Fall of Man Paradox
The Story
God warned Adam, "You may look upon any die with a finite number of sides, but you must never gaze upon the die with infinite sides, for doing so will lead to your certain death."
Yet the serpent, cunning in its ways, tempted Adam, suggesting, "Let's gamble with the die of infinite sides. Should you outroll me, divinity will be yours. But should your number be lower, death will be your fate. Rest assured, the dice are fair."
Faced with a seemingly even chance of victory, Adam agreed to the serpent's game, all the while keeping his eyes shut as he rolled his die to adhere to God's command. "Look at your die" urged the serpent, "and if your number displeases you, take mine, and I shall roll once more."
Considering the logic of smaller dice games, where checking one's roll can help one decide whether it's advantageous to swap, Adam reasoned it would be wise to apply the same logic to the infinite-sided dice game.
With trepidation, Adam opened his eyes to see a googolplex dots on his die. It dawned on him then: with his roll being finite, only a finite number of outcomes were lower, yet an infinite number were higher, making the serpent's probability of beating him exactly 100%.
Choosing life over death, Adam opted for the serpent's number, revealed to be 42. The serpent rolled again, extending the same offer. This cycle continued for hours until an epiphany struck Adam: by never concluding their game, he could elude death. Adam fled the Garden of Eden and took refuge in the comprehensibility of the finite world until, ultimately, his newfound finiteness overtook him.
My questions
Is this truly a paradox? If not, why not? If the probability of winning falls from 50% to 0% as described, at what point does this shift occur? Considering that observing his die leads to the same conclusion in every case, he could have reached that conclusion without ever looking at his die. This scenario seems to indicate a problem with the concept of an infinite-sided die, possibly even suggesting that such a die cannot exist. What are your thoughts?
God warned Adam, "You may look upon any die with a finite number of sides, but you must never gaze upon the die with infinite sides, for doing so will lead to your certain death."
Yet the serpent, cunning in its ways, tempted Adam, suggesting, "Let's gamble with the die of infinite sides. Should you outroll me, divinity will be yours. But should your number be lower, death will be your fate. Rest assured, the dice are fair."
Faced with a seemingly even chance of victory, Adam agreed to the serpent's game, all the while keeping his eyes shut as he rolled his die to adhere to God's command. "Look at your die" urged the serpent, "and if your number displeases you, take mine, and I shall roll once more."
Considering the logic of smaller dice games, where checking one's roll can help one decide whether it's advantageous to swap, Adam reasoned it would be wise to apply the same logic to the infinite-sided dice game.
With trepidation, Adam opened his eyes to see a googolplex dots on his die. It dawned on him then: with his roll being finite, only a finite number of outcomes were lower, yet an infinite number were higher, making the serpent's probability of beating him exactly 100%.
Choosing life over death, Adam opted for the serpent's number, revealed to be 42. The serpent rolled again, extending the same offer. This cycle continued for hours until an epiphany struck Adam: by never concluding their game, he could elude death. Adam fled the Garden of Eden and took refuge in the comprehensibility of the finite world until, ultimately, his newfound finiteness overtook him.
My questions
Is this truly a paradox? If not, why not? If the probability of winning falls from 50% to 0% as described, at what point does this shift occur? Considering that observing his die leads to the same conclusion in every case, he could have reached that conclusion without ever looking at his die. This scenario seems to indicate a problem with the concept of an infinite-sided die, possibly even suggesting that such a die cannot exist. What are your thoughts?
Comments (377)
"Infinite" is not a number as an amount of sides but is that which cannot be gotten to since it never completes, but to play along, the die is ever becoming more of a higher and higher resolution smoother sphere.
If the numbers cannot manifest, if they aren't allowed, then the dice cannot be infinite.
I guess you could use compact symbols though but you'd come up against some limitation of discernment, where what conveys one number is indistinguishable from what conveys another. The capacity for subtle discernment would have to be infinite also.
The serpent would end up disagreeing with you what number is on the face of the dice and you'd have to bring in a third party (God, if you trust him) to verify.
No, the die in the story remains unchanging and complete with a side for every natural number. This distinction matters because the potentially infinite die you describe doesn't concern me.
I concur that this narrative couldn't unfold in our physical reality, but your argument doesnt address the core of the paradox. The inclusion of God and the Garden of Eden in the story was specifically to lift us beyond our finite limitations.
Just as a polygon with infinite sides of equal length would be empirically indiscernible form a perfect circlethis irrespective of how close one observes it, say with a quantum microscopeso too would a die with infinite sides of equal length be empirically indiscernible from a perfect sphere, even when analyzed with a quantum microscope.
Hence, having the numeral 42, or any other, viewable on any one of the die's infinite sides would be a logical impossibility considering the nature of empirical realityan empirical reality which both the serpent and Adam utilize to look at the die.
So no, no paradox.
We are just concerned with the finite numbers that appear on both dice and which is greater than the other. Sounds like 50/50 chance.
Once the player knows what the serpent threw, there will be an infinite amount of numbers greater than that number and a finite amount less than, and therefore chance of winning against the serpent approaches 0. The limitation of possible representation is still an issue here; there must be a finite maximum magnitude for the game. Infinity cannot be a stipulation.
You could fix the problem with doing away the limit of natural numbers and including the negative integers to infinity. Chance of 50/50 seems locked in here, even after the player becomes aware of the serpent's number, unless the chance is actually undefined here. Maybe it's undefined for the natural numbers also.
If the serpent pulled 42 on a number line of integers what is the chance of throwing a higher or lower number? Does it even make sense to ask?
Your argument merely skims the surface, focusing only on the impossibility of the story occurring in our physical world. Let's remember, this is a die created by God, and the story takes place in the Garden of Eden, where ordinary physical laws don't hold. Nonetheless, I'll engage with your point since you're considering mine.
The die doesn't have to be polyhedral; it simply needs to be fair. Imagine a line segmented as follows: the first half represents roll 1, the first half of the remaining line represents roll 2, the first half of what's left represents roll 3, and so forth.
If we roll this line into a faceted arc, we achieve an object with infinite sides. All that's left is a little divine magic to ensure it rolls fairly, and extraordinary vision for the players to discern the minuscule markings on those higher rolls.
Do you believe that the set of all natural numbers exists? Is it reasonable to stipulate its existence? If infinite sets can exist in some realm, why can infinite dice not exist in God's realm?
Quoting Nils Loc
Addressing a problem by focusing on a completely different issue is like sweeping the original problem under the rug.
Quoting Nils Loc
But at what point does the math fail? The calculations don't seem to get much simpler.
In that case, Ill let this thread be. But I confess, to me, this conflux of mathematical thought with magic and extraordinary vision not bound by empirical sight so far equates to the question of: What defined result obtains when one divides three and a quarter invisible unicorns by zero? Just saying.
Why not? In a recent post to Javra, I described an object of finite size with infinite sides that could function as a die. Adam could certainly hold it in his hand. We would just need to trust God to ensure its fairness. Just to clarify, I'm not aiming for a religious debate here. I mention God in the story simply because it seems to me that objects with infinite properties could only exist within a supernatural realm.
No problem, and thank you for the discussion. I will say that, in my view, the conflux of mathematics and magical thinking was formalized by Georg Cantor and has been nearly universally adopted in modern mathematics. If you believe that infinite sets cannot exist, then I am preaching to the choir.
Quoting keystone
Just to be clear: when it comes to the realm of pure mathematical thought, I can see it both ways and so remain on the fence; but when it comes to empirical reality as in the form of a die, a member of the just stated choir I am. :grin:
Interesting way of expressing the issue, btw.
Yes, this post. To me, the story presents a paradox because the probability of Adam winning appears to drop from 50% to 0% simply by observing his die. Yet, observing his die provides no new information that he couldn't have figured out beforehand. So, does looking actually make a difference, and if so, why? Or was he doomed to lose from the beginning?
Although I presented the 'paradox' in the form of a story to make it engaging, it could just as easily have been described purely as a mathematical problem within the realm of abstract mathematical thought. Indeed, when you questioned the feasibility of constructing such a die, it seemed you were addressing the narrative element of the paradox, leaving the core mathematical issue untouched.
Infinity is an unreasonable stipulation and whether or not numbers "exist" is not all that relevant to me. The natural numbers definitely have a limit of physical representation and there'd be an infinite amount of numbers beyond that limit.
Your message begins stating that "infinity is an unreasonable stipulation" and concludes by stating that there are an "infinite amount of numbers". It seems that you're open to infinities in some contexts. Let's place this paradox in that context where you welcome infinities.
Well, the issue to me was and remains the conflux of pure mathematical thought and empirical reality. There often (although not always) is no tidy cohesion between the twoas exemplified by an infinite-sided die whose individual sides need to be looked at to be discerned. One can simplify the very same issue by addressing the purely mathematical concept of a perfect circle and its nonexistence in empirical reality.
As an aside, I am aware that without the concept of a perfect circle no such thing as quantum mechanics could obtain, for the latters uncertainty principle is necessarily dependent on pi, which can only obtain in a perfect circle (that, again, can only be non-empirical). But the basic discrepancy between pure mathematical thought of a perfect circle and its nonexistence in physical, empirical reality remains. This and related subjects, btw, being why Im on the fence regarding purely mathematical objects of thought being of themselves existents: e.g., if a perfect circle does not exist, then this directly entails that so too is quantum mechanics a fiction and, yet, QM works rather well to explain what physically, and hence empirically, is. I acknowledge this is a deviation from Cantor, but to me it speaks to the same issue: the relation, if any, between pure mathematics and the empirically perceivable physical world.
In our physical universe, a perfect circle cannot exist, nor can pi be fully represented as a decimal number with infinite digits. However, the universe does permit the complete equation of a circle and the full definition of pi as an infinite series, both expressed using a finite number of characters. The real question isn't whether quantum mechanics is fictional (an idea that seems absurd), but rather if quantum mechanics employs the infinite-rooted objects themselves or merely the finite descriptions of these objects. Or perhaps more crucially, the question is whether we need to assert the existence of the infinite-rooted objects at all, or if we can manage perfectly well with just the finite descriptions.
Interesting perspective! My own slant is that in physical reality there can be no absolutely integral physical being as is symbolized by the number one/1 when the issue is addressed more objectively: all individual objects are integral physical things only in relativistic terms, always permeated (by quanta, for one example) and hence entwined with what (we perceive and conceive to be) other and always in flux. Hence, our very understanding of mathematics in full is grounded in conceptual idealizations, rather than physical realities - starting with the idealizations of 0 and 1.
As regards infinities, if you happen to not be familiar with them, you may get a kick out of surreal numbers. They incorporate both infinite and infinitesimal numbers as well as real numbers. (While I'm no mathematician that can properly describe or implement them in mathematical formulations, surreal numbers have always held a rather aesthetic appeal to me.)
Doesn't the effectiveness of classical computers contradict your perspective?
Quoting javra
I have a tough time accepting the use of limits to define reals as numbers. The idea of applying limits upon limits is even more difficult for me to accept. But like you, I'm not a mathematician.
You could only make a good argument that it drops to 0% if every number on the die had an equal probability - but the die you laid out does not have that feature.
In the game with an infinite-sided die, suppose Adam rolls a 1000. What are his chances of winning? Consider the infinite set of larger numbers the serpent could roll. The probability of Adam winning is exactly 0%. However, a 0% probability of winning does not imply that victory is impossible for him. Instead, it means that he will almost surely lose. This distinction from measure theory opens up a whole new can of worms.
Quoting flannel jesus
When I mentioned that the dice with infinite sides are fair, I was specifically referring to each side having an equal chance of being rolled. After all, God is fair. :P
To be laconic about things: no. But then many an ontological perspective would need to be addressed to back this up - perspectives that would amount either to an objective idealism or, else interpreted, to a neutral monism.
As it is, as previously attempted, I'll try to bow out of this thread's discussion. But it was good discussing with you.
How? You said 1 takes up a full half of the die - 1 is less than 1000, and the other guy has a 50% of rolling a 1. He has a 25% chance of rolling 2, so that's already up to a 75% chance to win now. Where are you getting 0 from?
Then why did you point me to your post where 1 takes up half the space on the die :P
In my initial post I wrote "Rest assured, the dice are fair."
In the very post you mention I wrote "All that's left is a little divine magic to ensure it rolls fairly, and extraordinary vision for the players to discern the minuscule markings on those higher rolls."
Let's set aside the question of how God could make fair infinite-sided dice and just assume that he can.
Now, putting trivial matters aside, do you understand that Adam's probability of winning becomes exactly 0% once he sees his roll? If you disagree, what do you calculate his probability to be?
This interesting puzzler has a clear and unambiguous mathematical resolution.
First, the standard axioms of probability were laid down in 1933 by Russian mathematician Andrey Kolmorogov.
Kolmogorov is also the originator of the definition of randomness as incompressibility, currently being discussed in https://thephilosophyforum.com/discussion/15105/information-and-randomness
Kolmogorov's axioms of probability state (paraphrasing the Wiki exposition):
1) The probability of an event is a nonnegative real number.
2) The sum of the probabilities of all possible events is 1. This make sense, right? On a standard 6-sided die, each face has probability 1/6, and the probabilities of all the faces add up to 1.
3) If you have countably many mutually independent events, the probability that one of them occurs is the sum of the probabilities of the events.
This rule is designed to accommodate the following situation. Suppose we randomly pick a real number from the unit interval [0,1]. Endpoints don't matter for this discussion. We want to be able to say that the probability of choosing a number in [0, 1/2] is 1/2; and the probability of choosing a number in [1/2, 3/4] is 1/4; and the probability of [3/4, 7/8] is 1/8, and so forth. In other words, the probability of choosing a number in an interval is the length of the interval.
