing ball eventually ends up on the ground.
ppps -- And:
[quote=SEP]
For this reason, Earman and Norton conclude with Benacerraf that the Thomson lamp is not a matter of paradox but of an incomplete description.
[/quote]
But this is exactly what I've been saying all along.
So you are holding a position that many (some, several, whatever) philosophers disagree with, and you are arguing that I should break with them and come to your side. But the SEP article definitely shows that some philosophers are in complete agreement with the ideas I've been expressing all along.
I have always accepted this; it's the reason that the supertask is proven impossible.
A lamp being off must always precede it being on, and so the sequence off, on, off, ..., on cannot model a lamp.
Quoting fishfry
It's not a clock. It's a screen that displays the time that the button was most recently pushed.
If the button is pushed at 23:00 then the screen will display 23:00 until the button is pushed again at 23:30, and then the screen will display 23:30 until the button is pushed again at 23:45.
The button is not pushed at midnight and so the screen will not display 00:00 at midnight, even though 00:00 is the limit of the sequence.
Quoting fishfry
A supertask is any infinite succession of tasks performed in finite time. Having that task be to push a button is as good a task as any, regardless of what pushing the button actually does.
Quoting fishfry
The lamp has two buttons. One button turns the lamp on and off, the other does nothing. The same mechanism is responsible for pushing both buttons. If it is possible to have pushed one of these buttons an infinite number of times then it is possible to have pushed the other button an infinite number of times.
If Thomson's lamp proves that it is impossible to have pushed the button that turns the lamp on and off an infinite number of times then it proves that it is impossible to have pushed the button that does nothing an infinite number of times.
And we can replace pushing a button with literally any other task and the same conclusion follows. Therefore, Thomson's lamp proves that all supertasks are impossible.
At exactly midnight, there is no time interval or delay prior to it that doesn't jump us back into the sequence. This is because midnight is a limit point of the sequence.
But I've already discovered that Benacerraf and others have viewed this problem exactly as I do. So my position is valid, or you should take your objections up with them. The problem does not defined the lamp state at midnight and it can be anything you like.
Quoting Michael
Yes I agree with your point that if we look at the lamp between two consecutive inverse powers of 2, the state is determined by the earlier point.
But you are trying to apply that same reasoning to the limit point, and you can't do that. Every interval around the limit point necessarily contains all but finitely many members of the sequence. There is no tiny little interval before midnight where the sequence has ended, leaving the lamp in a particular state. That's just a mistaken intuition on your part.
Quoting Michael
Your logic is bad. I shouldn't eat one of those poisonous Japanese pufferfish, but that doesn't mean I should turn down a nice tunafish salad sandwich.
But now that I think of it, I no longer even accept that the lamp is "impossible" via your logic. You just haven't got a good intuition for the limit of a sequence. Any step backward, no matter how tiny, jumps over all but finitely many members of the sequence.
So if the lamp is on at midnight, there IS a prior moment at which it was off, and vice versa. Just not an immediately preceding point, because there is no such thing.
Quoting Michael
Yes, there are as many even inverse powers of 2 as odd ones.
Quoting Michael
It proves no such thing. And having discovered that Benacerraf totally agrees with me, I don't have much reason to continue to assert the same points I've made over and over. Clearly, reasonable people agree with my point of view. Benacerraf is still alive, perhaps you can take this up with him.
Quoting Michael
If repeatedly asserting a falsehood were proof, we'd all be geniuses.
As a side note, youve proven that reasoning with you about how infinity works is a supertask.
Quoting Michael
Now this conversation can resume towards its ever ever-approaching, but never accomplished, conclusion.
I addressed his paper a month ago. See also here for a more formal argument.
As for the screen, it can only ever display the time that the button was most recently pushed. If the button is only ever pushed at 23:00 then at midnight the screen displays 23:00. Notice that the rule about how the screen works still applies at midnight, long after the button has been pushed. This rule also still applies at midnight if the button is only ever pushed at 23:00, 23:30, 23:45, and so on.
And because the button is not pushed at midnight the screen cannot display 00:00 at midnight, even though 00:00 is the limit of the sequence.
Quoting fishfry
This is why the supertask is impossible. A lamp can only be on if something causes it to change from off to on. This is as true at midnight as it is at any time before and after midnight. Its a fundamental fact about lamps. Therefore a lamp cannot be modelled by the sequence off, on, off, , on.
I understand how infinity works. The problem is that fishfry doesn't appear to understand how a lamp works.
Thats what I was saying about the lamp a couple weeks ago and you were disagreeing with me and telling me how time works and how lamps work.
If we are using a limit within which there is an infinitely decreasing time interval, or infinitely decreasing distance interval, then, regarding a question about some state of time or some state of position at the limit, no answer to the question is deducible from the premises.
Thats it. No one can give a crap about the lamp or time or lights on or or off at 2 minutes, at 7 minutes, because it is nowhere in the universe of the premises. You dont even need to fix the limit at precisely 2 minutes. It can be slightly before two minutes and the premises will still not admit of the existence of that limit.
Any discussion of the bottom of an infinite staircase, of the state of the lamp at two minutes, or whether Achilles can beat the tortoise, cannot be deduced from the premises.
Its like we are approaching agreement, but you are quibbling about half of what I say, and then we approach agreement again and you quibble about half again.
Or you are very simply not explaining yourself.
You clearly understand the facts - time in minutes and seconds, halving fractional functions, the difference between on and off. You do understand infinity.
But you seem to disagree with how these facts must clearly relate to one another in order to make some point about completing super tasks and contradiction.
If you are saying the lamp must be on or off at two minutes because thats how time works, and thats how lamps work, then you are not recognizing that no answer to the question is deducible from the premises.
You are assuming something that you are not stating, not stating something clearly, not defining supertasks, or wrong about one or more of these things if you think the lamp has to be on or off at 2 minutes because thats how lamps work.
And because lamps are either on or off at all times, but you cant deduce the state from the premises, we dont have a contradiction. The premises of switching a lamp on and off at ever decreasing intervals of time simply do not speak to the state the lamp will be at when two minutes or more elapse. It just means that from your infinite fractional function no answer to the question is deducible from the premises.
Not only is no state deducible from the premises, no state is consistent with the premises.
The lamp is turned on if and only if the button is pushed when the lamp is off.
The lamp is turned off if and only if the button is pushed when the lamp is on.
Neither of these are possible at t[sub]1[/sub] if a supertask has been performed between t[sub]0[/sub] and t[sub]1[/sub]. But the lamp must be either on or off at t[sub]1[/sub]. This is the contradiction.
But no state being consistent with the premises doesnt create a contradiction. At two minutes, the lamp becomes a duck and flies away. That doesnt contradict the premises, because the premises never touch the lamp at two minutes. Two minute lamp is utterly a new whole scenario, needing new premises to start being discussed at all.
Yes they do. These are our background premises:
P1. Between 00:00 and 23:59, nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 00:00
P5. The lamp is either on or off at 12:00
Only then do we consider the manner in which the button is pushed.
If the button is only ever pushed at 11:00 then the lamp is on at 12:00.
If the button is only ever pushed at 11:00, 11:30, 11:45, and so on, then the lamp can neither be on nor off at 12:00. This is a contradiction.
Note especially P1 which is where fishfry and Benacerraf go wrong.
I think you mean between 22:00 and 23:00, assuming the two minute mark is 0:00.
Why do you need to redefine the premises? We are getting nowhere over and over again.
It time zero and two minutes later that are the limits. Two minutes later is a theoretical, because it is never actually reach by halving the prior interval starting at zero time, one minute time, one and half minutes time, one and three quarters, etc.
Precisely two minute lamp is outside the scenario. Period. Whatever state or non-state you assign or cant assign to it, is not a function of the half timed lamp switching scenario. You are ducking the issue.
No I don't. I mean exactly what I said.
Quoting Fire Ologist
I'm not redefining the premises.
Before we even consider when the button is pushed, it is implicit in the thought experiment that the lamp both existed (as a lamp) before we start pushing the button and continues to exist (as a lamp) after we finish pushing the button, and that for its entire existence nothing other than pushing the button controls the lamp.
The button pushing is just one aspect of the wider environment. You and fishfry seem to ignore this wider environment, treating the supertask as nothing other than some infinite sequence of values.
THERE IS NO AFTER WE FINISH PUSHING THE BUTTON!!
We are supposed to be pushing the button at half of the prior interval. This is infinite. If you end up at 2 minutes, if you finish, youve failed the thought experiment or added some new premise.
There is according to those who claim that supertasks are possible. They claim that within two minutes I can finish pushing a button an infinite number of times, allowing me to then carry on my day.
Do you not understand the discussion we're having?
I am not going to add confusion and complexity to this by starting to discuss super tasks.
I do not claim it is possible to complete this task. Task speak is about physical, finite things like lamps and switches, and actually switching the lamp and marking each time it switches as at one minute and at a minute and a half.
We are not in the physical world. We are using physical world pictures to demonstrate a purely theoretical, mathematical function. To hell with any introduction of actual tasks and actual lamp states at actual times on actual clocks.
If you make a a mark at one distance or one time period and call it 1 and then make a second mark at a further distance or further time and call it 1.5 and then use this pattern to make a further mark at 1.75, you can continue this exercise if you are so inclined to do so infinitely and you will never mark 2.
I agree actual tasks defined as finishing or completed that include functions involving infinitely available steps are absurd.
The odd thing is, I think somewhere in here we are seeing the same thing, just not saying what we see so the other sees that we are seeing the same thing.
You need to give a little bit to me to bring me to see your point because nothing Im saying seems refutable, and isnt being refuted by you.
Is this a joke?
Im just trying to get to one minute with you, step down the first step.
The concept of the super task is not essential to anything Im saying.
Is there any same page or common ground you see in anything Im saying?
I still dont know your point if your point is based on refutations of the basic things Im saying.
Find the common ground so we can walk together or take me to your point. Why do we need to talk about this under the concept of supertasks if you think Im missing something?
Do you mean after we finish pushing the button because people get tired and time presses on? Or do you mean after weve pressed the button an infinite number of times? Because there can be no such time, and it would certainly not arrive at two minutes.
Supertasks are the topic of this discussion. They are what the rest of us have been arguing about for 26 pages.
You could perhaps start with the SEP article.
Ok, so supertasks cant be completed.
Did you think I was saying anything to the contrary?
And supertasks didnt come up until later in the post and really another way of incorrectly claiming there is anything relevant to the lamp problem at two minutes.
Thomsons lamp is a thought experiment designed by Thomson to prove that supertasks are impossible, with Thomson being the person who coined the term supertask.
So what am I getting wrong with you? Why are you arguing with me?
Do you think supertasks can be completed?
Do you think there is finishing in an infinite task?
No. Im trying to prove that they are impossible, as clearly shown in this post.
I agree they are impossible. So why do you disagree with the other things Im saying?
Am I saying annything inconsistent with the fact that completing a supertask is impossible?
You claimed that the supertask described in Thomsons lamp does not entail a contradiction. It does, as shown by that previous post.
This contradiction proves that supertasks are impossible.
Im claiming that because supertasks are impossible, anything posited about some state of affairs after they are completed is irrelevant and non-existent in relation to the task, so it cannot be evaluated for its contradiction or otherwise.
You should look up refutation by contradiction.
Look, if in an argument, the premise includes a task that cant be completed, and the conclusion includes completion of that task, then there is a contradiction, and this contradiction refutes the conclusion. If that is the sticking point then you are right, and I was wrong.
Thats not enough though.
In the supertasks article, they mention a hotel with a countably infinite number of rooms. Right there, at the premise, what does countably infinite point to? Thats nonsense. Thats a square circle. We dont get out of the gate. The infinite is by definition uncountable, so any conclusion based on a premise that includes the countably infinite could be said to be irrelevant, because you cant create a logical connection between the conclusion and a nonsensical, unimaginable thing like the countably infinite, whether that relation is said to be contradictory or otherwise.
An infinite number of stairs. How is that possible to imagine, to state as a premise, to even picture?
The arguments in these thought experiments dont get off the ground once you think them through. We have to give the premises some credibility as containing complete thoughts in order to move towards a conclusion. I grant you that if you give the premises credibility (somehow), the conclusions of these arguments may be contradictory, but that is only because of us accepting a picture of infinite stairs, or a race where one of the participants must take infinitely smaller steps.
Ive addressed all of these premises before. There is no half until after there is a whole. You dont travel half a distance first then travel the second half and thereby complete the whole. To call a distance half you first call another distance whole and then cut it in half. The whole always comes first. So when Zeno says Achilles must first travel half, he forgot that Zeno already accounted for the whole so he could claim whatever shorter distance to be some fraction in relation to that whole. So Zeno, like Thompson, and the others, tried to make mathematical relations between numbers correspond to the physical relations between objects. They dont.
Like you said, you cant have an irrational number of apples in your fridge, you cant have a countably infinite number of hotel rooms, or switch a light on and off every half the interval of time prior, or travel any fraction of distance without knowing the denominator (whole distance) first.
See countable sets.
Hotel rooms is nonsense then. There is no such set imaginable.
Just because in mathematics we can distinguish natural numbers from irrational numbers and real numbers, and place certain references to infinite sets in relation to these other concepts, doesnt mean we can imagine countably infinite hotel rooms. You see the hallway there with the room numbers? Or do you see part of the hallway? What floor are we on, if it matters?
If this is a mathematics conversation then why are we ever referring to stairs, lamps, hotels, switches, starting lines at races??
I have tracked down but not yet had the time to read Benacerraf's famous paper on the subject,
Tasks, Super-tasks, and the Modern Eleatics (pdf link)
I'll let you know if I find anything of interest. I'm happy to find that my take is officially legitimate.
2-dimensional sensors are placed after 100m, 150m, 175m, and so on. These sensors approach the 200m finish line but importantly no sensor is placed on the 200m finish line.
When each sensor is passed the distance is displayed on a screen, as well as the time that the sensor was passed. This display remains until the next sensor is passed. Nothing else controls the behaviour of the screen.
Say we run at a constant speed. We pass the 100m sensor at 12:00:10, the 150m sensor at 12:00:15, the 175m sensor at 12:00:17.5, and so on.
When we reach the 200m finish line at 12:00:20, what distance and what time is displayed on the screen?
There is no sensor on the 200m finish line and so neither 200m nor 12:00:20 will display, and the screen won't turn into a plate of spaghetti.
When I first saw the phrase "countably infinite", I thought that was absurd, and I still think it is an unfortunately ambiguous description of what it means. I would put it this way - any (finite) part of the infinite set can be counted, even though the whole of the set cannot be counted in one go. But I think that Wikipedia also puts it in a reasonably clear fashion.
[quote=Wikipedia;https://en.wikipedia.org/wiki/Countable_set]Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements[/quote]
Though I would have said "even though the counting may never finish due to an infinite number of elements."
Quoting Fire Ologist
I read somewhere that Hilbert never discussed his hotel after the casual mention of it in a paper, even though it provoked enormous discussion. I'm pretty sure he invented it only to help people realize what infinity means. All these cases play in the border country between the mathematical and the physical; they are entirely imaginary (not in the sense that they are possible, but only in the sense we can imagine impossibilities). Consequently, the normal rules of possibility and impossibility are suspended and people think the fact that they can in some sense imagine them means that they are, in some sense, possible.
Quoting Fire Ologist
I agree with that and for that reason think that to say that the conclusion or limit of the set can be anything at all is misleading. In a convergent series, specifying the limit is essential to defining the series. But that doesn't mean that the function that generates the set can generate it's own limit. In fact, if it could, it wouldn't be an infinite set.
The other problem is that Zeno, and most people since him, lose track of the difference between an analysis and a dissection. If I measure the length of the race-track as being 10 units long, the race track is completely unaffected; my measurements are an analysis. If I then cut the race-track into 1-unit lengths, the race track is affected, and what I have done is to dissect it. One can analyze the race track in terms of a given convergent series. But that does not exclude other analyses, including my analysis of it as 10 units long.
My final issue with Zeno is that he forgets that if Achilles is travelling at a constant speed, he will take less and less time to travel each segment of the series, approaching an infinitesimally small time as the segments become smaller and smaller. And, of course, since he can cover a segment in an infinitesimal amount of time, covering an infinite number of them in a finite time becomes less of a problem.
So you never finish placing the sensors. The race never starts. You cant ask what the screen will display. Its not a math problem about distance and time.
Or its just the same math problem as the other paradoxes, that are only really intriguing because of all the people in the stands watching that tortoise go!
For the sake of the argument the sensors just exist at their locations. We don't have to place them. The thought experiment is only to examine the internal consistency of continuous space and time, not the practicality of carrying out the experiment.
Locations are in physical space. This isnt a math problem yet.
Quoting Ludwig V
A "countable" set is one that can be placed into a one-to-one correspondence with the counting numbers - integers. Why do mathematicians define "countable" that way? Well, think of how we actually count. You look at a herd of sheep, say, and go "1, 2, 3, ..." The number at which you stop is the number of sheep in the herd. What you have done is you placed the set of sheep into a one-to-one correspondence with a subset of integers. Or you may have used your fingers or beads or scratch marks - either way, counting comes down to placing a set of an unknown size into a one-to-one correspondence with a set of a known size.
Counting infinite sets is just a natural extension of the same idea: you try to establish a one-to-one correspondence between some infinite set and the set of integers. You do not need to actually count every member of the set one by one, you only need to establish a procedure of how you would do it, or even just prove that such a procedure exists. If you can do this, then you have established that your set has the same "size" as the set of integers.
Anyway, this is just a specialist term. It doesn't have to "make sense" to be cogent and useful.
There isn't a problem with specialist terms. But "cogent and useful" is both cogent and useful as a definition of "make sense". I would rather not have to try to find another definition. "Cogent and useful" can mean different things in different contexts.
Quoting SophistiCat
I don't disagree. But half the problem, for us ordinary folk, is understanding that procedure, especially if, as in this case, it can't actually be carried out. The difficulty is understanding the difference between "and so on" as laziness, when it could be carried out, but one is too lazy or busy to actually do so, and "and so on" in the context of a mathematical induction, when it can't. In the background, I understand, there are people who have doubts about the validity of mathematical induction.
Quoting Fire Ologist
I think that's too simple. It's about the applied math. The issue is about applying the math to physical space (and time). After all, there is no problem about applying ordinary arithmetic to these situations.
Quoting Michael
Sometimes, I am so slow I cannot believe it. The answer to the question is available, if only you would apply ordinary arithmetic to the problem. All the paradox proves is that an analysis in terms of a convergent series does not apply to the question.
Quoting Michael
It is as well not to confuse the conclusion you want to draw from the analysis with the point of the thought experiment. After all, Zeno did not draw your conclusion from it. Nor do I.
I was referring to how folk who are unfamiliar with specialist terms that are based on words in the ordinary language try to make sense of those terms: they interpret them in light of the more familiar senses of the words. Naturally, this doesn't always work. Misinterpretations happen even in familiar contexts, and they are all the more likely in an unfamiliar domain. And as with neologisms, some just aren't going to like the coining for one reason or another, even when they understand the context. But that alone shouldn't be a barrier to understanding and accepting specialist terms.
Quoting Ludwig V
Yes, ultra-finitists reject mathematical induction as a proof method, but that is a rather extreme position.
Then what is the answer?
Assuming you maintain a constant speed, you will pass 200m at 12:00:20, as you point out. That is also the point I was after.
One might think that the screen will display the penultimate distance and time. But that is not defined. Nor is the distance and time before that defined. Nor is the distance and time at any other stage defined. You had this discussion a while ago, as I remember it. You can't count backwards because the argument doesn't give you the information you need to do so.
Ordinary arithmetic will [s]not[/s] give you an answer at the limit, or at any other specified stage counting forwards, but not at any stage defined by reference to the last stage of the series. I didn't read the case you proposed carefully enough. I had in mind the Achilles case. My bad.
Quoting SophistiCat
Thanks for the confirmation. I don't think that position is at all plausible. But are there any non-extreme positions around this topic?
Quoting SophistiCat
I think most people in this day and age can cope with specialist jargon. Many of them speak one of the many jargons available. There's an additional problem here, that the context is so startlingly different from common sense.
To avoid the problem , you just assume the impossible. There is a limit to the number of sensors which can exist in that space, depending on the size of the sensors, Because a sensor takes up space. Or, are you assuming that an infinite number of sensors can fit in a finite space?
They can if they are infinitely small. Is it possible that you can imagine that? Is there any argument that will settle the issue either way?
The sensors are two dimensional with a width and height but no length. If spacetime is continuous and infinitely divisible, as is assumed, then an infinite number of two dimensional sensors can fit within finite space.
But if you prefer then we can stipulate that only one sensor exists at a time, the next placed only when the previous has been passed.
A thought experiment like this is perfectly appropriate in philosophy. See for example Bernadete's Paradox of the Gods which is similar in kind to mine.
Obviously, a sensor cannot be infinitely small. It consists of components.
Quoting Michael
That is not necessarily the case. A sensor is a material object, space and time are not material objects. There is no necessity that the limitations of a material object are the same as the limitations of space and time. In the end, it's all conceptual, and the problem is in making the conception of an object consistent with the conceptions of space and time.
What these "supertasks" show us is that there is a disconnect between the conceptual structures of mathematics and the concepts of the empirical, natural philosophy, (science). The problem is compounded when mathematicians assume that their conceptions are objects, and these supposed objects get integrated into the work of scientists so that the boundary between the two incompatible conceptual structures is lost. This is the case in quantum physics, where the influence of mathematics allows for a non-dimensional object in the physical world, virtual particles. The purely imaginary concepts of mathematical objects is allowed to penetrate the theories of physics to the point where physicists themselves cannot distinguish between the real and the imaginary.
Quoting Michael
Come on Michael. Fire Ologist explained the problem with "placing", and you said, we could assume that they are already placed. Now I show you the problem with "already placed", and you say we can assume placing. What's the point in switching back and forth, when both are shown to be problematic? Move along now.
Quoting Michael
Sure, and the purpose of such thought experiments is to determine the underlying conceptual problems. If someone denies that the exposed conceptual problems are problems, then the purpose of the thought experiment is defeated.
We can assume that they simply exist in their places or we can assume that they are placed just before the runner reaches the next designated distance.
Quoting Metaphysician Undercover
I agree. I am trying to prove this by accepting the assumptions of those who believe in supertasks and then showing that their assumptions entail a contradiction. This is how refutation by contradiction works, and is going to be more convincing than an argument that denies their assumptions outright.
Yes. I am neither physicist nor mathematician, and I'm not sure that a bystander like me has a proper basis for an opinion. But after the discussions on this thread, I understand the point.
I think there's another bugbear at issue here - the idea that whatever can be imagined is at least logically possible.
Quoting Michael
Why don't you just stick to the mathematics? If we ask about any specific stage of the series, we can calculate exactly what time, as you show:- Quoting Michael That corresponds to what your screens will show. That's all perfectly clear and correct.
It is also perfectly clear that we cannot place screens at each stage, even with your modifications, because we cannot complete the series. That is reflected in the fact that we cannot calculate the distance and time of the last stage, or the penultimate stage, or the one before that.
These so-called thought experiments are just distracting fairy tales.
As Fire Olo pointed out, if they are placed, you never get finished placing them, if it were the case that you could carry out what is prescribed. So the runner can never get past them all. And if they already exist in their places, there is the problem I pointed to, the sensors, being material objects cannot physically fit in the space as prescribed.
Quoting Michael
I know that's what you're trying to do, but you haven't succeeded in that way. And I think you misunderstand where the true contradiction lies, and that's what misleads you into thinking that you ought to be able to prove some other contradiction.
The contradiction is actually within the assumptions which you accept. As I've said since the beginning, the contradiction is between the premises of the prescribed supertask, and your assumption, that the amount of time which serves as the limit which the supertask approaches, will actually pass. In other words, if you accept that the prescribed supertask can carried out, than you must deny the possibility that the limiting amount of time will ever pass. The supertask makes it impossible for that amount of time to pass. And, vise versa, if you accept that the limiting amount of time will pass, then you deny the possibility of carrying out the supertask. It's actually quite simple, and Fire Ologist demonstrates a very clear understanding of this situation, where the two conceptual frameworks ( the conditions of the supertask, and the condition of the limiting amount of time passing) are simply incompatible.
Quoting Ludwig V
It's better stated that distinct things which are logically possible, may be mutually exclusive. So we might allow that whatever is not self-contradicting is logically possible, but one logical possibility might be incompatible with another. When logical possibilities are incompatible, there is not necessarily one specific method which we would use to choose one over the other. For example, we might choose the most useful one, or we might choose the one which is most consistent with empirical observation. The two are not always the same, and that appears to be the issue here. Infinite divisibility is probably the most useful, but it is incompatible with empirical observation, as these paradoxes show.
That seems reasonable. But the question arises whether we can imagine something that is logically impossible. Philosophical practice says no, we can't (thought experiments) and yes, we can (reductio arguments). I suppose if two contradictory statements follow from a single premiss, we can conclude that the premiss is self-contradictory. But then, that's not always obvious, as in this case.
Quoting Metaphysician Undercover
I'm not convinced of that. I think that the confusion develops from not distinguishing between "+1" as a criterion for membership of the set of natural numbers and as a technique that enables to generate them in the empirical world.
When we consider the first use, we think of the entire set as "always already" in existence; when we consider the second, we get trapped by the constrictions of time and space in the world we live it. The difficulties arise because it seems on the one hand that we can never specify the entire set by means of applying the algorithm and yet we can prove statements that are true of the entire set. This oscillation between the abstract and timeless and the concrete and time/space bound is very confusing, and, what's worse, it (the oscillation) encourages us to think that an infinite series can be applied to the physical world in just the same way as an ordinary measurement.
I'm channelling Wittgenstein here. I don't think finitism can make sense of this, but I'm deeply sympathetic to his approach to philosophy.
That's all wrong, of course. It's only an attempt to point towards an approach.
I believe this involves the distinction between imagining and saying. We can say contradictory things like "square circle", but can we imagine such things? Imagining involves a sense image, and this is where the difficulty arises because imagination defers to empirical data. So mathematics uses a technique where terms are defined, and the sense image is not necessary. For instance, a nondimensional point, infinite divisibility, etc.. These things cannot be imagined.
So the issue is not whether things can be imagined, but whether they can be defined so as to coherently fit into a conceptual structure without contradiction. In this way mathematics removes itself from imagination, and the empirical world associated with it.
Quoting Ludwig V
I don't see the relevance of "+1". The supertasks described here involve an endless division, not adding one in an endless process. These two are completely different. The formula for "+1" involves no limitations of space or time, so there are no restrictions and it can simply continue forever, without any inconsistency with empirical observation. The supertasks however, start with a defined space and time, and start dividing that specified section.
It is this, the idea of dividing a definite section of space and time, indefinitely, which creates the problem. What i think, is that the assumptions which provide for a definite section, also deny the possibility of indefiniteness. So for example, assume "one hour". To validate this measurement a beginning and end point is required. The assumed beginning and end point allow for the specified "one hour" and these points cannot be arbitrary because "one hour" is an empirically defined period of time. If the points are not arbitrary, they must inhere within time itself, therefore possible division would be dependent on those points, and could not be indefinite. In other words, a "definite section" relies on nonarbitrary points, but this is incompatible with infinite divisibility.
At first sight, that seems to be true. But is the impossibility of imagining a round square based on trying to imagine such a thing and failing? We frequently (in the context of sf fiction, for example, imagine faster-than-light travel between the stars. What picture could possibly constitute imagining it? Or consider @Michael's two-dimensional sensors?
(I won't bother with the psychologists' empirical claim that people differ in the extent to which they actually make a picture when they imagine something.)
Quoting Metaphysician Undercover
But people frequently disagree about whether a specific proposition is self-contradictory and/or incoherent or not - as in this thread.
Quoting Metaphysician Undercover
The problem for me, then, is that I do not see a relevant difference between "+1" and "
Quoting Metaphysician Undercover
... so you don't see a relevant difference, either. I agree with you that the problem arises in applying mathematics to the physical world, specifically to space and time.
Quoting Metaphysician Undercover
But if that's your problem, you ought to have a difficulty with "+1", because there are an infinite number of non-dimensional points between my left foot and my right foot whenever I take a step. Or are you thinking that "+1" involves adding a physical object to a set of physical objects?
If you don't have a problem with that, I can't see why you have a problem with a infinite convergent series.
There are real practical difficulties with the idea that a cheese can be cut up into an infinite number of pieces (which could then be distributed to an infinitely large crowd of people). I don't deny that. But dividing the space that the cheese occupies into an infinite number of pieces is a completely different kettle of fish. It doesn't involve cutting anything up and, hopefully, not imagining cutting anything up either.
I would not call that "imagining". Like the "round square" it's simply a case of saying without imagining. An author can say that the space ship moves from here to there in a time which implies faster than the speed of light, but to imagine faster than the speed of light motion requires imagining a material body moving that fast. That body moving that fast, could not be seen, and therefore cannot be imagined.
Quoting Ludwig V
Much of what is said in this thread, and supertasks in general, involve this problem, saying things which cannot be imagined. It's easy to say things which cannot be imagined, and we justify these things through logical possibility, but when logical possibility conflicts with empirical principles, then we have a problem.
Quoting Ludwig V
They are completely different principles. You're comparing apples and oranges, and saying 'my comparison is relevant because they are both fruit'. We are not talking about "our number system" in general, because that is not the problem. We are talking about a very specific problem which is infinite divisibility, not the general "fruit" (number system) but the specific apple (infinite divisibility).
Quoting Ludwig V
The problem is exactly what @Michael has been insisting on, the assumption that space and time are continuous. This supports the principle of infinite divisibility. The problem though is that space and time are conceptions abstracted from empirical observation, how material things exist and move, and the (unimaginable) mathematical conception of an infinitely divisible continuum is not consistent with the empirical data. Hence the Zeno type paradoxes.
Quoting Ludwig V
Why does "+1" need to imply anything other than counting? There is nothing between one and two in the act of counting, yet they are distinct. We need to account for that distinction. What separates one from two? When we describe this principle of separation we also provide ourselves with the basis for division.
Quoting Ludwig V
The problem is not "infinite convergent series". That is a misrepresentation which has occurred over and over again on this thread. The "infinite convergent series" is a particular mathematical idea which has emerged from a proposed solution to the problem of infinite divisibility. The problem arises when people believe that the infinite convergent series is the necessary outcome of the problem of infinite divisibility instead of seeing it as one possible representation.
Quoting Ludwig V
Why do you say this? The cheese is an imaginable, sensible object. The conception of "the space that the cheese occupies" is completely dependent on, and therefore abstracted from that empirically observed cheese. Why therefore, do you conclude that we can do something more with the space than we can do with the cheese?
OK. That seems clear enough for now. I won't argue about words.
Quoting Metaphysician Undercover
What empirical data do you have in mind?
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
You seem to be saying in the first quotation that the assumption that space and time are continuous gives rise to the problem of infinite divisibility and in the second that the problem of infinite divisibility gives rise to the problem of infinite convergent series. I must be misunderstanding you. Can you clarify?
But I agree with you that the convergent infinite series is a possible representation of certain situations. (I would call it an analysis, but I don't think the difference matters much for our purposes.) All I'm saying is that it doesn't give rise to any real problems unless you confuse that representation with the cutting up of a physical object.
Quoting Metaphysician Undercover
Because the cheese is a physical object and the space is not an object and not physical. You seem to be saying the same thing here:-
Quoting Metaphysician Undercover
By the way, nobody is worrying about the fact that we cannot picture an infinitely divisible continuum.
Quoting Metaphysician Undercover
And when we describe the principle of distinction between non-dimensional points on a line, we find that our counting is endless. The surprise is entirely due to mistaking non-dimensional points for a physical object - thinking that we can separate them, rather than distinguish them.
Although I don't agree there is a problem with "infinite divisibility", another procedure you might mention is described by Tannery's theorem, which concerns series in which each term changes as the series progresses. In the extreme case, a series in which each term converges to zero as described will itself converge to zero. I.e., infinite summable to zero.
(I extended this idea to composition theory some time back Generalizations . . .)
Although you and I don't agree on the soundness of established mathematics, I do enjoy reading what you have to say.
Quoting Ludwig V
There can be no counting to begin with.
S is countable if and only if either S is finite or there is 1-1 correspondence between S and the set of natural numbers.
And in that mathematical context it is not the case that 'infinite' and 'uncountable' mean the same.
Technical fields of study often have special definitions. Quibbling about that is pointless.
The physical evidence, is that we cannot just keep dividing something forever. There is nothing which provides us with the capacity to keep dividing it. The continuum of mathematics is not consistent with any sense evidence.
Quoting Ludwig V
I would say more, that the assumption of infinite divisibility gives rise to the idea of continuity, and the idea of continuity supports the idea of an infinite convergent series. So, to state it simply, the infinite convergent series is the result of, or produced by, the way that calculus deals with continuity. And we need to deal with continuity because we assume that some things (space and time) are infinitely divisible.
The root problem, I claim, is infinite divisibility. From this is derived the concept of "continuity", "continuum", and calculus deals with the continuum by applying the infinite convergent series. Since infinite divisibility is a bogus concept, the whole thing is a problem.
Quoting Ludwig V
It's not an analysis, but a hypothesis. Infinite divisibility is a theory. And, it does give rise to real problems, as is the case when the representation is not true,
Quoting Ludwig V
The point is that "space" as a concept, and "time" as a concept, are both derived from our experiences of sensing the world. Kant was wrong to say that these concept are somehow prior to, as necessary for sense experience. Since these concepts are derived from our experiences, then whenever they differ, or are incompatible with our experiences, they are faulty.
I recognize and uphold the difference between the physical things, and the concept, but I also affirm that when the concepts do not conform, there is a problem. So, cheese is like any other physical object, it is not infinitely divisible. The concept of "space" allows for infinite division, but that's inconsistent with the physical world, which "space" is supposed to provide a representation of, so there is a problem.
Quoting Ludwig V
Speak for yourself.
Quoting Ludwig V
What do you mean? What is this difference between distinguishing and separating?
Quoting jgill
Well.do you know of anything that's actually infinitely divisible?
Quoting TonesInDeepFreeze
Th Fourier transform and the resultant uncertainty principle.
I'm surprised. Could you explain why?
Quoting Metaphysician Undercover
That's odd. The surfaces of the objects around me look as if they are continuous.
Quoting Metaphysician Undercover
You said:- Quoting Metaphysician Undercover
Quoting Michael
Only if space is infinitely divisible and they are not physical sensors. And you say in the quote below that a sensor is a material object.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
What do you mean by "actually"? Take any natural number. It can be divided by any smaller natural number. The result can be divided by that same number again. Without limit.
Quoting Metaphysician Undercover
Whenever concepts are defined in relation to each other, they can be distinguished but not separated. Distinguishing is in the head, separation is in the world. Examples of inseparable distinctions are "up" and "down", "north" and "south" (etc.), "convex" and "concave", "clockwise" and "anti-clockwise", "surface" and "object" (in cases such as tables and chairs).
Isn't that surface itself an edge, a discontinuity? And isn't it true, that what you see (sense) is actually a discontinuity, and you think it to be a continuous surface? I suppose, that you might think that within the confines of the edge, there is continuity, but look closer, and you'll see colour changes, texture changes, and other deformities which indicate discontinuity within the surface.
Quoting Ludwig V
Is this directed at me, or Michael? I maintain that a sensor is a material object consisting of components. The proposition of a non-physical sensor is incoherent.
Quoting Ludwig V
What I meant by "actually", is what can be carried out in practise. Your example is theory. Anything is infinitely divisible in theory. You see an object and theorize that it can be endlessly divided. But practise proves the theory to be wrong. That is the point in me asking the question of whether you know of anything which is actually infinitely divisible. This would provide evidence that the theory is not false. However, in reality, all attempts have led to a falsification of that theory of infinite divisibility. So what you have stated as your example is just a false theory.
Quoting Ludwig V
Aren't you making a category mistake here? If separation is in the world, and distinguishing is in the head, then your examples up/down etc., are examples of distinctions, not separations. It is a category mistake to talk about these as "inseparable" by the terms of your definitions, separable and inseparable would apply to the category of things in the world, while distinguishable and indistinguishable apply to what's in the head.
Here is your statement again. I'll ask the question in a different way, using the definitions you've provided.
Quoting Ludwig V
What is "the principle of distinction between non-dimensional points on a line"? How would you distinguish one point from another, except by location? But location is dimensional, and actually a separation. There actually is no distinction between one point and another, they are all exactly one and the same, by definition, each point is the very same point as another point, if they are supposed to be things. The only thing which makes them not the same is a dimensional separation, the idea that they are supposed to be at different locations in the world.
H'm. I thought you would throw the results of sub-atomic physics at me - that apparently solid object is mostly empty space. But you are right. You are also right that the surface of an object is a discontinuity - a border - between the object and the rest of the world. But my point is that you cannot peel the surface of an object off, in the way that you can peel a skin off it. We can distinguish between a surface, with all its irregularities, and the object, but we cannot separate them.
Quoting Metaphysician Undercover
I'm sorry. I get confused sometimes about who said what. I'm glad we agree on that.
Quoting Metaphysician Undercover
You are right. I should have put the point differently - something along the lines you used.
Quoting Metaphysician Undercover
So we are closer than we seem to be. The difference between theory and practice is well enough known. It is unusual to say that difference proves theory to be wrong. I would be happy to say, I think, that Zeno's application of the theoretical possibility of convergent series to time and space and the application in Thompson's lamp is a mistake. But calculus does have uses in applied mathematics, doesn't it? I imagine that physics will come up with some interesting ideas about time and space; at the moment it all seems to be speculation, so I'm suspending judgement about that.
Quoting Metaphysician Undercover
Non-dimensional points which have a dimensional separation? H'm. But then a boundary (between your property and your neighbour's) doesn't occupy any space, even though it has a location in the world and will consist of non-dimensional points.
Real numbers are uncountable.
I see. Why can't I count with natural numbers?
"principle of distinction between non-dimensional points on a line" does not specifically speak of natural numbers. Language play.
You had me going there. :smile:
So if I had said "And when we describe the principle of distinction between non-dimensional points on a line, we find that our counting with natural numbers is endless", you would have agreed?
No. If "the points on a line" correspond to integers or rational numbers, yes. Way too vague.
Fair enough. Should I be talking about a bijection between the non-dimensional points on a line and the set of integers?
You can use that term, but only if you are more specific about "points on a line" and specify natural numbers or rational numbers corresponding to these points. That's it.
Do you mean "bijection"?
Quoting jgill
Do you mean how they are to be identified?
separation
dimensional separation
dimensional point
non-dimensional point
Why do you say this? Doesn't science proceed through the falsification of theories?
Quoting Ludwig V
You don't go as far as me then. I say that infinite divisibility is a mistake.
Quoting Ludwig V
Of course, we need to distinguish between truth as our goal, and pragmatics, which doesn't have any specific goal. Usefulness is relative to the goal, and the goal could be anything. I don't deny that calculus is extremely useful, but that usefulness may be misleading relative to the goal of truth.
Quoting Ludwig V
Don't you agree, that this is the only way in which one point may be distinguished from another point, through a spatial, or dimensional, separation?
Quoting Ludwig V
This is the way I understand boundaries between two pieces of private property. The boundary exists in theory as lines of two dimensions, or sometimes even three dimensions because elevations need to be accounted for. In theory, the line occupies no space. In practise though, the boundary becomes a fence, a disputed sliver, or something like that, and ends up actually occupying space.
So if you and I had a shared boundary, on paper the boundary would be described as occupying no space, you on one side, I on the other, and that would be the theoretical boundary. But in practise, there would be an area, known as the place of the boundary. Even if there is pins, and we stretched a string from pin to pin, the string occupies an area. And so does the pin occupy an area.
That is, for what natural number n is it the case that 1/(2^n) is not a number?
/
"non-dimensional points [...] dimensional separation"
Maybe start by defining 'non-dimensional point' and 'dimensional separation'.
/
If the discussion is about points in ordinary real 2-space or real 3-space then points are distinguished by being a different ordered tuple.
In 2-space, the point
In 3-space, the point
If a particular line, say the ordinary horizontal axis, then <0 x> differs from <0 z> iff x not=z.
This is not the least bit baffling.
That is an empirical matter:
For all x and y, x-y = the unique z such that y+z = x.
For all x and y, x/y = the unique z such that y*z = x.
What would be the definitions when there is a smallest number p?
Note that in informal contexts, we use a notion of 'undefined'. But in a fully formalized theory, we don't allow 'undefined' as it would violate the definitional criteria of eliminability, which is crucial for the requirement that the syntax be recursive.
Then no, there is no smallest number.
But then I'm not sure what relevance this question has to the matter at hand?
It's not that complicated. Imaginary numbers have a use even in electrical engineering but I cannot have an imaginary number of apples in my fridge.
There's nothing wrong with maths, just sometimes an improper use of it. There is no smallest number, but if paradoxes like Zeno's and Thomson's are valid then it would suggest that there is a smallest unit of space and/or time and that this isn't just a contingent fact about the physics of our world but something far more necessary.
I understand the view that there is no smallest number but that there are smallest distances and durations. But I am asking whether some people here do believe there is a smallest number.
I'm not claiming that the implications are or are not complicated.
The analogy with imaginary numbers and apples is amiss in this regard: Yes, apples are counted by integers, not imaginary numbers, so indeed imaginary numbers are not the correct kind of number to count with. But distances and durations are measured by real numbers, so smaller and smaller real numbers are not a difference in the kind of number.
Yes. The trouble is that the inapplicability of convergent series in certain situations does not, for my money invalid them in all situations.
Quoting Metaphysician Undercover
Well, it would be interesting to know what your criterion of truth is in mathematics, if a calculation procedure is effective and useful.
Quoting Metaphysician Undercover
I agree with everything you say.
Quoting TonesInDeepFreeze
Thank you. That's very clear.
Quoting Michael
I'm puzzled. I thought you thought that Thompson's paradox was flawed and therefore invalid - as Thompson did, didn't he?
Quoting TonesInDeepFreeze
That seems to be the result of some recent research. But I don't think it applies to mathematics as such, and perhaps one ought to wait and see whether anything else emerges from research.
No, I think (as did he) that it successfully shows that supertasks are not possible.
Quoting TonesInDeepFreeze
Yes. If space and/or time being infinitely divisible entails that supertasks are possible, and if supertasks being possible entails a contradiction, then it is proven that space and/or time are not infinitely divisible.
Quoting TonesInDeepFreeze
In this case the mistake is in the application of transfinite numbers.
[EDIT: I may have erred in the paragraph above. At least in the first part of Thomson's paper, he does not say that infinitely many tasks may not be completed in finite time. Rather, he says only that it is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.]
I think "valid paradox" is, at best, ambiguous and confusing.
That's a good question. As a relative simpleton who's been trying his best to follow along with the recent arguments, I would ask: why could you not take the smallest number (my rudimentary mind imagines something along the lines of 0.0[insert a bajillion zeroes here]1) and divide that by 2? And divide even that by 2? And so on? We can't "run out" of numbers, per se. For that is there design. Though I'm sure there reaches a "hard point", a threshold if you will where a certain degree of number fails to appear or exist anywhere in the known universe thus ceases to become of use or mention. I'm sure there's a term for that- somewhere. :chin:
So his paradox shows that the time between each task in a sequence cannot in principle be modelled by a geometric series, e.g. where the first task takes 1 minute, the second 30 seconds, the third 15 seconds, and so on, because if it were possible then it would be possible for an infinite number of tasks to be performed within 2 minutes.
I'd need to check Thomson's paper again to ascertain whether that properly describes the particulars of his view.
But, yes: The infinite sequence of durations converges to 0. And the marked time converges to 2 minutes. But it seems that Thomson is saying that it is in the nature of tasks that there is not an infinite sequence of them such that they are all completed in finite time.
If there is a greatest divisor, then there is a greatest natural number, call it 'g'. So then what is g+1? If one says addition is not allowed with g as a summand, then one needs to come up with a different definition of addition, which becomes very very complicated if we wish to still have addition in a formal theory.
Also quite inelegant. If, for arbitrary example, we say that g = 66589080980923842343287098023450390811321445645098760011287390453735490233999934393 is the largest natural number, then naturally a person would want to say, "My, that's awfully specific for mathematics that we would like to be most general."
Moreover, why should numbers be limited to only how many particles there are? Such a limitation would preclude the natural human inclination to ruminate on such things as, "Suppose we made a mistake and there are actually twice that number of particles" or even with g as all the possible finite combinatorial arrangements of the particles. Well, that entitles us to talk about numbers larger than g.
Ultrafinitists are welcome to try to convince me, but I am a tough customer when it comes to giving up my natural prerogative to add 1 to any number.
Yes, that's what I thought. I think the concept of a valid paradox is a bit confusing.
Quoting Michael
But space or time being infinitely divisible does not entail that supertasks are possible.
Quoting Michael
That's the first I've heard of any use of transfinite numbers in this thread. I don't think they are relevant - more, I very much hope they are not relevant.
Quoting TonesInDeepFreeze
How is that possible? Infinite means without limit.
Surely, the number of tasks you can complete in a given time depends on how long they take. If you want to perform an infinite number of tasks in a limited time, just define a task that takes the appropriate amount of time. in the puzzle, each task takes less time to perform, without limit. The trouble with Thompson's lamp is that no switch can function in an infinitely small time.
Quoting TonesInDeepFreeze
I'm sure it could count as a human right. But can we also stand up for the right to form the inverse of any natural number? (For clarity, forming 1/2 from 2, 1/3 from 3 and so on. (I'm not sure whether 0 or 1 need to be included here.)
If we're talking about an infinite number of tasks being performed then we are talking about a transfinite number of tasks being performed.
Quoting Ludwig V
I'm not talking about physical possibility. But even then, if space and time are infinitely divisible then motion is a physically possible supertask.
Quoting Ludwig V
Then forget the word "paradox". If Thomson's argument is valid then it proves that supertasks are impossible.
Smaller numbers are not needed. This idea simply produces unnecessary complications. When you divide one thing into two, you get two things, not two halves. The idea that you get two halves when you divide one thing into two, rather than getting two new whole things, causes the problem being discussed in this thread
Quoting TonesInDeepFreeze
Yes, I do believe that the mathematics needs to be changed, for the reason given above. The issue, (as I stated earlier in the thread), is that division presupposes an entity or object to be divided. And, divisibility is dependent on the type of thing to be divided. Therefore, when it comes to division one standard does not fit all things, and the principles of division must be specifically designed for the different type of things to be divided.
Maybe some mathematicians like to think that "a number" is a type of thing, or object, and that there is no limit to the way that this type of object may be divided. But I think that's just a mistaken idea.
I've been thinking about this. My comment on this was wrong. Of course, one cannot complete infinitely many tasks in a finite time. "Complete" does not apply to infinite series, by definition.
On the other hand, what counts as one task. If one takes three steps, one completes three tasks. But that distance can be analysed in many different ways, so it could be represented as one task, or many.
Quoting Michael
It depends on how you choose to analyse it.
Customer: I'd like a cherry pie, divided in two. I'm going to give one half to my niece and that other half to my yoga teacher.
Baker: That makes no sense. I would have to cut the pie, and then there would be two different things, not two halves of the same pie.
Gina: Excuse me?
Baker: You heard me.
Gina: I heard you. But I don't understand.
Baker: What don't you understand about the fact that when you slice something apart, there are never halves of anything, only two new things? If you still don't get it, then I suggest you read my posts at the 'Phil's Ossify For 'Em' website. It's a philosophy place where I write my posts showing that all of mathematics is wrong.
Gina: Well, I've studied philosophy and mathematics, and have not read anything like what you're saying.
Baker: Exactly. If you want to know what's really up, you have to come to me for it.
Gina: I just want you to cut the pie in half and put the separate halves in separate boxes. If necessary, I'll pay extra for you to do that.
Baker: There is no money in the world that would permit me to cut a pie in half. It's not a matter of money. It's that it is metaphysically, ontologically and mathematically impossible to do. Now I can slice a pie. But I can't call the pieces halves or sell them as halves. I can only sell you two new things that are not to be referred to as halves of anything. So would you like one new thing in its box and another new thing in its box?
Gina: Yes please.
Baker: Fine. But not if you call them halves!
Gina: Okay, I promise not to do that.
Later that day Gina arrives home and talks to her husband Ralph and son Timmy:
Gina: I had the most peculiar conversation with the new owner of the bakery downtown. He insists that he can't sell me two pie halves but only two different things that are made of pie.
Ralph: Yes, I know. He is a bit odd. Last week I asked for a baker's dozen of bagels and he gave me only twelve even though I reminded him that a baker's dozen is thirteen. He said that is a contradiction in terms and instead I need to ask for thirteen at the same price as for twelve and he'd do that. So I said, okay, just give me one more bagel to add to the twelve, since 12+1 is 13. Then he went into thing about how 12+1 is not 13, that numbers aren't even things of any kind, and that people are all wrong about the law of identity when they say things like "1+1 is 2". He even said that '1 is 1' is false because the first symbol '1' is not the second symbol '1'. Very strange fellow.
Timmy: I talked with him too. He has all kinds of very strong opinions about math, but he doesn't know anything at all about.
If the poster can't answer that question, then he lacks basis for his dogma that every set has an inherent ordering, which goes to the heart of his bizarre imaginings about mathematics.
I'm going to have a cup of tea. I shall divide it into parts 1/2, 1/4, 1/8 ... so it will last for as long as I want it to.
Did Thomson make that argument? Was that part of his answer to the paradox?
Everyone knows that tea is taken at at the tea time hour and that one is not to dawdle still drinking it, not even hypothetically, not even gedankenishly, past the tea time hour.
Not in mathematics.
The lamp puzzle doesn't require anything to occur in an infinitely small amount of time.
And I don't think the discussions about a switch being moved or any aspect of the agency by which the light changes are relevant. It is missing the point to quibble about the mechanics of how the light changes. We need only take it for granted that it does change at the rate stated in the puzzle.
A deep reading of American history reveals that the right to the arithmetic operations was to be enshrined in the Bill of Rights. But it failed to pass because the mid-Atlantic states feared that too much public exercise of arithmetic would allow citizens to become too number savvy and that would hamper the sports betting industry that was legal back then, especially in New Jersey.
The only infinite number in the puzzle is the domain of the sequence.
Quoting Michael
So it seems your analogy is between misuse of imaginary numbers and misuse of infinite numbers.
I might not put it that way. But I do understand Thomson's point that there cannot be infinitely many task steps executed in a finite duration.* And I understand the different argument that the puzzle may dissolve if we allow that there is a shortest distance and shortest duration.
* EDIT: As I mentioned in an edit a few posts ago, I may have erred. At least in the first part of Thomson's paper, he does not say that infinitely many tasks may not be completed in finite time. Rather, he says only that it is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.
Where does such a claim about mathematics even come from? What actual piece of written mathematics is claimed to say such an unfocused thing?
Rather, division of real numbers (which is the subject here) is merely by definition:
for y not= 0, x/y = the unique z such that z*y = x. And that is based on having previously proven that, for y not= 0, and for any x, there is a unique z such that z*y = x.
I know of no mathematician who wrote gibberish saying that "there is no limit to the ways objects may be divided".
An infinite series that has a sum (some might say the series is the sum) requires first having an infinite sequence (each entry in the sequence is a finite sum) that converges, and the sum is the limit. The sequence whose entries are 0, 1, 0, 1 ... does not converge. However, whatever you mean by 'complete', there are infinite series that have a sum.
An infinite sequence is a function whose domain is an infinite ordinal.
The infinite sequences in this context are:
(1) The function that maps n to to 1/(2^n)
(1) The function that maps n to either 0 or 1 (off or on) depending on whether n is odd or even
Yes. I discovered that after the tea-time hour, it turns into a grumpy tortoise.
Quoting TonesInDeepFreeze
Since then, however, it has been discovered that citizens will still get themselves into a hopeless muddle even if they practice all day. So the betting industry is safe.
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Yes. I was careless.
Quoting TonesInDeepFreeze
But here's my problem. If I take one step, do I execute one task, or many? The argument of the paradox is that in order to take my step, I either must execute infinitely many tasks in a finite duration or fail to complete (or even begin) my step. I maintain that the issue is about how you choose to represent my step, and representing my step as composed of infinitely many segments is only one of many representations.
Quoting TonesInDeepFreeze
I think everyone agrees that there's misuse of something going on here. There's disagreement about what is being misused and how.
Quoting TonesInDeepFreeze
I never meant to deny that.
No, he only argued that "talk of super-tasks is senseless."
I simply use this as a refutation by contradiction. If spacetime being infinitely divisible entails the metaphysical possibility of supertasks and if supertasks are metaphysically impossible then spacetime is not infinitely divisible.
Although on this point there is perhaps some ambiguity in the phrase "infinitely divisible," a point that Thomson addresses:
So I suppose what I mean to say is that within some finite section of space and time there cannot be an infinite succession of halfway points.
In my mind this requires that there be some smallest unit of space and time, but I'm open to the possibility that it doesn't, and that I'm simply failing to fully grasp the notion of infinity. Perhaps this is related to the fact that [math]\mathbb{N_0}[/math] is infinite but that every [math]n\in\mathbb{N_0}[/math] is finite.
Although as I suspect that some quantum theory of gravity will be correct, and as such quantum theories require discrete spacetime (as far as I understand), I think that it's at least true that there is some smallest unit of space and time, even if it's not necessarily true.
Quoting TonesInDeepFreeze
That really depends on what you mean by "object". If you mean "physical object" then mathematics doesn't say anything about them at all, and whether or not some physical object is infinitely divisible is a matter for empirical investigation.
As it stands one cannot have half a photon (or any point particle) and if spacetime is quantized then there is a limit to how far one can divide some section of it into two.
The pie is cut in half and placed in two different wrappings, as two different things, going to two different places. That these two different things were at one time united in a single source is irrelevant to the fact that after division they are two distinct things with two distinct centres of gravity.
The fact that we call the two things "halves" is just a feature of common vernacular. We know that they are not really each exactly a half, by any strict logical principles. It's just an approximation. Likewise, if we measure two things as 420 kg, we say that they are "the same weight", even though there is a discrepancy of a few grams here or there.
You argue by equivocation, confusing common vernacular with the logic of mathematics. If 1/2 in mathematics was allowed to be imprecise, as the baker cutting the pie in half is allowed to be imprecise, the problem of the op would not arise.
This is the issue which Ludwig and I have been discussing. "One half" in practise does not have the same meaning as "1/2" in theory.
I hate to be difficult, and I'm not really disagreeing, just amplifying. But I would like to add that if the pipe is cut in half lengthways, neither half is a (newly individuated) pipe. You have two gutters (or that is what I call them). And that if I paint half the pipe blue and half red, the halves do not become objects in their own right, but remain halves of the same pipe, even though they are of different colours.
Why do you say that the two gutters are not distinct objects. A gutter is an object. Or do you not think so? Even if we call them halfpipes, the two are still separate objects. And if we can't even think of a name for them, we acknowledge that they are no longer united as one object, but are now two separate objects. Division of an object does not necessarily produce two new objects that are of the same type as the original, in fact it often does not. So there is no need to think that dividing a pipe ought to make two pipes. However, dividing an object in two always produces two new objects (as well as the waste material). The "waste" becomes an important feature often overlooked in systems theory and the conservation energy, as energy lost to inefficiency or entropy. But the fact of waste in any act of division nullifies the validity of the supertask.
Quoting Metaphysician Undercover
Perhaps I wasn't clear. I never intended to say that. I thought this was crystal clear:- Quoting Ludwig V
Quoting Metaphysician Undercover
It depends what you mean by "divide" and by "object".
Quoting Metaphysician Undercover
This is not exactly wrong, but requires that you recognize that "division" and/or "object" may change their meaning in some contexts. That's why I said:-
Quoting Ludwig V
I said that the two painted halves do not become objects in their own right, meaning separate, distinct objects. You may argue that this is not dividing the pipe, or that each half becomes a distinct object. I don't mind what you choose. This shouldn't be too difficult for you, since you said earlier:- Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Oh, there's no doubt that no-one could actually cut the pipe into halves, and then divide one of the halves into halves ad infinitum. But painting the pipe shows that it depends what you mean by "divide" and/or "object". You could say that painting the pipe is a theoretical, not a practical division; that would be a bit at odds with ordinary language, but we are not speaking ordinary language here.
But maybe we could notice that we are not actually cutting the pipe, but imagining cutting the pipe. So perhaps we could imagine cutting the pipe without any waste?
Oh, I misunderstood. You are painting the pipe without dividing the pipe. I guess we don't mean the same thing with "divide". I think of "divide" as "separate or be separated into parts; breakup; split". This is why I say that to divide something in two makes two distinct objects, because they are separated, each with its own centre of gravity.
Quoting Ludwig V
I would say that painting a pipe two different colours is not a case of dividing the pipe. To use your terminology, you are distinguishing two halves without separating them. This does not qualify as "dividing". When I look at an object I can distinguish different parts of the object, and even draw lines on its surface, and all this is done without dividing the object.
OK. So I guess measuring an object would count as "distinguishing different parts" of it even if the line that I draw does not correspond to any pre-existing difference or discontinuity in the object.
"Thomson says [...] there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time."
I may have erred there. At least in the first part of Thomson's paper, he does not say that infinitely many tasks may not be completed in finite time. Rather, he says only that it is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.
The puzzle supposes infinitely many segments. Of course, if you deny the supposition of the puzzle, then it may be easy to dispense the puzzle. But one may wish not to take the easy way out but instead grapple with the puzzle under the suppositions it makes. Of course, one can hold that there are empirical or even physics theoretical bases to hold that there is a finite upper limit to how many times distance and duration may be divided, but one might still wish to drive a harder bargain, which is that the question is not that of physical but rather of logical possibility.
Where in the paper does Thompson say that?
Quoting Michael
I had in mind not 'object' but that 'ways of dividing' is vague. Or maybe the poster just refers to the fact that there are many different divisors. And, yes, among the reals there are uncountably many divisors ("ways to divide").
This is less directed to you than it is directed to the poster who has ridiculous, ignorant ideas about mathematics. These are points that are too terribly obvious to anyone of even barely adequate intelligence and education, but the poster drags discussion down to the level that these things need to be made explicit:
Regarding numbers, when we say 'divide x by y to get z', that is an instruction for a procedure. That procedure merely upholds that x/y is the unique number z such that y*z = x. That does not imply, for example, that from the number x we to create two other numbers - one x/2 and another x/2. Rather, x/2 is itself a number. There are not two x/2, each one a separate object made by dividing x.