On the one hand, the chance of choosing in [0,1] must be 1. And we know from the theory of infinite series that 1/2 + 1/4 + 1/8 + ... = 1. Countable additivity lets us compute the sum of the whole by adding up the countably infinite collection of individual probabilities.
A probability distribution is uniform if each event has an equal probability. For example a standard dice roll is uniform, because each face has probability 1.
A familiar example of a non-uniform probability distribution is he bell curve, or Gaussian distribution. The total area under the curve is 1. But events near the middle have a much higher probability than events near the left and right tails.
Now we are in a position to prove the crux of the matter.
There is no uniform probability distribution on a countably infinite set of events.
Why is this? Say we have a countably infinite set of events {E1,E2,E3, }.
Suppose we have a uniform probability distribution on this set of events. What must be the probability of each event? We have two cases:
- The probability of each event is 0. But then by countable additivity, the probability that any event at all happens is 0. Contradicts the axiom that says the total probability of the entire set of events must be 1.
- The probability of each event is some tiny positive number. But then the total probability is infinity, no matter how small each probability is. Another contradiction.
Conclusion: It is not possible to place a uniform probability distribution on a countably infinite set. In particular, there is no uniform probability distribution on ?
.
With this knowledge in hand, we can analyze this intuitively appealing but sadly meaningless statement:
Quoting keystone
We can see what's wrong with this. First, you haven't got any kind of probability distribution at all. You're thinking that there are infinitely many events, with each event getting one vote. But that adds up to infinity. It's not a probability space. Remember the probabilities of each face on a six-sided die add up to 1. The same principle must apply to an infinite-sided die. So the logic of this paragraph makes no sense.
Secondly, you are trying to compare one set of nonexistent probabilities to another. Since there is no uniform probability distribution on ?, you are trying to reason about a nonexistent mathematical entity.
So that's the answer to the puzzler. There is no uniform probability distribution on the natural numbers that allows the quoted paragraph to have any meaning at all.
Even God can not place a uniform probability distribution on the natural numbers.
Now you COULD conceptually throw all the natural numbers in a bag and reach in and select one. But you could NOT then try to use mathematical reasoning on that situation. That's the flaw in the paradox. You can't reason mathematically about a uniform probability distribution on the natural numbers, because there isn't any such thing. You can't add the individual probabilities, because adding probabilities only applies to well-defined probability spaces, and you haven't got one. Even God does not have a uniform probability distribution on the natural numbers.
For additional insight, I'll give two variations on the game.
* An example of a non-uniform probability distribution on ?
For simplicity, let the natural numbers be 1, 2, 3, ... In other words I'm not considering 0 just to make the argument a little simpler; else I'd be off-by-one throughout.
Let the probability of the number n be [math]\frac{1}{2^n}[/math].
.
For example the probability of 1 is 1/2. The probability of 2 is 1/4. The probability of 3 is 1/8, and so forth.
The probabilities add up to 1, and countable additivity is true, so we have a probability distribution. It just doesn't happen to be uniform, since each event has a different probability.
Now, let's play the game. The serpent randomly obtains a number, let's say it's 5.
What should you do? The probability that you get 1, 2, 3, or 4 is 1/2 + 1/4 + 1/8 + 1/16 = 15/16.
The probability that you also get 5, tieing the game, is 1/32.
And the probability that you get a larger number is the tail of the distribution, which adds up to 1/32.
So you should switch.
It's clear that the second player is at a big disadvantage half the time. If the snake plays first, Adam should always switch.
The only time this fails is if the snake picks 1. That happens half the time, since 1/2 is the probability of randomly picking 1.
Then when you pick, half of the time you'll tie, by picking 1. The other half you win, by picking a larger number.
But if the snake picks any other number besides 1, which happens half the time, you should switch.
* Example two: A die with the cardinality of the reals that has every real number between 0 and 1 on some face, and no two faces have the same number.
In this case, say you randomly pick a number between 0 and 1/2. Chances are, the snake has a large one. You should switch.
If your number is greater than 1/2, you should keep your number.
If you get exactly 1/2, it doesn't matter what you do.
In conclusion:
* There are variations on the game that make sense and that have a strategy that confers an advantage on Adam.
* In the game as stated in the OP, it's meaningless to try to reason probabilistically. And you certainly can't try to count "finite versus infinite," since the sum of the probabilities of the events is infinite to start with. Probability spaces must have probability 1. So says Kolmogorov.
Let me know if this was sufficiently clear, and please let me know if there are questions.
Quoting keystone
I hope you can see now that there is no sensible way at all to apply any numerical probability to the events in this game. Any number you pick violates the axioms of probability.
The potential infinity realm can still use reasoning.
But that requires a currently bounded scenario.
So, you loaded the question and that is not nice.
In any case the example is horrific as well with each side being half the pervious. It is not neat. It's not even really that interesting. The sides should be of the same length. And since infinity extends in both directions, or all directions, and not just one direction your arbitrary single bound of natural numbers is yet another nonsensical limit that does not help in any way. The absolute value and zero included as a set is more tenable as an infinity and it ruins your nonsense. Lead be thou gold!
I know you are but what am I, ... infinity!
Quoting fishfry
Are you suggesting that the gambling event can occur but that we can't discuss it in mathematical or probabilistic terms? That's hard to accept. Even if we set aside mathematical reasoning, can you truly say that you have no opinion on whether Adam should exchange numbers with the serpent?
Quoting fishfry
I agree with your point and also agree with Kolmogorov's axioms. However, I think the flaw in your argument lies in presenting a false dichotomy by suggesting that there are only two possible scenarios in the game.
1) The probability of each event is 0.
2) The probability of each event is some tiny positive number.
Quoting fishfry
Let me propose a third, perhaps controversial, scenario:
3) The probability of each event is 1.
Of course, this is only feasible if there exists just one natural number, meaning that when you deal with the set of all natural numbers, you are essentially dealing with a singularity where every natural number is identical. While this notion may seem preposterous, similar issues emerged with calculus, which were resolved using limits. For instance, finding the tangent by dividing by zero results in a singularity, yet one can sensibly approach a zero denominator. In a similar vein, I argue that dealing with the set of all natural numbers also results in a singularity, but probabilities can be sensibly managed by approaching an infinite set. In other words, infinite sets as completed objects do not truly exist.
Although my proposed resolution has significant implications, I believe none of these are insurmountable.
What do you think?
I don't think you understand my position. I'm playing in the "paradise" which Cantor created (involving infinite sets) not because I believe in it but because I want to convince others that it's a mirage (at least in my view).
Quoting Chet Hawkins
Isn't the concept of an infinite-sided die that could fit in your hand intriguing? Its impossible to construct a die with finite volume if you insist that all sides must be of equal length.
Quoting Chet Hawkins
Do you not believe in natural numbers being bounded by 0 (or 1) on one end? And regarding time, isn't it widely believed that time had a beginning (meaning one boundary of time is t=0)?
You need a non-standard probability theory to express the idea of an infinitely sided fair die, but the idea only makes sense for Dedekind-finite dice.
Rather than assigning a real number to a set of outcomes to denote the probability of the set, which doesn't work in the case of infinitely sided dice, we can in the case of a fair and infinite die eliminate the distinction between sets of outcomes and their probabilities, because in this case the probability of a set of outcomes is equivalent to the set itself.
So let P(N) = N
- We simply drop the normalisation factor since it is a constant, and let N directly denote both the set of natural numbers and the probability of choosing a natural number (i.e the value of probability one) .
Let N/a denote the set of natural numbers that is the complement of the subset of natural numbers a. Then
P(N/a) = N/a
P(a) = a
P(a OR N/a) = N/a + a = N
- If N/a is cofinite, meaning that it's complement is finite, then its interpretation as a probability value is larger than the probability value for any finite set of naturals, but is nevertheless smaller than N.
- if a is finite but non-empty, then it's interpretation as a probability value is smaller than the size of any cofinite set and any infinite set, but is nevertheless larger than zero.
- If both N and a are infinite, then their corresponding probability values are of intermediate magnitudes between the two previous cases.
I think this is pretty much all that is needed for a basic non-standard probability treatment of a fair infnitely sided dice, and the resulting measure is both countably additive and normalizes to 'one'.
Lastly, the assumption of Dedekind finiteness is important, because we don't want to derive
P(N) < P(N).
So the intuition stated in the OP, that switching is always the best decision, is represented in terms of the magnitudes of cofinite sets of natural numbers as always being larger than the magnitudes for finite sets, but without succumbing to the false conclusion that the probability of getting a better result is 1, which is a 'bug' of classical probability theory caused by it's insistence upon using standard models of arithmetic.
I'm quite fond of this potential infinity solution and believe it may be the correct direction to pursue.
However, the die in the paradox possesses an actually infinite number of sides (the set of sides is Dedekind-infinite). What more needs to be said to argue that such a die cannot exist?
That's why I gave the example of throwing all the numbers into a bag and picking one. You remember those big rolls of tickets that movie theaters used to use, do they still have cardboard movie tickets? I haven't been to a movie theater in a while.
Of course God, being God, has an infinite roll of tickets. He has an angel separate them all at the perforations and throw them into a big fish bowl. Then Adam reaches in and pulls out a ticket.
I can't think of any reason this is conceptually impossible. It's perfectly clear.
So what is it that we can't do? We can't do probabilistic reasoning about the ticket draw until we make all of our assumptions extremely clear. What's a probability, are you allowed to add and compare them, and so forth. Mathematically you can't do that without violating Kolmogorov's axioms.
There are other variations. You can drop countable additivity and replace it with finite additivity, which is strictly weaker but obviously not so problematic.
The issue isn't whether some alternate concepts and rules of probability might save the situation.
The issue is whether you can be super-duper crystal clear about the concepts and rules you are using. That's a standard you haven't met. The googolplex example is very murky, you are counting one point for each number and then comparing a finite total to an infinite one. But then your probability space has a total measure of infinity. That's not in and of itself illegal. It's just not Kolmogorov. So you have to tell me exactly what are the rules of your system of probability spaces of infinite measure.
Quoting keystone
I go back and forth on whether that's a meaningful question. I think I just convinced myself you're right. Once the serpent chooses, it's a heck of a lot more likely the next number will be in the unbounded segment, even if we can't formalize what we mean. You just convinced me. My intuition agrees with yours. You're right, it's a paradox.
Quoting keystone
I hope it's clear that this is Kolmogorov's idea and not mind. But surely you're not suggesting that there could be negative probabilities. That seems like it would take us far afield. Also, probabilities don't have to be tiny. Kolmogorov's first axiom on Wiki is that probabilities are nonnegative real numbers. It's then a consequence of the total measure being 1, and the finite additivity (implied by countable additivity) that together show that each probability must be between 0 and 1 inclusive.
Quoting keystone
Your next paragraph is a bit out there. But by the end I sort of think I might know what you are getting at.
Quoting keystone
Ok this is concerning. First, I don't know what a singularity is in math or set theory. It's a term from cosmology. There are singularities in math, but they are not to be found among the natural numbers. So you are tossing in a buzzword that should cause you to try to be much more precise about the idea you're getting at.
Quoting keystone
That's the problem. It's not even preposterous. It's not a well-formed idea. All the natural numbers are the same number and they form a singularity? I'm sorry, there is not an idea there at all.
And besides, by the axiom of extensionality, a set can't have two elements that are the same. So you are conceptualizing a set {x} where x is some particular natural number or all the natural numbers, and you are calling it a singularity.
I hope I am being clear that the informational content of this idea is nil. I have no idea what you are trying to say.
Quoting keystone
I think you are making an analogy where there isn't one.
Quoting keystone
Yes, the limit of the difference quotient was 0/0 and Newton got the right answers but could not find the right formalism. Can't blame him, it took almost another 200 years. I take your point but you are a long way from your {x} "singularity," can you see that?
Quoting keystone
I'm not feelin' it. There are singularities in analysis, such as the blowup of 1/x at x = 0. In complex analysis they classify singularities in terms of how bad they are.
But singularities in the natural numbers? I must insist on your clarifying this point. It means nothing as far as I can understand.
Quoting keystone
I don't know what that means.
Quoting keystone
Uh oh. You contradicted the game. You can't make a random choice from a bag that never contains all the numbers. God's fishbowl contains ALL the tickets. A completed set of tickets, a completed set of natural numbers.
If you reject completed infinite sets, you can't play the game in the first place. Right?
Quoting keystone
I don't believe you expressed a coherent idea. I don't mean to say that as a reflection on you. I only intend to convey my own state of mind. I do not understand your idea. All the natural numbers are the same and you call them a singularity and suddenly you don't believe in infinite sets. You have lost me.
Quoting keystone
It occurs to me that perhaps you're getting at infinitesimal probability theory. After all what we'd really like to do is assign the probability [math]\frac{1}{\infty}[/math] to each natural number in such a way that the infinitesimals add up to 1, if we could only figure out how to make that rigorous.
There are researchers working in that area, but it's a bit arcane and I don't know much about it. I found a link.
https://www.journals.uchicago.edu/doi/full/10.1093/bjps/axw013
The first problem is one logical inconsistency. In Kolmogorov's treatment, the axioms exclude the proposition; if one introduced such a die as a new axiom, the system wouldn't be consistent. Whereas in my above (very rough) proposal, A Dedekind infinite set is measured directly in terms of its definition rather than in terms of it's cardinality,but which in turn implies that it has lower probability than its subsets, violating additivity, (Here I am assuming that we want to use standard rules for mapping distributions from one set to another. I'm not actually sure if there might be some other workaround than banning Dedekind-infiniteness).