However, when mathematics is applied, then a material object may have x number of units, such as x number of square inches, or x number of pounds, or x number of grains of salt. Then, when we say, for example, "divide the object in half", we mean that we will have two different objects, each with x/2 square inches, or x/2 pounds, or x/2 grains of salt, as the case may be.
Those two different senses of "divide" are not in contradiction when a reasonable person considers the two different contexts.
The poster claimed that I equivocate about this. On the contrary, I am clear of quite clear of the distinction and none of my comments employ any equivocation regarding it.
Quoting Michael
The paradoxes discussed don't require splitting such things as photons. And one may take it as a premise or as an established fact that there is a shortest distance and a shortest duration. But perhaps one may also logically take a hypothetical premise that that is not the case.
Oh please! Of course it is understood that crumbs fall off and that we can't ensure that a knife cut will result in two pieces that are very much more than quite roughly the same weight. But that has no bearing on the principle of division. A person who drags in the fact that material objects are only measured approximately as if that refutes anything in a discussion such as this one is a person who has the mentality of a juvenile.
Moreover, the notion of dividing ad infinitum, such as in the paradoxes does not even depend on always dividing by halving or any precise number at all. The different divisions could be by different divisors, while still we get smaller and smaller distances or times. The poster brought in a big fat stupid red herring with his pointless point that measurements of material objects are not exact.
I don't understand this at all.
Quoting TonesInDeepFreeze
The problem is, that when we divide a material object such as a pie, in half, it is never a perfect division with two perfectly equal parts. There is always some degree of approximation. Dividing in half in mathematics is perfect, no approximation. Therefore "half" has a different meaning in the theories of mathematics, from the meaning it has in practical usage. So your example of dividing a pie, which necessarily involves an approximation, to demonstrate "half" in the context of theoretical mathematics, which is necessarily a division of perfect precision, is simple equivocation.
Quoting TonesInDeepFreeze
This is exactly why your example of dividing a pie in half, is a case of arguing through equivocation.
And the poster again states his big fat stupid red herring about approximation, thus deserving a restatement by me:
"Of course it is understood that crumbs fall off and that we can't ensure that a knife cut will result in two pieces that are very much more than quite roughly the same weight. But that has no bearing on the principle of division. Moreover, the notion of dividing ad infinitum, such as in the paradoxes does not even depend on always dividing by halving or any precise number at all. The different divisions could be by different divisors, while still we get smaller and smaller distances or times." The poster argues like a child.
Just two of many now:
(1) The thought experiments don't depend on exactitude of division. Only a person who hasn't thought about the matter would overlook that one divisor might be 2 and the next divisor 2.1 and the next divisor 10, and we'd still have diminishing lengths.
(2) After several iterations of the challenge in threads, the poster still won't say what he supposes is the inherent ordering of, for example, the members of the Beatles, as lack of facing that challenge illustrates that the poster cannot sustain his attack on the notion of the extensionality of sets, either in everyday life or in mathematics, as the principle of extensionality is basic to the mathematics he presumes, without any education in the subject, to refute.
Quoting TonesInDeepFreeze
It is taking this hypothetical premise that there is no smallest unit of space and time that gives rise to such things as Zeno's Paradox, Bernadete's Paradox of the Gods, and Thomson's lamp.
So if these arguments prove that contradictions follow if we assume that there is no smallest unit of space and time then as a refutation by contradiction it is proven that there is some smallest unit of space or time.
Sticking to the supposition of this puzzle creates confusion. The only possible solution is to look at it differently, not being hypnotized by 1/2, 1/4, ..... But I accept that it is your choice.
Quoting TonesInDeepFreeze
That's right. The difficulty is, I think, the assumption that "divide" means exactly the same thing in all contexts, taking the case of cutting something into pieces as the model. It obviously doesn't apply to numbers, or to space or time.
Quoting Michael
No, it's confusing theory with practice, abstract with concrete and not understanding that infinity means endless (but not necessarily limited)
Combined with another premise, it yields a paradox.
Thomson outlines the paradox:
Premise 1. To compete the main task requires completing infinitely many subtasks.
Premise 2. Infinitely many subtasks cannot be completed.
Conclusion. The main task cannot be completed.
But the main task can be completed. So one of the premises must be false.
One may say that Premise 1 is false. But that is a kind of "jumping the gun" when we would want to examine whether Premise 2 is false and Premise 1 is true. To say that Premise 1 is false on the grounds that it yields a falsehood skips that it's not Premise 1 alone that yields a falsehood, but rather it's the conjunction of Premise 1 and Premise 2 that yields a falsehood.
So Thomson does not right away say that it is Premise 1 that is false but rather he examines arguments for and against both premises.
Also Thomson argues:
"It is conceivable that each of an infinity of tasks be possible (practically possible) of performance [...] To deny [that] is to be committed to holding what is quite absurd, that for any given kind of task there is a positive integer k such that it is conceivable that k tasks of the given kind have been performed, but inconceivable, logically absurd, that k + 1 of them should have been performed."
That is, if there is a finite upper bound to how many times division can be executed, then there is some finite number k such that division can be executed k number of times, but division cannot be executed k+1 number of times. But why? Again, it's a leap to say that the answer is that otherwise there would be the false conclusion of the paradox, as the false conclusion might stem from the other premise of the paradox. Also, keeping in mind that Thomson is not arguing about the fact of the matter as to divisions of whatever, but rather about conceivability.
And I would consider that it's reasonable that the problem need not be contained to what happens to be true or false per some theory of physics, but rather that the problem of tasks may be abstract so that we may wish to resolve the paradox while granting the logical possibility or logical impossibility of the premises.
And Thomson's first point is one of logic, whatever the truth of falsity of the premises:
It is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.
Quoting Ludwig V
Who makes that assumption?
Quoting Ludwig V
There are different philosophical notions of infinity. But I don't know what specific approach to the paradoxes you think depend on a certain notion of infinity that, on its own terms, carries a misunderstanding. At least in context of mathematics that is mentioned in the puzzles, we have definitions:
S is finite if and only if there is a 1-1 correspondence between S and some natural number.
S is infinite if and only if S is not finite.
S has an upper bound in S if and only if there is a member of S that is greater than or equal to all members of S.
S has an upper bound if and only if there is an x such that x is greater than or equal to every member of S.
S has a lower bound in S if and only if there is a member of S that is less than or equal to every member of S.
S has a lower bound if and only if there is an x such that x is less than or equal to every member of S.
g converges to L if and only if for any d, there is an n such that for all k greater than or equal to n, |g(k) - L| is less than d.
If g converges to L, the L is the limit of g.
So, regarding the infinite sequences in Thomson's lamp problem:
The range of the sequence 1, 2, 3 ... has no upper bound in the range, but it has an upper bound as far as ordinals go (the least upper bound is wu{w}). ['w' for the set of natural numbers and 'u' for union]
The sequence 1, 2, 3 ... does not converge.
The range of the sequence 1, 1/2, 1/4 ... has no lower bound in the range, but it has a lower bound (the greatest lower bound is 0).
The sequence 1, 1/2, 1/4 ... converges to 0.
The range of the sequence 1, 0, 1 ... has an upper bound and a lower bound in the range.
The sequence 1, 0, 1 ... does not converge.
Those are just the ordinary mathematical definitions, which (in some sense) we don't evaluate as true or false but rather they're just stipulative definitions, and as such, they don't carry any "misunderstandings".
Thank you for the quote. His statement of his view that the notion of super-tasks is nonsense is wedged in a chain of reasoning, and somewhat hedged by saying 'if'. It's odd the way he slipped it in rather than stating it more centrally, since it would seem to be the most important conclusion in the paper. But for practical purposes, yes, we would take him as holding that notion of super-tasks is nonsense. Though it is not clear to me how he reached that conclusion (but I admit not really following certain parts of the paper).
But two points he does stress are the faulty inference from "no finite upper bound to the number of subtasks that can be completed" to "there is a completion of an infinite number of subtasks" and (2) that there are good grounds for doubting the premises of the paradox and that if there is misunderstanding in in those grounds than they are found in the "mathematical solutions" too (if I'm not mistaken, he has in mind that those faults are an incorrect understanding of the notion of an infinite sum).
Quoting Michael
My reaction to that is the same as I wrote above to Ludvig V.
Yes it surely does have bearing on "the principle of division". If, the principle of division indicates that when we divide a 6 kg item into two equal parts, we will have two pieces of 3 kg, yet when we repeatedly carry out the procedure, the closest we can get is two parts of 2.99999 kg, then the principle of division is proven to be false. It would be a theory which has been falsified by empirical evidence.
I predict that the crank will just say his incorrect argument over again.
And notice no response from the crank to the point that "there is no finite bound to the number of calls to division" does not require that each division be by the same divisor, as the paradoxes discussed don't require that the divisors all be the same.
And still interested in what is supposed to be the inherent ordering of a set such as the set of bandmates in the Beatles. If that question can't even be addressed by the crank then his claim that sets have inherent order (thus that the mathematical notion, and even the everyday notion, of sets is wrong) is not sustainable.
So from this we make the following argument:
P1. If the first task is performed at 11:00, the second at 11:30, the third at 11:45, and so on, then infinitely many tasks have been completed by 12:00
P2. It is impossible to have completed infinitely many tasks
C1. Therefore, it is impossible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on.
So what prevents the performance of the first task at 11:00, the second at 11:30, the third at 11:45, and so on? One answer is that it is metaphysically necessary that time is discrete with some smallest unit of time. At some point the time between task n and task n+1 is equal to the time between task n+1 and task n+2.
Is there another answer that allows for continuous time?
(2) C1 doesn't follow from P1 and P2. And it contradicts the point you quoted. Again, even if there is no completion of all of infinitely many subtasks, it is not entailed that there is a finite upper bound to how many may be completed, so, a fortiori, it is not entailed that each of the subtasks is not completed.
(3) Thompson argues, "It is conceivable that each of an infinity of tasks be possible (practically possible) of performance [...] To deny [that] is to be committed to holding what is quite absurd, that for any given kind of task there is a positive integer k such that it is conceivable that k tasks of the given kind have been performed, but inconceivable, logically absurd, that k + 1 of them should have been performed."
(4) If I'm not mistaken, Thompson recognizes physical possibility and logical possibility, which are at least fairly well understood, but he doesn't mention metaphysical possibility. That's not to say that the notion of metaphysical possibility should be ruled out, but only that it requires explication.
P2 is what Thomson tries to prove by introducing his lamp. Having performed infinitely many tasks entails a contradiction (the lamp must be either on or off but cannot be either).
Quoting TonesInDeepFreeze
It seems to be a straightforward modus tollens.
If A then B
B is impossible
Therefore, A is impossible
As far as I can tell, you're committing the very conflation that Thompson warns about and as you quoted my paraphrase of it.
Think of the difference between "For all, there exists" and "There exists for all". Even if there does not exist a completion of all the subtasks, it does not follow that there there does not exist a completion of each of them. Just like that there is no completion of all of infinitely many additions but there is a completion of any finite number of them.
Where is the conflation in my argument? I'll set it out more clearly:
P1. If (A) the first task is performed at 11:00, the second at 11:30, the third at 11:45, and so on, then (B) infinitely many tasks have been completed by 12:00
P2. B is impossible
C1. Therefore, A is impossible
The argument is certainly valid; it's modus tollens. So are you saying that A doesn't entail B or are you saying that B is possible?
The conflation is just as I stated it.
But your new argument is quite different from the previous argument, and, as far as I can tell, it does not suffer the conflation.
But your new argument is not modus tollens.
Modus tollens is:
A -> B
~B
Therefore ~A
Your new argument ['p' for 'possibly'] is:
A -> B
~pB
Therefore ~pA
The antecedent and consequent of the conditional are A and B; they are not ~pA and ~pB.
I'm not sure whether the argument is modally valid (I'm very rusty in modal logic).
/
I'm not firmly opining as to whether A implies B nor as to whether B is possible.
First, though, what sense of 'possible' is meant? Thomson discusses physical possibility and logical possibility. If I'm not mistaken, he doesn't mention metaphysical possibility. Of course, discussion doen't have to be limited to Thomson's context, but 'metaphysical possibility' requires even more explication.
Anyway, I tend to favor that A is correct. But I tend to think Thomson may well be on the right track when he argues that it is not logically impossible to execute infinitely many steps in finite time.
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Thank you for the clarification. I must admit, I was a bit puzzled by "bound". I'm used to "limit". This clarifies something that was puzzling me - how one could describe the relationship between the 0 and 1 to the steps of the series. This seems to work very well.
Quoting TonesInDeepFreeze
Are you suggesting that it might be the case that all of infinitely many tasks can be completed? What would the last task be?
On the other hand, perhaps we should accept that when Achilles catches the tortoise or finishes the race, he has completed all of infinitely many tasks. That might need some explaining, though, wouldn't it?
Quoting TonesInDeepFreeze
There is the possibility that he doesn't recognize metaphysical possibility. Not everyone does.
Quoting Michael
Here's a thought. When we define the series, we have defined each and every step in the series (or, if you prefer, each and every member of the relevant set.) Defining a step is a task. Suppose we do that at 10:55. So we completed infinitely many tasks, not only before 12:00, but before 11:00. So it is not impossible to complete infinitely many tasks before 12:00.
What's more, adding together the first step and the second step is a task. I can calculate the limit (sum) of the infinite series, well before 12:00.
Obviously, I haven't defined the last step, or the penultimate step, but that wasn't the challenge.
Or have I misunderstood the mathematics?
I made no judgement on that. Again:
Suppose there can be no completion of all the tasks. That does not entail that there is a finite upper bound to how many task can be completed. That is, suppose for some finite n you say that no more than n tasks can be completed. But n+1 tasks can be completed without contradicting that there can be no completion of all of them. So there is no n that is a finite upper bound to how many tasks can be completed.[s]And your statement (at least as you wrote it) was that none of them could be completed, which is even more wrong.[/s] [strikethrough in edit; the remark pertains to a different poster.]
To see that explained again, see Thomson's paper.
Quoting Ludwig V
Indeed. Hence 2500 years of philosophers, mathematicians and scientists talking about it.
I like to keep the word 'series' for sums per convergences, and the word 'sequences' for sequences.
It is not the case that when we define an infinite sequence we must individually define each entry in the sequence. Example:
Definition of sequence S:
The domain of S is the set of natural numbers. For every natural number n, S(n) = n+1.
That's a finite definition (all definitions are finite) of an infinite sequence.
Then let's rephrase P1 as a question.
If the first task is performed at 11:00, the second at 11:30, the third at 11:45, and so on, then how many tasks are performed by 12:00?
Quoting TonesInDeepFreeze
This is where Thomson's lamp comes in. His argument is that if B is performed then a contradiction follows; the lamp can neither be on nor off at 12:00 but must be either on or off at 12:00. Therefore B is proven impossible.
Quoting TonesInDeepFreeze
I'm pretty sure it is. But if you prefer:
P1. If (A) it is possible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on, then (B) it is possible for infinitely many tasks to be performed by 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A
Quoting TonesInDeepFreeze
See subjunctive possibility:
I am claiming that supertasks are more than just nomologically impossible. They are either logically impossible or metaphysically impossible. I say the latter simply because it's the weaker claim.
In the quotation in that message, I made no statement. I just asked a question.
Quoting TonesInDeepFreeze
Thank you for the answer to my question. It is very helpful.
Quoting TonesInDeepFreeze
I would have thought that "for any finite number of tasks, there may be a completion of all the tasks" does not imply that there may be a completion of all of infinitely many tasks and does not imply that there may not be a completion of all of infinitely many tasks.
Quoting TonesInDeepFreeze
I agree with what you say. But, nonetheless, we have defined each entry in the sequence. Or is there an entry that is not defined? I can't think of one.
My indirect response is:- I'm sorry. I should have explained.
I was thinking about "there is no finite upper bound to how many task may be completed in finite time." It occurred to me that that depended on how long each task takes. (And that depends on what the task is.)
Clearly, if each task takes 1 second, there is a limit to the number of tasks that can be completed in 2 minutes. But how long does it take to add 1 to a given number? Does it take less time or more to add 1,000,000 to a given number? How long does it take to switch Thompson's lamp on or off? How long does it take to divide a given number by 2 or 10?
And then,the proof that sqrt(2) is irrational is also a proof that no rational number is sqrt(2)? Do I have to show separately and individually that each rational number is not sqrt(2)? I think not, but I have proved, of each rational number, that it is not sqrt(2). Is this one task, or many? How long does it take, either to prove that sqrt(2) is irrational or to prove, of a specific rational number, that is not sqrt(2)?
Quoting TonesInDeepFreeze
Certainly. But there are some points I am not clear about.
Quoting TonesInDeepFreeze
Yes, but how long did it take? Have you not defined each individual member of the sequence and all the members of the sequence? Which members of the sequence are not defined? How many tasks have I completed?
Quoting TonesInDeepFreeze
Thank you. I struggle with that difference. I'm not sure that everyone is consistent. What term do you use for a member of the sequence. People seem to by using "stage" or "term". Then there's the difficulty that "0, 1, 0, 1, ...." has, in one way, two members, each of which occurs repeatedly, So what do we call the first "0" as distinct from the second "0"?
Quoting TonesInDeepFreeze
Yes. They're still talking about Epimenides the Cretan (and variants), as well. I can't deny they are both fascinating and annoying.
quoted by @Michael
Which means that metaphysics does not have an authoritative definition that we can all use for communication purposes, apart from the recognition that it involves logic.
For that reason, I prefer to stick to logical and physical possibility. There at least appears to be some consensus about the use of those terms, even though philosophers will, from time to time, quarrel about their definitions as well.
I've moved your discussion on set ordering and the meaning of equality to this discussion.
Thank you sir.
But in that example, his argument includes the premise that there is a state at 12:00 and that that state must be determined by an immediate predecessor state but that there is no immediate predecessor state.
And an earlier point:
Quoting Michael
As I mentioned, C1, as you wrote it, is a non sequitur. That it is impossible for infinitely many tasks to be performed in finite time does not entail that there is a finite upper bound to how many tasks may be performed in finite time, let alone that each of the tasks is impossible to be performed. But maybe you didn't mean C1 as you wrote it.
Quoting Michael
It would be difficult not to say that denumerably (thus infinitely) many are performed. But I am reserving full committment to that, as there may be a finer analysis depending on complications in the notions of 'task' and 'performed' and perhaps in the inscrutabiliy of the overall hypothetical context.
Quoting Michael
That's okay (though, personally for me, modulo the residue of doubt that I mentioned about P1). But I wouldn't take P2 as a given without justification.
P2 is what Thomson's argument tries to prove. The lamp must be either on or off at 12:00, but if the button is pushed an infinite number of times between 11:00 and 12:00 then the lamp can neither be on nor off at 12:00. This is a contradiction. Therefore the button cannot be pushed an infinite number of times between 11:00 and 12:00.
Quoting TonesInDeepFreeze
This is the argument I am making:
P1. If (A) it is possible for a button to be pushed at 11:00, 11:30, 11:45, and so on, then (B) it is possible for a button to be pushed an infinite number of times between 11:00 and 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A [from P1 and P2 via modus tollens]
P3. If (C) it is possible for time to be continuous then A
C2. Therefore, it is not the case that C [from C1 and P3 via modus tollens]
C3. Therefore, it is necessary for time to be discrete [from C2]
My mistake. Just now I put an edit note in the post.
Quoting Ludwig V
Right.
Quoting Ludwig V
I am using 'define' in the exact sense of making a mathematical definition. A definition is a single formula. It follows from the definition of the sequence that each entry in the sequence is "determined" (for lack of a better word), but that is not to say that each has been individually defined.
Quoting Ludwig V
I was speaking in the context of the completion times halving.
Quoting Ludwig V
Less time or more time than what? And what is meant by "time to add a number"? Does it mean number of steps in some given adding algorithm?
Quoting Ludwig V
The problem presented states the increasing rate of alteration. Maybe you're asking about how long it takes to jab the button. I don't find that to be relevant, since whatever times it takes, we need only assume that it happens within the durations given in the problem. For that matter, I don't think the particulars about buttons, jabbing, or especially about human acts such as fingers reaching to touch a device are relevant, as the problem could be entirely abstract, as what is essential only is that the lamp goes on and off at the increasing rate mentioned, or, for that matter, it's not essential even that it's a lamp or any other particular device (could be clown klaxon going off an on for all it matters) as long as there are alternating states, whatever they may be.
Quoting Ludwig V
For all practical purposes, yes.
Quoting Ludwig V
You proved a universal generalization:
For all x, if x is rational then it is not the case that x^2 = 2.
Then, with universal instantiation, for any given rational number, for example, 1.4 we prove that it is not the case that (1.4)^2 = 2.
Quoting Ludwig V
Which sequence? There are different sequences involved in the puzzles here.
An infinite series that has a sum (some might say the series is the sum) requires first having an infinite sequence (each entry in the sequence is a finite sum) that converges, and the sum is the limit. The sequence whose entries are 0, 1, 0, 1 ... does not converge. However, whatever you mean by 'complete', there are infinite series that have a sum.
An infinite sequence is a function whose domain is an infinite ordinal. *
* In discussion about the task problems, we are modifying a bit by not having 0 in the domains of the sequences. That is, we index starting with 1 rather than with 0.
A function is a certain kind of set of ordered pairs.
The members of a function are ordered pairs.
The domain of the function is the set of first coordinates of the ordered pairs. We call members of the domain "arguments for the function".
The range of the function is the set of second coordinates of the ordered pairs. We call members of the range "values of the function". I also call them "entries".
We say that an argument maps to a value.
The infinite sequences in this context are:
(1) The function that maps every natural number n>0 to to 1/(2^n)
This is the function {<1 1/2> <2 1/4> <3 1/8> ...}
(1) The function that maps every natural number n>0 to either 0 or 1 ("off or on") depending on whether n is odd or even
That is the function {<1 0> <2 1> <3 0> ...}
Quoting Ludwig V
0 is 0. There are not different 0s. But with the function just mentioned, and 0 occurs as values for different arguments of the function. 0 is the value at the arguments 1, 3, 5 etc.
As I mentioned, that is a premise that you don't include in your own argument. As I mentioned:
"his argument includes the premise that there is a state at 12:00 and that that state must be determined by an immediate predecessor state but that there is no immediate predecessor state."
I can't imagine anyone denying that there is no immediate predecessor state, but some partisans who don't accept the argument deny that the state at 12:00 must be determined by an immediate predecessor state. So you must include the premise that the state at 12:00 must be determined by an immediate predecessor state.
If you don't mean "Therefore, it is impossible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on" then it should be considered scratched.
The division of time mentioned in the thought experiment doesn't require continuousness of time; it only requires density time (via the density of the rationals).
Continuousness implies density, but density does not imply continuousness. So banning continuousness does not ban density. But you need to ban density of time.
You should have instead:
P1. If (A) it is possible for a button to be pushed at 11:00, 11:30, 11:45, and so on, then (B) it is possible for a button to be pushed an infinite number of times between 11:00 and 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A [from P1 and P2 via modus tollens]
P3. If (C) it is possible for time to be dense then A
C2. Therefore, it is not the case that C [from C1 and P3 via modus tollens]
C3. Therefore, it is necessary for time to not be dense [from C2]
But:
(1) We may doubt P2.
(2) You have an unstated premise on which P2 is based, viz. that that state must be determined by an immediate predecessor.
(3) We may doubt P3.
The lamp exists at 12:00 and as per the laws of excluded middle and noncontradiction is either on or off.
Given the way lamps work, or at least the lamp in this example, if the lamp is off and the button is pushed then the lamp is turned on and if the lamp is on and the button is pushed then the lamp is turned off.
Nothing other than pushing the button can affect the lamp.
Scenario 1
The lamp is off at 10:00 and the button is only pushed at 11:00. Therefore the lamp is on at 12:00.
Scenario 2
The lamp is off at 10:00 and the button is only pushed at 11:00 and 11:30. Therefore the lamp is off at 12:00.
Scenario 3
The lamp is off at 10:00 and the button is only pushed at 11:00, 11:30, 11:45, and so on. Therefore the lamp is ? at 12:00.
No substitution for "?" is either derivable or consistent. Therefore the button-pushing procedure in scenario 3 is proven impossible in principle.
Quoting TonesInDeepFreeze
I don't know what this means.
Quoting TonesInDeepFreeze
I do mean that.
While you were posting, I revised my post to better explain about continuousness and density.
Quoting Michael
That is not at issue. Rather, as I've said twice now, at issue is whether its state at 12:00 depends on there being an immediate predecessor state. Thomson assumes that it does.
Quoting Michael
Not just that it was off and then turned on, but rather that it was off at time t1 and on at time t2. That is, that it's not just a matter of the lamp having been off previously but rather that there is an off state that is an immediate predecessor of the on state and that that extends to 12:00 too so that for the lamp to be on at 12:00 there must be an immediate predecessor state in which the lamp was off, mutatis mutandis for the lamp being off at 12:00. Thomson mentions this. It's a premise that needs to be stated.
Quoting Michael
(1) The first task is impossible to be performed. The second task is impossible to be performed. The third task is impossible to be performed ...
Quantified:
For all tasks, there is not a performance of any of them.
I think that is not what you mean.
(2) It is not possible for there to be a single performance of all the tasks.
Quantified:
There is not a performance that performs all the tasks.
I surmise that is what you mean.
I wouldn't write "it is impossible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on" because it can be understood in sense (1).
It is not possible for the first dancer to do a flip today, for the second dancer to do a flip tomorrow, and so on.
I would take that to mean that none of the dancers can do a flip on their appointed day.
I also revised my post after posting it.
The three implicit premises are:
1) The lamp exists at 12:00 and as per the laws of excluded middle and noncontradiction is either on or off.
2) If the lamp is off and the button is pushed then the lamp is turned on and if the lamp is on and the button is pushed then the lamp is turned off.
3) Nothing other than pushing the button can affect the lamp.
It therefore follows that if the lamp is on then it is on only because it was off and the button was pushed to turn it on. It makes no sense for a lamp to be on without having been turned on, and in this specific case it can only be turned on by pushing a button when it is off.
Therefore if the lamp is on at midnight then it is on only because it was off and the button was pushed to turn it on. This is impossible if the infinite sequence 11:00, 11:30, 11:45, ... models the times that the button is pushed.
Quoting TonesInDeepFreeze
I am saying that it is impossible in principle for the infinite sequence 11:00, 11:30, 11:45, ... to model the times that the button is pushed.
That seems okay at face value. But since you've put the argument in a list, I'd make explicit all the premises.
I'd like to read Benacerraf's paper that disputes that there can't be a state at 12:00, and the papers mentioned in Thomson's paper for required context.
I just found this when reading of p-zombies on SEP:
They hold there is an identity that is metaphysically necessary, and it is metaphysically necessary because it is a a posteriori necessity.
I wandered over and happened to notice this remark. I can say something about it. I was pointed at Benacerraf's paper a few weeks ago, and since I had developed the same solution, I felt vindicated and at the very least sane. I know Benacerraf from his great paper, What Numbers Could Not Be, and has a lot of credibility with me.
So I feel qualified to explain his point.
Here is Benacerraf's argument in my math-y conception. It's the same argument.
The lamp problem is best modeled as a function defined on the ordinal [math]\omega + 1[/math], with output in the set [math]\{0, 1\}[/math]. (I don't recall if Benacerraf explicitly uses this mathematical approach but his argument is the same ).
[math]\omega + 1[/math] can be visualized as the sequence
0, 1, 2, 3, ...; [math]\omega[/math]
where 0, 1, 2, 3, ... ranges through all the natural numbers; and there is a "point at infinity" after them all, traditionally called [math]\omega[/math] in this context. It can of course be formalized as an ordinal but that's not even important in this context.
If it makes anyone happy in the context of this problem, we can model it as
1/2, 3/4, 7/8, ...; 1
where the values are times getting ever closer to 1, with 1 being the mathematical limit, as well as the time on a clock. Or 1/2, 1/4, 1/8, ...; 0 where the times are seconds before midnight, and 0 is exactly midnight.
I analogized that last idea to Cinderella's coach, which is a beautiful coach at 1/2, 1/4, 1/8, ... seconds before midnight; but becomes a pumpkin at midnight. It's not usually thought of as such, but Cinderella's coach is a supertask puzzle. If Cinderella checks her watch at each time 1/2, 1/4, etc., it's always a coach. How does it become a pumpkin?
It's the same mathematical insight that applies to most of these supertask puzzles. Sequences don't ever "reach" their limit, but every sequence can be assigned an arbitrary terminal state. Sometimes the terminal state is the limit of the sequence, and sometimes it's not. You can always assign a terminal state because it's just a mathematical function. It's a function defined not on [math]\omega[/math], as infinite sequences usually are; but on [math]\omega + 1[/math], to include a terminal state.
We can use this structure to model the lamp. For each natural number 0, 1, 2, 3, ... the output alternates on/off, which we can denote as 1, 0, 1, 0, ...
So the lamp problem defines the function for each of the inputs 0, 1, 2, 3, ... but does not specify the value at [math]\omega[/math]. Therefore we are free to define it any way we like.
Both outcomes are entirely consistent with the premises of the problem.
Additionally, unlike in some supertask puzzles, in this case there is no natural or preferred solution. That's because in neither case can the terminal value be the limit of the sequence.
Contrast this, for example, with the sequence 1/2, 3/4, 7/8, ...; 1. That notation says that there's an infinite sequence; and after the entire sequence, we have a value, 1.
Now just as before, the assignment of the value of the terminal state is arbitrary. But in this particular case, the value 1 is natural, in the sense that 1 is the mathematical limit of the sequence. This idea of adjoining the limit of a convergent sequence to the end of the sequence is a convenient and natural formalism.
With the lamp, there is no possible way to assign a terminating value that makes any particular sense. Instead, absolutely any answer will do. On, Off, or as I facetiously said earlier, a plate of spaghetti; to emphasize the arbitrariness of the choice.
I've visualized the lamp problem that way for a long time. And when I found out that Benacerraf gave the same argument (minus the spaghetti), I was happy.
That's Benecerraf's argument in my words. Here's his paper.
Tasks, Super-Tasks, and the Modern Eleatics
Quoting TonesInDeepFreeze
I was thinking of a term that would apply to sequences in general.
Quoting TonesInDeepFreeze
I think I can work with that. But I see that you used "entry" elsewhere. That's simpler.
Quoting TonesInDeepFreeze
I didn't realize that is the context. Then most of what I said is irrelevant.
Quoting TonesInDeepFreeze
I agree with all of that. There is a twist, of course. The sum is not the total addition of all the entries, but the limit of the total addition of all the entries. The total addition of all the entries up to a specific point will converge on/with the sum.
Quoting TonesInDeepFreeze
I agree with all of that. That's why I ended up formulating the problem in terms of 1 and 0 alternating. Unfortunately, that doesn't resolve everything. But I need time to work out how to articulate this reasonably clearly. Sorry. I will get back to you.
Quoting Lionino
Well, I don't want to quarrel about a name. Call it metaphysics or call it a posteriori necessity, my account of the problem derives from Wittgenstein, Ryle and Toulmin. That's what matters.
Quoting Ludwig V
The same comment applies to much of the discussion in this thread as well. Ryle seems to have thought it applies to all philosophical problems, but I wouldn't go that far.
Quoting Michael
My problem is that I don't understand what carrying out a supertask is. So I can't even think about the consequences of carrying one out. @TonesInDeepFreeze, This is the best that I can do to articulate what bothers me.
The first sentence is true and is the proof that "supertasks are senseless" (as Thomson says).
The second sentence is false. As mentioned several times, the implicit premises are that the lamp continues to exist (as a lamp) at midnight and that nothing other than pushing the button can turn the lamp on or off.
We are being asked about the causal consequence of having performed a supertask. Your introduction of magic or God or some other entity or event at the end is a red herring.
This is abundantly clear with the pseudocode I offered a month ago:
The logic of this does not allow for echo isLampOn to output true or false or 'a plate of spaghetti'.
Your "solution" is to inject some additional code after while (true) { ... }, assigning some arbitrary value to isLampOn, but in doing so you are no longer addressing the problem as posed.
@TonesInDeepFreeze asked for Benacerraf's argument, of which I provided my own version without any change in meaning. I also linked Benacerraf's paper per Tones's request. That was my motivation for posting.
Quoting Michael
This is a red herring of no relevance. I've responded to the button pushing argument as many times as you've mentioned it. If "nothing other than pushing the button turns the lamp on or off," then at midnight, the button pusher pushes the button and turns the lamp on or off, per your premise. The terminal state is On or Off. In either case "the button was pushed" since you insist on it. No matter the final state, the button was pushed, and your premise is satisfied. Neither On nor Off is to be preferred since the sequence 1, 0, 1, 0, ... has no limit. Thompson himself makes this observation in his paper.
I did enjoy our earlier conversation, but no additional typing on my part could be of any use to you. If it could be, it would have already been.
I regard Thompson's lamp as a solved problem. I'll leave it at that, and refer you to my existing posts on the subject.
He doesn't push the button at midnight. He only pushes it at 23:00, 23:30, 23:45, and so on. This is an explicit premise of the problem.
Also, pushing the button will only turn the lamp on if it was off and so to say that the button was pushed at midnight to turn it on is to say that the lamp was left off after having performed the supertask, which is false.
I appreciate your enthusiasm, but surely you can see that nothing new has been said between us for a long time. We could just reread each other's old posts. I hope you'll forgive me for declining to engage. If you had better insight into infinite sequences and their limits, you wouldn't be troubled by what happens at midnight. I say that because the code you posted exactly reflects your misunderstanding of the nature of infinite sequences. Think of Cinderella. Why does the coach turn into a pumpkin? It does so because that's the premise of the story. The lamp is in some state at midnight because you say it is. And it doesn't matter which.
I understand how infinite sequences and limits work, as did Thomson. That is why I understand that an infinite sequence of button pushes before midnight is inconsistent with the lamp being either on or off at midnight.
The problem is that you seem to fail to acknowledge how lamps work.
Lamps that switch state in arbitrarily small intervals of time? I missed that day in lamp school.
The lamp is as much a fairy tale as Cinderella's coach, which is why I use that example. It's a magic lamp. You are the one invoking magic, and then acting like it's a real lamp to try to argue a pointless point. That's another source of your confusion.
"How lamps work?" Lamps don't work that way! Maybe YOU missed that day in lamp school.
This is the assumption we allow for to examine the possibility of supertasks.
But it is still the case that the lamp cannot arbitrarily be on (whether at midnight or any other time). It can only be turned on by pushing a button when it is off. You continually ignore this fact when you talk about the mathematical value ?.
Jeez I didn't even finish editing before you snapped back. Have a nice evening. You are trolling me now. Doing a good job of it.
Sure.
P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 22:00
P5. The lamp is either on or off at midnight
C1. If the button is only ever pushed at 23:00 then the lamp is on at midnight
C2. If the button is only ever pushed at 23:00 and 23:30 then the lamp is off at midnight
C3. If the button is only ever pushed at 23:00, 23:30, 23:45, and so on ad infinitum, then the lamp is ? at midnight
It is essential to note that in each case we are being asked about the causal consequence of having pushed the button one or more times.
No substitution for "?" in the consequent of C3 is either derivable from or consistent with the premises and the antecedent of C3. Therefore we must accept that it is impossible in principle to have performed the supertask.
The following pseudocode provides a demonstration of C3 and is the correct way to interpret the logic of Thomson's lamp:
We are being asked what value is output by the line echo isLampOn (which allegedly will run after two minutes). The problem is that it cannot be true and cannot be false but cannot be anything other than true or false.
Benacerraf's solution is akin to injecting some additional code after while (true) { ... }, assigning some arbitrary value to isLampOn, but in doing so he is no longer addressing the problem as posed.
Infinite density of time. What could that possibly mean? "Density" implies two distinct substances, as a ratio (weight per volume for example). Are you suggesting that time consists of two distinct substances, in a relation which is infinitely dense? That seems absurd.
I know your sophistry TIDF. When cornered in an argument, instead of retreating, you create an absurd concept, by combining words in an incoherent, irrational way, hoping that the incoherency will go unnoticed. This would allow you to slither away through the crack between the infinitely dense particles of time. This is better represented as the irrational space between the rational numbers, and the slippery sophist slides out of that possible world, through the irrational exit.
I don't think it is obvious what label we should have for this phenomenon. But I do think there are problems with the proposal. For the record:-
Quoting Michael
Why don't you just run the code and see?
You call it a pseudo-code, but one assumes that you could turn it into real code.
Because in reality a computer cannot perform two consecutive operations within 10[sup]-44[/sup] seconds.
But we don't need to run the code. We can understand the logic of it and so know that there is no consistent answer. It cannot be true, cannot be false, and cannot be anything other than true or false.
Fair enough. But that's a much better description, IMO. What more is there to say?
The matter under consideration is whether time can be divisible ad infinitum.
Density is a property of orderings. An ordering is dense if and only if between any two points there is another point. If time is divisible ad infinitum, then the ordering of points of time is dense.
I have not taken a position on the matter of whether time can be divisible ad infinitum. I only pointed out that, pertaining to the lamp puzzle, Michael's argument is not to refute continuousness but rather to refute density (to be clear, Michael is not the crank).
But, of course, the crank in his unfocused and ignorant way just sees some words on a screen, doesn't know what they mean and then lashes out wildly.
No, the infinite sum is the limit of the sequence of the finite sums.
Yes. Sorry. Careless.
Yes, the usage of the term falls out of the traditional. If we want to really go back, metaphysik? is whatever Aristotle wrote (catalogued as such) after his physics. The crux is that when we are talking about metaphysical possibility, it is not automatic that we are talking about possibility in metaphysics, but something related, though different altogether, that makes a noun by itself, metaphysical-possibility, or new(-)metaphysics as you say likewise logical-possibility does not immediately overlap with possibility in logic, if such a phrase even makes sense immediately. Case in point, the arrangement of games, as described in Toulmin's article, seems to have nothing to do with 'metaphysics' as traditionally used.
Quoting Ludwig V
"Metaphysical possibility" is sought as distinct from logical and physical possibility. Using the same Venn diagram I started the thread with, metaphysically contigent must also be logically contigent (Earth is the third planet from the Sun), but it doesn't inherently matter whether it is physically necessary (electric permeability ?) or contigent (kangaroos are digitigrades). Metaphysically contigent includes, at least, the set of all physical contigency. Strictly speaking, and using the definition I tried to work with, metaphysical contigency is everything where there is a possible world where X is not the case; those would also be synthetic contigencies especially if we take analytic statements to be necessary, otherwise further definition is needed.
Quoting Ludwig V
Some philosophers make away with both the a posteriori / a priori and analytic/synthetic distinctions, Kripke gave a strong blow against the idea that a posteriori propositions are always contigent. I don't feel informed enough yet, if ever, to make a statement about it. I take the traditional view and accept all those distinctions.
Yes. Exactly.
Quoting Lionino
I'm afraid I couldn't follow your account of this. I'll have to take another look at it later on. But I'm not sure that the project of trying to articulate the Venn diagram is necessarily the best way to go. It may be constraining, rather than guiding, your thinking.
Quoting Lionino
Yes, but are the philosophers who want to make synthetic necessity among them?Quoting Lionino
I'm afraid that I was brought up in the tradition that says that the analytic/synthetic distinction is the only one that means anything. My dilemma is that can see some sense in the point that that one only applies to language, where a priori applies to knowledge and contingent applies to "the world".
As usual, you're making things up to suit your purpose. "Dense order" is a property of the elements of sets, commonly numbers, and never "points". Points do not make up the elements of a set, neither does time make up the elements of a set. You continue in your sophistry, taking a definition from set theory which is specifically applicable only to the elements of a set, and applying this definition to points and time. TIDF, the epitome of a sophist, taking a term with a specific articulated definition, designed for a very specific application, and applying it somewhere else, where it is not suited. How do you propose that a multitude of "points" could be the distinct elements of a set?
I've been thinking about your button. I was going to try to address your argument on your own terms, but first I thought I'd go back to Benecerraf's paper to review his argument, which is the one that makes sense to me. He directly addresses some of your concerns.
A note on notation. You like 12:30-1:00 and I prefer 1/2, 3/4, 7/8, ... We can convert between my notation and yours as 12:30 + that fraction of 30 minutes. So my t = 1/2 is your 12:45. Hope that's acceptable. Benecerraf uses my notation, or rather he says that [math]t_0[/math] and [math]t_1[/math] are the initial and terminal times of the lamp. He doesn't go into any more detail, but this corresponds to my idea of 1/2, 3/4, 7/8, ...; 1 as the times at which we observe the lamp.
I find it helpful to analogize with the mathematical sequence 1/2, 3/4, 7/8, ..., which converges to 1.
Now to Benecerraf's argument.
He posits that two individuals, Aladdin and Bernard, each "perform the supertask." At time [math]t_1[/math], the terminal state, Aladdin finds the lamp on, and Bernard finds it off.
Benacerraf argues that neither outcome is inconsistent with the rules of the problem, for the reason that the rules are defined at each of the times t = 1/2, 3/4, 7/8, etc., but NOT at t = 1.
He writes: [Benecerraf's italics as indicated, but I didn't mark up all of them]
[quote=Benecerraf]
I submit that neither description [Aladdin or Bernard's - ed] is self-contradictory, or, more cautiously, that Thomson's argument shows neither description to be self-contradictory (although possibly some other argument might).
According to Thomson, Aladdin 's lamp cannot be on at [math]t_1[/math], because Aladdin turned it off after each time he turned it on. But this is true only of instants before [math]t_1[/math]! From this it follows only that there is no time between [math]t_0[/math] and [math]t_1[/math] at which the lamp was on and which was not followed by a time also before [math]t_1[/math] at which it was off. Nothing whatever has been said about the lamp at [math]t_1[/math] or later. And similarly with Bernard's lamp. The only reasons Thomson gives for supposing that his lamp will not be off at [math]t_1[/math] are ones which hold only for times before [math]t_1[/math]. The explanation is quite simply that Thomson's instructions do not cover the state of the lamp at [math]t_1[/math], although they do tell us what will be its state at every instant between [math]t_0[/math] and [math]t_1[/math] (including [math]t_0[/math]). Certainly, the lamp must be on or off at [math]t_1[/math] (provided that it hasn't gone up in a metaphysical puff of smoke in the interval), but nothingf we are told implies which it is to be. The arguments to the effect that it can't be either just have no bearing on the case. To suppose that they do is to suppose that a description of the physical state of the lamp at [math]t_1[/math] (with respect to the property of being on or off) is a logical consequence of a description of its state (with respect to the samne property) at times prior to [math]t_1[/math]. I don't know whether this is true or not, and in section II I shall briefly investigate some matters that bear on this issue. But, true or not, the argument is invalid without the addition of a premise to that effect.
[/quote]
I could not have said that any better. Though I'd have inserted some paragraphs. I left it as is since that's what he wrote.
I note in passing that he's anticipating my plate of spaghetti. When he says, "... the lamp must be on or off at [math]t_1[/math] (provided that it hasn't gone up in a metaphysical puff of smoke in the interval) ..." he is making the point that the terminal state of the lamp is entirely arbitrary. There are no constraints on the terminal state in Thomson's description of the problem, not even the necessity of being either off or on.
He also denies that the terminal state of the lamp is a logical function of the prior states. He is directly addressing your belief that the terminal state must somehow arise from what's gone before. He says that without additional assumptions, the argument is invalid.
Can you respond to Benecerraf's argument? Clearly he is responding to Thomson's version, and perhaps you have additional assumptions.
My sense of your view is that you intuitively wish that the limit 1 had an immediate predecessor among 1/2, 3/4, 7/8, ... But of course it doesn't. Limits are like that. They don't have immediate predecessors. In fact if you plotted the points 1/2, 3/4, 7/8, ..., and 1 on the number line, you can see that any step left from 1, no matter how tiny, necessarily jumps over all but finitely many of the terms of the sequence.
It's true that if the terminal state is on, then it was off at some time in the past. And if it's off, then it was on at some time in the past. Benecerraf points that out as if anticipating your thought process.
I address it all here. P1 is an implicit premise in Thomson's argument. He is asking "what happens to a lamp if we push its button an infinite number of times?", not "what happens to a lamp if we push its button an infinite number of times and then some arbitrary thing happens to it?".
If your only way to make sense of a supertask is by introducing God or magic to fix the problem at the end then you haven't made sense of a supertask at all. You might as well try to resolve something like the grandfather paradox by doing the same. Time travel into the past isn't physically possible but granting its possibility for the sake of argument doesn't then entail that anything goes, but that seems to be your approach to this issue.
P1 says, "Nothing happens to the lamp except what is caused to happen to it by pushing the button"
I do not know if that is something Thomson said or if you added that.
But in that case, then you are subject to Benacerraf's point. You haven't said ANYTHING about the terminal state. I don't know if a button is pushed or not at the terminal time. Who says it's not? You've just got your imagination running away with you about things that don't make sense.
If nothing happens to the lamp unless the button was pushed; then at the terminal time, if anything at all has happened to it -- turned on, turned off, turned into a plate of spaghetti, or as Benecerraf himself allows, just disappeared entirely -- then a button was pushed. If you say so.
There's no contradiction. You said if something happened, a button was pushed. Ok. Something happened and a button was pushed. What of it?
Quoting Michael
This is something you're adding, I don't know where you're getting it.
P1 is your own rule. If something happened to the lamp, a button was pushed. So at the terminal state if ANYTHING happened, a button was pushed. You're right. P1 is satisfied. And the lamp is on. Or the lamp is off. Or the lamp has turned into a plate of spaghetti.
Quoting Michael
I'd invite you to engage more deeply with Benacerraf's argument. I was indeed gratified that he even made my point about the spaghetti. "... provided that it hasn't gone up in a metaphysical puff of smoke in the interval ..." The terminal state is arbitrary.
C3 says it's not. If the button is only ever pushed at 23:00, 23:30, 23:45, and so on ad infinitum, then ipso facto the button is not pushed at midnight.
Quoting fishfry
Your arbitrary stipulation that the lamp is on or off at midnight is inconsistent with P1-P4.
The lamp can only ever be on iff the button is pushed when the lamp is off to turn it on. The lamp can only ever be off iff either it is never turned on or the button is pushed when the lamp is on to turn it off. Midnight is no exception.
Quoting fishfry
That seems to be true, so Benacerraf is right.
Doesn't it follow that both outcomes are consistent with the rules of the problem?
If both outcomes are consistent with the rules of the problem, doesn't that imply that they are not self-consistent (contradict each other)? If so, Michael is right.
But if they contradict each other, doesn't ex falso quodlibet applies (logical explosion)?
The logical explosion implies your conclusion, that justifies your plate of spaghetti, doesn't it? So you are right.
End of discussion? Maybe.
The rules must be consistent with each other where they apply. The problem is that the rules don't apply to the limit, because the limit is not generated by the function, that is, it is not defined by the function.
The limit is defined, however, as part of the function, along with the starting-point and the divisor to be applied at each stage. In that sense, they are all arbitrary. But the idea that they could all be replaced by a plate of spaghetti is, I think, I mistake. Don't we need to say that these numbers are not defined by the function, but are assigned a role in the function when the function is defined, which is not quite the same as "arbitrary"? The range of arbitrary here, has to be limited to natural numbers; plates of spaghetti are neither numbers nor, from some points of view, natural.
It's not true, and so he's not right.
These are our premises before we even consider if and when we push the button:
P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 10:00
From these we can then deduce:
C1. The lamp is either on or off at all t[sub]n[/sub] >= 10:00
C2. The lamp is on at some t[sub]n[/sub] > 10:00 iff the button was pushed at some t[sub]i[/sub] > 10:00 and <= t[sub]n[/sub] to turn it on and not then pushed at some t[sub]j[/sub] > t[sub]i[/sub] and <= t[sub]n[/sub] to turn it off
C3. If the lamp is on at some t[sub]n[/sub] > 10:00 then the lamp is off at some t[sub]m[/sub] > t[sub]n[/sub] iff the button was pushed at some t[sub]i[/sub] > t[sub]n[/sub] and <= t[sub]m[/sub] to turn it off and not then pushed at some t[sub]j[/sub] > t[sub]i[/sub] and <= t[sub]m[/sub] to turn it on
From these we can then deduce:
C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00
C6. If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum, then the lamp is neither on nor off at 12:00 [contradiction]
Benacerraf's "solution" is inconsistent with C2 and C3.
You missed out "The lamp is either on or off at all times."
The laws of noncontradiction and excluded middle are implied.
Clearly not by you. Could've easily included it. So why didn't you?
Surely you must believe your interlocutor has a point in his statement other than that of a any random dreg of society: "you're wrong".
He has a point, a truth and testimony, an entire world he wishes you to experience
So we can agree that the consequent is false. Ex falso quodlibet, so a plate of spaghetti.
Yes by me. I didn't include them because they're implied. But if you insist on making it explicit then I will.
Yes. And therefore the antecedent is necessarily false. The supertask is metaphysically impossible.
I don't know about metaphysically possible or impossible. Logically impossible, certainly. So what are you arguing about?
That the supertask is metaphysically impossible.
Is is not the case that "logically impossible" implies "metaphysically impossible"?
Yes.
So what are you arguing about?
I have already said. I am arguing that the supertask is metaphysically impossible.
You were when you said this:
Quoting Ludwig V
Benacerraf is not right. His stipulation that the lamp is on (or off) at t[sub]1[/sub] is inconsistent with the premises of the problem.
As you understand it, of course. And, make no mistake, I would be among the first to swear your understanding and take of things is valid, but also the first to defend the idea that things can change.
Surely some things must remain consistent, for is this not the basis of sanity, after all?
But there still remains some outlier arguments you believe to be irrelevant, or in your words "implied". This is all the opposing party wishes to acknowledge, I do believe.
In simple terms, perhaps, a remote possibility exists, the individual who declared such a statement, has different views and relevant perceptions toward said statement than you hold. Nothing more. Simply the remote possibility. Not that one is "wrong" but simply there is more to the intended truth than one currently perceives. Surely such a thing is possible, no?
And that is understood. Well and good. This is, after all, the only rational non-ill-intended platform of debate or intellectual opposition. Reasonable inquiry itself, even.
I suppose, to simplify, how would you best explain the reasoning behind your recent argument/reply/rebuttal or opposing position to someone who is either: unaware of the argument, that is to say, does not have the fortune to be in the posses ion of the knowledge you hold (yet is capable of so). Surely your questioner is of sound and rational mind and morality. So, if he is incorrect, and you assert the position you remain the opposite, how does one become equals on the same page of truth and rationale you see?
I can't preempt someone's disagreement. If someone wants to argue that my conclusion is false then they need to tell me which step in the argument they disagree with, and why. Only then can I respond.
Surely there are many ways to assist a friend that their current action or belief that will assuredly guide future actions are "lacking", that is to say, have reasonable methods of improvement not currently taken.
Things are not alwaysfalse or valid. Binary enslavement. It is not always "my way or the highway", that is to say, perhaps one there is more that one can contribute to a certain goal or ideal than can be immediately ascertained. Is this false or true?
I'm really sorry, but my fat thumb syndrome struck and my last message got posted before I had finished with it. This version is finished.
Quoting Michael
Benecerraf's sentence is not exactly that:-
Aren't you forgetting tertium non datur?
The rules of the problem stipulate whether the lamp is on or off at 11:00, 11:30, 11:45, and so on ad infinitum, but not whether it is on or off at 12:00.
I grant you that they will tell you whether it is on or off at any specific time before 12:00, but they does not tell you whether it is on or off at 12:00.
The lamp is either on or off at t[sub]1[/sub].
But if the button is pushed at t[sub]1/2[/sub], t[sub]3/4[/sub], t[sub]7/8[/sub], and so on ad infinitum, then the lamp is neither on nor off at t[sub]1[/sub]. This is the contradiction.
Not quite. The lamp is not defined as on or off. It's just that the rules don't apply at 12:00. But tertium non datur does apply. So it must be (either on or off).
Yes they do. P1-P3 are always true. C1-C3 follow from P1-P4 and explicitly apply at all times >= 10:00.
The fact that the conjunction of these premises with the performance of a supertask entails a contradiction is proof that the supertask is impossible, not proof that we can dispense with the premises at 12:00.
Sorry. I have something else to do. I didn't expect to convince you, but our discussion has helped to confirm my opinion.
I'd find it helpful if you would write down a complete description of your version of the problem in one place, rather than pointing me to P1 here and C3 there. Just write down a complete description of the problem for my reference please.
Also, can you please note any divergences between Michael's lamp and Thomson's.
Quoting Michael
Mine, yes. But Benacerraf's also. I wish you'd carefully study his argument and respond to it. I mention this "appeal to authority" because Benacerraf is a big time mathematical philosopher, not just some anon rando such as my humble self.
Quoting Michael
Benacerraf addressed this point. Your assumption about what happens at the limit point is either (a) not part of Thomson's original formulation; or (b) logically inconsistent.
Seems that way to me as well. @Michael is wishing that a limit point had an immediate predecessor, but it doesn't, and he's stuck there.
Quoting Ludwig V
Yes, Benacerraf and I conclude that.
Quoting Ludwig V
Why? I drive down the road and come to a fork. One day I turn left. Then next day I drive down the same road and turn right.? What logical inconsistency do you see to there being multiple possible outcomes to a process that are inconsistent with each other, but each consistent with the rules of the game?
Quoting Ludwig V
Not at all. What logical inconsistency is there if I turn left today and right at the same junction tomorrow?
Is geometry inconsistent because it allows both Euclidean and non-Euclidean geometry? They are inconsistent with each other; but they are each consistent in themselves.
Quoting Ludwig V
As far as I'm concerned this is a solved problem. And I do believe that if @Micheal would stop falsely imagining that there's an immediate predecessor to a limit point, he'd agree.
Quoting Ludwig V
Yes. Benacerraf explicitly says that. He says it's a mistake to assume that the terminal state bears any "logical relation," his words, to the preceding sequence of states.
Quoting Ludwig V
Benecerraf explicitly says: "... Certainly, the lamp must be on or off at t1
(provided that it hasn't gone up in a metaphysical puff of smoke in the interval) ..."
In other words he is making the the point that for all we know, the lamp is not even constrained to be either on or off at the terminal state. And why should it be so constrained?
I am indeed gratified to see Benacerraf making this point.
Quoting Ludwig V
Why? It's an imaginary lamp to start with. I have previously analogized it to Cinderella's coach, which turns into a pumpkin at exactly midnight, but is a beautiful jewel-encrusted coach at and time prior to midnight.
What rule of the problem constrains the terminal state of the lamp? Thomson gives no such constraint. On the other hand @Michael may indeed have added his own such constraint, making the problem impossible. So Michael has not solved Thomson's lamp, he's solved Michael's lamp, by adding a condition that isn't present in the original problem.
These are our premises before we even consider if and when we push the button:
P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 10:00
From these we can then deduce:
C1. The lamp is either on or off at all t[sub]n[/sub] >= 10:00
C2. The lamp is on at some t[sub]n[/sub] > 10:00 iff the button was pushed at some t[sub]i[/sub] > 10:00 and <= t[sub]n[/sub] to turn it on and not then pushed at some t[sub]j[/sub] > t[sub]i[/sub] and <= t[sub]n[/sub] to turn it off
C3. If the lamp is on at some t[sub]n[/sub] > 10:00 then the lamp is off at some t[sub]m[/sub] > t[sub]n[/sub] iff the button was pushed at some t[sub]i[/sub] > t[sub]n[/sub] and <= t[sub]m[/sub] to turn it off and not then pushed at some t[sub]j[/sub] > t[sub]i[/sub] and <= t[sub]m[/sub] to turn it on
From these we can then deduce:
C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00
C6. If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum, then the lamp is neither on nor off at 12:00 [contradiction]
Benacerraf's "solution" to arbitrarily stipulate that the lamp is on/off at 12:00 is inconsistent with C2 and C3. Even our hypothetical lamp cannot spontaneously and without cause be on.
His "solution" doesn't even answer the question, which asks what happens to the lamp if we push its button an infinite number of times. We want to understand the causal consequence of having performed the supertask. Having the lamp spontaneously and without cause be on after having performed the supertask does not tell us what having performed the supertask entails.
You have a hidden element here, known as freedom of choice. The "multiple possible outcomes" are only the result of this hidden premise, you have freedom to choose. That premise overrules "the rules of the game", such that the two are inconsistent. In other words, by allowing freedom of choice, you allow for something which is not "consistent with the rules of the game", this is something outside the rules, the capacity to choose without rules.
None. I'm afraid I'm indulging in double-think in this discussion. I can't make sense of the imaginary lamp. Either it is just a picturesque way of dressing up the abstract structure of the mathematics or it is a physical hypothesis. Some time ago I asked @michael why he didn't just run his computer program. He replied that a computer couldn't execute in the programme in less than some minute fraction of a second, so it wouldn't give an answer. Which was the answer I expected. The computer program was just another way of dressing up the mathematical structure. So I translate all talk of the lamp into abstract structure in which "0, 1, 0, 1, ..." is aligned with "1, 1/2, 1/4, ...".
Quoting fishfry
I agree. But I have some other problems about this. I'll have to come back to this later. Sorry.
It is the point of this thread as briefly stated in the OP itself https://thephilosophyforum.com/discussion/14855/metaphysically-impossible-but-logically-possible/p1
As to the issue with Venn diagrams, that is something that other users articulated too. Nevertheless, I think this stems from the definition of metaphysics we are going with. In some thread me and Josh had a short debate about it. If we see metaphysics as broadly synonymous with ontology, the Venn diagram linked there seems to be unproblematic: surely we agree that every physical possibility is also a logical possibility, so one encompasses the other; I think we can also agree that if something is ontologically possible, it is also logically possible, so one encompasses the other; using an etymological fallacy, metaphysics (ontology) is everything beyond physics, so metaphysical possibility encompasses physical possibility.
But it is mostly a musing of mine. There is a lot of talk about the spheres of possibility. I don't expect others to read it fully, but all I can do is recommend the following article: https://plato.stanford.edu/entries/modality-epistemology/ The beginning is the most important part of the article anyway.
Quoting Ludwig V
I don't get it. There is something missing in this phrase.
Quoting Ludwig V
I don't disagree, especially when names such as "logically-possible" seem to actually be conceptually/analytically-possible instead of anything to do with logic. The SEP article claims that the epistemology of possibility is the primordial issue at stake in issues such as mind-body dualism and the Berkeley's argument for Idealism. Though this could be true in a sense, I myself think that the issue is a semantic one and, if it is a metaphysical issue in any capacity, it is a derivative issue, not a primordial one just like p-zombies is completely derivative from dualism/physicalism.