The second problem is one of motive. Is the motive good enough? Consider what it means to say that the Natural Numbers are Dedekind infinite. In type theory, it refers to an object N with an arrow
1 + N --> N that has an inverse ( here 1 denotes zero, and + indicates disjoint union, and the arrow is the successor function). A standard computational reading of this arrow is that it conveys the fact that one can count upwards from zero to an arbitrary finite number of one's choosing and then count downwards to return to zero. In a nonstandard reading one is also allowed to count from an arbitrary position that cannot be reached from zero. But in either case, the arrow doesn't have the extensional significance that set theorists like to assume. That is to say, the arrow doesn't imply that "every member of the natural numbers exists prior to it being counted" , rather the arrow is used to construct as many members as one desires. In summary, we can say that Dedekind-infiniteness is a type of rule that can be used to generate Dedekind-finite extensions of any size that can be freely extended as and when one desires, by applying the rule once more.
In the case of an infinitely sided die, if the die can only be rolled a finite number of times, then its trajectory of outcomes is equivalent to the trajectory of some Dedekind-finite die that by definition is guaranteed to possess an arbitrary but finite number of unrolled sides after the final roll of the die. Is rolling the die a Dedekind-infinite number of times extensionally meaningful? Not according to the functional interpretation of Dedekind-infiniteness, which deems the previous analysis sufficient for the philosophical analysis of the fall of man paradox.
https://www.quora.com/Whats-the-craziest-mathematical-fact-you-know/answer/Brian-Overland-1?ch=15&oid=164968393&share=3b2e848f&srid=o1wj&target_type=answer
The whole answer is related to this thread, but the second to last paragraph is of particular interest to me in regards to this theoretical dice:
This leads me to think that maybe, after all, it's not safe to assume god could prepare such a dice.
And here's a stack exchange conversation on why.
https://math.stackexchange.com/questions/14777/why-isnt-there-a-uniform-probability-distribution-over-the-positive-real-number
I wouldn't venture to disagree with Kolmogorov on this matter. Although it's not an easy undertaking, particularly for me, I find it relatively easier to contest the notion of infinite sets than to formulate rules for a system of probability spaces with actually infinite measure.
Quoting fishfry
I'm not saying I've convinced you, but no oneespecially a mathematicianhas ever responded like this to my mathematical/philosophical thoughts. It makes me feel a bit less out of my mind. Thank you.
Quoting fishfry
You're likely familiar with the Principle of Explosion, where a single contradiction can undermine an entire logical system. I have a different take on what 'explosion' actually means, perhaps because I hold consistency paramount. Let's consider my system of arithmetic, which starts with universally accepted statements such as:
Statement 1: 1+0=1
Statement 2: 1+1=2
Statement 3: 1+2=3
Statement 4: 1+3=4
These statements are not in question. Now introduce the following into the system:
Statement 5: 1+2=2
To maintain logical consistency in this updated system, our only choice is to accept that 0=1=2=3=4=... Realizing this, the system remains consistent but becomes trivial and loses all distinction. This situation resembles a singularity, where distinctions that exist in more sensible systems dissolve.
Moreover, dividing by zero (a classic error leading to mathematical singularities) can yield absurdities like Statement 5. You likely have come across arithmetic tricks using division by zero to demonstrate fallacies like 1=2. (EXAMPLE)
Returning to the infinite-sided dice game, consider the successor function S(). Some statements would be:
Statement 1: S(0)=1
Statement 2: S(1)=2
Statement 3: S(2)=3
Statement 4: S(3)=4
I'm not suggesting these statements are incorrect or trivial. However, if we theoretically extend this pattern infinitely, insisting on a complete sequence of natural numbers, then we must accept 0=1=2=3=4=5=... In an infinite set, natural numbers lose their distinctiveness. Even if different sides of a die show different numbers of dots, in an infinite scenario, every roll results in a tie because all numbers effectively become one. The real twist in the story is that Adam lost everything for nothingthe game invariably ends in a tie but Adam never lets it end, a truly cunning maneuver by the serpent.
Quoting fishfry
Does it still seem like the information content of my idea is nil?
Quoting fishfry
Let me give it another go. In calculus, we handle singularities by using limits to approach (but never actually reach) a singularity. We can apply a similar principle here. If I roll a 42, my probability of winning can be illustrated as follows:
Number of faces, Probability of Winning
42, 1
100, 0.42
1000, 0.042
10000, 0.0042
As the number of faces approaches infinity, my probability of winning approaches zero. However, it never actually reaches zero because we never consider a truly infinite-sided dieit simply doesn't exist.
Quoting fishfry
In my initial example with four arithmetic statements, they seemed meaningful, right? Each one features a unique set of type characters, creating the impression of distinct statements. However, as I explained, in the context of an inconsistent system, they lose significance. We might as well condense them into a single statement: 0=1=2=3=.... The situation is similar with the concept of an infinite die. Each face of the die appears different, suggesting a variety of numbers, but upon closer examination, we realize the distinctions are superficial. The dots essentially hold no value. We might as well be dealing with a die that has only one face.
Quoting fishfry
I subscribe to the concept of completed infinite sets, but with a twist: I believe they encompass just one unique element. As a related example, when natural numbers are defined as nested sets of empty sets, I don't perceive an infinite collection of distinct objects; instead, I see a single entity: the void - emptiness.
Some may argue that this perspective is naïve, but I disagree. Conventionally, we begin with natural numbers and develop our systems upward from there. I contend that this approach is fundamentally backwards, though that's a conversation for another time.
Quoting fishfry
Ew. Actual infinitesimals are no better than actual infinities.
Quoting keystone
Quoting keystone
Wow, if the last time you went to a theater they were still using those raffle-like ticket stubs, you've missed out on quite a few great theater experiences. You definitely need to see the next Avatar movie in the theater in 3D.
Are you certain? By definition, a roll of tickets that has no end can't be completed (for that would mark the end of the roll) attempting to do so is akin to trying to create a married bachelor. Nevertheless, I agree that God could do it, though it would mean losing the distinction between numbers in the first example and words in the second.
But you are. You are trying to develop a probability measure on [math]\mathbb N[/math], which contradicts the Kolmogorov axioms. So embrace your infinite-measured probability distribution on [math]\mathbb N[/math] and see if you can make it work.
Quoting keystone
Correct. It's difficult to make your idea work.
But you are really conflating two entirely different topics. One, your puzzler is interesting. But two, you're promoting some sort of finitistic argument that's entirely beside the point.
Quoting keystone
You convinced me.
Quoting keystone
That's the second time in two days that I've been called a mathematician here. If only. It was once a failed dream. I studied math in school many years ago and keep up a bit online.
Quoting keystone
You're welcome.
Quoting keystone
This entire passage is insane. I only say that to clarify that when you said that I have made you feel less insane, that is not my intention. My purpose in posting was to note that there is no uniform probability measure on the naturals. I offered no opinions on the sanity of any member of this forum. Would you like me to? Oh my.
Quoting keystone
That last bit about your version of the principe of explosion, and your idea of infinite sets containing only one element, has made things considerably worse.
Quoting keystone
You reject the modern theory of limits? This is a real problem. The Adam puzzler is genuinely interesting. You are confusing the issue by rejecting infinite sets. If there are no infinite sets there's no game in the first place. What do you mean an infinite sided die doesn't exist? A googolplex-sided die doesn't exist either, but we can still put a uniform probability measure on it. Nothing in math "exists" in a physical sense.
Quoting keystone
Yes, you're defining the names of the Peano natural numbers.
Quoting keystone
Nutty.
Quoting keystone
So you're NOT promoting a finitistic agenda.
Quoting keystone
1 is a finite number.
Quoting keystone
The set {{{{}}}} is a singleton. It has cardinality 1. It's finite. You are making no sense.
"The void?" Did Nietzsche say that when you look into the empty set, the empty set always looks back? Maybe he should have. You're just waving your hands and not saying anything meaningful.
Quoting keystone
What?
Quoting keystone
Actually I was trying my best to be charitable. With your talk about Newtonian infinitesimals and 17th century calculus, I thought nonstandard probability theory was exactly what you were getting at. If not, then I can't conceive of any referent for your explanations.
Quoting keystone
Ties seem incidental to the problem. I'd just ignore them.
Quoting keystone
I'm absolutely baffled by it. I mean that. I can not find any referent to your ideas. I don't know what you are talking about. Singleton sets are not infinite, for example. They have cardinality 1. Every set may only be nested downward a finite number of times, so there is nothing to be said about them.
Quoting keystone
Fine. Then your total probability space has measure [math]\infty[/math]. Nothing inherently wrong with that, you just have to try to make it work.
Quoting keystone
All true for any finite number of faces.
Earlier you tried to make some sort of limit argument, but that doesn't work. If you assign probability 0 to each natural number, countable additivity requires that the total probability is 0.
Quoting keystone
Sticky floors and noisy patrons? I watch at home these days. I hadn't gone to a live theater in many years, then I went once for a film I absolutely had to see, and then I remembered why I don't go to theaters.
Quoting keystone
I didn't see the last one in any dimension. Not my cup of tea. Just checked the IMDB page. Maybe I'll watch it sometime.
Quoting keystone
I don't see why not. Hilbert's hotel is completed. That's the entire point of the exercise. Conceptualizing infinite sets. I visualize God's ticket roll just like Hilbert's hotel. Hilbert's movie theater.
If you reject infinite sets that's perfectly legitimate, but it's an entirely different discussion.
Your OP contemplates randomly choosing an element of [math]\mathbb N[/math]. If you reject the mathematical existence of infinite sets, you have no [math]\mathbb N[/math] and you have no game.
Quoting keystone
I just don't follow your idea that all the natural numbers are actually one single number, yet the set containing it is infinite, yet you don't believe in infinite sets.
Anyway I may have lost a paragraph or two in my trying to respond to several comments at once, so let me know if I missed anything.
Whatever your point or vision is about the infinite singleton, I can't figure it out, nor how it would bear on the problem in the first place.
[EDIT: Sorry this chart is incorrect. I will repost a corrected chart in the coming days. Removing now to avoid confusion.]
[CHART RETRACTED]
I believe this probability chart captures the all of the essentials of the infinite dice game and yet I do not see how it violates Kolmogorov's treatment. Can you explain?
(cc @fishfry)
Love it.
Quoting sime
But with this approach, probabilities can only be considered in retrospect, which seems insufficient.
Thanks for sharing that. The Quora post is definitely relevant. However, I believe the chart I just shared demonstrates a different perspective.
I didn't understand the picture.
I just came here late at night to reply before bed but I'll stand by for further developments.
For what it's worth, the fact that we can't put a uniform probability measure on the natural numbers doesn't mean they have to be "all the same number." They're all different numbers. And I can't understand the idea you're getting at.
You know, here's a variation that challenges the strategy of Adam always taking the serpent's number.
As you know, the rational numbers are countably infinite. Now suppose we play the same game, but with rationals. We can even use the same die. All we need to do is repaint the faces, replacing natural numbers with rationals.
But now, whatever number Adam picks, there are infinitely many numbers smaller, and infinitely many larger. There is now no sensible strategy at all.
Suppose we play the game in the unit interval of reals, which has total measure 1.
If we were playing with a real-number sided die, there's an obvious strategy. I gave this example earlier. If Adam's number is less than 1/2, he should switch; else not. Because the measure, or probability, of an interval of real numbers is its length.
But if we now play the game with the rational numbers in the unit interval, that no longer works. The unit interval of rationals has measure 0 (because all countably infinite sets have measure 0).
If Adam picks, say, 1/googolplex, a tiny number, the serpent's number has probability 0 of being to the left, and probability 0 of being to the right.
But if we conceptually filled in the rationals with the rest of the reals, then we can assign sensible probabilities and Adam should switch.
I do believe this is a pretty good paradox or at least a highly counterintuitive situation.
Note please that we can analyze this situation, and be completely confused by it, without the need to deny infinite sets. That seems to complicate the issue. Once we are thinking about choosing a random natural, we are already contemplating making a selection from an infinite collection, whether or not we call it a set.
Ok I'll leave all this to you.
I took the idea to mean that the faces of an infinite die isn't a well-ordered set, unless the Axiom of Countable Choice is assumed. If this axiom isn't assumed, then the sides of the die can only be ordered in terms of their order of appearance in a sequence of die rolls, which implies that unrolled sides are indistinguishable.
The natural numbers are well ordered in their usual order.
Any countably infinite set can be well ordered simply by bijecting it to [math]\mathbb N[/math] and using the order induced by the bijection.
Quoting sime
No choice is needed to well-order the natural numbers or any countably infinite set.
Quoting sime
I don't follow this. The real numbers in their usual order are not well ordered, but they are certainly distinguishable.
You don't need to well order a set to distinguish its elements. The elements of a set are all distinct from each other by the the axiom of extensionality. A set is completely characterized by its elements. There are no duplicates.
Are you saying the remaining five faces on a standard 6-sided die can't be distinguished if we've only rolled it once?
Yes, that is true, by Peano's inductive construction of the natural numbers. And a well-order is usually assumed for an infinite sided die, in spite of its construction lacking an inductive specification (for which side should be assigned what number?) - So the assumption of a well-ordered infinite sided die that lacks an inductive definition is the same as a countably infinite set of objects equipped with the axiom of countable choice.
On further reflection the infinite sided die shouldn't need a choice axiom in its construction (e.g a sphere can be painted by working clockwise and outwards from a chosen pole - since there is an algorithm choice isn't needed). But then what of the idea of rolling said die an actually infinite number of times? That surely is equivalent to choice, assuming the rolls are random.
Is it possible you're misunderstanding what the axiom of choice says? It's surely not contradicted or made irrelevant by painting a sphere.