On the talk of necessity and contigency, I recently came across something interesting, "inus" conditions, where necessity and sufficiency blend:
(1) We do not make explicit the premise that the state at 12:00 is determined by an immediate predecessor state.
That tends to favor Benacerraf.
(2) We do make explicit the premise that the state at 12:00 is determined by an immediate predecessor state.
That tends to disfavor Benacerraf.
But there is no immediate predecessor state to the state at 12:00, so I find it difficult to conceive also requiring that the state at 12:00 is determined by an immediate predecessor state that does not exist.
I understand the idea that the domain is w+1 and I too mentioned it a while back.
You're putting the cart before the horse.
Before we even consider if and when we push the button it is established that the lamp can only ever be on if the button is pushed when the lamp is off to turn it on. The lamp cannot spontaneously and without cause be on.
Our problem is that if (a) a supertask is performed and if (b) the lamp is on after we stop pushing the button then (c) the lamp is spontaneously and without cause on.
@fishfry and Benacerraf fail to acknowledge that (c) is impossible, proving that (a) and (b) are incompatible.
If (a) is true then (b) is false. If (b) is true then (a) is false. The same reasoning holds when we claim that the lamp is off after we stop pushing the button.
We just need to say that the infinite sum is the limit of the sequence of finite sums.
But meanwhile there is the crank who posts pure garbage as he shows that he is not even following along in the discussion. In the course of exercising his personal umbrage, he terribly fouls up the subject matter and the path of the conversation here.
The comments below should not have to be belabored, but the crank's trash talk should not be left standing.
The problem under discussion itself is couched in terms of moments* in time and durations of time that are indexed numerically. And the problem supposes that the durations may be divided and that there are infinitely many divisions. *And it is common for people to refer to 'points in time' pretty much in the same sense of 'moments in time'. Again, l have not taken any position about the nature of time or the common understanding of time represented as points on a line. I am only taking the problem as it is presented. The premises and rubrics in the problem are not ones that I have endorsed nor disputed.
Michael claimed to prove that time is not continuous. I pointed out that, for his own purposes, he needs to prove that time is not dense. Again, I haven't taken a position on that question.
The problem presupposes that points in time are ordered, such that 11:00 precedes 11:30 precedes 11:45, etc. A dense order is one in which between any two different points there is yet another point. And the ordering of time in which the thought experiment supposes that time can be divided ad infinitum is a dense order. So, my remark about density is that for Michael's own purpose, he should be claiming to refute that time is dense in that sense and not the weaker claim that time is not continuous. Again, I haven't taken any position of my own about time; I have only taken the problem and poster's arguments on their own terms.
But the crank lashes out with stupid strawmen, getting the path of conversation abysmally mixed up. He irrationally snarls with nonsense projected onto me such as "time consists of two distinct substances", "infinitely dense particles", "the irrational space between the rational numbers".
The crank is a bane.
A horse can push a cart, not only pull it.
Quoting Michael
I haven't refused that. But I suggest that 'immediate predecessor' is a good way of couching the matter. Then, we may consider that the problem itself is impossible in the sense that it requires:
(1) a state requires an immediate predecessor state
(2) there is a state at 12:00
(3) there is no predecessor state to the state at 12:00
Yes, that's the basic argument I've been making. The lamp being off must always precede it being on. Therefore, the lamp cannot be modelled over time by the infinite sequence off, on, off, ...; on.
That seems to drawing an inference from an impossibility.
If we agree that (1) (2) (3) are together impossible, then we can infer anything from the assumption that they are possible.
If you're referring to the principle of explosion, then sure.
The point though is that Thomson shows that the lamp can neither be on nor off after having performed the supertask, which is a contradiction, and so that the supertask is impossible in principle to perform.
But, if I am not mistaken, your argument comes down to: From the assumption that (1) (2) (3) are together possible, we infer that time is not infinitely divisible, not merely that a certain supertask is impossible.
More like:
P1. The lamp being off must always precede it being on.
C1. Therefore, the lamp cannot be modelled over time by the infinite sequence off, on, off, ...; on.
P2. If time is infinitely divisible then the lamp can be modelled over time by the infinite sequence off, on, off, ...; on.
C2. Therefore, time is not infinitely divisible.
Although on this latter point perhaps this argument is more compelling.
We reject that it is possible for (1) (2) (3) to hold together. So we can reject (1) and be left with a consistent set of two premises. So it is not ruled out that it is possible that there are denumerably many alternating states and that time is infinitely divisible.
I don't understand your argument, or at least I don't think you understand my argument. Before we even address the infinite divisibility of time, the full argument related to Thomson's lamp is presented here.
Do you disagree with anything said there?
You can couch the hypothetical situation with whatever premises you like. In that sense it's not a matter of me agreeing or disagreeing. And the part about Benacerraf doesn't seem relevant to my previous post, since I am not using referencing Benacerraf's argument in my previous post.
Meanwhile, I don't know what is not understandable about my previous post.
Thank you.
To reject (1) is to claim that the lamp can spontaneously and without cause be on at 12:00. This is impossible. The lamp can only be on at 12:00 if the button was pushed when the lamp was off to turn it on.
We must reject (3), and so reject the possibility of the supertask.
It rejects that having an on/off state is determined by an immediate predecessor state, so when we reject that premise, it is not ruled out that the state at 12:00 is determined in another way. (Yes, this is similar to Benacerraf.)
Quoting Michael
You're reiterating a premise that we are free to reject since it is impossible given the other premises. Just reiterating it like that is begging the question in this context.
It's not a premise. It's a conclusion derived from our premises. See the argument above.
No, I've linked you to the argument. You can read the premises there.
C1 is a premise. It is the premise that the lamp has only two states. But that's not a substantive problem; only that I'm mentioning that it is a premise.
C2 and C3 together seem to include "only by an immediate predecessor". But I don't see a valid inference for them from the premises. It seems to me that the premises don't preclude that the button can be pushed at 12:00 without there being an immediate predecessor state.
Thank you.
Its not, its a valid inference from the premises.
As per P4, the lamp starts off. As per P2, pushing the button will turn it on. As per P3, pushing the button again will then turn it off. As per P2, pushing the button again will then turn it on. And so on ad infinitum. And as per P1 there's no third party magic or God or gremlin that can cause the lamp to vanish in a puff of smoke or turn into a plate of spaghetti. So the lamp can only ever be either off or on. There's no mechanism by which the lamp can be anything else.
Quoting TonesInDeepFreeze
Firstly, as per the antecedent of C6, the button is not pushed at 12:00. Thomson is addressing what happens if the supertask is performed, not what happens if the supertask and some independent subsequent task is performed.
Secondly, pushing the button at 12:00 will only turn the lamp on if the lamp is off when the button is pushed. So to say that the button is pushed at 12:00 to turn the lamp on is to say that if the button is not pushed at 12:00 then the lamp would be off at 12:00. But that's not possible because of C3.
Possible outcomes can indeed be inconsistent with each other. But if they are inconsistent with each other, they can't both be actual at the same time. You can't drive down the road and turn left and right at the same time.
Quoting fishfry
There is more to this than meets the eye, I think. Benecerraf's quotation is somewhat hedged. And "for all we know" hints at unexpressed complexities, I'm interested in all that. See below.
Quoting Lionino
I was commenting on Quoting Lionino
I'll try again. "Is it the case that all the philosophers who want to make away with those distinctions the same as those who want to define synthetic necessary truths"
I looked again at Benecerraf's article and found what I was looking for. His position is much more nuanced than I thought. Selective quotation is not ideal, but my summary would likely be worse. So here goes:-
[quote=Benacerraf on Supertasks - The Journal of Philosophy, 1962, p. 781] A "swindle" has taken place, and we have been the victims. Somehow, all was going along swimmingly, and suddenly we find ourselves drowning in contradiction with no idea of how we got there. We are told that the concept of a super-task is to blame, but we are not told what about it has such dire consequences. We are sufficiently sophisticated mathematically to know that the concept of infinity is not at fault (or if it is, a lot more than the future of super-tasks is at stake).[/quote]
[quote=Benacerraf on Supertasks - The Journal of Philosophy, 1962, p. 783/4] I suspect that, by and large, it is principally compound expressions that suffer the fate I attribute to 'completed infinite sequence of tasks' ..... What seems most notable about such compounds is the fact that one component (e.g., 'infinite sequence') draws the conditions connected with its applicability from an area so disparate from that associated with the other components that the criteria normally employed fail to apply. We have what appears to be a conceptual mismatch. Sequences of tasks do not exhibit the characteristics of sequences that lend themselves to proofs of infinity. And since there seems to be an
upper bound on our ability to discriminate (intervals, say) and none on how finely we cut the task, it appears that we should never be in a position to claim that a super-task had been performed. But even if this is true, it only takes account of one kind of super-task, and, as I argue above, it hardly establishes that even this kind constitutes a logical impossibility. [/quote]
[quote=Benacerraf on Supertasks - The Journal of Philosophy, 1962, p. 784] To look at the matter diachronically and therefore, I think, a little more soundly, we can see our present situation as akin to that of speakers of English long before electronic computers of the degree of complexity presently commonplace when confronted with the question of thinking robots (or, for that matter, just plain thoughtless robots, I suspect). They were as unthinkable as thinking stones. Now they are much less so. I am not sure that even then they constituted a logical contradiction. However, I would not resist as violently an account which implied that the expression 'thinking robot' had changed in meaning to some degree in the interim. Viewed as I suggest we view them, questions of meaning are very much questions of degree-in the sense that although relative to one statement of meaning there may be a more or less sharp boundary established, no statement of meaning (viewing things synchronically now) is uniquely correct. Other hypotheses, and therefore other lines may be just as reasonable in the light of the evidence. The statement of the meaning of a word is a hypothesis designed to explain a welter of linguistic facts-and it is a commonplace that where hypotheses are in question many are always possible.[/quote]
[quote=Benacerraf on Supertasks - The Journal of Philosophy, 1962, p. 784]Therefore, I see two obstacles in the way of showing that supertasks are logically impossible. The first is that relevant conditions associated with the words and the syntactic structure involved must be found to have been deviated from; and it must be argued that these conditions are sufficiently central to be included in any reasonable account of the meaning of the expression. The second is simply my empirical conjecture that there are no such conditions: that in fact the concept of super-task is of the kind I have been describing above, one suffering from the infirmity of mismatched conditions.[/quote]
The bolded sentence expresses my preferred diagnosis. (Which, by the way, is channelling Ryle. I think Benecerraf must have know that - look at the date of the article.) In the light of the various further supertasks that have been developed, a conclusive refutation seems as unlikely for the supertask problems as it is for the Gettier problems. But this is a good candidate.
I repeat: Please post a complete description of the problem.
In P1 I have no idea what "the lamp" is. It's perfectly clear that you are adding hidden assumptions to make your view of the problem work out. I'm trying to get you to make these assumptions explicit.
So I ask you to please post a complete description of the problem, as if I'd never heard of it.
That is far from "established." That's why I keep asking you to write out a complete description of the problem, as if I've never heard of it before. That's the only way to make plain the unspoken assumption's you're adding that are not in Thomson's original formulation.
Benacerraf makes this point. He specifically says that Thomson's argument is invalid, unless additional assumptions are made. You have clearly made additional assumptions, and I'm asking you to make them plain by writing out a complete description of this problem.
Same way I see it. The sequence 0, 1, 0, 1, ... has no limit, so one terminal state is as good as any other.
Quoting Ludwig V
This is regarding the puff of smoke or the plate of spaghetti. And that's why I mention Cinderella's coach. Nobody ever complains about that. Why is the lamp constrained to be off or on, when it's a fictitious lamp in the first place?
Quoting Ludwig V
Different copies of the lamp, or same experiment run at different times. Why is this unclear? Benacerraf didn't say Aladdin and Bernard each see different things at the same time with the same lamp. Why are you objecting to an argument nobody made?
Didn't follow that.
The claim was that multiple possible outcomes of a process is inconsistent. Not so. Each outcome is consistent with the rules of the problem. There's nothing inconsistent about a lamp being on sometimes and off other times.
The claim was directed at your example of choosing a direction at a fork in the road. The only way that you could have multiple possible outcomes is by assuming a principle that overrules the rules, i.e. transcends the rules. Freedom of choice, allows you to choose rather than follow a rule. If your example is analogous, then multiple possible outcomes being consistent with the rules, implies that choice is allowed, i.e. the rules allow one to transcend the rules. Strictly speaking the actions taken when the rules are transcended are not consistent with the rules, because these actions transcend the rules. The rules may allow for such acts, acts outside the system of rules, but the particular acts taken cannot be said to be consistent with the rules because they are outside the system.
The terminal state of the lamp is not defined, so it may be on or off. What on earth is wrong about that? If you flip a coin it might be heads or tails. That doesn't mean it can't be both at different times. Why is Benacerraf's point confusing to people?
(1) Why not use 11:00 rather than 10:00? Usually the problem concerns 11:00 to 12:00, which is tidy for the halvings of the durations. (I'll use 11:00.)
(2) We could do without pushing a button and even the lamp. We could couch it in more abstract terms.
(3) Your premises have only one necessary condition for the lamp changing state, which is P1.
This is consistent with your premises:
The lamp is off at 11:00. The button is pushed at 12:00 and the lamp goes on.
So I think you've left out a lot of what you need in your premises.
It doesn't matter to me what the lamp is.
I can regard the problem abstractly, in terms just of:
time
two states
sufficient and necessary conditions for changing states
Mentioning a lamp and it being on or off and a button that is pushed are, for me, all just visualization aids that are dispensable. Moreover, they seem to interfere sometimes when people get hung up on how to relate such a hypothetical lamp and button with actual lamps and buttons.
Yes, another one of @Micheal's conceptual confusions. That's why I mention Cinderella's coach. The lamp is a fairy tale, and it's a fallacy to try to reason about how it works.
[s]I'll try this:
Suppose:
There are two states F and N.
At any moment either F is active or N is active and not both.
For every natural number n>0, there is a time T(1/(2^n)).
At time T(0), F is the active state.
For every natural number n>0, the active state changes at time T(1/(2^n)).
The active state changes only at time (T(1/(2^n)) for some natural number n.
Question: What is the active state at time T(1)?[/s]
I don't see it as a confusion of Michael. He is only rendering Thomson's setup. And I don't see Michael getting tripped up by the metaphorical use of a lamp and button. And I don't see Thomson as getting tripped up either.
P1-P4 are our premises. C1-C3 follow. And then C4-C6 follow.
P1 is implicit in Thomson's argument. Using the principle of charity you should infer it. As neither you nor Benacerraf have done so I have had to make it explicit.
As a comparison, consider the following:
The lamp is off at 10:00. The button is pushed 10[sup]100[sup]100[/sup][/sup] times between 10:00 and 10:01. Is the lamp on or off at 10:02?
Any reasonable person should infer that nothing else happens between 10:01 and 10:02. Even though this is a physically impossible imaginary lamp, and even though I haven't told you what happens at 10:02, it is poor reasoning to respond to the question by claiming that the lamp can turn into a plate of spaghetti. The correct answer is that because 10[sup]100[sup]100[/sup][/sup] is an even number, the lamp will be off at 10:02.
There is no Supreme Button Pusher arbitrarily willing the lamp to be on or turning it into a pumpkin. There is only us pushing the button once, twice, or an infinite number of times, where pushing it when the lamp is off turns the lamp on and pushing it when the lamp is on turns the lamp off.
Yes, this is where we have C4 and C5:
C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00
It is only when we change from having pushed the button a finite number of times to having pushed the button an infinite number of times that we are met with a contradiction:
C6. If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum, then the lamp is neither on nor off at 12:00
For every [math]n \in \mathbb{N_0}[/math] it is possible to have pushed the button [math]n[/math] times (and every [math]n \in \mathbb{N_0}[/math] is finite), but it is not possible to have pushed the button [math]\aleph_0[/math] times.
This doesn't make sense. Each flip of the coin is an individual act, and it has a single outcome. Once the outcome is achieved, that outcome stands until there is another flip. The outcome "can't be both at different times", because a different outcome requires a different flip. However, there can be different outcomes from different flips.
I think the problem is precisely that there is nothing to constrain the lamp and we want to find something. In theory, we could stipulate either - or Cinderella's coach. But we mostly think in the context of "If it were real, then..." Fiction doesn't work unless you are willing to do that. It's about whether you choose to play the game and how to apply the rules of the game.
Quoting fishfry
This seems to be more in tune with common sense, for what it's worth. The question is, why? I think it is because of the dressing up of the abstract structure. We assume the lamp has existed before the sequence and will continue to exist after it. So the fact that the sequence does not define it does not close the question and we want to move from the possible to the actual. But it is not clear how to do that - and we don't want to simply stipulate it. Perhaps that's because defining the limit of the convergent sequence as 1 - or 0, which have a role in defining the sequence in the first place, invites us to think in the context of the natural numbers (or actual lamps), whereas defining ? as the limit of the natural numbers does not.
Quoting Metaphysician Undercover
I hope you meant that actions taken outside the system are neither consistent nor inconsistent with the rules. Could we not express this by saying that the rules don't apply, or that it is not clear how to apply the rules, in the new context?
Quoting Michael
No, but it might be the case that common sense reasoning doesn't apply or is misleading in the context of infinity.
Quoting Michael
You can think about us doing that, but you can't limit our thinking to that context. That's where the problems start.
Quoting Michael
Us doing this is not an empirical possibility, so there can't be any causal consequences. But I think you mean to ask what outcome there can be if we think only in that context. Sadly, that context doesn't give us an answer - except possibly that the state of the lamp is both on and off or neither on nor off.
The lamp is off at 10:00. I push the button at 10:01, turning the lamp on. Is the lamp on or off at 10:02?
The correct answer is "on".
You don't get to invent your own premises and stipulate that some magical gremlin turns the lamp into a plate of spaghetti at 10:02. In doing so you are no longer addressing the thought experiment that I have presented.
Quoting Ludwig V
Neither is pushing the button 10[sup]100[sup]100[/sup][/sup] times within one minute, but we are still able to reason as if it were possible and deduce that the lamp would be off when we finish. That's just how thought experiments work.
Your thought experiment, your rules. But whose thought experiment is Achilles' race and Thompson's lamp? I had the impression that they are Zeno's or Thompson's. What if there's something wrong with them, such as they contradict each other or lead to a self-contradictory conclusion?
Quoting Michael
True. I wrote carelessly. What deduction do you make when you think about pushing the button after an infinite sequence, which is defined without completion, of button pushes within one minute. Oh, wait, I know.
Quoting Michael
You know perfectly well that's self-contradictory, so necessarily false. Ex falso quodlibet otherwise known as logical explosion. Or your deduction is wrong. (But I don't think it is wrong - or at least, not any more wrong than the spaghetti).
Not my rules. Yours.
Yes, this is how refutations by contadiction work.
I'll quote Thomson for you:
This is what proves that "super-tasks are not possible of performance". Any completed sequence of tasks is necessarily finite.
Quite so. And the phrase "completed sequence of tasks" is self-contradictory. So what do we need your argument for?
Did you mean that the phrase "completed infinite sequence of tasks" is self-contradictory? If so then yes.
Quoting Ludwig V
To prove it. Those like Benacerraf and fishfry either claim that it isn't self-contradictory or that it hasn't been proven to be self-contradictory.
You are right, of course. I'm glad you could decipher what I meant to say.
Quoting Michael
Benacerraf's position is a bit more complicated than that.
[quote=Benacerraf on Supertasks]Thomson is ... successful in showing that arguments for the performability of super-tasks are invalid and ... nevertheless his own arguments against their possibility suffer the same fate. [/quote]
Quoting Michael
Thanks for clarifying that you meant self-contradictory. I've been wondering what your conclusion contradicted.
I'm not sure whether that doesn't amount to a contradiction or whether it is an entirely distinct issue. But it seems like that if that's the case, one doesn't get as far as a contradiction.
But this implies that one has to consider what sense can be attributed to, for example, "complete" in the context of its use. It is already clear, isn't it, that the meanings of various words have to change to be applied in the context of "infinity". Just because we understand what it is to complete a finite sequence, it doesn't follow that we understand what it is to complete an infinite sequence. "Countably infinite sequence", for example, doesn't mean what you might think it means. Each and every term in the sequence can be counted, but you couldn't complete a count of all of them. In finite cases, counting each one ends up counting all. "Larger" and "smaller", if I've understood right, have to be re-defined as well.
For me, the most persuasive example is that I want to be able to say that as Achilles passes the winning post, he is completing Zeno's sequence. Or, at least, I think it is reasonable to say it. That task, of course is very different from Thompson's lamp because the sequence on/off,... doesn't have a limit, so I don't see that working in that case.
No, I mean they are inconsistent. To be consistent with the rules is to act according to the rules. Actions which are outside of the rules are not according to the rules, therefore they are inconsistent with the rules.
Believe @Michael has different assumptions as Thomson, but I haven't gone back to check Thomson's original formulation.
Not so. What lamp? Describe the full setup.
Quoting Michael
Not possible, else Benacerraf's objections to Thomson's formulation would apply.
Quoting Michael
My charity ran out long ago regarding this subject. The lamp is a solved problem.
As a comparison, consider the following:
Quoting Michael
Perfectly clear that you have stated nothing about 10:02. For all we know it turns into a pumpkin.
Quoting Michael
Ah, hidden assumptions. Argument by "any reasonable person." You're wrong. Nothing was said about 10:02.
Quoting Michael
Benacerraf himself anticipated the spaghetti, saying the lamp might vanish in a puff of smoke.
Quoting Michael
Us. You and me? Human beings pushing a button in arbitrarily small intervals of time.
Nonsense. You're just typing in nonsense. Us pushing the button?
Can't you see why I'm demanding that you write out, in one place, your entire description of the problem. That way you would be able to catch yourself making stuff up as you go.
You're agreeing with me. Play the lamp game twice. Sometimes the terminal state is on, other times it's off, other times the lamp turns into a plate of spaghetti, or "vanishes in puff of smoke" as Benacerraf says.
Which is why I endlessly challenge @Michael to make his version explicit once and for all. And why he won't.
Quoting Ludwig V
What justifies such an assumption with regard to an entirely fictional lamp, coach, or pumpkin?
Quoting Ludwig V
Neither 0 nor 1 is the limit of the sequence of alternating 0's and 1's.
Quoting Ludwig V
[math]\omega[/math] can be defined such that it is the limit of the sequence of the natural numbers.
The premises don't not specify that the button is ever pushed.
The premises do not specify that there are only two states, unless, in this very hypothetical context we are clear that 'Off' is defined as 'not On', though it does seem reasonable that that is implicit.
Of course, w is a limit ordinal, and it is the ordinal limit of the sequence of all the natural numbers.
But, just to be clear, we still need to prove that there exists a set such that every natural number is a member of that set, since that set is the domain of the aforementioned sequence.
wut? axiom of infinity. what's wrong with you tonight? you just blasted out six mentions of me, one more mindless than the next.
(edit) Some of them were compliments. Those weren't mindless!
The axiom of infinity is how we prove that there is a set that has every natural number as a member.
From the axiom of infinity, we derive that there is a unique least inductive set.
Then we define: w = the least inductive set.
Then we prove that all the natural numbers are members of w.
My point is that we have to be careful in thinking of making this definition:
w = the limit of the sequence of all the natural numbers
since the domain of that sequence is the set of natural numbers, which would already have to have been defined, and so we would have already defined w.
w = the ordinal limit of the sequence of all the natural numbers*
is a theorem, but it would be tricky were it a definition.
* Given a reasonable definition of 'limit of sequence of ordinals'.
/
This is yet another instance of you lashing out against something that I wrote without even giving it a moment of thought, let alone maybe to ask me to explain it more. Your Pavlovian instinct is to lash out at things that you've merely glanced upon without stopping to think that, hey, the other guy might not actually being saying the ridiculous thing you think he's saying. Instead, here you jump to the conclusion that "there's something wrong" with him.
Quite so. Except I thought that it had actually been done.
Quoting fishfry
Quite so. That's why I specified "convergent sequences". (I don't know what the adjective is for sequences like "+1" or I would have included them, because they also have a limit.) "0, 1, ..." is neither. Does the sequent 0, 1, ... have a limit - perhaps the ?th entry?
Quoting TonesInDeepFreeze
Yes. My only point was that it is not a natural number, whereas 1 and 0 are. Hence, although both are limits of their respective sequences, as 1 or 0 also are, 1 and 0 are used in other ways in other contexts. This makes no difference to their role in this context and does not affect their role in other contexts, but does affect what we might call their meaning. ? is not used in any other context - so far as I know.
Quoting fishfry
I agree that we can agree not to ask questions about the lamp outside the context of Thompson's story. But I'm not sure that an assumption really requires a justification. But, for the sake of argument, if I'm telling you a story about a real ball and the shenanigans the prince got up to, you would make that assumption. So if I'm pretending to myself that Cinderella's ball actually happened, I will make the same assumption. This is one reason why I prefer to stick to the abstract structure and shed the dressing up.
Quoting fishfry
Can I ask what your solution is? Just out of interest.
Quoting Metaphysician Undercover
But actions which are outside of the rules are not contrary to the rules, so they are consistent with the rules. However, on thinking about it, I think my answer it that it depends on the rule. Sometimes the rule means that actions that are not permitted are forbidden and sometimes the rule means actions that are not forbidden are permitted. And sometimes neither.
Quoting Michael
Quite so. But how does it help when we are thinking about an infinite sequence? As I understand it, the point is that the sequence cannot define it's own limit. (If it could, it would not be an infinite sequence). The limit has to be something that is not an element of the sequence. It has to be, to put it this way, in a category different from the elements of the sequence. (I'm trying to think of a self-limiting activity, but my imagination fails me. Perhaps later.)
That's precisely the problem. Both of these things are true:
1. The lamp can never spontaneously and without cause be on
2. If the supertask is performed, and if the lamp is on after the performance of the supertask, then the lamp being on after the performance of the supertask is spontaneous and without cause.
Therefore we must accept that the supertask cannot be performed.
And even if we were to grant an alternate account that allows for the lamp to spontaneously and without cause be on, doing so does not answer Thomson's question. He wants to know what the performance of the supertask causes to happen to the lamp. Having some subsequent, independent, spontaneous, acausal event after the performance of the supertask does not tell us what the performance of the supertask causes to happen to the lamp. It's a red herring.
As per P1, the lamp cannot spontaneously and without cause turn into a pumpkin, and there cannot be a god or wizard or gremlin magically turning the lamp into a plate of spaghetti.
And then as per P2, P3, and P4, pushing the button can never cause the lamp to vanish in a puff of smoke.
So the lamp can never turn into a pumpkin. It can never turn into a plate of spaghetti. It can never vanish in a puff of smoke. It can only ever be either off or on.
Before we even consider a supertask, do you at least understand that if the button is pushed to turn the lamp on (and then not pushed again) then the lamp stays on?
Do you at least accept these?
C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00
Quoting fishfry
I did so. It's here.
If the button is never pushed then as per P1 and P4 the lamp will forever be off, consistent with C1.
Quoting TonesInDeepFreeze
Yes, "off" means "not on". The lamp's bulb is either emitting light or not emitting light.
See also the first part of my response to fishfry above.
Not quite. If the last stage of the supertask was on, it is not on spontaneously and without cause.
The problem is that whether the supertasks can be performed is not really the issue. The issue is about how to perform a thought experiment - how much of reality you can import into the story. The state of the lamp, and even its existence, is not defined after the limit. So we can fill in the blanks. You prefer common sense to fantasy, but the story is fantasy, so common sense is not necessarily appropriate.
If you are accepting that the button can be pushed an infinite number of times in a finite time, you have already abandoned causality. I could add a premiss that the lamp cannot switch from one state to another in less than a nano-second, and prove that supertasks cannot be performed. Would that convince anyone? I tried that a long time ago and was put right in short order. You earlier brusquely told me that the reason you didn't run the program was that the computer couldn't process the information fast enough.
A supertask has no last stage. Again to quote Thomson, "I did not ever turn it on without at once turning it off [and] I did in the first place turn it on, and thereafter I never turned it off without at once turning it on."
Therefore, if the lamp is on after having performed the supertask then the lamp being on has nothing to do with me having pushed the button to turn it on. The lamp being on would be spontaneous and without cause, which just isn't possible given our premises.
Quoting Ludwig V
Yes, it is. Thomson's argument attempts to prove that supertasks cannot be performed.
I'm sorry. There is a serious typo in The first sentence of my reply to you. It should have read:-
Quoting Ludwig V
I hope it makes better sense now.
I refer to the last stage only because the question presupposes it. That presupposition is false, of course, which is why there is no answer to the question.
So I don't see the point of your argument here. It's about something else.
Benacerref claimed that the supertask being performed and then the lamp being on is not a contradiction. I am trying to prove that it is (or rather that Thomson already proved this).
The lamp cannot be on after the performance of the supertask and cannot be off after the performance of the supertask precisely because there is no final button push and because the lamp cannot spontaneously and without cause be either on or off.
The pseudocode I provided before shows this. Its logic does not allow for echo isLampOn to either be determined to output true or false or to arbitrarily output true or false. Therefore, we must accept that it is impossible in principle for while (true) { ... } to ever complete.