Quoting sime
In the problem posed by the OP, there are exactly two rolls: one by the serpent, and one by Adam. Not sure what rolling infinitely many times has to do with this.
Quoting sime
Why?
In the Peano axioms I can invoke the successor function infinitely many times, but not only isn't there a need for any choice axioms, there isn't even a concept of sets.
I've given much thought to your critiques of my proposed resolution, and largely, I find myself in agreement with you. While I believe I'm onto something profound, my arguments have been somewhat muddled, and I've mistakenly mixed up the concepts of the null set with infinite sets. I aim to refine my approach moving forward.
Let me attempt another explanation, starting from a different angle.
SETUP PART 1
In the game, Adam can be in one of three states: win, lose, and undecided. He starts in the undecided state (i.e., before the dice are thrown or while they are still in motion) and transitions to either win or lose once the dice stop and their values are observed. Here are the abbreviations I'll use:
For instance, if Adam rolls a 10 and the serpent rolls a 4, we record Adam's history as (W or L) -> (W).
SETUP PART 2
What I should have clarified in my original post is that the fairness of the game extends beyond just the dice; it includes the rolling process itself. You cannot simply place the die on the table with the number 6 facing up just because you desire that outcome. The die must be tossed to allow each face some time facing upward before settling on a number. The more faces a die has, the more it needs to bounce around to ensure fairness.
MY PROPOSED RESOLUTION
There are 4 distinct groupings of game histories based on the number of faces on the dice.
Previously, I incorrectly conflated the null set with infinite sets. It was largely because I incorrectly conflated 0 faces with infinite faces because their histories both summarize to (W or L). However, I failed to appreciate that their histories are fundamentally differentone doesnt begin, while the other never ends.
Thus, my answer to the paradox is that the narrative isnt fairly told because when Adam opens his eyes, he should see the dice still in motion. In such an undecided state, it doesnt matter whether he chooses to switch rolls with the serpent or not.
As long as the roll can't be completed, there is no paradox. This raises a more significant question: what, if any, endless processes can be completed? If supertasks are unachievable, does this imply that infinite objects are also impossible?
ASIDE
SETUP PART 1 may seem superfluous but inclusion of this undecided state is extremely important to my approach to resolving paradoxes. Take a look at my recent post about the Unexpected Hanging Paradox. I believe the universe uses this same approach to avoid paradoxes/singularities, but in physics speak this (W or L) state would be called a superposition.
Now on to some of the other discussion points....
Quoting fishfry
I do not reject the value of limits and their importance at making calculus rigorous, however I interpret them to describe a journey not a destination. In other words, when I consider the limit of 1/x at x = 0, I do not see a need to say that the there is a destination at x=0 corresponding to number called infinity but rather I see an unending journey to increasingly and unboundedly larger function values as we approach x = 0. While you may agree that there is no destination in this case, we would end up disagreeing on a lot of other limits where the limit is a real.
With my view, reals retain all of their value in calculus, they just aren't numbers in the sense that rationals are numbers. In summary, I think that limits, the reals, and calculus represent significant achievements, but they require a fresh philosophical interpretation.
Quoting fishfry
The most effective way to discredit a system is to follow it to its logical conclusion and show that this outcome is absurd. Using this strategy, I am entirely justified in assuming that infinite sets and the game exists. With this in mind, I'll retract my stance on infinite sets and strive to discuss the topic in more agnostic terms.
Quoting fishfry
I have concerns with this hotel.
Quoting fishfry
I think the dice would keep bouncing around and so Adam's status would remain undecided (W or L).
Quoting fishfry
It's impossible to place uncountably many numbers on countably many sides. It seems like you're venturing into the realm of the Dartboard Paradox. If every point has a probability of zero of being hit, how could any point on the dartboard possibly be hit? Additionally, how does this reconcile with Kolmogorov's axiom that the sum of the probabilities of all possible events must equal 1? That being said, I do see the value in Measure Theory and the concept of probabilities on continua. Those aspects make sense to me.
After reading this, I feel I must have totally misunderstood your post. I thought you were trying to put a sensible probability measure on [math]\mathbb N[/math] that formalizes the obvious intuitive correctness of Adam always switching. But now it seems you are more interested in the physical sense of rolling an infinite die. I thought that got dispensed with early on. Of course this problem is only an abstract thought experiment, there are no infinite-sided dice. But physical infinite-sided dice seems to be what you are interested in. I am confused.
Quoting keystone
There is no die bouncing around. This is not a physical experiment. There is no physical infinite-sided die. That's why I suggested God's fishbowl, containing a countable infinity of identical movie theater tickets.
If you are concerned with bouncing, then we are not even having the same conversation, and never were.
Quoting keystone
No friction? A spherical ball bearing, machined to be as perfect a sphere as engineers can create in this world, set rolling on a perfectly smooth steel floor, will eventually come to a halt with exactly one point at the highest height. So you are proposing a physical experiment, but with alternate physics.
What are the rules for your alternate physics? Why is there no friction?
Quoting keystone
Can't parse this.
Quoting keystone
I am certain we have never been having the same conversation. I thought you were interested in putting some sort of measure on the set of naturals that makes Adam's switching strategy rational. But apparently not.
Quoting keystone
What do you mean by object? Physical objects? Those are impossible, unless you believe in the eternal inflation theory of cosmology.
In math, we have infinite objects all the time. Even finitists, who deny the axiom of infinity, still allow for the endless collection of the succession of natural numbers. They just don't allow it to be called a set.
Quoting keystone
A superposition is just a linear combination of states, in principle no more mysterious than the fact that the point (1,1) in the plane is the linear combination (1,0) + (0,1). You are throwing spaghetti at the wall now.
Quoting keystone
The formal definition of a limit, the epsilon-delta definition, is perfect rigorous and leaves no room for metaphysical ambiguity.
Quoting keystone
You don't believe in the real numbers? Ok. I don't think it would be productive for me to respond to this paragraph here. But if you start a thread entitled, "Do the real numbers exist?" I could talk about that all day. Yes they have mathematical existence. And unless there is a great leap forward in physics someday, they do not have physical existence.
Quoting keystone
Why? Just use the theater ticket metaphor, and label each ticket with a rational number, using your favorite bijection between the naturals and the rationals.
I am baffled that you are hung up on the idea of a physical infinite-sided die; and that in your conception, such a die would bounce forever, violating the known laws of physics that include friction.
I am also chagrined that I misunderstood your OP so completely that I jumped in at all. We have not been having the same conversation since the beginning.
Quoting keystone
Just as there is a conceptual countably infinite-sided die, there is a conceptual real-number sided die.
But unlike the naturals or the rationals, there is indeed a uniform probability measure on the unit interval of real numbers. As I indicated a while back, Adam should switch if and only if he rolls a real number between 0 and 1/2. And there is a rigorous mathematical basis for that conclusion.
Quoting keystone
Every point on the dartboard has probability zero. The total probability is 1, assuming the dartboard has area 1. That's consistent with Kolmogorov's axioms.
Quoting keystone
Kolmogorov requires only countable additivity. That's the point. Nobody knows how to logically account for the fact that uncountably many zero-area points can sum up to a positive area. We just accept it, and we have many formalisms to express it. The length of the unit interval of the real numbers is 1, even though each point has length 0. It's a philosophical mystery, but a mathematical fact.
Quoting keystone
I'm glad if I said something you found useful.
Why is there gravity but no friction in your alternative physics? Why wouldn't the die just float up into the air?
I'll close with this xkcd, which I just ran into yesterday.
https://xkcd.com/704/
I agree with you that a uniform probability measure on N is impossible. I think the resolution to this paradox lies elsewhere.
Quoting fishfry
While I initially used the rolling of the die to visually express my idea, your critique concerning the non-existence of perpetual motion machines suggests that my approach failed. Let me therefore explain in broader terms:
Objects - My consideration isn't just for objects that exist within our physical universe, but extends to those that could exist in a simulation or program. For example, a die with a googolplex sides is conceivable. Likewise, numbers and sets are considered objects in this context. Whereas a married bachelor is a contradiction so I say that it cannot exist. If one could prove a similar contraction about surreal numbers then I would say that they do not exist.
Process - This term refers to tasks or algorithms that perform an operation with objects (or other processes). For instance, rolling a die or executing a random number generator are both processes.
This distinction is important, as sometimes one can easily mix up the two. For example, the python code that defines the random number generator is an object. The execution of the code is a process. The random number outputted by the program is an object.
Peeling away the story elements of the paradox, it involves an infinite object (N ) and an infinite process (random number generator operating on N) [As you noted, establishing a uniform probability measure on N is unfeasible in any scenario, which implies that a random number generator targeting N would indefinitely continue without halting.]
I see two possible resolutions to the paradox:
1) The game never starts because infinite objects don't exist.
2) The game never ends because infinite processes never terminate.
Quoting fishfry
I'm not suggesting that labeling the undecided state as (Win or Lose) is enigmatic. However, the notion of a superposition of multiple states isn't generally embraced by mathematicians and philosophers. If it were, why wouldn't we resolve the Liar's Paradox by accepting (True or False) as its core solution, or use (Alive or Dead) to solve the Unexpected Hanging Paradox, as I have previously proposed?
Quoting fishfry
My argument is that limits correspond to processes, not objects. I know textbook problems are often handpicked where shortcuts can be used to determine the limit (e.g. L'Hopital's Rule). In such a case, you can exibit your work (the object) and you're set. Seems like an object, right? However, the vast majority of limits don't allow for shortcuts and involve the unending work of narrowing epsilon further and further (let's put a pin on this idea of shrinking intervals). There's no complete object you can exhibit and say that that's the limit. The best you can do is work through the unending process. That's why I believe that fundamentally limits correspond to processes.
So do I believe in pi and all of it's usefulness? Yes, BUT I believe it corresponds to a process. Just as I believe 4 - 4/3 + 4/5 - 4/7 + 4/9 - ... describes a process not an object.
Please consider my version of the Stern-Brocot Tree:
There's a lot to unpack here, so let me explain in detail.
The yellow tree primarily represents an extended version of the original Stern-Brocot Tree, now including negative values. Each vertex on this tree corresponds to a rational number, identified by the sequence of left and right turns taken from the top to reach it. The value corresponding to a vertex is calculated by taking the mediant of the vertices above. For example, the number 1/2 is represented by the path 'RL', and -2 by 'LL'. What's fascinating is that every rational number eventually will appear, in reduced form, exactly once on this tree, and they are organized in increasing order from left to right. For instance, the second row lists the numbers [-infinity, -2, -1, -1/2, 0, 1/2, 1, 2, +infinity], incorporating the rational numbers from previous rows such as [-1, 0, 1].
However, this tree, having no endpoint, excludes real numbers, yet intriguingly, it feels as though the reals should be represented here too. For example, the golden ratio, if it were on this tree, would appear at the vertex RRL (where the underline indicates repeating). But such a vertex would exist at row infinity and no such row exists. This raises the question: if RRL isn't an 'object' in the tree, then what is it? I hope you see where I'm going with this...
Switching gears in the spirit of measure theory, which efficiently handles intervals rather than points, let's consider the blue lines in my diagram. Here, instead of the tree branching, each subsequent row splits into intervals. For instance, 'RL' corresponds to the interval (0,1), and 'LL' to the interval (-infinity, -1). Now, consider the following:
[] = (-infinity, +infinity)
R = (0, +infinity)
RR = (1, +infinity)
RRL = (1, 2)
RRLR = (3/2, 2)
...
Those are the intervals corresponding to the first few digits of RRL. In my perspective, the sequence RRL represents a never-ending descent through the rows, marked by a continually narrowing interval between rational numbers. If this 'tree' had an endpoint, the interval would eventually shrink to a point, specifically the golden ratio of approximately 1.618033988749... However, the absence of a bottom means we're perpetually left with an interval. This illustrates that real numbers are better understood as unending processes that involve ever-decreasing intervals, rather than as objects fixed on the tree.
Quoting fishfry
This is paradox screaming at us telling us that we're missing something. And at the heart of the issue is our belief that calculus is a study of objects (real numbers as if they were vertices on the tree), not processes (reals as if they described an endless journey down the tree corresponding to ever shrinking intervals).
Quoting fishfry
I'm really enjoying our discussion and finding it incredibly beneficial. Thank you for your patience and the knowledge you share. I feel very lucky to have you sticking around.
Quoting fishfry
Love it. XKCD rocks.
I'll close with a quote from Niels Bohr:
"How wonderful that we have met with a paradox. Now we have some hope of making progress."
Perhaps, but you are kind of all over the map in what follows.
Quoting keystone
I didn't critique perpetual motion. I just asked, since you seem to have gravity but not friction, what are the rules of your physics?
Objects Quoting keystone
The business with the processes and objects doesn't seem to bear on the problem at hand.
Quoting keystone
There are no infinite processes. You stick your hand into God's fishbowl and pull out a ticket and read the number. I don't understand why you're attacking the premises of your own problem. Conceptually, we pick an arbitrary natural number. That's very straightforward. You're just confusing yourself by going into all these different directions.
Quoting keystone
I don't think pop quantum theory is helpful here. I was only pointing out the superpositions are not all that mysterious. They're just linear combinations in the state space.
Quoting keystone
I don't think discussing the foundations of calculus is all that helpful either. I really think you have a lot of things in your mind and you're just tossing them out. There is an interesting problem that you originally posed, but this is going nowhere.
Quoting keystone
That's fine. It's perfectly ok to identify computable real numbers with the algorithms that generate them.
However, noncomputable real numbers exist, and they do not have algorithms.
Quoting keystone
I fail to see the relevance. This is just not helpful. Not to me, anyway. You seem to want to discuss the nature of the real numbers, but that's very far afield from the original question.
Quoting keystone
If so, fine. But far afield again. Real numbers are not vertices on the tree, all the vertices are rational.
Quoting keystone
I'm glad to help. I wonder if we could have a more focussed conversation. The bit with the Stern-Brocot tree threw me for a loop. I have no idea where you were going with that. Wasn't there a thread about that on his board a while back? Here it is.
https://thephilosophyforum.com/discussion/14273/real-numbers-and-the-stern-brocot-tree
Is your concern with the nature of the real numbers? That's really got nothing to do with the original post, which is trying to find a logical basis for Adam's strategy of always switching.
You know, there's a thing called the counting measure. The Wiki article's not very good, gets too technical. The idea is that the measure of a set is its number of elements, or infinity if it's infinite.
In this case, the serpent's choice partitions the natural numbers into a left-hand segment, with finite counting measure; and the right hand segment, with infinite counting measure. In that sense, the right-hand segment is always larger, and that "explains" the strategy.
The problem is that counting measure is not a probability measure because the total measure's not 1. But I think that's a sensible way to resolve the problem.
Now you're introducing narrative elements into our discussion, mentioning God and fishbowls. If we assert that God can do anything, then we could just as easily conclude that God can define a uniform probability measure on N and leave it at that. However, there are limits to even what God can do. As a programmer would understand, creating a true random number generator is incredibly challenging. While theoretically, you might write such a program (using finite lines of code), in practice, it would run indefinitely without halting. Could God create a random number generator for N that actually stops? Or does his magic only work when we talk informally about fishbowls?
Quoting fishfry
I don't think you're truly entertaining my propositions. Did you understand what I was saying?
Quoting fishfry
While I would really like to continue this tangential discussion, there's no point in addressing this (and other tangential) comments if you aren't going to read my responses simply because they don't directly relate to the original post.
Quoting fishfry
I would have appreciated your specific insights on this topic if you had engaged more sincerely in this tangential discussion.
Quoting fishfry
My main concern revolves around the concept of completed infinities. R, N, and the process of generating a random number on N all inherently involve completed infinities. They are interrelated. Now, consider this 'paradox':
God created a married bachelor and declared he would kill the man at noon if he was married. Is the man alive at 12:01?
There are different ways to approach this paradox. One method is to seek a logical explanation for God's decision on whether or not to execute the man. Alternatively, and just as validly, one can challenge the premise itself. You are not allowing for this possibility, which seems unfair.
Quoting fishfry
This definitely aligns with Adam's reasoning. However, as you pointed out, the counting measure is not a probability measure, which I find problematic. Regarding the specific paradox, at what point would it be prudent for him to swap rolls with the serpent? Does this decision occur the moment he opens his eyes and makes an observation? What if he only pretends to open his eyes? What if he makes an observation but totally forgets what he observes? What if he keeps his eyes closed, but an ant sees his roll? What if God is watching? What if God sees the roll and informs Adam that he saw his roll but doesn't say what it was? Counting measure does not offer an answer to these questions.
Or will you instead chose not to answer these questions related to observation and simply say that pop quantum theory is not helpful here?
For what it's worth, here's how I would construct a random number generator on N in our physical universe:
1) Employ a quantum event that has a 50% chance of yielding 1 and a 50% chance of yielding 0.
2) Assign the outcome to the first digit of a binary number1 for a result of 1 and 0 for a result of 0.
3) Continue this process for each subsequent digit.
Two key observations:
1) There is one potential issue with this approach. It's remotely possible that the latter output could be an infinite sequence of 1's. If, hypothetically, this program could be executed as a supertask (completing in finite time), it might return infinity, which does not belong to the set of natural numbers.
2) The program never halts. If you stop it prematurely, you haven't encompassed all natural numbers. Since the program is intended never to halt, it avoids the theoretical problem of returning infinity, rendering the aforementioned flaw negligible.
If we're discussing fishbowls, I'd argue that when God reaches into the bowl and selects the top ticket, it's an unfair draw. He should shuffle the tickets first. However, when dealing with an infinite pile, the shuffle would never conclude. Let's set aside the fishbowl analogy and turn our focus to programming, which offers a more tangible approach to discussing random number generation on N.
Let's reframe this discussion in terms of my concepts of objects and processes:
1) The random number on N (i.e., the output of the RNG function) - an object that cannot feasibly exist.
2) The code defining the RNG function - a finite object that exists.
3) The process of executing the code to completion - an infinite process that cannot be completed.
In mathematics, there is a tendency to treat the output (1) as the fundamental element. However, I contend that the actual code (2) deserves our primary attention. This shift focuses on the tangible aspects of mathematical constructs rather than on abstract, unattainable outputs.
I think this is all going south.
Quoting keystone
I don't find your propositions entertaining.
[Hey man sorry, with a straight line like that I could not resist!]
Quoting keystone
About the real numbers? It just seemed like a change of topic. I didn't feel like engaging on the subject.
Quoting keystone
Noncomputable numbers directly bear on your idea of using computations to define real numbers. Most real numbers can not be so characterized.
I am not sure why you feel I'm obligated to follow you far afield from the OP. I only jumped in because you were speculating on a uniform probability measure on the natural numbers, and I just happen to know the factoid that there can't be one. I thought I was helping.
Quoting keystone
I don't know much about the Stern-Brocot tree. You can get the same result using the completed infinite binary tree, in which there are countably many nodes, yet the real numbers are encoded as paths through the tree.
Quoting keystone
Arguing finitism is a lot different than what you presented in the OP. I just don't care to follow all the changes of subject.
Quoting keystone
Why are you throwing this out there? I'm not going to respond. I don't have to respond. Let me just note that several dozen other currently active members of the forum didn't respond either. Why aren't you unhappy with them?
Quoting keystone
I suggested counting measure since it bears on the OP. Can't imagine what opening his eyes has to to with this. I'm feeling a little backed into a corner and no longer enjoying this.
Quoting keystone
I was enjoying this a lot more when you wrote ...
Quoting keystone
I'm sorry I can't be more helpful. You have many questions. I can't respond to them all.
As far as my remark about pop quantum, I meant it. You just pulled superposition out of the air. It has nothing to do with any of this.
Clearly my responses are making you unhappy and that's not my intent. But I don't have the inclination to discuss all of the varied subjects you brought up. Really, I said everything I had to say in my first post.
What makes you think that wasn't determined at the moment of the big bang?
Quantum events have certain probabilities of being observed in certain states. We don't know for sure whether they're actually deterministic or not.
Quoting keystone
For all you know, this procedure is entirely deterministic. Just as flipping a fair coin is random for practical purposes, but is clearly deterministic if we only knew the physical variables precisely. The same might be true for subatomic events. We don't know.
Quoting keystone
Return infinity? Supertasks? I can't respond to any of this.
Quoting keystone
You've lost me again.
Quoting keystone
It's a conceptual thought experiment. If you don't like the metaphor, forget it.
Quoting keystone
You can't prove that there is any such thing as a true RNG.
Quoting keystone
I've already agreed that it's perfectly fine to identify a computable real number with the algorithm, or the set of all algorithms, that generate it. But that misses all the noncomputable real numbers.
Quoting keystone
Ok, you're arguing a constructivist viewpoint. Interesting subject, but far removed from the subject at hand.
But you have me so confused.
On the one hand, you are asking about a totally abstract thought experiment where God rolls an infinite-sided die.
And in the next breath, you say you're only concerned with the "tangible" aspects of math. Which surely precludes God rolling dice, right?
Likewise, and thanks.
Quoting keystone
I did suggest I was trying to wrap it up. And then I asked you a question! "Do I contradict myself? Very well then I contradict myself, (I am large, I contain multitudes.)" -- Walt Whitman
So even though I'm trying to wrap this up, I do have a question.
You said you were interested in the "tangible" aspects of mathematics.
But your original question is totally intangible, concerning God rolling an infinite-sided die.
So: tangible or abstract?
ps -- I just had a thought. It's a semantic solution.
If instead of choosing a random number, what if we just choose an arbitrary one? That conveys the same conceptual scenario, but without invoking all the mathematical and philosophical context of randomness.
1) Tangible and possible - for example, a horse.
2) Tangible and impossible - such as a black hole as described by Relativity, with a singularity at the center.
3) Abstract and possible - like the number googolplex.
4) Abstract and impossible - such as a four-sided triangle.
Our physical universe, though entirely described by mathematics, appears to have circumvented singularities. Why not look to it for inspiration? In physics, breakthroughs often occur when one identifies something tangible and impossible and rethinks our understanding to shift it to tangible and possible. This approach has driven many major advancements in the frontiers of physics, which is why numerous eminent minds are engaged in quantum gravity research.
The next significant breakthrough in mathematics could occur when someone pinpoints what is currently abstract and impossible yet accepted within modern mathematics, and the community transforms it into something abstract and possible. The arithmetization of analysis is an excellent illustration of such a transformation. While I deeply appreciate the value of what is abstract and possible (acknowledging that mathematical truths are both beautiful and useful), much of it surpasses my grasp, so I can't personally revel in it. However, what really captures my interest is the pursuit of the abstract and impossible in mathematics. Personally, I view it as the most important, thrilling, and accessible area to engage in at the moment. Although most impossibilities in mathematics have been resolved (no serious mathematician is exploring four-sided triangles), I believe paradoxes like the ones we discuss suggest that some impossibilities still remain.
To summarize my interests:
1. Tangible and Possible - This is my day-to-day work as an engineer. I thoroughly enjoy the innovations that stem from exploring this domain, especially my computers.
2. Tangible and Impossible - The physics community already excels in this area. They are actively working to resolve the impossibilities in their theories. Yet, there are still opportunities to influence through philosophical interpretations of quantum mechanics.
3. Abstract and Possible - Mathematicians excel in this field, continually advancing our understanding and capabilities.
4.Abstract and Impossible - Typically, those who challenge the established norms here are labeled as cranks. There is a significant opportunity for philosophers of mathematics to make strides in this area. This is where my interest lies, in exploring and potentially reshaping the abstract impossibilities that still exist in mathematics.
With this in mind, we seem to disagree on whether the paradox I propose is abstract and impossible or abstract and possible. It might be an exaggeration, but from my perspective, this disagreement translates to me seeing it as crucial, whereas you might view it as merely an interesting concept, but nothing more.
Additionally, I believe I have the beginnings of an idea that could transform it from abstract and impossible to abstract and possible. This concept also holds the potential to resolve many other persistent paradoxes, such as the Liar's Paradox, the Dartboard Paradox, and Zeno's Paradox. Yet, I find myself struggling to even convince you that the paradox, which appears possible from a conventional standpoint, is actually abstract and impossible.
What do you think about this?
Perhaps my next paradox will make a stronger impression. Even though this conversation might conclude, please keep in mind that I'm always open to picking it up again if you're interested.
Quoting fishfry
It appears that an arbitrary number would be relevant in discussing the potential outcomes of Adam's story before or after the event has occurred. However, for the story to progress as it unfolds, in Adam's 'present' a random number would need to be selected. Please correct me if I'm misunderstanding your point.
Wow this was a good post. I understood everything you're saying and I agree with much of it. Even in parts where I disagree, we're still talking about the same thing. Thanks for this.
Quoting keystone
I see your point.
Quoting keystone
This is different than the others. A four-sided triangle is impossible simply by virtue of the meaning of the words. I thought that since you called googolplex abstract and possible, then you would use the transfinite ordinals and cardinals as examples of abstract and impossible things.
Small quibble anyway.
Quoting keystone
Agree.
Quoting keystone
OMG my thoughts exactly. The analogy is non-Euclidean geometry, which was thought to be a mathematical curiosity with no practical use when discovered in the 1840s, and then becoming the mathematical formalism for Einstein's general relativity in 1915.
My candidate for the next breakthrough like this is the transfinite cardinals, the higher infinite. Nothing more than a mathematical curiosity today, but in 200 years, who knows
Quoting keystone
I don't share your enthusiasm for logical paradoxes as the fulcrum for the next scientific revolution, I do agree with your point.
Again I don't like four-sides triangles or married bachelors as examples, because those are only based on the meaning of the words. Like jumbo shrimp, or Kosher pork.
Quoting keystone
Quoting keystone
As a longtime student of crankology, I disagree. Alternative and novel ideas don't make one a crank. It's a certain lack of the logic gene or a certain basic misunderstanding of the nature of proof and logical argument that separates the cranks from the merely novel thinkers.
Quoting keystone
Ok. I just don't know if the standard logical paradoxes are that important, but time will tell.
Quoting keystone
I'm not really on board with your terminology, so I can't agree or disagree.
I have not realized earlier that you are not interested in the interesting question of choosing an arbitrary natural; but rather trying to link this to some kind of paradox. But the relation's a stretch. I still don't see the connections that you've tried to make with dice that roll forever (why gravity but no friction in your world?), quantum physics, and various other topics.
Quoting keystone
It's a cute problem, but as I indicated originally, it has a mathematical answer, which is that there's no uniform probability on the natural numbers.
I don't think it has any sigificance beyond that, but of course that's a matter of opinion and not fact, so we can agree to disagree on that.
Quoting keystone
What you call are abstract and impossible are just word meanings like married bachelor. There's nothing of real interest.
Quoting keystone
Much ink spilled over the years on this, but just not an interest of mine. Personal preference.
Quoting keystone
This is a genuine paradox of interest. How does a collection of sizeless points make up a length or an area? We have mathematical formalisms but no real explanation.
There's really nothing to be done about the basic paradox.
For what it's worth, Newton thought of lines as being paths of points through space, so there's no real paradox if you assume space is like the real numbers. Which it almost certainly isn't.