And so we must accept that it is impossible in principle for a supertask to be performed.
Thank you. That is much clearer.
If you want to include a wider, more commonsensical context, you could think that a lamp does not spring in to existence at the beginning, or disappear in a puff of smoke after the limit (12:00 or 2:00 or whatever it is). Nor does time stop. But in that case, you can confidently say that its status cannot be determined, with the implication that you need to wait to find out what its status is.
But once you've gone down that road, there are other things you might need to bring in, such as the time it takes for the lamp to transition from one state to another. Then the scenario falls apart - the experiment cannot be conducted.
Quoting Michael
Nor is it. He talks about two instances of the game, and either outcome would be consistent - on its own. But they contradict each other and that's the problem. I don't rate that "refutation" any more than you do.
But, to be fair, he does grant that Thomson's demolition of the arguments for supertasks are valid. It's just his argument against that he takes exception to. It's interesting, though, that neither he nor Thomson considers the other solution - including the limit in the series. [quote=SEP on Supertasks]The price is that the final state will not be reached from the previous states by a convergent sequence. But this by itself does not amount to a logical inconsistency.[/quote]
An interesting indeterminate comment. But I think that the impossibility of the final cycle before the limit does put paid to it. It's all about what "complete" means in the context of infinity. Benacerraf, it I've read him right, allows that Achilles can be said to complete infinitely many tasks in a finite time, but argues (rightly) that Thomson's lamp is a different task and suggests to me that he is inclined not to allow that conclusion in that case.
Use of language. When a mathematician says, "X can be done," that's just as good as doing it. There are many jokes around that idea.
There's a formalism or concept called the order topology, in which you can put a topological structure on the set 0, 1, 2, 3, ..., [math]\omega[/math] such that [math]\omega[/math] is a limit point of the sequence, in exactly the same way that 1 is the limit of 1/2, 3/4, 7/8, ...
A topological structure is an abstraction of expressing closeness with open intervals, as in the real numbers. The point is that [math]\omega[/math] is the limit of 0, 1, 2, 3, ... in exactly the same sense as "abstracted freshman calculus," if you think of it that way.
Quoting Ludwig V
No. 0, 1, 0, 1, ... does not have any limit at all. And we can even prove that. Note that it has two subsequences, 0, 0, 0, ... and 1, 1, 1, ,,, that each have respective limits of 0 and 1.
Now it's a theorem that if a sequence converges, all of its subsequences must converge to the same limit. Makes sense, right? A convergent sequence "squishes down" to near the limit.
So a sequence like 0,1,0,1 ... that has two subsequences with different limits, proves that the sequence can not have a limit.
Also, I don't think there even is a name for an arbitrary termination value for a non-convergent infinite sequence. Like
1/2, 3/4, 7/8, ...; 47
In this case 47 is still the value of the "extended sequence" function at [math]\omega[/math]. I call it the terminal state.
I've never seen anyone else use this idea as an example or thing of interest. It doesn't have a name. But to me, it's the perfect way to think about supertasks. The terminal state may or may not be the limit of the sequence; but it's still of interest. It could be a lamp, or a pumpkin, or it could "disappear in a puff of smoke."
Question: Do you put the same constraint on Cinderella's coach? Why or why not? Want to understand your answer.
Regarding the rest of it, I'm lamped out, so I will not debate your ideas further. We have heard each other's talking points multiple times at this point. At least I've got a big time philosopher on my side. How cool is that, right? To actually have professional vindication for a personal idea. I've gotten more than my money's worth from this conversation.
If you feel like answering whether you put the same constraint on Cinderella's coach, I'd be intereted to know. Can't respond anymore to the rest of it. When I get to the point that I haven't typed any words on the subject that I haven't typed before, that's how I know I"m done with that topic.
Thanks for the chat plus any Cinderella comments.
I don't understand your question.
Asking me why I'm using P1 as a premise is as nonsensical as asking me why I'm using P2 as a premise. They are just the premises of the thought experiment. The intention is to not allow for the lamp to be off, for the button to be pushed just once, turning the lamp on and then for the lamp to be off.
We are trying to understand what it means to perform a supertask, and so we must assert that nothing other than the supertask occurs. There are no spontaneous, uncaused events. If we cannot make sense of what the performance of the supertask (and only the supertask) causes to happen to the lamp then we must accept that the supertask is metaphysically impossible.
I thought that might be your answer. Perhaps we shouldn't pursue the jokes, though.
It's called a performative speech act. Do you know about them? Very roughly, the saying of certain words is the doing. The classic example is promising. A particularly important - and complicated - variety of speech act is a definition. Particularly interesting cases are the definition of rules. (Well, definitions are always regarded as rules, but there are cases that are a bit tricky.)
The relevance is that I'm puzzled about the relationship between defining a sequence such a "+1" and the problem of completion. Each element of the sequence is defined. Done. (And an infinite number of tasks completed, it seems to me). But apparently not dusted, because we then realize that we cannot write down all the elements of the sequence. In addition to the rule, there is a distinct action - applying the rule. That is where, I think, all the difficulties about infinity arise. We understand how to apply the rule in finite situations. But not in infinite situations. Think of applying "countable" or "limit" to "+1". The concept has to be refined for that context, which, we could say, was not covered (envisaged) for the original concept. (By the way, does "bound" in this context mean the same as "limit"? If not, what is the difference?)
Quoting fishfry
Oh, yes, I get it. I think.
Forgive me for my obstinacy, but let me try to explain why I keep going on about it. I regard it as an adapted and extended use of the concept in a new context. (But there are other ways of describing this situation which may be more appropriate.) My difficulties arise from another use of the "1" when we define the converging sequence between 0 and 1. It seems that there must be a connection between the two uses and that this may mean that the sense of "limit" here is different from the sense of ? in its context. In particular, there may be limitations or complications in the sense of "arbitrary" in this context.
Quoting fishfry
I thought so. So when the time runs out, the sequence does not? Perhaps the limit is 42.
Quoting fishfry
So we say that all limited infinite sequences converge on their limits. Believe it or not, that makes sense to me. Since it is also an element of the sequence, it makes sense not to call it a limit.
Quoting fishfry
I have completist tendencies. I try to resist them, but often fail.
That's exactly right. And as I told Michael, way back, in the beginning, if the rules allow for zero importation from the "real world", (which is distinct from the possible world of the supertask), then the allotted amount of time known as the limit, can never pass. And the supertask dilemma never gets off the ground. When Michael insists that the duration must pass, "reality" is needlessly being imported into the thought experiment. We have two incompatible possible worlds.
That the allotted amount of time must pass, if true, is enough evidence to reject the supertask as impossible. However, if we attempt to prove that the amount of time must pass, we run into problems, like those exposed by Hume, namely a lack of necessity in the continuity of time.
I'm glad you agree. And you are right to go on to consider choices we could make.
Quoting Metaphysician Undercover
That's interesting. Do you mean a proof that the amount of time must pass in reality, or a proof that the amount of time must pass in the story? If the former, then we do have a problem. But if the latter, I would argue that the amount of time must pass in order for the conclusion to be drawn. Actually, if the task is suspended before it is concluded for any reason, no conclusion can be drawn either way. So I would think that we have to say that the passing of time is a presupposition of the problem. So I wouldn't use this case as an argument against the infinite divisibility of time (or space, in the case of Achilles). (Actually, following our earlier argument, I'm inclined to see that as a mathematical or conceptual proposition, rather than a fact about the real ("physical") world.)
There's a principle here, that we are willing to import any presuppositions of reality (common sense reality) that are needed to make the argument work, in the sense of drawing a conclusion. But that is limited to what I call presuppositions.
There is another presupposition. There is a presupposition that real people are reading the story and arguing about it - and making choices about how much reality to import.
We can, of course, import whatever we want, in one sense. The issue might then arise whether the new version of the story is the same story or a new one.
It's complicated.
In the thought experiment, the allotted amount of time cannot pass. The switch must complete an infinite (endless) cycle of on/off before the allotted time can pass. The endless cycle cannot ever be finished, so there will always be more switching to do, and the allotted amount of time will never pass. This is just like Zeno's thought experiment, Achilles will never pass the tortoise, because there will always be more distance to cover first..
Now, we add a bit of "reality". Achilles will pass the tortoise, the allotted amount of time will pass. That is reality So we see that what we take for "reality", is inconsistent with, or contradicts what the thought experiment asks us to consider.
We'd think that the rational human being ought to choose "reality" over the ideas of the thought experiment, then we'd reject the nonsense. But this "reality" is concerned with "what will happen", and Hume's problem of induction applies. How do we know that there will be a tomorrow? Because there always has been in the past. How do we know there will be a next hour? Because there always has been. How do we know that there will be a next moment? Because there always has been. However, "because there always has been" does not provide proof that there will continue to be into the future.
Therefore "reality" concerning "what will happen" is lacking in certainty, due to the problem of induction. And, theoretically, a system which prevents the allotted amount of time from passing, through a mechanism similar to the one of the thought experiment could possibly be arranged. Imagine that what is really represented is a continuous slowing down of "our time". Imagine that the mechanism is in a different time frame, so that in the different time frame, the switching on/off is at a constant speed. From our perspective, the switching appears to get faster and faster, but what is really happening is that our time is passing slower and slower in comparison to the other time frame. As it slows more and more and more, it approaches a complete stop, without ever reaching that complete stop, so that every tiny fraction of a second which goes by in our frame, is extremely long in the other. Then it is actually going so slow in comparison to the other time frame, that a very large number of switching can occur in a very short time, and so on as it approaches an infinite amount.
Quoting Metaphysician Undercover
OK. I'm with you that far. Comment:-
That's exactly what Zeno and Thomson want us to do. I guess the complications come in when we want to resist their conclusion, without resorting to "It's just a silly story". We could dismiss lots of perfectly respectable fiction on that basis, of course. But no-one worries about the implausibility/impossibility of the events in "Start Trek" or "Star Wars" or "The Hitchhiker's Guide..". That's where the thought experiment isn't a piece of fiction like a fantasy. Aesop's Fables are also not just a piece of fiction; we are meant to draw conclusions about how to live our lives from them. So "It's just a silly story" is not playing the game. This story wants us to draw a conclusion about how reality is.
Quoting Metaphysician Undercover
Yes. What you are doing is applying the actual context (reality) of the story, but instead of drawing on "common sense", drawing on philosophy. That seems to be not unfair, given that Zeno drew a rather radical philosophical conclusion in direct contradiction with "common sense". (He doesn't even have the grace to compromise by dismissing change as an illusion.) Thomson is different because all he wants to conclude is that supertasks are impossible. That's one thing I've never grasped - If supertasks were possible, what philosophical conclusions would follow?
So, instead of rejecting the idea that time is infinitely divisible, you are turning to Hume and arguing that anything can happen. Maybe you are on stronger ground here. I think some people would feel that you are importing more reality than the rules allow. But I can't be dogmatic about that because I don't really know what the rules are - and I'm certainly not going to argue with Hume - perhaps I'm just shirking a long complicated argument, because I don't think he's right, even though he has a point.
Quoting Metaphysician Undercover
Yes. I don't know how this would play with actual Relativity Theory. But in any case, I don't think that resolves the problem. Why? Because it doesn't actually get Achilles to the finishing line. In the case of Thomson's lamp, it doesn't get to the crunch point when the time runs out. In other words, it postpones, but doesn't resolve, the issue.
For what it's worth, I thought for a while, that one could argue that Achilles always has plenty of time to get to the finishing line, because he passes each stage more and more quickly. But crossing the finishing line requires him to pass the last stage, and there isn't one. That's why I insist that the convergent sequence is not about space or time, but about the analysis of space and time. (I do realize that there's some difficulty understanding that distinction, but I'll assume it for the moment, if I may.) I found something very like that point in Benacerraf's article. [quote=Benacerraf on Supertasks p. 766]If we have made a continuous uninterrupted journey from A to B we can be said to have covered all the stretches described in the first premise; that is, our motion can be analyzed as covering in turn AA', A'A", etc. (his italics) [/quote]
I agree, and the conclusion which needs to be drawn is that there is inconsistency between what we observe as the reality of space and time, and the way that we portray space and time through the application of mathematics.
Quoting Ludwig V
I brought all that in, to say, as I've been saying since the beginning of the thread, that the rejection of the fiction, because it is inconsistent with "reality", isn't that simple. What we know as "reality" has the problem of lacking in certainty. This is what Plato demonstrates in how to deal with Zeno's problems. We cannot simply refer to what we know as "reality", because this is based on sense perception, and the bodily senses are proven to mislead the mind. This is why that sort of "paradox" persists, we dismiss them with reference to "reality" and they go away, that is until skepticism about "reality" reappears.
The issue here is that we really know very little about the nature of the passing of time.
Quoting Ludwig V
The point, was not to resolve the problem, but to demonstrate how the "unreal" situation described could actually be real. So I was showing how the lamp could actually switch on and off infinitely, in the described manner, such that the allotted amount of time would not ever pass, and how it is possible that Achilles might never pass the tortoise. It is an example of time slowing down, and approaching a complete stop. Instead of the action of the lamp switching speeding up, think of the passage of time as slowing down, so it appears like the action is speeding up. Then the point which marks the limit, midnight or whatever never comes
Quoting Ludwig V
I agree with this, but I'd describe it as how we apply mathematics to space and time.
Ok.
Quoting Michael
Pending either of us having anything new to say, I am out of this conversation.
Quoting Michael
"We" does not include me. I regard Thomson's lamp as a solved problem. When you say "there are no spontaneous, uncaused events," you are ignoring the physically impossible premises of the problem. Pushing a button in an arbitrarily small time interval to activate a circuit that likewise switches in an arbitrarily small time interval is already a spontaneous, uncaused event. That's why I commend to you the parable of Cinderella's coach.
Quoting Ludwig V
The jokes illustrate the principle. The mathematicians takes the kettle off the stove and places it on the floor, reducing the problem to one that's already been solved.
Quoting Ludwig V
That tingled the circuit in my memory bank. Searle's doctoral advisor Austin talks about speech acts, and I believe Searle does too. That is everything I know about it. Not really clear what it's about.
Quoting Ludwig V
Well I'm not sure I see what those examples are driving at. Speech where the speech is also an act. So, "It's raining out," is not a speech act, because I haven't done anything, I've only described an existing state of affairs. But telling you how the knight moves in chess (example of a rule] is a speech act, because I've brought the chess knight into existence by stating the rule. Something like that?
Quoting Ludwig V
It's very simple. First, by "+1" do you mean Peano successors? You used this notation several times in what follows and I am not sure I know exactly what you mean.
In Peano arithmetic (PA), we generate all the natural numbers with two rules:
* 0 is a number; and
* If n is a number, then Sn is a number, where S is the successor function.
We can use these two rules to define names like 1 = S0 and 2 = SS0 and so forth, and then use the successor function to define "+" so that we can prove 2 + 3 = 5 and so forth.
There is no "completion" of the sequence thereby generated, 0, 1, 2, 3, 4, ...
In particular, there is no container or set that holds all of them at once. The best we can do is say that there are always enough of them to do any problem that comes up in PA.
That gives you one logical system, PA, that has a certain amount of expressive power. We can do a fair amount of number theory in PA. We can NOT do calculus, define the real numbers, define limits, and so forth.
In PA we have each of the numbers 0, 1, 2, 3, ... but we do not have a set of them. In fact we don't even have the notion of set.
Next step up is set theory, for example ZF, that includes the axiom of infinity. The axiom of infinity actually defines what we mean by a successor function for sets; and says that there is a set that contains the empty set, and if it contains any set X, it also contains the successor of X.
This gives you something PA doesn't: A "container that holds all of 0, 1, 2, 3, ... at once, in fact not just a container, but a set, an object that satisfies all the other axioms of ZF.
We can then show that the axiom of infinity lets us construct a model of PA within ZF; and we take that model to be the natural numbers.
The tl;dr is this:
PA gives you each of 0, 1, 2, 3, ...
ZF with the axiom of infinity gives you {0, 1, 2, 3, ...}; that is, all the marbles AND a bag to put them in.
Hope that wasn't too much information, but it's the way to think of "potential" versus "completed" infinities, which are philosophical terms that don't really find use in math.
Quoting Ludwig V
Mathematical sequences and supertasks are two entirely different, but strongly related, ideas.
There is no time in mathematics. But supertasks are all about time. That's where a lot of the confusion comes in. Supertask discussions talk about time, which is a physical concept; but then examples like Thomson's lamp posit circuits that can change state in arbitrarily short intervals of time, which is a decidedly NON-physical idea. It's a fairy tail (under currently known physics). That's where much of the confusion comes in.
So I hope that you can separate out these two concepts. Are you asking about mathematical sequences, such as 1/2, 1/4, 1/8, ... that have the limit 0? That is a completely understood subject in math.
Or are you imagining that someone "speaks these fractions out loud" in their corresponding amount of time, thereby "saying them all in finite time?" This is a totally nebulous, made-up conceptual fairy tail that is the cause of much confused thinking among philosophers.
Quoting Ludwig V
This is actually not much of an objection. It is far too weak. We cannot write out all the terms of any sufficiently large FINITE sequence, either. You can't write out the numerals from 1 to googolplex in y you lifetime at one number per second. It would take longer than the age of the universe.
So you are not making any substantive objection.
In PA the numbers are conceptually created one at a time, but they're really not, because there is no time. 0 is a number and S0 is a number and SS0 is a number, "all at once." You can call that completion if you like.
In ZF, it's more clear. There is a set that contains 0, 1, 2, 3, ... You can give the set a name and you can work with it.
But either way, your concept of completion involves time; and as I've noticed, that involves CONFUSING mathematical sequences, about which we have perfect logical clarity; with supertasks, about which we have much pretentious confusion.
Quoting Ludwig V
No, that is something you are bringing to the table, but that I don't think is correct. There's no distinct action of applying the rules.
In the PA incantation: 0 is a number and Sn is a number if n is; that creates all the numbers. There is no time involved. Time is a factor that you are letting confuse you.
Quoting Ludwig V
We understand how to apply successors perfectly well in the infinite situation. In fact the rule that "If n is a number, then Sn is a number," is an instance of induction, or its close relative, recursion. These things are perfectly well understood.
Quoting Ludwig V
You are making this up out of some level of confusion involving time. Time is not a consideration or thing in mathematics. All mathematics happens "right here and now."
I am trying, I don't know if I'm getting through or not, but I am trying to get you to separate out your naive notion of timeliness in mathematics, with mathematics. Time matters in physics and in supertask discussions. It's important to distinguish these related but different concepts in your mind.
Quoting Ludwig V
Good question. A bound and a limit are two different things. A couple of examples:
* Consider the set {1/2, 1/4, 1/8, ...}.
-43, that is negative 43, is a lower bound of the set. 0 is the "greatest lower bound," a concept of great importance in calculus.
* But here's a more interesting example. Consider the sequence 1/2, 100, 1/4, 100, 1/8, 100 ...
It has two limit points, 0 and 100. But it has no limit, because the formal definition of a limit is not satisfied. To be a limit the sequence has to not only GET close to its limit, but also STAY close.
0 and 100 would be the greatest lower bound and the least upper bound, respectively.
Now I know this was too much info!! This is just technical jargon in the math biz, don't worry about it two much. But bounds and limits are different concepts. Limits are more strict.
Quoting Ludwig V
Maybe that bit about the order topology was a little too much. My only point is that there is a mathematical context in which omega as the limits of the natural numbers is the same as calculus limits. That's all I need to say about that.
Quoting Ludwig V
This didn't parse, I don't know what you are referring to. What is "it" and "this situation." Nevermind I'll work with the rest of the text.
Quoting Ludwig V
This is a little convoluted and confused. What converging sequence between 0 and 1? Say we have the sequence 1/2, 1/4, 1/8, ... for definiteness.
We can think of this as a FUNCTION that inputs a natural number 1, 2, 3, ... and outputs [math]\frac{1}{2^n}[/math]. I'm starting from 1 rather than 0 for convenience of notation, it doesn't matter.
Now in order to formalize where the limit 0 fits into the scheme of things, we can say that the limit is the value of that function at the point [math]\omega[/math] in the EXTENDED natural numbers
0, 1, 2, 3, ...; [math]\omega[/math]
Those are NOT the natural numbers. I've stuck a conceptual "point at infinity" at the end. I hope this is not confusing you. Tell me what your concerns are.
Quoting Ludwig V
The "termination state" is 42. 42 is not the limit of the sequence 0, 1, 0, 1, ... The word limit has a very technical meaning. It's clear that the sequence does not "get near and stay near" 42.
That's why for purposes of analyzing supertasks I am DEFINING the phrase "termination state" of a sequence to be a value "stuck at the end," but that is NOT NECESSARILY A LIMIT.
I hope this is clear. The termination point is arbitrary, it can be 42 or a pumpkin. But in no case are those values limits in the calculus sense.
Quoting Ludwig V
Hmmm. "Limited" is not a term of art in this context. Given a sequence, it either converges to a limit or it doesn't. A convergent sequence of course converges to its limit, but this is a tautology that follows from the definition of convergence to a limit. A convergent sequence converges to its limit, but that doesn't really any anything we didn't already know.
Quoting Ludwig V
Glad it makes sense, but the limit is NOT repeat NOT part of the sequence.
It's part of what I'm calling the extended sequence, with the limit or termination point stuck at the end. But that is my terminology that I am making up just for these supertask problems.
Hope that's clear.
When I write my semicolon notation: 1/2, 1/4, 1/8, ...; 0
that is a fishfry-defined extended sequence. The sequence is 1/2, 1/4, 1/8, ..., and the limit is 0.
I use this notation to describe the termination state of a supertask: on, off, on, off, ...; pumpkin
The sequence is the on/off part; the pumpkin is the termination state.
Hope this is getting clearer.
Quoting Ludwig V
I don't even know what that means :-) What are completist tendencies?
You would hate the rational numbers then. They are not complete. For example the sequence 1, 1.4, 1.41, 1.412, ... where each term is the next truncation of sqrt(2), does not have a completion in the rationals.
The real numbers are the completion of all the sequences of rationals. That's how we conceptualize the reals.
Well I wrote a lot, let me know if any of this was helpful and let me know what's still troubling you.
tl;dr to this entire post:
Mathematical sequences are clear and rigorous. We have a fully worked out theory of them.
Supertasks are nebulous and vague. Reason: There is no time in math. Time is a concept of physics. And Supertask problems always involve physical impossibilities, like flipping a lamp in arbitrarily small intervals of time. That's the source of all the confusion. Supertasks are fairy tales, like Cinderella's coach; and you can no more apply logic to a supertask problem than you can to the coach turning into a pumpkin.
I traced back to your mention of the axiom of infinity, and I still fail to see the relevance of the remark in context. I apologize for lashing out regardless. "wut" is a standard Internet location, and though it carries a bit of snarkitude, it's not considered overly aggressive in the scheme of things. Just an expression of puzzlement.
wut?
Quoting fishfry
My response was to 'what's wrong with you tonight?', not so much to 'wut?'.
Convenient for you now to self-justify by highlighting 'wut?' and not 'what's wrong with you tonight?'.
There was nothing wrong with what I posted that night. You just snapped-at as if there were, when actually the problem is that you, as often, reply to your careless mis-impression of what is written rather than to what is actually written.
Hey, I get your whole "Aw shucks, I'm just a scorpion who's gonna do what a scorpion's gonna do. I don't mean nothin' by it" routine. But it doesn't mean jack to me as far as feeling any less right in answering right back.
Ah. The what is wrong with you and not the wut. I can see that now that you mention it.
I am terribly sorry to have offended you once again.
You first claimed that I was offensive to you. So I pointed out that you don't realize how offensive you often are. So I just gave you that info. I don't sweat being offended in posts. But you carelessly misconstrue what I've posted, and claim I've said things I haven't said, and write back criticism of my remarks by skipping their substance and exact points. And that is what I post my objections to.
Meanwhile, what you say about my posting style is rot. You say it's too long. But you also say it doesn't explain enough. Can't have it both ways. And I do explain a ton. But, again, I can't fully explain without having the prior context back to chapter 1 in a text already common in the discussion. And l explain somewhat technically because being very much less technical threatens being not accurate enough. Meanwhile, your own posts are usually plenty long, so take that tu quoque.
I've not gone back to review all that's been said in this thread, and I need to catch up to your replies, but starting again from the beginning of your argument.
I surmise that the reason you put your argument in numbered steps is so that it can be seen to be airtight.
Is your argument intended to be Thomson's argument?
You have mentioned different conclusions you draw:
(1) The conditions (the premises) of the lamp are inconsistent.
(2) Supertasks are impossible. (But can we infer from the impossibility of Thomson's lamp that all supertasks are impossible?)
(3) Time is not continuous. (I've suggested that what you actually seem to dispute is that time is not densely ordered (infinitely divisible), which is a stronger claim.)
(4) Benacerraf is wrong.
Here's Thomson's statement of the problem:
"There are certain reading-lamps that have a button in the
base. If the lamp is off and you press the button the lamp goes
on, and if the lamp is on and you press the button the lamp goes
off. So if the lamp was originally off, and you pressed the
button an odd number of times, the lamp is on, and if you
pressed the button an even number of times the lamp is off.
Suppose now that the lamp is off, and I succeed in pressing the
button an infinite number of times, perhaps making one jab
in one minute, another jab in the next half-minute, and so on,
according to Russell's recipe. After I have completed the whole
infinite sequence of jabs, i.e. at the end of the two minutes, is
the lamp on or off? It seems impossible to answer this question.
It cannot be on, because I did not ever turn it on without at
once turning it off. It cannot be off, because I did in the first
place turn it on, and thereafter I never turned it off without at
once turning it on. But the lamp must be either on or off. This
is a contradiction."
Here's your presentation:
Quoting Michael
I want to get back to looking at this more closely, but in the meantime, do you consider your presentation equivalent with Thomson's statement of the problem?
When possibility is part of the analysis, the analysis can get complicated. We should be careful that our inferences regarding the modalitiy are proper.
Are you just committed to picking fights with me? I've apologized several times tonight, for sins real and imagined. And some cosines too. Enough bro'.
I won't argue with that. For some reason, I've never been able to get my philosophical head around that topic. Just like Augustine, all that time (!) ago.
Quoting Metaphysician Undercover
I was going to reply that slowing down isn't stopping. I didn't realize that the slowing down was a convergent series. Perhaps slowing down can be stopping.
Quoting Metaphysician Undercover
Well, we could if we wanted to do. But why would we want to? Apart from the fun of the paradox. Mind you, I have a peculiar view of paradoxes. I think of them as quirks in the system, which are perfectly real and which we have to navigate round, rather than resolve. Think of the paradoxes of self-reference. Never permanently settled. New variants cropping up.
I don't ask for apologies. But it's okay if you want to give them. But you embed into your apologies yet more items that I feel deserve response. Your apologies themselves are snarky; "sins imagined" e.g. I don't even object to snark, except it's your way of ostensibly apologizing while still turning it back on me.
If I misconstrue someone's math or philosophy points, especially to mischaracterize them, then if the person calls me on it or I discover it myself, before posting back to that person again, I should post my recognition of my mistake. That's my ethos. Yours might be different. But I will stick with my prerogative to reply when I like.
And to answer your question: No, I definitely do not have any interest in "picking fights" and I find no value in fighting for the sake of fighting. But I do find value in posting disagreements and corrections, whether regarding the math and philosophy or regarding the personal specifics of the posting interchanges. In various thread, you have posted a lot of inaccuracies and misconceptions about math, and now lately about me. I respond to that.
Ok no more snark.
Interesting. I hope I didn't bury the lede. I'm not all up about sarcasm. Rather, what I find important is (1) striving not to misrepresent a poster's remarks and to stand corrected when it is pointed out that one has; and (2) not to argue by ignoring key counter-arguments and explanations; not to just keep replying with the same argument as if the other guy hadn't just rebutted it.
Yes.
No I'm not. I accept that one of the premises of the thought experiment is physically impossible. That doesn't then mean that we cannot have another premise such as "there are no spontaneous, uncaused events".
You seem to think that because we allow for one physical impossibility then anything goes. That is not how thought experiments work.
It is physically impossible for me to push a button 10[sup]100[sup]100[/sup][/sup] times within one minute, but given the premises of the thought experiment it deductively follows that the lamp will be off after doing so. Your claim that the lamp can turn into a plate of spaghetti is incorrect.
In many cases of common language usage, "slowing down" is stopping, but that implies the end, not yet achieved. The point is that "stopping" is distinct from "stopped'. And if the slowing down never reaches the point of being stopped, then the term "stopping" is not justified. The convergent series is misrepresented as "stopping", because the end of "stopped is never achieved.
In the modern physical world of relativity, "stopped" is arbitrarily assigned according to an inertial reference frame. This implies a sort of equilibrium, or stability within that specific reference frame, but it's highly unlikely that it is a true case of "stopped", more likely very slow movement, misrepresented as "stopped". We like to round things off.
Quoting Ludwig V
Why would we want to? Because we are philosophers seeking knowledge. Understanding is the primary objective. I look at such paradoxes as indications of a lack of understanding. The principles applied do not adequately map to the reality which they are being applied, this is a failure of our knowledge. Then we need to subject all the principles to skeptical doubt, to determine the various problems. We could just live with quirks in the system, but that's unphilosophical. Knowledge evolves, and that evolution is caused by people attempting to work out the quirks in the system.