In fact I would venture to say that the ultimate nature of space or spacetime is nothing at all like the mathematical real numbers.
Quoting keystone
Already resolved mathematically by the theory of infinite series, and physically by the fact that motion is possible. Also just not a major interest of mine.
Quoting keystone
Well I've already noted that your definition of abstract/impossible is only about word games like married bachelor. I would say your definition of abstract/impossible is not fully thought out.
But what you have failed to convince me of is that "the paradox" -- which one of many that you've discussed?? -- is important, either in general or especially to me.
I've seen all the paradoxes but they don't hold central interest for me.
Quoting keystone
I agree with you that some of these seeming paradoxes might be the key to future insights. But surely not semantic jokes like married bachelors or four-sided triangles. Those aren't paradoxes and they're not of interest at all IMO.
Quoting keystone
I saw the other thread, that looks like a variant of Thompson's lamp or any of several other similar puzzles. In Thompson's lamp, the final state is not defined so you can make it anything you want it to be. It's not as interesting to me as it is to others I suppose.
Quoting keystone
My point is that "arbitrary" works just as well, without carrying all of the context of randomness. But if it's not helpful, then nevermind on that.
Bottom line I agree with you that things that seem useless today, like transfinite cardinals, may someday be useful to physics, as non-Euclidean geometry became.
And I agree that the dartboard paradox shows (to me) that the physical world is highly unlikely to be accurately modeled by the mathematical real numbers.
But the other ones, Thompson's lamp and the staircase and so forth, arise from the fact that the final state is simply not defined.
Great! It does feel nice to feel heard.
Quoting fishfry
I didn't bring up transfinite numbers as examples of abstract impossibilities because I knew you might disagree. However, you're right that my initial example was trivial. Let's consider a non-trivial one: "This statement is false." This paradox challenges classical logic by appearing both true and false simultaneously. Yet, consider the profound influence it has had. This paradox sparked the development of numerous non-classical logics. Reflect on its siblings like "the set of all sets that do not include themselves" and "the formula with Gödel number ___ cannot be proved". Dismissing such a seemingly abstract and impossible statement would have deprived us of significant philosophical and mathematical advancements. And I believe the revolutionary impact of this paradox is far from over.
Quoting fishfry
I love this example.
Quoting fishfry
Fishfry called it here first :)
Quoting fishfry
I think you meant to say 'the next mathematical revolution'. Paradoxes, or singularities, have been and continue to be pivotal in sparking scientific revolutions.
But yes, the lack of enthusiasm applies to you and pretty much everyone else. Unfortunately, I lack the mathematical prowess needed to convince you to listen.
Quoting fishfry
You're generally correct, but there are exceptions like Norman Wildberger. I hope that one day AI can help cranks build a more compelling argument because I think they aren't completely mistaken.
Quoting fishfry
Its tempting to just snip off the loose thread and assume everything is fine. After all, how much damage could a small loose thread really do to your sweater, right? Abstract impossibilities are such rare gems I'm saddened that we don't value them.
Quoting fishfry
I believe it's impossible to choose an arbitrary natural number in N. I understand that my earlier questions about the impact of observation seemed aggressive, so let me answer them instead and see if you have any comments. Before the dice stop rolling, Adam has a 50% chance of winning. Once Adam sees his roll, his chances drop to 0%. If Adam forgets his roll, his chances go back to 50%. If only God sees the roll, God knows Adam's chance of winning is 0%, but Adam still believes it's 50%. If God reveals that he saw the roll, both are aware that Adam's chance is 0%. It's pretty wild, isn't it? Even if we find a way to choose an arbitrary natural number in N, the situation remains just as bizarre. Declaring that there's no uniform probability on the natural numbers is not an answer. It's akin to dismissing "this statement is false" as an invalid statement that can be ignored because it doesn't fit into classical logic. You're snipping off the exposed part of the loose thread.
Quoting fishfry
Fair.
Quoting fishfry
I appreciate your acknowledgment that mathematical formalisms don't provide an explanation. However, I strongly disagree with the notion that nothing can be done about it. It just seems you might not be interested in an informal solution, and if that's your stance, I'm a little sad but it's a reasonable one to hold.
Quoting fishfry
I don't understand.
Quoting fishfry
I would venture to say that the ultimate nature of spacetime is very much like the objects that real numbers are intended to model (continua).
Quoting fishfry
I strongly disagree on the topic of infinite series, but I wont delve into it since it doesnt seem to interest you. Zeno wasn't attempting to prove that motion itself is impossible; rather, he aimed to demonstrate that motion, as understood by the prevailing theories of his time, was impossible. This serves as a prime example of a concept once believed possible, which he identified as both a tangible impossibility and an abstract impossibility. The paradox remains unresolved to this day.
Quoting fishfry
I think we're both at fault here. I haven't explained my perspective well, and you haven't been entirely open to hearing it.
Quoting fishfry
All of the major mathematical paradoxes today share a common theme: superposition. The liar's statement is (true or false), Thompson's Lamp is (on or off), the staircase (exists or doesn't), Icarus is (alive or dead), and the state of Adam's game is (win or lose). Unfortunately, I suspect you might dismiss this entire explanation as lacking substance.
Quoting fishfry
The universe must ultimately settle on a state because something has to occur. Are you suggesting that God simply flips a coin? All signs, including those from quantum physics, indicate that we need a new state for Thompson's Lamp upon completion of the supertask, one that goes beyond just being (on) or (off).
@fishfry: 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.
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...
What thread are we on? Did you combine this one with the other one? Maybe you could ask a mod to do that for you.
I wasn't saying that this is "the issue", only that it is the logical outcome. 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.
Quoting keystone
No, the end is not reached, as explained above. This is what Andrewk neatly explained in the other thread. That a minute will pass, and the end will be reached, is a presumption outside the prescribed scenario. You are assuming that from some other principles.
All of my responses were to messages on this thread!
@fishfry
Oh you're right...this got messed up. Let me reach out to the moderators. Sorry!
Oh YOU messed the threads up? I apologize to the moderators, whose names I have taken in vain. :-) said jokingly of course
Quoting fishfry
No it wasn't me. That was the Canadian in me saying sorry!
Quoting keystone
This statement was incorrect. I said it not knowing that the threads got mixed up.
Before making such a statement, we'd need to define what we mean by "the ground". Very difficult, because it needs to be a specific point that is infinitely far below the top of the stairs.
I think we could construct an imagined world where we could make such a definition, using a concept like the first infinite ordinal ?, as described here.
The ordinals deal only with whole numbers, whereas we want fractions too, as we're measuring distance. I expect we could extend the ordinals to include fractions, simply by interpolating between successive ordinals. But there may be an obstacle I'm missing.
The "ground", thus defined, is a point that cannot be reached from the stairs, being infinitely far below it. Similarly, you cannot reach the stairs from that point, as every stair is infinitely far above it. That's why the man on the "ground" can't see any stairs as described in the OP story. They are all too far away above him.
By making such a definition, we are essentially dividing our thought-experiment-world into two parts, neither of which can reach the other.
I've delete my message from here and posted it in the correct thread.
@Metaphysician Undercover, @fishfry, @andrewk, please move your related posts as well if that's not too much trouble.
Oh well then now I'm thoroughly confused. It's fitting to be down a rabbit hole, given the nature of the topic.
ps -- Oh I see what happened. I posted a response about omega sequences that should have been over in the stairway thread. I moved it over there.
https://thephilosophyforum.com/discussion/comment/898761
I'll try to get to your other points later.
If you and I agree on something but I just don't allocate it the same percentage of my overall interest and passion as you do, that's ok, right? We basically agree on Zeno, I just don't give it much thought. I've given it some thought over the years. But I truly never cared about it in the sense that you do. And I hope you can make your peace with that, because you seemed to be saying that you wanted to convert me not only to your point of view, but also to your level of passion. And that may not be productive.
Time is valuable, and it's perfectly fine for you to express that you're not interested in continuing our conversation; we can leave it at that. If you choose to end the discussion but also mention that you agree with me, that's a nice extra, though not necessary. Regarding converting you to my point of view, I do want to do that and will seize any opportunity that comes up. I thought that since you provided your resolution to Zeno's paradoxes that you invited further discussion, but it seems I may have misinterpreted your intentions.
I'm perfectly happy to continue the conversation. I'm only saying that you might be disappointed if you hope to convert me to your degree of passion, even on items where I agree with your point of view.
I'm sure poor old Zeno is getting a sufficient workout in the staircase thread.
Great. And if it seems like you're no longer making debatable points or asking questions, I'll take that as a hint that the conversation has reached its end. :D
Quoting fishfry
Might? As in there is still a chance? [said like a clueless teen not getting the hint from repeated rejections from his crush. Lol.]
Quoting fishfry
Yeah, let's keep Zeno to that thread. I'm glad to see you couldn't resist joining in, though. :)
Believe so.
Quoting keystone
No chance.
Quoting keystone
Didn't do any good, nobody understood a word I said.
I think I understand what you said; I just have some issues with your perspective.
Max Planck once said "a new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it." Certainly, I hope you have a long and fulfilling life, but your response brought this quote to mind.
Your argument is that Zeno's paradox is so new and revolutionary that I'm too old to see it?
Zeno lived 2700 years ago (5th century BCE according to SEP). So your argument fails.
I'll assume that your wish for my death did not come out the way you meant it. Way over the line.
But in what way could any living human be too old to understand Zeno's ancient paradoxes? Your analogy is totally flawed.
What perspective do I have and why on earth are you going on about it like this?
I apologize if it seemed like I was implying anything about wishing for your death; that was not my intention at all. I specifically expressed a desire for you to have a long life. My main point was about the acceptance of new ideas, highlighting that they often gain traction because a new, possibly more open-minded audience emerges over time. The longevity of those holding old beliefs isn't the crucial factor.
Quoting fishfry
Zeno was significant, but the concepts and solutions I'm advocating are not entirely his ideas.
Quoting fishfry
I believed we agreed to confine the Zeno discussion to the Staircase thread, which is why I was vague here. However, I offered detailed criticisms of your perspective in that other thread. I'm not trying to be cagey.
Ok fine.
Quoting keystone
In what sense do you regard Zeno's paradoxes as new ideas? That doesn't make sense.
There are no new original records of Zeno's paradoxes so they are not new ideas. However, I think that Zeno's paradoxes remain unsolved, and I have an original perspective that resolves these and many other paradoxes in a way that they no longer seem contradictory. I sense you can tell I'm enthusiastic about this viewpoint, but it seems you aren't interested in delving into or critiquing it. Perhaps after considerable reflection, you've already formed your opinion on these issues and don't find additional discussion worthwhile. That's completely acceptable.
So your point was that if everyone older than you dies, you'd win the argument?
Your use of Planck's quote makes not a lick of sense. He was talking about older scientists not being able to get on board with radical new ideas accepted by younger ones. But there's no radically new theory of Zeno that old scientists are rejecting, except for your own personal theory, which as far as I can tell you have not clearly articulated. So it's a failed analogy.
Quoting keystone
I'm pretty sure I haven't heard a clear statement of your idea. I could not repeat your idea back to you even to disagree with it. Perhaps I missed it. I'd be happy to critique your idea if you stated it clearly. [If you did state it clearly and I missed it, my apologies]. I could argue your thesis (if I knew what it was) all day, without ever having much interest in the subject. Well maybe not, but I'd kick it back and forth a little.
Maybe you can state your thesis so I know what revolution will ensue if everyone older than you would only die already.
Quoting keystone
I have no idea what point you are trying to make. You started this thread with a paradox of probability, which has a solution that's mathematically correct but not intuitively satisfying, namely that there's no uniform probability on the naturals.
Then you changed the subject to encompass many other ideas I consider irrelevant to the OP (quantum, etc) and I lost the ability to follow your thinking entirely. So yes, I don't have much interest in talking about Zeno's paradoxes of motion; but more to the point, I have no idea what is your grand new thesis that the old folks just don't get.
Fine. What matters is that you're being very generous with your time to me and I offended you. I don't want to waste the time I have with you arguing over this. Again I'm sorry and I grant that you're entirely right on this. I hope we can put to rest this specific topic.
Quoting fishfry
I've been sharing aspects of my perspective here (but I feel like you never read it, perhaps because it seemed tangential), and other details have emerged in the Staircase thread. Nevertheless, I haven't presented it as a complete picture. Should we continue such a discussion in this thread, which has become like our private chat room, or would you like me to start a new thread?
Apology completely accepted. I'm a little hypersensitive in general. No worries as they say.
Quoting keystone
I may have misunderstood a lot because I was focussed on the probability aspect. I'm not reading most of the posts in the Staircase thread and it's all over the map at this point.
Quoting keystone
Ok, that's fair. Would be happy to chat about your idea if you present it.
Quoting keystone
This thread's fine. The Staircase thread's hopeless, way too many side issues. It's nice and peaceful in here.
Agreed. Okay, let's begin!
It's elementary
Even if you believe that the foundations of mathematics and our understanding of continua is rock solid, you must acknowledge that it confounds many people. Take, for instance, the difficulty in convincing a child that 0.999... equals 1, or the prominance of Cantor cranks. By contrast, I believe children would grasp my concept more easily because it is fundamentally simple, albeit it requires adopting a different viewpoint towards the foundations of math. To use an analogy, my perspective is less like a target that's difficult to hit and more like one that's difficult to spot.
Why I believe it's important
The validity of my ideas is still up for evaluation, but if they prove to be correct, deep truths often end up having practical relevance, even if their complete implications are not immediately apparent. Nevertheless, I am convinced that my theories could enhance mathematics education, resolve many paradoxes, and shape our understanding of reality, particularly in the context of physics. Ironically, coming from an engineer, I don't anticipate any significant impact on applied mathematics, as practitioners in such fields typically do not focus on the foundational aspects of math. I also want to clarify that my work is not meant to suggest that previous efforts by mathematicians were wasted.