Quoting fishfry
The objects that constitute both Euclidean and non-Euclidean (the unending many of them) spaces are abstract and both exist. Those objects may be applied in our scientific theories because a description of these objects can also describe some phenomenons in the real world. The problem is how do we get knowledge of these objects, if they are not physical? That is Benecerraf's problem.
Quoting Ludwig V
This is not one of those cases. The world here is meant by everything that is not created by the mind (realism X anti-realism), not just what is located in space-time (physicalism).
I am so appreciative that you straightened me out on this extensionality thing that I can't argue with you about anything. I accept all your criticisms. You say I've done these things and I don't deny them. I make no defense nor explanation.
I do have a sarcasm gene and that rarely works online. You'd think I'd learn.
I respectfully leave this conversation. We've said it all. i've enjoyed our chat.
Yes I see what you meant. Thanks.
Then you can't argue with me that you can argue with me.
It's very helpful, so that's fine. I get my revenge in this post.
The system is not helping me here, because it invites me to link to specific comments, but I'll do my best to make clear what I'm responding to.
Quoting fishfry
:grin:
Perhaps that's why philosophers keep tripping up on them. It is well known that they don't notice what's on the floor - too busy worrying about all the infinite staircases and the fall of man.
Quoting fishfry
That was not a very well thought out remark. I would certainly have hated them in the long-ago days when the Pythagoreans kept the facts secret so that they could sort it out before everyone's faith in mathematics was blown apart. But now that mathematicians have slapped a label on these numbers and proved that they cannot be completed, I'm perfectly happy with them.
Quoting fishfry
Yes. Austin invented them, Grice took them up, Searle was the most prominent exponent for a long time, although he has now moved on to other things now. It's a thing in philosophy For me, it's a useful tactical approach, but a complete rabbit-hole as a topic.
Quoting fishfry
Something like that. The initial point was to establish that there are perfectly meaningful uses of language that are not propositions (i.e. capable of being true or false), in the context of Logical Positivism. I doubt that you would welcome a lot of detail, but that idea (especially the case of the knight in chess) will be at the bottom of some of the later stuff.
Quoting fishfry
It was very helpful to me. I have doubts about the terminology "potential" vs "completed", but the idea is fine. I particularly liked "don't really find a use in math".
Quoting fishfry
Too much or not. It helped me. Someone else started talking about bounds and I couldn't understand it at all. I may not understand perfectly, but I think I understand enough.
Quoting fishfry
I know that. It's not a problem. If I said anything to suggest otherwise, I made a mistake. Sorry.
Quoting fishfry
... because "1/2, 1/4, 1/8, .." gets near and stays near 0. Yes?
Quoting fishfry
I understand that distinction.
Quoting fishfry
Quoting fishfry
Many of my notions are naive or mistaken. But this separation is my default position. I'm not making an objection, but am trying to point out what may be a puzzle, which you may be able to resolve. On the other hand, this may not be a mathematical problem at all.
Quoting fishfry
There are other ways of putting the point. What about "Mathematics is always already true"? Or mathematics is outside time? Or time is inapplicable to mathematics?
But your example of making a rule in chess. Note that as soon as the rules are made, we can starting defining possibilities in chess, or calculating the number of possible games and so forth. It's as if a whole structure springs into being as we utter the words. So a timeless structure is created by our action, which takes place in time. Isn't that at least somewhat like a definition in mathematics? And the definition is an action that takes place in space and time.
More difficult are various commonplace ways of talking about mathematics.
Quoting fishfry
Quoting fishfry
Quoting fishfry
Quoting fishfry
Quoting fishfry
At first sight, these seem to presuppose time (and even, perhaps space) Perhaps they are all metaphors and there are different ways of expressing them that are not metaphorical. Is that the case? I recognize that I may be talking nonsense.
Quoting Lionino
So when you use the appropriate sense of the "world", and say that realism is true of the world, you are saying that realism is true of some parts of the world - the abstract parts?
I strongly approve of defining the context in which one is using "real" or "realism", but using it of the world, defined as everything that exists independently of the mind, you are simply re-asserting the basic thesis that both geometries are true independently of the mind. Since they are both true in the abstract world, but not simultaneously in the physical world, would it not be helpful to add that explanation?
Yes, I agree with that. I was suggesting that a slowing down according to a convergent series might count as stopped, since it would never reach the limit or "0".
Quoting Metaphysician Undercover
If you are right about relativity, I wouldn't disagree.
Some time ago I mentioned time dilation in relativity theory in this regard.
Correct. Which is why I acknowledged your complaints and said nothing else. If I did, you'd complain that I was minimizing my apology by contextualizing it, either with snark or denial.
So I didn't even apologize. I acknowledge your complaints and I stand mute. I have nothing to say at all.
Glad to know. Revenge? What do you mean? By writing a long post? Well I write long posts but prefer when others write shorter ones. I haven't solved this dilemma yet.
Quoting Ludwig V
Not sure what you mean. I generally quote the whole post then stick in quote tags around the specific chunks of text I want to respond do.
Quoting Ludwig V
I don't know many philosopher jokes.
Quoting Ludwig V
Sorry maybe I was off track about the rationals.
Quoting Ludwig V
Yes he got in trouble for harassing his female doctoral students.
Quoting Ludwig V
Ok. Why did you bring it up relative to math? Oh I remember. "Let x = 3" brings a variable x into existence, with the value 3. So statements in math are speech acts, in the sense that they bring other mathematical objects into existence. I can see that.
Quoting Ludwig V
Ok. Not entirely sure where you're going.
Quoting Ludwig V
Ok I was only trying to be philosophical. Aristotle (I think) made the distinction. It doesn't come up in math, nobody ever uses the terminology. But the way I understand it is that Peano arithmetic is potential and the axiom of infinity gives you a completed infinity.
Quoting Ludwig V
Ok, bounds. They're just the shoulders of the road. Thing's you can't go past. Guardrails.
Quoting Ludwig V
That was about limits versus "termination state." I should emphasize that limits are perfectly standard mathematical terminology. But "termination state" is my own locution for purposes of talking about supertasks. The termination state is like a limit in the sense that we can conceptually "stick it at the end" of an infinite sequence; it just doesn't have to satisfy the definition of the limit of a sequence. Like 1/2,/ 3/4, 7/8, ...; 42
The semicolon notation is my own too. I don't think mathematicians talk about supertasks. They're more of a computer science and philosophy thing.
Quoting Ludwig V
I am not aware of what problem or puzzle you are expressing.
Quoting Ludwig V
The subject matter of mathematics does not speak about time. That's different than saying "math is outside of time," although it's kind of related. Physics talks about time, and physicists use math to model time, but that is a very different thing.
It's the difference between a loop in math versus programming.
In math when we say that 1/2 + 1/4 + 1/8 + ... = 1, we mean "right now," though even that is a reference to timeliness. The equality "just is."
But in a programming language when we write a loop that keeps adding each term to a running total, that notation stands for a physical process that takes place in a computing device and requires time and energy to execute, and produces heat. A programming loop is a notation for a physical process.
Quoting Ludwig V
Ok. So when I write down the rules of set theory, I instantiate or create all the complex world of sets as studied by set theorists. And you speculate that this might be an event that takes place in time.
There is another point of view. The structures of the sets were there. Mathematicians discovered the structures. So the discovery of set theory is historically contingent and takes place in time, around 1874 or so with Cantor's first paper on set theory. But the sets themselves, the structures of set theory, are eternal!
In other words this is the old "invented or discovered" question of mathematical philosophy.
Now chess, I think we can agree, was invented and not discovered. But math is somehow different. Math is somehow wired into the logic centers of our minds, and perhaps the universe.
Quoting Ludwig V
Are you referring to what I just talked about?
Quoting Ludwig V
I cannot fathom what you might mean. A sequence does not approach its limit in time. The limit of 1/2, 1/4, 1/8, ...is 0 right now and for all eternity. The fact is inherent in the axioms of set theory, along with the usual constructions and definitions of the real numbers and calculus. In that sense the fact "came into existence" when Newton thought about it, or maybe when Cauchy formalized it, and so forth.
But the history of our understanding of the fact is not the same as the fact itself. The earth went around the sun even before Copernicus had that clever idea. Likewise every convergent sequence always converged to its limit, independently of our discovery of those limits, and our understanding of what a limit is.
Is this your point of contention or concern? That you think that time is hiding in there somewhere? I profoundly disagree. You greatly misunderstand mathematics; or you have an interesting and original philosophy of mathematics; if you believe there's time hiding inside mathematics.
Quoting Ludwig V
If I am understanding you, you think time is somehow sneakily inherent in math even though I deny it.
Have I got that right?
It was merely a math quip.
Quoting jgill
I either skimmed past it or forgot it. Sorry. Not having been trained for it, I wouldn't want to comment on it. But it is that left field plausibility that I always appreciate.
Yes. I was saying in a complicated way, that a long post is not, for me, a bad thing.
Quoting fishfry
That's a useful tactic. I shall use it in future.
Quoting fishfry
He did indeed. It was very common back in the day. It was disapproved of by many, but not treated as unacceptable. I don't think anyone can really understand how horrible it is unless they've actually experienced it.
Quoting fishfry
Exactly. There's a lot of refinement needed. But that's the basic idea. What those objects are is defined entirely by their use in mathematics.
Quoting fishfry
I was just being pedantic. It was a thing in the era before Descartes &c. But I understood that the distinction was "potential" and "actual". Nonetheless, the idea of a "completed" infinity catches something important.
Quoting fishfry
That's a very helpful metaphor.
Quoting fishfry
Yes.
Quoting fishfry
Nor can I. That's the problem.
Quoting fishfry
That's the starting-point.
Quoting fishfry
Why is this a problem? The traditional view is that mathematics, as timeless, cannot change. Our knowledge of it can, but not the subject matter. (Strictly that rules out creating any mathematical objects as well, but let's skate over that.) "A sequence does not approach its limit in time" makes no sense.
I may be about to solve my own problem. That doesn't mean that raising it with you is not helpful.
We have to accept that a sequence approaching its limit is not like a train approaching a station. The train is approaching in space and time. But you can't ask what time the sequence left its origin and when it will arrive at its limit.
You can call the sequence approaching its limit a metaphor or an extended use. The train approaching the station is the "core" or "paradigm" or "literal" use. The sequence approaching its limit is a different context, which, on the case of it, makes no sense. So we call this use is extended or metaphorical.
We can explain the metaphor by drawing a graph or writing down some numbers and pointing out that the different between n and the limit is less than the difference between n+1 and the limit is less and that the difference between n and n-1 is greater.
And so on.
Quoting fishfry
Yes. I realize this is border country. Godel seems to live there too.
Quoting fishfry
If PA here is first order, then PA does not have a predicate 'is a number' nor those axioms.
Quoting fishfry
Just to be clear, that occurs in set theory, not in PA. In PA, '+' is not defined. It is primitive.
Quoting fishfry
Of course, that's correct regarding PA.
Quoting fishfry
Right.
Quoting fishfry
Right.
Quoting fishfry
The axiom of infinity does not define anything, including the successor operation.
The successor operation only requires pairing and union:
Df. the successor of x = xu{x}.
That is logically prior to the axiom of infinity. Then the axiom of infinity only says that there is a set that has 0 and is closed under successor.
Then we prove that there is a unique set that is a subset of all sets that have 0 and are closed under successor.
Then we define w = the set that is a subset of all sets that have 0 and are closed under successor.
Quoting fishfry
Not "and". All it says is what you said after the "and": "there is a set that contains the empty set, and if it contains any set X, it also contains the successor of X".
Quoting fishfry
Rather than "the model" I would say "the standard model". There are other models too. And models not isomorphic with the standard model.
Quoting fishfry
Quoting fishfry
Both are right, and well said. In both PA and Z without infinity (even in Z with the axiom of infinity replaced by the negation of the axiom of infinity), we can define each number natural number, and in Z we can prove the existence of the set of all and only the natural numbers.
Oh I see. I prefer shorter posts so I don't get lost in the quoting!
Quoting Ludwig V
Yes, I just highlight the whole post and say Quote.
Quoting Ludwig V
"Back then" wasn't that long ago, this scandal's just a few years old IIRC. I didn't follow the particulars. I'm sure it's just as common today. Or maybe not as 1950's, say. Thinks have changed since then. Still. The male-female thing, those are very deep energies being played with. You are not going to stamp it out with rules. You can change the form in which the scenarios are played out. What Eric Berne called the games. Games People Play, remember that?
Quoting Ludwig V
There's another funny thing that goes on. Sometimes you make a definition, and it DOESN'T bring a mathematical object into existence. For example we write down the axioms for a group. But we have no idea if there are any groups, or if the axioms are perhaps vacuous. So the next step is to exhibit some groups, like the additive structure on the integers. This is a very common pattern in math: Make a definition, then show that there's something that satisfies the definition!
Quoting Ludwig V
Right, potential/actual versus potential/completed. I've heard them both. Since they don't come up in math I kind of use them interchangeably. But they probably have more specific technical meanings or contexts I don't know about. But for me, I just regard the axiom of infinity as the sharp boundary between the two concepts. Induction on the one hand, versus a "completed loop."
Quoting Ludwig V
Shoulders of road. Bounds. Glad that helped.
Quoting Ludwig V
I quoted that entire exchange. I can't fathom your meaning. But you say you can't either.
Time is a concept in physics. You can see that, right? Math is outside of time. It doesn't describe or talk about time, though it can be used by physicists to model time. And then again, we have no evidence that the mathematical real numbers are even a decent model for time. The real numbers are continuous, but nobody knows if time is.
If you can give an example of what you are thinking, that might be helpful.
Quoting Ludwig V
Ok. You seem to be agreeing with me.
Quoting Ludwig V
And now you're not agreeing. The word "approach" is colloquial. It is not intended to evoke images of panthers stalking their prey, or arriving at your destination in a car. Not at all. It's just the word we use for the limiting process. But 1/2 + 1/4 + 1/8 + ... IS 0; it does not "become" or "approach" zero. It's this language ambiguity that is the source of so much online confusion about the subject. See any .999... = 1 debate. You'll hear that .999... "approaches but does not reach" 1. But the sum is exactly 1 nonetheless, by virtue of the definitions. The math is designed to make it work out.
Quoting Ludwig V
I'd like to know what the problem is, regardless. If for no other reason than to make sure you're understanding it correctly!
Quoting Ludwig V
Correct. It's a shame we use the word "approach," because many are confused by that.
Quoting Ludwig V
Right. Trains are physical objects. Numbers in a sequence are mathematical abstractions. They don't live in the physical world.
Quoting Ludwig V
I think it's just a confusing use. When mathematicians use the word approach, in their minds they already have the full context of the theory of limits. So they are not confused. But non-mathematicians hear the word and associate it with is everyday context. No harm is done, till these misunderstanders show up to .999... = 1 threads online. Then we get problems.
Quoting Ludwig V
With supertasks? I don't think so.
I would organize Thomson's argument differently from the way he organizes it. Near the end of his argument, he says "But the lamp must be either on or off." But he's actually invoking a premise. It is natural to regard the lamp as being either Off or On and not both, but in this highly hypothetical context, it would be good to say that as an explicit premise.
Then Thompson invokes infinitely divisible time. But not as a premise. I would include it as a premise. The advantage of doing that is that then the premise is explicitly a candidate for rejection to avoid the contradiction.
I simplified the language of your conclusions (we don't need all those tn, ti, tj and inequality symbols), and I don't think you need the conclusions to be biconditionals to derive that the lamp is neither Off nor On.
('r' for 'revised')
Premises:
rP1: At all times, the lamp is either Off or On and not both.
rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.*
*The pushing of the button and the change are together instantaneous, and the button can be pushed only once in any moment. This is not needed except to simplify the argument (especially to state rC1, rC2 and rP6).
rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.
rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.
rP5: The lamp is Off at 10:00.
Conclusions:
rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.*
*Notice that T1 and T2 are in chronological order.
rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.*
*Notice that T1, T2 and T3 are in chronological order.
Premise:
rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.*
* We could easily make that mathematically rigorous.
Conclusion:
rC3: The lamp is neither Off nor On at 12:00. Contradicts rP1.
QUESTION: How do you state the arguments for rC1 and rC2 from the premises rP1-rP5?
I think that this is what the so-called "paradox" of supertasks is all about. What is revealed is that at least one or the other, space or time, or both, must not be continuous. I think that's what @Michael has been arguing since the beginning. Tones attempted to hide this behind sophistry by replacing the continuity of the real numbers with the density of the rational numbers.
The real issue is that if one of these, space or time, is not continuous, then it cannot be modeled as one thing. There must be something else, a duality, which provides for the separations, or boundaries. But I don't think anyone has shown evidence of such a duality, so we have no real principles to base a non-continuous ordering system on.
Quoting fishfry
I'd say this is similar to Tones' use of "identity" in set theory. We take a word, such as "approach", which clearly does not mean achieving the stated goal, and through practise we allow vagueness (to use Peirce's word), then the meaning becomes twisted, and the use of the word in practise gets reflected back onto the theory. So we have the theory stating one thing, and practise stating something different, then the meaning of the words in the theory get twisted to match the practise. Practise says .999... is equal to 1, so "approach" in the theory then takes on the meaning of "equal". Practise says that two equal sets are identical, so "equal" in the theory takes on the meaning of "identical". These are examples of how theory gets corrupted through practise when the words are not well defined.
In the case of platonism, yes. In the case of immanent realism, it would be true of some (physical) parts of the world. Now, a finer distinction: in the case of psychologism/conceptualism, true about our minds (so mathematics would reduce, prima facie, to psychology if psychologism is true).
Quoting Ludwig V
I don't know about neaty gritty details of platonistic matemathics if mathematics is true of the physical world too or rather only applies to it , if such shenanigans are even developed that deeply, but Euclidean geometry applies to a car going from the theater to the restaurant (the surface of the city is flat), non-Euclidean to an airplane going around the Earth (spherical geometry) or things interacting in space-time (hyperbolic geometry).
There is no sophistry on my part. And no "replacing". I merely pointed out that proving that time is not continuous does not prove that time is not densely ordered (or infinitely divisible).
And the crank is so ignorant and mixed up about this very thread that he wildly infers that my remarks about the thought experiment vis-a-vis Michael's version of it entail that I have myself made certain claims about time beyond that not-continuous does not imply not-dense.
The garbage posting crank doesn't know what he's talking about, regarding continuity or denseness, or me. He is a bane upon reasoned inquiry.
/
The crank is on about .9... Whatever he's trying to say, in his usual thought salad way, we should at least recognize that the notation '.9...' is informal for the limit of a certain sequence.
Meanwhile he has his own utterly mixed up notions about what 'identical' and 'equal' mean. But he hasn't the least reasonability even to understand that his own having notions about what words should mean doesn't entail that everyone else is wrong for using the words both in their ordinary English senses and also in stipulated mathematical senses. He does not understand even the notion of stipulative definition, just as, in another instance, he does not understand even the difference between use and mention.
The crank falsely rails on and on against mathematics and modern logic, even though he has not read page one in a textbook in the subject. As he serves as a textbook example in crank sophistry.
I don't say that selecting and organizing the quotations is easy. It fits better with the fact that I tend to get slabs of time when I can pursue these discussions but in between, I'm not available at all. So the quick back and to is more difficult for me.
Quoting fishfry
I didn't mean to imply that they were living together. That would be .... interestingly mnd-boggling.
Quoting fishfry
Don't get me started. What particularly annoys me is that so many people seem absolutely certain that they are right about that. I think it is just a result of thinking that you can write probability = 1, when 1 means that p cannot be assigned a probability, since it is true. A friend once conceded to me that it was a degenerate sense of probability, which is like saying that cheese is a degenerate form of milk.
Quoting fishfry
Since my earlier comment on this,
Quoting Ludwig V
I've discovered that potential infinity is the definition of the sequence and actual infinity is the completion of the sequence. So "potential" and "completed" can be fitted together after all.
Quoting fishfry
I think I shall stick to my view that defining an infinite sequence or getting a beer from the fridge is the completion of an infinite number of tasks. I don't think it gives any real basis for thinking that supertasks are possible.
Quoting fishfry
You notice that maths outside time is metaphorical, right? I prefer to say that time does not apply to maths, meaning that the grammatical tenses (past, present and future) do not apply to the statements of mathematics. I like "always already" for this. There is a use of language that corresponds to this - the "timeless present". "One plus one is two" makes sense, but "One plus one was two" and "One plus one will be two" don't.
Quoting fishfry
Yes. But there are complications. How does math apply to the physical world?
Quoting fishfry
We have a choice between insisting that Non-Euclidean geometries are not created but discovered and insisting that they are not discovered but created - though they exist, presumably, forever. But if we create them, what happens if and when we forget them?
Quoting fishfry
As I said before there are a number of ways to describe this. They're all a bit weird.
Quoting fishfry
It sounds as if you are saying that "approach" is a simply two different senses of the same word, like "bank" as in rivers and "bank" as in financial institutions. An old word given a new definition. Perhaps.
Quoting fishfry
That's a very neat definition. I'll remember that. But you can see, surely, how difficult it is to shake off the picture of a machine that sucks in raw materials and spits out finished products. But actually, you are describing timeless relationships between numbers. Or that's what you seem to be saying.
Quoting TonesInDeepFreeze
I don't really understand this. If the lamp is neither off nor on at 12:00 (and still exists) then it must be in a third state of some kind. Or do you mean that it is not defined as on or off, which leaves the possibility that it must be in one state or the other, we just don't know which.
Quoting Lionino
I don't get the difference. If mathematics applies to the physical world, surely it is true of it?
Quoting Lionino
Yes. Different geometries apply in different contexts. That's only a problem if you think that just one of them must be absolutely true, which appears to be false.
Everybody agrees that mathematics applies to the physical world, but nominalists will broadly say that 2+2=4 is not about the world, so it is not true of it.
By the premises, there is no third state. Indeed, even if not a premise but a definition:
Df. 'On' means 'not Off'
there is no third state.
Quoting Ludwig V
No, Thomson's argument is: The premises entail that at 12:00 the lamp is neither Off nor On, but the premises also include the stipulation that at all times the lamp is either Off or On, so the premises are inconsistent.
Quoting Ludwig V
No, it's not a matter of knowledge. Rather, at 12:00 the lamp is neither Off nor On, which contradicts that at all times the lamp is either Off or On.
Who says anything about probability when merely mentioning that .9... = 1.
We prove that .9... = 1, from the definition of the notation '.9...'.
'.9...' stands for the limit of a certain sequence, and that limit is 1.
Anyone is free to regard '.9...' with a different definition and to get different results accordingly. But in context of the ordinary mathematical definition, we prove that .9... = 1.
The adjective 'is potentially infinite' has no mathematical definition that I know of, including in alternative theories.
The adjective 'is infinite' is defined in mathematics.
The adjective 'is actually infinite' has no mathematical definition that I know of, including in alternative theories, unless it means simply 'is infinite'.
'is potentially infinite' is a notion about mathematics.
'is actually infinite', if not meaning simply 'is infinite', is a notion about mathematics.
Relax! I was talking about the traditional Aristotelian approach to infinity which was orthodox before Descartes but not since, so far as I know. Though I have since seen someone apparently still using the terms in Two Philosophers on a beach with Viking Dogs
Quoting TonesInDeepFreeze
Yes, I didn't think of the possible application of that idea to this discussion. I've only ever encountered it in the context of probability.
Quoting TonesInDeepFreeze
That's interesting. Can you refer me to a source?
Quoting TonesInDeepFreeze
I'm sorry. It's probably not worth pursuing, but I was struck by the point that "at all times the lamp is either Off or On" appears to be true while "the lamp is neither Off nor On" appears to be false, by reason of a failed referent. It's true by definition that a lamp is either off or on, so if some object is capable of being neither off nor on is not a lamp. The story is incoherent from the start. We cannot even imagine it.
This issue was actually resolved a long time ago by Aristotle, in his discussions on the nature of "becoming". What he demonstrated is that between two opposing states (on and off in this case), there is a process of change, known as becoming. This process is the means by which the one property is replaced by the opposing property. If we posit a third state between the two states, as the process of change, then there will now be a process of change between the first and the third, and between the third and the second. We'd now have five distinct states, and the need to posit more states in between, to account for the process of change which occurs between each of the five. This produces an infinite regress.
So what Aristotle proposed is that becoming, as the activity which results in a changed state, is categorically different from, and incompatible with states of being. Further, he posited "matter" as the potential for change. "Potential" refers to that which neither is nor is not. As what may or may not be, "potential" violates the law of excluded middle. So in the example, when the lamp is neither on nor off, rather than think that there must be a third state which violates the excluded middle law, we can say that it is neither on nor off, being understood as potential. This is the way that I understand Aristotle to have proposed that we deal with such activity, which appears to be unintelligible by violation of the law of excluded middle, neither having nor not having a specified property. The unintelligibility is due to a thing's matter or potential.
I won't refer you to a source.
I'll refer you to this:
Definition: .999... = lim(k = 1 to inf) SUM(j = 1 to k) 9/(10^j).
Let f(k) = SUM(j = 1 to k) 9/(10^j).
Show that lim(k = 1 to inf) f(k) = 1.
That is, show that, for all e > 0, there exists n such that, for all k > n, |f(k) - 1| < e.
First, by induction on k, we show that, for all k, 1 - f(k) = 1/(10^k).
Base step: If k = 1, then 1 - f(k) = 1/10 = 1(10^k).
Inductive hypothesis: 1 - f(k) = 1/(10^k).
Show that 1 - f(k+1) = 1/(10^(k+1)).
1 - f(k+1) = 1 - (f(k) + 9/(10^(k+1)) = 1 - f(k) - 9/(10^(k+1)).
By the inductive hypothesis, 1 - f(k) - 9/(10^(k+1)) = 1/(10^k) - 9/(10^(k+1)).
Since 1/(10^k) - 9/(10^(k+1)) = 1/(10^(k+1)), we have 1 - f(k+1) = 1/(10^(k+1)).
So by induction, for all k, 1 - f(k) = 1/(10^k).
Let e > 0. Then there exists n such that, 1/(10^n) < e.
For all k > n, 1/(10^k) < 1/(10^n).
So, |1 - f(k)| = 1 - f(k) = 1/(10^k) < 1/(10^n) < e.
(I saw an argument in a video that is much simpler, but I didn't get around to fully checking out whether it's rigorous. But arguments that subtract infinite rows are handwaving since subtraction with infinite rows is not defined.)
I was quite relaxed when I provided the information.
Quoting Ludwig V
The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. You can go back to the argument to witness it step by step. Best is to read Thomson's paper that is not long and not abstruse, free to download.
I agree with everything you wrote in this post.
Two presentations that are equivalent.
I would like to know how C2 and C3 are derived in Michael's version. That is rC1 and rC2 in my version.
But we get them anyway from my premise rP6 (the antecedent of Michael's C6). I'll show that presentation too (PRESENTATION2). And I think it is closer to Thomson's argument.
MICHAEL'S PRESENTATION
P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 10:00
From these we can then deduce:
C1. The lamp is either on or off at all tn >= 10:00
C2. The lamp is on at some tn > 10:00 iff the button was pushed at some ti > 10:00 and <= tn to turn it on and not then pushed at some tj > ti and <= tn to turn it off
C3. If the lamp is on at some tn > 10:00 then the lamp is off at some tm > tn iff the button was pushed at some ti > tn and <= tm to turn it off and not then pushed at some tj > ti and <= tm to turn it on
From these we can then deduce:
C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00
C6. If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum, then the lamp is neither on nor off at 12:00 [contradiction]
TONESINDEEPFREEZE'S PRESENTATION:
Premises:
rP1: At all times, the lamp is either Off or On and not both.
rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.
rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.
rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.
rP5: The lamp is Off at 10:00.
Conclusions:
rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.
Premise:
rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.
Conclusion:
rC3: The lamp is neither Off nor On at 12:00. Contradicts rP1.
Again, I don't know how we derive Michael's C2 and C3 (my rC1 and rC2). But we don't need them anyway:
TONESINDEEPFREEZE'S PRESENTATION 2:
Premises:
rP1: At all times, the lamp is either Off or On and not both.
rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.
rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.
rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.
rP5: The lamp is Off at 10:00.
rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.
Conclusions:
rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.
rC1: The lamp is neither Off nor On at 12:00. Contradicts rP1.
So, we don't have to be concerned whether rP1-rP5 entail rC1-rC3. Rather, we see easily that rP1-rP6 entail rC1-rC3. It's a clean and correct inference that way.
So, unless we do have a proof of Michael's C2 and C3 from his P1-P4, he has his argument out of order: we need my rP6 in the premises. And that seems to be flow of Thomson's argument too.
I've bowed out of the supertask discussion, having not typed anything new in weeks. It would be inappropriate for me to comment on anything @Michael said, since he'd then be obliged to reply and we'd be right back in it again. @TonesInDeepFreeze merely made the point that 1/2, 1/4, etc only requires the rational numbers. Perfectly sensible observation.
Quoting Metaphysician Undercover
I'm fully supertasked out.
Quoting Metaphysician Undercover
Let's keep the Infinity theory in that thread. Well I'm not a moderator here so nevermind, do what you like. I prefer not to engage in these thread-hijacking points here. Every discipline has its terms of art, which confuse non-practioners. When a doctor tells you your liver is "unremarkable," that's great news and not an insult.