How I'm going to share my ideas
I understand that for an idea to gain acceptance in the mathematical community, it needs to be formalized. I'm just not there. I don't have a formal paper to share with you, but instead, I plan to share my ideas gradually, in a manner akin to our ongoing discussions. Just as we can introduce children to the basic concepts of Cartesian coordinate systems without heavy formalities, I hope you can allow me the same flexibility in explaining my ideas with a similar level of informality.
Mathematical terminology often comes with preconceived notions; for instance, mentioning a continuum might lead you to assume I am discussing real numbers. To avoid these assumptions and start with a clean slate, I'll be using a 'k-' prefix in front of familiar terms (like k-points, k-curves, k-continua, etc.). By the end of our discussions, I hope you'll not only find my approach more appealing but also recognize that it aligns with the mathematics that applied mathematicians have been practicing all along. At that point, it may be justified to remove the 'k-' prefix.
Thoughts?
Agreed. Okay, let's begin!
It's elementary
Even if you believe that the foundations of mathematics and our understanding of continua is rock solid, you must acknowledge that it confounds many people. Take, for instance, the difficulty in convincing a child that 0.999... equals 1, or the prominance of Cantor cranks. By contrast, I believe children would grasp my concept more easily because it is fundamentally simple, albeit it requires adopting a different viewpoint towards the foundations of math. To use an analogy, my perspective is less like a target that's difficult to hit and more like one that's difficult to spot.
Why I believe it's important
The validity of my ideas is still up for evaluation, but if they prove to be correct, deep truths often end up having practical relevance, even if their complete implications are not immediately apparent. Nevertheless, I am convinced that my theories could enhance mathematics education, resolve many paradoxes, and shape our understanding of reality, particularly in the context of physics. Ironically, coming from an engineer, I don't anticipate any significant impact on applied mathematics, as practitioners in such fields typically do not focus on the foundational aspects of math. I also want to clarify that my work is not meant to suggest that previous efforts by mathematicians were wasted.
How I'm going to share my ideas
I understand that for an idea to gain acceptance in the mathematical community, it needs to be formalized. I'm just not there. I don't have a formal paper to share with you, but instead, I plan to share my ideas gradually, in a manner akin to our ongoing discussions. Just as we can introduce children to the basic concepts of Cartesian coordinate systems without heavy formalities, I hope you can allow me the same flexibility in explaining my ideas with a similar level of informality.
Mathematical terminology often comes with preconceived notions; for instance, mentioning a continuum might lead you to assume I am discussing real numbers. To avoid these assumptions and start with a clean slate, I'll be using a 'k-' prefix in front of familiar terms (like k-points, k-curves, k-continua, etc.). By the end of our discussions, I hope you'll not only find my approach more appealing but also recognize that it aligns with the mathematics that applied mathematicians have been practicing all along. At that point, it may be justified to remove the 'k-' prefix.
Thoughts?
I have never expressed, nor do I presently hold either of those beliefs.
Quoting keystone
I cannot take responsibility for the execrable state of math education, or frankly education in general these days.
Quoting keystone
Lotta fluff so far. "Where's the beef?" (*)
Quoting keystone
More marketing fluff. I'm regretting this already.
Quoting keystone
I fear that you're going to wave your hands and present a lot of mathematically naive ideas, and I'm going to find myself back in the .999... wars and all the rest of it. And I'm frustrated that you wrote so many words here without saying anything at all.
Quoting keystone
If not, then what?
Quoting keystone
You're going to rework the whole of mathematics? I'm getting a sinking feeling.
Quoting keystone
I regret encouraging this conversation.
Quoting keystone
I apologize for the negativity, but you didn't say a thing yet. And you apparently have some kind of grand unified theory of math that you're going to wave your hands at while I attempt to be open-minded.
I am open-minded, and want to hear your ideas, but you don't seem to be interested in presenting them. I was hoping for a paragraph, but you gave me what looks to be the introduction to a very long and very frustrating exposition.
If you have a paragraph or two that I can sink my teeth into, by all means present it.
If what you've got is as vague as this post, then I humbly apologize for encouraging you to aim this at me. On the one hand I'm curious as to what you are talking about, and on the other, well ... you just haven't said anything but you're promising to say way too much.
Can you boil down what you want to say in a couple of clear paragraphs? Without the marketing about how it's revolutionary and will be understandable to children?
I do welcome your thoughts, but I encourage you to get to the point and try not to turn my curiosity into frustration. Clearly this post sent me over that line.
Here is my response to this post in a nutshell.
(*)
https://www.youtube.com/watch?v=Ug75diEyiA0
I agree, I just wanted one post to set the stage before I get into it...
Quoting fishfry
Well, how much beef can one actually put in a paragraph? Have you ever sunk your teeth into an abstract?
Anyway, I don't want to write another long post. My first real post will come tomorrow...I got consumed by the Staircase post this evening...
I await your next missive with both curiosity and trepidation. Is it wrong for me to encourage you on the one hand, then give you a hard time the next? I'm conflicted.
I don't want you to go easy on me. I pride myself in my ability to correct my trajectory in the face of new evidence/feedback.
Standing by for something specific. And if it's not too much to ask, can you keep it short? I myself tend to write long-assed posts. I should take my own advice.
Should I abbreviate my explanation, you might resort to conventional thinking to bridge the gap, which could lead to misunderstanding. My goal is to present my ideas with such simplicity and clarity that youll effortlessly grasp them, swiftly perceiving their evident truth. So I hope that quick instead of short is acceptable.
So here goes
PART 1 - The Elastic Ruler
THE MATERIALS
Everything that follows takes place within my abstract sandbox. I intend to construct a ruler in an unconventional manner, starting with an abstract elastic band (i.e. a k-topological object). The graphics that follow are an imperfect representation of what's truly happening my sandbox. For example, my abstract elastic band is a one-dimensional object with the property of perfect elasticity (i.e. it can stretch infinitely far), unlike its physical counterpart. It's crucial that you not ascribe physical characteristics to elements within my sandbox. So, resist the urge to suggest that my elastic band is made up of indivisible atoms or exists amidst a quantum foam.
THE TERMINOLOGY
I am going to cut the elastic band a couple of times. At this point we can begin to label regions.
Notice that both the void and the elastic band segments display a sense of duality in that they are defined in relation to one another. They are both important.
While we must remember that we're talking about cuts to an elastic band, the gap size is unimportant. As such it is much cleaner to make the cuts 0-width in illustrations as depicted below.
Let us use the following terminology:
THE RULER
I will take the elastic band and cut it 3 times, adding k-number and k-interval labels as depicted below.
And there we have it, an elastic band ruler!
USING THE RULER
Test one: Measuring a pen in my sandbox. Conclusion, pen is 1 unit long.
Test two: Measuring a pen again in my sandbox with ruler stretched. Conclusion, pen is 0.5 units long.
Test three: Measuring a screwdriver in my sandbox with ruler slightly stretched. Conclusion, screwdriver is 1 unit long.
Test four: Measuring a screwdriver in my sandbox with ruler stretched more. Conclusion, screwdriver is 0.5 units long.
Employing this ruler as demonstrated could yield highly questionable outcomes, like equating 0.5 with 1, suggesting all objects are of the same size, or that an object's size is not consistent with itself, among others. This would appear to render the ruler quite ineffective, wouldn't it? Yet, let's explore the subsequent experiment.
Test five: Achilles and the tortoise live on the elastic band. Achilles is at (0,0.5) and the tortoise is at 0.5.
As the elastic stretches, the positions relative to one another remain constant, thus allowing the ruler to accurately depict their placement.
CONCLUSION
1) Elastic rulers are useful only when applied internally (i.e. locally).
2) Elastic rulers are not useful, and in fact misleading, when applied externally (i.e. globally).
3) IF it can be proved that there will always be abstract objects that exist outside of the ruler (similar to how there is no set of all sets), then it would follow that the existence of a Universal Elastic Ruler (one that can measure everything) is not possible.
Are you with me? I know this seems extremely basic (and perhaps inconsequential), but I'm laying the groundwork for a more consequential idea so I hope you stick with me.
Should you abbreviate it, I might have a chance at reading it.
My first impression -- again, forgive me, but I'm finding some virtue in just telling you the truth of my own experience of seeing this -- my first impression is the overwhelming passion that you have toward this subject; passion that is admirable in you, but makes me reluctant to even try to engage. Pictures, and rulers, and pencils ... it's a little ... off putting. That I would be engaging with someone too obsessed for their own good. I would feel that I need to tread cautiously.
Now I do want to try to give this a fair reading. I have a few other mentions to get to tonight and I'll put this aside for later. But I must say one thing. It is impossible to prove anything mathematically using physical constructions. There's no way you and I are even having the same conversation, if you think your opening salvo should involve pencil and paper and scissors and ... is that a screwdriver? I don't have to come down to the basement to see this, do I? Why are you closing that door behind me ...? Aiiiiiyyyyy.
Quoting jgill
I wish I was up there with you.
I know that, and I explicitly stated that the photographs are not perfect representations of the abstract objects which I'm actually talking about.
Quoting fishfry
I'm certainly not too obsessed with this. This topic has sat dormant in the back corner of my mind for years before it resurfaced with my unexpected hanging paradox paradox post a few weeks back. Should this discussion not lead anywhere significant, it'll probably return to its quiet corner. Nevertheless, I'm keen to hear your candid thoughts. Whether they nudge me towards new insights or help me lay these concepts to rest, I am welcoming of both possibilities.
Quoting fishfry
I would really appreciate that. I don't plan to have many photographs in my subsequent posts. This was just my way of laying the groundwork.
But it is no mirage. The same difficulty arises in math with limits and with the repeating value of constructs like Pi. These are NOT mirages. They are actual and demonstrable within reality. So much of reality answers to the limit functions that their utility and probable inclusion as meaningful and dependable is a great practice. If you wish to dismiss them, I must report that you'd need some fairly compelling, next to miraculous new ways of looking at the entire universe in order to approach success.
Quoting keystone
So YES, it is intriguing and also impossible. As for your second sentence, no, not at all. Unless you misstated what you were trying to say, all regular shapes of equal sides are easily of finite volume at any n where n = length of a side. {picks up D&D dice to prove it. Yup, finite volume. }
Quoting keystone
Belief in such a concept has no relationship to stating that the concept is not useful in the example given. In REALITY the infinity goes, BOTH, MULTIPLE, OR ALL; ways at the same time. So, offering examples that do not match reality is ... a mirage ... which I thought was what you were trying to avoid or point out.
Wow. Man. I read your post. I have no idea what you're talking about. A metric space has a metric. Some topological spaces are metric spaces. But a topological space without a metric can not have a sensible notion of distance. Topological spaces by definition are stretchable. But you can't measure distance consistently in them.
You have totally lost me. I don't know what point you are making.
Quoting keystone
You lost me totally. I have no idea what your point is other than that you're stretching a topological space and noting that there's no sensible notion of distance.
At this stage, I'm making such minor points that perhaps you are confused why it took me so many words (and pictures) to express it. If that is the case, my apologies.
I think what I'm trying to say is the following:
1) Topological spaces have no sensible notion of distance.
2) Topological metric spaces have a sensible notion of distance.
3) If you lived outside a topological metric space, you wouldn't be able to use it as a measuring tool on external objects (i.e. the metric qualities of the space are not applicable to objects outside of the topological metric space).
4) If you lived inside a topological metric space, you'd perceive it as a metric space, where the topological qualities aren't obvious in everyday experiences. For instance, if our world were a topological metric space and everything, including the space, ourselves and our measuring tools, suddenly grew twice as big, we wouldnt detect the change because all our measurements would scale up too.
5) If it is always possible for an object to exist outside of a topological metric space, it's notion of distance cannot be universally applied to all objects. I phrased this as, 'there cannot exist a Universal Elastic Ruler'.
6) I'm constructing a topological metric space from the ground up, rather than examining one that already exists in completion. So, in my example, it's a very crude ruler and there is no mention of real numbers. Does this qualify as a topological metric space?
Aside from the topological discussion, I also made the following point:
7) I'm treating continua as fundamental objects and points as emergent objects which become actualized when I make cuts. I've adopted the 'k-' prefix to denote this distinction, as it's common to encounter the reverse belief - that points are fundamental objects and continua are created by assembling infinite points. Perhaps you wouldn't characterize your viewpoint in these exact terms; you might regard points and continua as simply coexisting without one preceding the other. However, it's undeniable that the conventional approach primarily describes continua in terms of points rather than the reverse.
Is there disagreement or confusion on any of these points?
I'm still concerned about that screwdriver ...
Quoting keystone
Perfectly standard.
Quoting keystone
By the definition of a metric space, right? Also perfectly standard.
Quoting keystone
Yes ok.
Quoting keystone
Yes that's true.
Quoting keystone
Ok, but "universal elastic ruler?" That part I don't get.
Quoting keystone
Is it a topological space? Is there a metric? Then yes.
Quoting keystone
Emergent objects become actualized? Bit vague for me.
Quoting keystone
Losing me.
Quoting keystone
Ok.
Quoting keystone
Not much disagreement, only confusion about where this is all going. It's perfectly clear that some topological spaces are metrizable and others aren't.
I understand that as a trained mathematician, you have the ability to articulate complex ideas clearly using descriptive language. I admire that skill, but as an engineer, my strengths lie more in visual thinking. This is particularly true with mathematics, where I sometimes struggle to express my thoughts precisely in words. Consequently, I tend to rely on illustrations to communicate my ideas. I ask for your patience and flexibility in trying to understand the essense of my message.