Oh I see. That's what I like about discussion forums. You can pick up a topic weeks or even months later.
Quoting Ludwig V
Oh my. We must have a conversation about probability sometime. You're wrong about that. 1 is a perfectly sensible probability. But worse, probability 1 events may be false. For example if you randomly pick a real number in the unit interval, it will be irrational with probability 1, even though there are infinitely many rationals.
Quoting Ludwig V
1 is a perfectly sensible probability. Your friend is misinformed. As Mark Twain said, if you don't read the newspapers, you're uninformed. If you do read the newspapers, you're misinformed.
Quoting Ludwig V
It's mathematically unhelpful to think of a infinite sequence as the "completion of an infinite number of tasks." It leads to confusion. It's not how mathematicians think about sequences.
Quoting Ludwig V
No, it's literally true. Of course math as a human activity is historically contingent. But math itself speaks to truths that are outside of time.
Quoting Ludwig V
Ok, but IMO it's deeper than a semantic point.
Quoting Ludwig V
Right. We don't even have good words to talk about things outside of time. Timeless present is a pretty good phrase.
Quoting Ludwig V
As in Wigner's famous paper on the "unreasonable effectiveness" of math in the physical sciences. If math doesn't actually refer to anything, why's it so useful?
Quoting Ludwig V
I have no idea. I have myself argued from time to time that 5 was not a prime number before there were intelligent beings to observe that fact. I don't actually believe that, but I've argued it.
Quoting Ludwig V
This was in reference to 0, 1, 2, ... existing "all at once" in PA. What ways are there to describe this? Is it a timeless present? That's a great locution.
Quoting Ludwig V
Yes. "Approach" is a term of art in mathematics. It has a specific technical meaning that is unambiguous. It is not the same as the everyday meaning.
Quoting Ludwig V
Term of art. A lovely legal phrase. Lawyers commonly have to deal with the jargon of whatever discipline a a particular dispute is about.
Quoting Ludwig V
There's a class math majors take called Real Analysis, where they teach you all this; and after which you are forever clear in your mind about things that were formerly vague and fuzzy. Sadly nobody but math majors takes this class, leading to so much confusion.
Quoting Ludwig V
Yes, that's a "function machine," a visualization when we teach functions to high schoolers. And of course mathematical functions are routinely applied to real world processes. A vending machine is a function of two variables: put in money and push a particular button, and the appropriate product comes out.
So the picture, or visualization of a function as a process or a machine is perfectly valid. The mathematical abstraction that strips away the process or machine interpretation is for the purpose of clarifying our ideas.
Quoting Ludwig V
Yes. The elements of a sequence have a timeless relation to the index set 1, 2, 3, ...
I haven't yet read all of Benacerraf's paper, but at least where he disscusses Aladdin and Bernard, it seems to me that he's not addressing Thomson's problem but only offering a different problem that does have an easy solution.
With Thomson's problem we have:
If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
and
If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.
It seems to me that Benacerraf is skipping that condition. And so is the Cinderella example, which, if I'm not mistaken is a rewording of Benacerraf.
Next would be to examine whether your inference is correct that the problem shows that time is not infinitely divisible (or that it is not possible that time is infinitely divisible - and the modality there may make this more complicated). If I understand correctly, Thomson does't announce such a view about time, though, of course, what Thomson may believe doesn't determine our own conclusions.
@Michael's point, about which he and I disagree.
Quoting TonesInDeepFreeze
Don't believe so. But by expressing disagreement I invite rebuttal. I am supertasked out, really. Hope I have the strength to not get sucked in again.
If Benacerraf is not skipping the condition, then where does he recognize it? [EDIT: Actually he does address it, but, as far as I can tell, he gets it wrong when he addresses it.]
What essential difference is there between Aladdin/Bernard and Cinderella?
"A. Aladdin starts at to and performs the super-task in question just as
Thomson does. Let t1 be the first inistant after he has completed the whole
infinite sequence of jabs - the instant about which Thomson asks "Is the
lamp on or off? - and let the lamp be on at t1.
B. Bernard starts at to and performs the super-task in question (on an-
other lamp) just as Aladdin does, and let Bernard's lamp be off at t1.
I submit that neither description is self-contradictory, or, more
cautiously, that Thomson's argument shows neither description to
be self-contradictory"
But that contradicts:
If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
and
If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.
Benacerraf is saying the lamp gets switched in a way that is not possible given Thomson's conditions.
When we describe the events, we have to look closely and exactly at whether they may occur given Thomson's premises. We can't dissolve Thomson's argument merely by ignoring the premises of the argument.
/
Benacerraf:
"According to Thonmson, Aladdin's lamp cannot be on at t,
because Aladdin turned it off after each time he turned it on.
But this is true only of instants before tl!"
There he does recognize the premises, but, it seems to me, he mistakes them. The premises don't cover just what happens before 12:00. The premises state conditions that obtain at all moments whatsoever. The fact that certain conditions are specified for before 12:00 doesn't entail that all the rest of the conditions don't obtain at all times.
"Nothing whatever has been said about the lamp at t1
or later."
That seems to me to be incorrect. The premises state conditions that obtain at all moments whatsoever. The fact that certain conditions are specified for before 12:00 doesn't entail that all the rest of the conditions don't obtain at all times.
Benacerraf:
"The explanation
is quite simply that Thomson's instructions do not cover the state
of the lamp at t1, although they do tell us what will be its state at
every instant between to and t1"
The instructions don't need to specify what happens at 12:00. The instructions specify what happens at all moments and also what happens before 12:00, but what happens at 12:00 still must conform to the instructions that apply to all moments.
The issue is not that the instructions don't specify what happens at 12:00. The issue is that the instructions entail that at 12:00 the lamp is Off and at 12:00 the lamp is On. Thus the instructions are contradictory.
Please accept my regrets for not engaging. I have little interest in supertask puzzles in general, and this thread has long since exhausted any points I could possibly make.
The simple reasoning is that if time is infinitely divisible then pushing a button an infinite number of times within two minutes is theoretically possible. Pushing a button an infinite number of times within two minutes entails a contradiction and so isn't theoretically possible. Therefore, time is not infinitely divisible.
Although I think perhaps this variation of Zeno's paradox might be better at questioning the infinite divisibility of spacetime.
I'm not deeply versed in Aristotle, but my impression is that he did indeed resolve the issue, as it was understood in his time (and what more than that could he possibly resolve?). In doing so, he invented or discovered or recognized the concept of categories, which was a titanic moment in philosophy. It's a pity that there seem to be so many people around who are completely unaware of it.
Quoting Metaphysician Undercover
I think it would be more accurate to say "The apparent unintelligibility is due to a thing's matter or potential."
Quoting Metaphysician Undercover
I don't think that's quite right. It is true that if the lamp is on, it has the potential to be off, and if the lamp is off, it has the potential to be on. But that's not the same as having the potential to be neither off nor on. A lamp, by definition, is something that is on or off, but not neither and not both. There are things that are neither off nor on, but they are not lamps and the point about them is that "off" and "on" are not defined for them. Tables, Trees, Rainbows etc.
Quoting Metaphysician Undercover
I don't think that's quite right. The LEM does not apply, or cannot be applied in the same way to possibilities and probabilities. "may" does not usually exclude "may not". On the contrary, it is essential to the meaning that both are (normally) possible - but not both at the same time.
i.e. the lamp can't turn into a pumpkin.
i.e. the lamp is on if and only if the button is pushed (when the lamp is off) to turn it on (and not then pushed to turn it off).
Infinite divisibility doesn't entail a contradiction. Rather, infinitely divisibility along with the other premises entails a contradiction. Moreover, you are adding another premise (call it 'DT'): if infinite divisibility, then tasks can be performed at each of the infinitely many times. Therefore, we are entitled to question any of the premises, including the new one DT, not just infinite divisibility.
I'm not rejecting anything.
I'm saying:
(1) What is the proof of C2 and C3 from the premises? (Though we don't need it, if we adopt my rP6.)
(2) Instead of rejecting infinite divisibility, we may reject other premises instead.
That comment was directed at fishfry who claims that the lamp can turn into a pumpkin or spontaneously and without cause be on at 12:00.
Got it.
That is the advantage of platonism over nominalism on the matter of the application of mathematics. Each different nominalist program for mathematics will have a different solution for it Azzouni's solution being deflationary (dissolving the issue).
I personally think the issue is misguided.
I'm deeply flattered. But that is far too much for me to grasp in less than a month or two.
Quoting TonesInDeepFreeze
Perhaps it would serve our purposes. I could probably get the point even if it isn't completely rigorous.
But let me explain why I need convincing.
In my book 0.9 + 0.1 = 1 and 1 - 0.1 = 0.9 and so 0.9 does not equal 1. There's a similar argument for 0.99 and 1 and so on. So for each element of 0.99999....., I have an argument that it does not equal 1. However, I see that your proof involves limits and I know that in that context words change their meanings. So I'm curious.
Quoting TonesInDeepFreeze
Well, it seems clear that at any specific time, it will be on or off depending on whether the button has been pushed an even number of times or an odd number of times since 11:00.
So at each of the times specified in the sequence, it will be on or off depending whether the number of times it has been pushed since 11:00 is odd or even.
Quoting TonesInDeepFreeze
The contradiction is created here - specifically in the last two words, which make it impossible to know whether it has been pushed an even or odd number of times since 11:00.
Which one do you think should be rejected?
If I said anything about that, I would be way out of my depth. So I'm afraid I shall have to ignore it - until another time, maybe.
Quoting fishfry
.. in the context of probability theory, that may be so. But I'm interested in probability in the context of truth and falsity, which is a different context. So when you say that 1 is a perfectly sensible probability, are you saying that probability = 1 means that the relevant statement is true? (I don't want to disappear down the rabbit hole, so I just want to know what you think; I have no intention of arguing about it.
I think this is a misunderstanding of the problem.
Say we accept that Thomson's lamp entails a contradiction; the lamp can neither be on nor off at 12:00.
I take this as proof that having pushed a button an infinite number of times is metaphysically impossible.
You seem to take this as proof that having pushed a button an infinite number of times is metaphysically impossible only if the premises are true.
As an example, let's say that our button is broken; pushing it never turns the lamp on. In such a scenario we can unproblematically say that the lamp is off at 12:00. But this does not then entail that it is possible to have pushed the button an infinite number of times.
We can imagine a lamp with two buttons; one that turns it on and off and one that does nothing. Whenever it's possible to push one it's also possible to push the other, and so if it's possible to have pushed the broken button an infinite number of times then it's possible to have pushed the working button an infinite number of times. Given that the latter is false, the former is also false.
Having pushed a button an infinite number of times is an inherent contradiction, unrelated to what pushing the button does. Having the button turn a lamp on and off, and the lamp therefore being neither on nor off at the end, is only a way to demonstrate the contradiction; it isn't the reason for the contradiction.
Which is also why Benacerraf's response to the problem misses the mark.
The pseudocode I provided a month ago helps explain this:
isLampOn is only ever set to true or false (and never unset) but the echo isLampOn line can neither output true nor false. This demonstrates the incoherency in claiming that while (true) { ... } can complete.
Changing echo isLampOn to echo true does not retroactively make it possible for while (true) { ... } to complete.
Having pushButton() do nothing does not make it possible for while (true) { ... } to complete.
It is metaphysically impossible for while (true) { ... } to complete, regardless of what happens before, within, or after, i.e. neither of these can complete:
Code 1
Code 2
Here's how I look at it. I think that everyone will agree that a formula is not about anything specific and, in itself is neither true nor false. x + y = z doesn't make any assertions, until you substitute values for the variables. So 2 +1 = 4 is false, but 2 + 3 = 5 is true. So there's a temptation to think it must be true of something. Hence realism. But 2 + 3 = 5 is itself like a formula in that once we specify what is being counted, it does make an assertion about the world - 2 apples + 2 apples = 4 apples. It is true of the world. Of course, 2 drops of water plus 2 drops of water doesn't make 4 drops of water, (until we learn to measure the volume of water). The domain of applicability and truth is limited.
Ok forget that. But 0 and 1 are perfectly legitimate probabilities. After all if I roll a die, the probability is 1 that it will be either 1, 2, 3, 4, 5, or 6, Right? Nothing degenerate or unusual about that. And the probability is 0 that it will show 7.
Quoting Ludwig V
Are you talking about credence, perhaps?
Quoting Ludwig V
No. Only that the event is certain, in the finite case; or "almost certain" to happen, in the infinite case.
Can you give me an example of what you mean? What kind of statements are you applying probability to?
Quoting Ludwig V
Never mind probability. You just startled me by denying the legitimacy or sensibility of 0 and 1 as probabilities.
I don't quite get what you mean here. Let's say there's something about reality which appears to be unintelligible. If we assign a name to that aspect, aren't we saying that there is actually something there which is unintelligible, and we've named it. This is to take a step further than simply that it appears as unintelligible.
Quoting Ludwig V
The point is that potential defies the laws of logic. That's why modal logic gets so complex, it's an attempt to bring that which defies the laws of logic into a logical structure.
The point I made, derived from Aristotle, is that whenever the lamp switches from on to off, or vise versa, there is necessarily a period of time during which it is changing (becoming). In other words, it is impossible that the switch from one to the other is instantaneous, and this is proven logically. In this 'mean time', the lamp is neither on nor off, and this defies the law of excluded middle. Dialethists would hold that it is both on and off, defying the law of noncontradiction.
Aristotle uses the concept of "potential" to explain his choice for defying the law of excluded middle rather than defying the law of noncontradiction. For him, the concept of "potential" is required to explain how something changes form having x property (being on), to not having x property (being off). "Potential" is a requirement of such a change, the thing cannot change without having the potential for change. However, this is a temporal concept, and the conclusion is that actualization requires a duration of time. So there is always a period of time between having x property (being on), and not having x property (being off).
What the lamp problem does not take into account, is that period of time between being on and off, during which it is changing. Assuming that the amount of time required to change from on to off, and vise versa, remains constant, then as the amount of time that the lamp is on and off for, gets smaller and smaller, the proportion of the time which it is neither, gets larger and larger. So at the beginning, when the time on and off are relatively long periods, the time of neither seems completely insignificant. But as the off/on actualization rapidly increases, the time of being on and off soon becomes insignificant in comparison to the time of being neither. The time of neither approaches all the time
Quoting Ludwig V
You only say this, because the time of change in which the lamp is neither on nor off is so short and insignificant that it appears to be nil. Aristotle demonstrated logically that it cannot be nil. So when we say things like "lamps are a type of thing which must be on or off, and cannot be neither", this is a statement about how things appear to be, and this facilitates much of our talk about such things. But when we get down to the way that things actually are, the way that logic tells us they must be, we can see that this way of allowing appearances to guide our speaking is actually misleading.
Quoting Ludwig V
I don't understand this. If a thing neither has nor has not the specified property, the excluded middle principle is violated (unless it's an inapplicable category). Potential itself neither is nor is not, and that's why we say it refers to what may or may not be. So "may or may not be" refers to the property we judge as in potential, and this says it neither is nor is not attributable to the thing.
I don't proffer an opinion on that. But I can see that presumably the most likely candidate is "At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum." At least intuitively it is the ripest and lowest hanging fruit. Or put pejoratively, at least intuitively it is the sore thumb.
But logically we may reject any of them. None of them are logical truths (though, "At all times, the lamp is either Off or On and not both" would be logically true as a conclusion from defining 'On' as 'not Off'.
(1) If a set of premises G entails a contradiction, and for any member P of G we have that G\{P} does not entail a contradiction, then we are logically free to reject any member of G. None of the premises are logically true (except "Either On or Off and not both" as conclusion from a definition "'On' means 'not Off'"), so we can reject any of them. For example, we could reject "The lamp does not change from Off to On, or from On to Off, except by pushing the button."
(2) We still don't have a satisfactory definition here of 'metaphysically impossible'.
(3) For what it's worth, if I'm not mistaken, Thomson does not conclude the time is not infinitely divisible, but rather he weighs in against the notion of super-tasks.
As a coda to Thomson's argument, you rely on the premise "If time is infinitely divisible then the super-task is possible.' [not a quote of yours]
But we may reject that premise.
It seems to me that "the lamp super-task is executed" entails "time is infinitely divisible". But the converse - "time is infinitely divisible" entails "the lamp super-task is executed" - at least requires an argument.
And, it seems to me, that analysis is even more difficult because it involves modalities. First, we have to distinguish between "time is infinitely divisible" and "it is possible that time is infinitely divisible". Second, it's not really, "the lamp super-task is executed" but "it is possible that the lamp super-task is executed". The argument needs to checked whether the modal inferences are correct.
Quoting Michael
No, I'm not claiming that.
Quoting Michael
Of course.
Quoting Michael
Whatever the validity of that, I don't see the point of it here.
Quoting Michael
It's not a contradiction in and of itself. Rather, it is inconsistent with the other premises (especially that the infinite number of executions occurs in finite time).
Quoting Michael
You would need to tell me the difference between a demonstration of a contradiction and the reason for a contradiction. For example, if someone asks "What is the reason that the unrestricted comprehension schema with the separation schema yields a contradiction?" then my best response would be to show a demonstration.
Quoting Michael
As far as I can tell, it's off-base because it doesn't address the premises of the lamp.
And related arguments against Thomson are that it is not problematic that the premises don't provide for concluding whether the lamp is Off or On at 12:00. But that misses the point that it is not that it is problematic that the lamp's state is undetermined, but rather that Thomson's argument shows that the lamp is neither Off nor On. Not that it is undetermined what the state is, but rather that is determined that the state is neither Off nor On.
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I don't know enough about coding to have a comment on your pseudocode.
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A while back you gave the argument in quite succinct form:
"P1. The lamp is turned on and off only by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t0
P5. The button is pushed at successively halved intervals of time between t0 and t1
P6. The lamp is either on or off at t1"
But you add to that argument two things:
(a) We must reject P5.
But each of the premises is required for the contradiction, so we can reject any one of them rather than P5. Granted, P5 does stand out as the candidate we would intuitively reject, but it is not logically required that it is the one we reject. For example, famously:
(U) ExAy yex
(S) ExAy(yex <-> ~yey)
yield a contradiction.
But to dispel the contradiction it is not logically required that we reject (U) to keep (S) when we could reject (S) to keep (U). We may have reasons for preferring that we reject (U) rather than (S), but that is not a demonstration that (U) is logically impossible or even that it is false.
(b) If time is infinitely divisible, then the super-task may be executed.
But that is not logically true either. It may be the case that time is infinitely divisible but still the super-task cannot be executed. Moreover, the modality "may be" slips in there, so the argument requires that it is made clear that the modal inference is permitted.
An infinite sequence of operations is by definition an endless sequence of operations. An endless sequence of operations does not come to an end. That's what makes the premise of a supertask an inherent contradiction.
Having the operation be to push a button, and having this button turn a lamp on and off, is simply a way to make this inherent contradiction even clearer.
If you accept that this proves that this button cannot have been pushed an infinite number of times then what is the reasoning behind the claim that if some wizard steps in at 12:00 to magically turn the lamp on then this retroactively makes it possible to have pushed this button an infinite number of times? Let's even assume for the sake of argument that this wizard will only appear with a probability of 0.5, and that this is determined only at exactly 12:00, i.e after the performance of the supertask. It must already be possible for the supertask to be performed for him to even appear, and so his appearance cannot retroactively make the supertask possible, even if half the time it resolves the secondary contradiction regarding the state of the lamp at 12:00.
I don't think you're really grasping what distinguishes a supertask from an abstract infinite sequence.
I didn't say they end.
Quoting Michael
Again, the contradiction comes from the conjunction of the premises. It is not a given that it is a contradiction in and of itself that infinitely tasks are executed in finite time. It's quite unintuitive that infinitely many tasks an be executed in finite time, but to show that doesn't entail that it is a contradiction in and of itself.
Quoting Michael
I never claimed any such thing. It's a straw man, even if unintentional.
What I said is that if we drop the premise that the lamp is only turned on by the button, then we don't get the contradiction. The point of that is that we are not logically obliged to reject only one certain premise. Note that I am not committing the fallacy of not addressing Thomson's premises. Rather, I am pointing out that rejecting one of the premises to avoid contradiction must then allow rejecting any other premise to avoid contradiction.
When I saw that you had put care into your numbered arguments, I surmised that you were interested in pursuing rigor. So in that regard, I'm examining all your reasoning.
Again (new numbering):
(1) Since none of the premises are logically true, but they yield a contradiction, we may reject any one of them to avoid contradiction. We are not logically bound to reject the one that happens to be least intuitive.
(2) Infinitely divisibility of time does not entail executability of denumerably many tasks in finite time, even though, executability of denumerably many tasks in finite time entails infinite divisibility of time.
(3) We don't have a satisfactory definition of 'metaphysical possibility' here.
(4) The argument is more complicated than appears with only a cursory look, since it involves the modality of 'possible'.
(5) If I'm not mistaken, Thomson does not conclude that time is not infinitely divisible. So, heuristically, we may wonder why that is if your conclusion actually follows as ineluctably as you claim.
(6) I wonder why you don't note my point about continuousness and density, which I mentioned to help sharpen your argument.
A supertask is an infinite sequence of operations that ends in finite time.
Quoting TonesInDeepFreeze
One of the contradictions does; the state of the lamp at 12:00. This isn't the only contradiction. The other contradiction is the inherent contradiction of an endless sequence of operations coming to an end. The former is simply a tool to better demonstrate the latter.
Finding some way to resolve the former does not also resolve the latter.
Just in case you missed my edit to my previous post:
Let's even assume for the sake of argument that this wizard will only appear with a probability of 0.5, and that this is determined only at exactly 12:00, i.e after the performance of the supertask. It must already be possible for the supertask to be performed for him to even appear, and so his appearance cannot retroactively make the supertask possible, even if half the time it resolves the secondary contradiction regarding the state of the lamp at 12:00.
Quoting TonesInDeepFreeze
See here.
All I mean by it is that supertasks are more than just physically impossible. No alternate physics can allow for them.
As a different example, consider the grandfather paradox. I don't just take this as a proof that one cannot travel back in time and kill one's grandfather before one's father is born; I take this as a proof that one cannot travel back in time.
The premise of having one kill one's grandfather before one's father is born is just a tool to prove the impossibility, not the reason for the impossibility.
You answered pretty fast. That's your prerogative. But it make me wonder whether you're giving much thought to my remarks, as still it would be your prerogative not to. So I'll take the same prerogative.
The definition of a super-task is as you say. But your listed premises don't say anything about completion or ending.
Quoting Michael
The contradiction is: The lamp is either On or Off T 12:00 and the lamp is neither On nor Off at 12:00.
But that contradiction comes from a set of premises, each of which is not logically true, and dropping any one of the premises blocks deriving the contradiction. It would help if you would at least tell me that you understand that.
Quoting Michael
(1) The premises don't say it comes to an end. It would help if you would at least tell me that you understand that.
(2) It is begging the question merely to declare it is a contradiction that denumerably many tasks can be executed in finite time. Indeed, the argument itself doesn't declare that it is a contradiction. Rather, the argument derives a contradiction from that premise along with other premises.
Again, the example I gave:
AxEy yex
ExAy(yex <-> ~yey)
entails a contradiction, but it doesn't entail that either of the above is itself a contradiction.
Even most minimally:
P
~P
entails a contradiction, but that doesn't entail that either P or ~P is a contradiction.
It would help if you would at least tell me that you understand this.
Quoting Michael
There's no just in case that you missed my own post. You missed that I said I don't argue in any such way that is knocked down as a straw man as you have.
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The link doesn't go to a defininition. It merely says that metaphysical possibility may be logically possibility and that there's another notion that the article describes ostensively. So is it just the same as logical possibility, and if not what is a proper definition that is not merely ostensive?
Finding some way to resolve the former does not retroactively resolve the latter.
Quoting Michael
So, yes, clearly taking your prerogative to answer so quickly did result in your not even taking note of what I said, let alone taking a moment to understand it.
I am not "finding some way to resolve the former [to] retroactively resolve the latter." You can read my post again to see my explanation.
The infinite button pushes ends after two hours. That's the premise of Thomson's lamp (albeit minutes in his specific case). In his own words, "after I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off?".
If, as per the premise, I only push the button at 11:00, 11:30, 11:45, and so on ad infinitum, then I am no longer pushing the button at any time after 12:00. My infinite button pushes has allegedly ended.
The very thing we're discussing is the possibility of supertasks, i.e. can an infinite sequence of operations end in finite time?
Quoting TonesInDeepFreeze
That's one of the contradictions. If one drops or adds or changes any premises, e.g. by stipulating that the lamp spontaneously and without cause turns into a pumpkin at 12:00, then you have resolved the contradiction regarding the state of the lamp at 12:00, but doing so does not then allow for the possibility of supertasks; it does not resolve the contradiction in claiming that an infinite sequence of button pushes has come to an end.
Quoting TonesInDeepFreeze
It's simply true by definition. An endless sequence of operations cannot end. An infinite sequence of operations is an endless sequence of operations. An infinite sequence of operations cannot end.
Quoting TonesInDeepFreeze
I'm not the authority on the matter. I am simply arguing that supertasks are more than just nomologically impossible. I use the phrase "metaphysical impossibility" rather than "logical impossibility" simply because it's the weaker claim. Call it hedging my bets if you will.
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t0
P5. The button is pushed at successively halved intervals of time between t0 and t1
P6. The lamp is either on or off at t1
But actually, if we look at the mechanics of the inferences toward the contradiction, we see that we need that the lamp is never both on and off. So I would write.
P1. The lamp is turned on and off only by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t0
P5. The button is pushed at successively halved intervals of time between t0 and t1
P6. At all times the lamp is either on or off and not both.
Then we derive a contradiction:
C1. The lamp is either on or off, and the lamp is neither on nor off.
Now, what premise do we delete to avoid the contradiction? Since none of them are logically true, and each is needed in the derivation of the contradiction, we may delete any one of them.
P5 is an inherent contradiction, just as travelling back in time is an inherent contradiction.
The lamp being neither on nor off at t[sub]1[/sub] and killing one's own grandfather before one's father is born are secondary contradictions to prove the inherent contradictions.
The possibility of P5 does not depend on whether or not P1-P4 are true, e.g. having the button be broken does not make it possible to push the button an infinite number of times within two hours.
You're stuck thinking I'm making a certain kind of argument, but I am not. You're not thinking about what I've specifically written, as probably you take me to be making a version of other arguments around. If you're not going to take my arguments as given, then there's no rational inquiry to be had.
But I'll address these two:
Quoting Michael
But completion is not in your premises.
Quoting Michael
That is begging the question. It is begging the question to rule that there can't be denumerably many tasks executed in finite time. You haven't proved: If there are denumerably many tasks executed in finite time then there is an end to their executions. So you have to either prove it or add it as a premise.
Quoting Michael
It's not much of a claim if it is not defined.
This is my argument.
Notice the antecedent of C6: "If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum...".
If I am only ever pushing the button at these times then I am not pushing the button at 12:00 or at any time after 12:00. Therefore, my (infinite/endless) button-pushing has ended by 12:00.
That's what supertasks are. They are an inherent contradiction, irrespective of what pushing the button actually does. It is as metaphysically impossible to have performed a supertask on a broken button as it is metaphysically impossible to have performed a supertask on a working button. Having the button turn the lamp on and off, like killing my grandfather before my father is born, is just a means to better demonstrate this impossibility and not the reason for it.
You haven't paid attention to my answer to that. Now, you're arguing by mere assertion and repeated mere assertion.
Moreover, if P5 is deemed in and of itself contradictory, then we don't need an argument with other premises to derive a contradiction. If P5 is, from the outset declared a contradiction, then we don't have to bother with the bloody lamp at all.
You're not thinking about my points as it seems you just lump them in with other people's arguments. You are skipping key points. You argue by question begging and mere assertion. And now you link me to a post from eleven days ago that I had already addressed in detail.
As to C6, you put the halving ad infinitum as an antecedent in a conclusion, which is fine. But it's equivalent to just making it a premise. And I mentioned that a while ago, and mentioned why it is more stark to make it a premise, but you ignored that too. Your terse argument that I've recently mentioned and your argument eleven days ago are essentially the same: It doesn't matter whether you put the halving ad infinitum as an antecedent in a conclusion or as a premise - it's logically the same.
Anyway, we're going in circles as you skip my arguments while reasserting your own. When I saw that initially you were taking care to make numbered arguments, I got interested and thought you might be open to more scrutiny. But I can see you're not, as you only keep repeating assertions and not actually thinking about the replies to them. It seems your primary interest is to persist that you are rigtht and not to truly think through objections. So, bye for now.
Yes, it makes no difference if it's an antecedent in a conclusion or as a premise. Either way, the supertask is the completion/end of an infinite/endless sequence within finite time (e.g. I have stopped pushing the button by 12:00) and is an inherent contradiction, irrespective of what the task is. Having the task be to push a button that turns a lamp on and off is just a means to demonstrate the impossibility of a supertask and not the reason for its impossibility, and neither having the button be broken nor having the lamp spontantously turn into a pumpkin allows for the supertask to be possible.
No, it doesn't. It's called 'the deduction theorem'. For example:
P, Q, R |- S
is equivalent with
P, Q |- R -> S
Huh? I'm reiterating/agreeing with your claim that "it doesn't matter whether you put the halving ad infinitum as an antecedent in a conclusion or as a premise - it's logically the same"?
I was rushing. My mistake.