Quoting keystone
Yes, that's right.
Quoting fishfry
Instead of saying that there cannot exist a "Unversal Elastic Ruler" what if I say there cannot exist a "Universal Metric"?
Quoting fishfry
Think of it like this: a hole is an emergent property. To have a hole, you first need an object that can contain a hole. In this sense, the object is more fundamental. We begin with the object, which holds the potential for a hole. Then, once we make a cut, what we have is the same object, but now with an actual hole in it.
Quoting fishfry
If you return to my photographs, you will see that I start with a continous object and put cuts in it. I call those cuts points. Just as an object is more fundamental than the hole, with my view a continua is more fundamental than the cuts (i.e. points). I used k-continua and k-points instead of continua and points because I wanted to avoid a debate over what's more fundamental. In my sandbox the continua are more fundamental. If you want to grant me that, then perhaps we can set aside all this 'k-' terminology.
Quoting fishfry
Okay, this feels like progress. Let's iron out the points discussed above and then I'll give you more details on where this is going.
If it's not obvious, I want you to know that I really appreciate you sticking with me on this.
I just don't see where you're going with all this. You're pointing out that some topological spaces aren't metrizable. Right?
Quoting keystone
Oh but there is one. For any universe or set, define a metric as follows: d(x,y) = 1 if x and y are different, and 0 if x and y are the same. This is known as the discrete metric.
You can put the discrete metric on any space of points whatsoever.
https://en.wikipedia.org/wiki/Discrete_space
Quoting keystone
There's a whole SEP article on holes. Deep stuff.
https://plato.stanford.edu/entries/holes/
Quoting keystone
I don't know what's common. Does it matter?
Quoting keystone
I did not understand the photos.
Quoting keystone
Why is "what's more fundamental" important? Do you think I hold one view versus the other? What difference does it make?
Quoting keystone
Ok I'll keep going as long as I can, but I feel like I'm going down in warm maple syrup.
So far I've got the idea that you think objects are more fundamental than holes. I just don't see why you're telling me this. Did I argue the contrary at some point?
Back to the popcorn.
Quoting fishfry
No, I'm only talking about topological metric spaces. I'm pointing out that their metrics don't extend beyond their boundaries (meaning externally, they act like topological spaces without a metric), and internally, they have entirely geometric characteristics (meaning internally, they are indistinguishable from metric spaces without the topological aspects).
Quoting fishfry
Interesting! Let's treat the Discrete Metric as a trivial metric, and by Universal Metric I'm considering only non-trivial metric.
Quoting fishfry
Wow, it's a deeper topic than I imagined.
Quoting fishfry
It turns out the photos were more helpful to me than to you. You've helped me realize that what I'm actually discussing are metrics.
Quoting fishfry
There are two primary methods for creating core mathematical artifacts:
Bottom-up Approach:
Top-down Approach:
I've observed that orthodox mathematics predominantly favors the bottom-up approach. However, my informal exploration of the top-down method has revealed a perspective where everything seems to fit together perfectly, without any apparent disadvantages, paradoxes, or unresolved issues compared to the bottom-up view. I'd like to share this perspective with you, so you can either help identify any potential flaws (I don't want to waste my time on a dead end) or guide me further (for example, I've already learned from this discussion that I should be describing them as topological metric spaces rather than elastic rulers).
A metric space is typically just called a metric space. There aren't "nontopological" metric spaces. Any metric space can be made into a topological space by defining the open sets in terms of the metric.
Quoting keystone
This is kind of muddled. Typically we start with a set and put some structure on it -- a metric, a topology, whatever. It makes no sense to talk about "outside" the space till we say what set that is. For example, what's outside the real numbers. Well the complex numbers are, but so are all the animals on Old McDonald's farm. The complement of any set is the entire rest of the universe; and if you don't say what universe you're working in, you run in to the "set of all sets" paradox. The unrestricted complement of a set is not a set. So it would be good if you could clarify this point. What's outside your metric space of interest?
Quoting keystone
Metric spaces are indistinguishable from metric spaces, yes. But isn't that a trivial remark?
And as I said, you will have trouble rigorously defining what you mean by outside of your metric space, unless you first say what the enclosing set is. So please do. By analogy, if you wish to discuss what's outside the real numbers, you have to say if you're talking about the complex numbers, the quaternions, or everything in the entire mathematical universe, which turns out to not be a set. Because the set of all sets that don't contain themselves is a member of the "outside" of the real numbers. Hope I'm making this clear.
Quoting keystone
As it happens, the trivial topology is already defined as the opposite idea. The discrete metric has the most possible open sets. The trivial metric has the fewest open sets. Only the empty set and the entire space are open.
https://en.wikipedia.org/wiki/Trivial_topology
But you can't just eliminate the one metric that falsifies your idea, there could be other weird ones. You have to say exactly what you mean.
Also I have no idea what the "universal metric" is. You have not communicated that to me.
Quoting keystone
Holes are deep!
Quoting keystone
Ok.
Quoting keystone
You just ignored my comment and steamrollered over it. Why do I care which is more fundamental? I don't even know what that means. Sets are fundamental, then you add properties. That's how it works.
Quoting keystone
Sets are fundamental, not points. Elements of sets are sometimes called points, but it's possible to do set theory without elements! All you actually need is to describe the relationships among sets, without regard for the internal contents of the sets.
Quoting keystone
I can't imagine how you would get anything done that way. And you are not getting me to believe you have a coherent idea about it.
Quoting keystone
Starting from sets, yes. Lot of mindshare the past century and a quarter. There's also type theory, which I imagine you'd see as another bottom up approach. I don't know what a top down approach to mathematical ontology would look like.
Quoting keystone
Not to me. Maybe to you. You have not yet communicated to me what is a top-down development of math. How would you top-down construct or define the real numbers? Unless you mean axiomatically. Is that what you mean?
Quoting keystone
Where's the beef? That's handwavy, tells me nothing.
Quoting keystone
I'd like to hear it. What is a top-down construction of the real numbers? Of the integers? Of the number 6?
Quoting keystone
A metric space is a metric space. If you are interested in metric spaces there's a large literature on the subject.
I don't get the top-down idea. 'Splain me please.
Point taken.
Quoting fishfry
I need to bring this one picture back.
Based on this picture, what I want to say is that Achilles can occupy any position on the continuous line, but, for this specific example where the ruler only has a few tick marks on it, I'm limited to describing his location using one of five specific intervals:
I believe what I want to do is define a 2D metric space on set S={(0,0),(0,0.5),(0.5,0.5),(0.5,1),(1,1)} where each element is an ordered pair (x1,x2).
While I will eventually explore higher dimensional spaces, for now, let's say that my sandbox is limited to sets of ordered pairs of rational numbers.
Quoting fishfry
You're right. Scratch the Universal Metric. If my metric is |x2-x1| I want to say that there is no Universal Set (within my sandbox) for which my metric yields 0 across the board. This is yet another trivial conclusion since we know that rational numbers alone cannot model a continuum.
Quoting fishfry
Is it sets all the way down or do you eventually get to points? Anyway, you don't have to answer that question. I'm willing to agree that it doesn't matter which is more fundamental. What matters is what approach yields the most powerful math. Let's move on.
Quoting fishfry
I was hoping to get closure on the open topics first, but if you don't have any problems with this post then I think we're there. By the way, if you ever feel like my time is running out then please let me know and I'll plow through. But at the current pace I'm extracting a lot of value from our conversation.
Sorry what? We're doing Zeno now? I must pass on that.
Quoting keystone
I do not know what you are talking about now.
Quoting keystone
Lost me again. In a metric space the distance between two points is 0 if and only if they are the same point.
Quoting keystone
It's sets all the way down. In set theory everything is a set.
Points are just elements of a set. Sometimes a "point" in a function space can be a function. Sometimes a point is just a tuple of coordinates in Euclidean space. Points aren't fundamental. Perhaps you're thinking of Euclid's original formulation of geometry.
You are trying to invent something more powerful than contemporary math?
Quoting keystone
I'm fine.
By the way I wanted to mention that there are often two ways of describing a mathematical object, internal and external. For example we can define the real numbers internally, by building them up from the empty set to get the naturals, integers, rationals, and finally reals.
Or, we can define the reals as the unique Dedekind-complete totally ordered set. That characterizes the reals without bothering to construct them. Perhaps you're getting at this.
You also talked about cuts, and perhaps you're interested in Dedekind cuts, which are used to construct the reals out of sets of rationals.
https://en.wikipedia.org/wiki/Dedekind_cut
You seem to want to make points out of cuts in a line, but I don't see where you're going with that.
Indulge me in an analogy.
I see the Matrix (pictures):
You see 'Digital Rain' (sets):
Both perspectives accurately correspond to the simulation. So I agree that sets are fundamental, and I could even be convinced that digital rain is more fundamental than the Matrix. But Let's not go there. I'm specifically talking about the (continuous version of the) Matrix where I believe continua are more fundamental than points. But I don't even want to debate this further, I'd rather show you what could be done with a Top-down approach and let you decide.
I bring up the Matrix because, I want you to know that I recognize the unique purity and precision of the digital rain, but there are times, especially in discussions on geometry, when it's more effective to visually interpret the geometry from within the Matrix. Please allow yourself to enter the Matrix, try to understand my visuals, just for a little while. End of Matrix analogy.
Quoting fishfry
Okay, I lost you because I made a mistake. Let me try again:
Set: { (0,0) , (0,0.5) , (0.5,0.5) , (0.5,1) , (1,1) } where x1 and y1 in element (x1,y1) is a rational number
Metric: d((x1,y1),(x2,y2)) = | (x1+y1)/2 - (x2+y2)/2 |
Quoting fishfry
Upon further consideration, I've decided to significantly restrict my focus to a smaller enclosing set. I am now interested only in what I want to call 'continuous sets' which are those sets where, when sorted primarily by the x-coordinate and secondarily by the y-coordinate, the y-coordinate of one element matches the x-coordinate of the subsequent element. For example, we'd have something like:
Quoting keystone
You're right, |x-y| doesn't qualify as a metric. Let me try again. Forget about Universal Set. Instead, I aim to define a Continuous Exact Set. A set is defined as an exact set if all elements satisfy |x-y|=0. I propose that within my enclosing set, the only Exact Set is the trivial set, containing just one element. Once again, this isn't a groundbreaking revelation; I am simply emphasizing that rational numbers by themselves are insufficient for modeling a continuum.
Quoting fishfry
Zeno greatly inspires me, yet from my viewpoint, his paradoxes serve merely as an aside. I assure you, the core thesis I'm proposing is much more significant than his paradoxes. But to save me from creating a new picture, please allow me to reuse the Achilles image below as I try again to explain the visuals.
The story: Achilles travels on a continuous and direct path from 0 to 1.
The bottom-up view: During Achilles' journey he travels through infinite points, each point corresponding to a real number within the interval [0,1].
The top-down view: In this case, where there's only markings on the ground at 0, 0.5, and 1, I have to make some compromises. I'll pick the set defined above and describe his journey as follows:
(0,0) -> (0,0.5) -> (0.5,0.5) -> (0.5,1) -> (1,1)
In words what I'm saying is that he starts at 0, then he occupies the space between 0 and 0.5 for some time, then he is at 0.5, then he occupies the space between 0.5 and 1 for some time, and finally he arrives at 1.
Quoting fishfry
Inconsistent systems allow for proving any statement, granting them infinite power. While debating the consistency of ZFC is beyond my current scope and ability, my goal is to develop a form of mathematics that not only achieves maximal power but also maintains consistency. Furthermore, I aim to show that this mathematical framework is entirely adequate for satisfying all our practical and theoretical needs.
Quoting fishfry
I haven't studied his original work, so I can't say with certainty, but I don't believe I'm referring to Euclid's formulation.
Quoting fishfry
I'm familiar with these methods. I believe there is a bottom-up and a top-down interpretation of them. I'm not satisfied with the orthodox bottom-up interpretation of them.
Quoting fishfry
I'm getting there, and your feedback has been instrumental in enhancing my understanding of this 'digital rain'. Up until now, my approach has primarily been visual.
Aside: Please note that I will have a house guest for several days, which may cause my responses to be slower than usual.
This entire idea was completely lost on me.
Quoting keystone
Digital rain is more fundamental than the Matrix. That's very poetic.
Quoting keystone
You know, it might be better if you would write a draft then edit it. This seems like stream of consciousness. It has a groovy vibe to it but it doesn't say anything.
Quoting keystone
As it happens I hate that stupid movie. It's a kung-fu flick with silly pretensions to pseudo-intellectuality. Also someone did the calculation and it turns out that humans make lousy batteries. Very inefficient.
Where is the line between your indulging yourself, and your trying to communicate a clear idea to me?
Quoting keystone
No idea what you are trying to do, what you are doing, why you are doing it, and why you are telling me about it.
Quoting keystone
Like a triangular section of the plane? Why?
Quoting keystone
I just don't know what you're doing. You seem to be having fun, and I don't mind because this like taking a rest after the mortal combat of the staircase thread.
Quoting keystone
Wasted on me, hope you got something from it.
Quoting keystone
No idea, eyes glazed long ago.
Quoting keystone
Quite a tall order.
Quoting keystone
Well Euclid considered points fundamental, along with lines and planes. But modern set-theory based math takes sets as fundamental. In fact there is nothing other than sets. You start with the empty set and the rules of set theory and build up everything else.
The word point is only used casually, to mean an element of some set, or a tuple in Euclidean space, or a function in a function space.
Quoting keystone
I'm just throwing things out that seem related to what you're saying.
Quoting keystone
I'm very glad I can help. What is the digital rain? Do you remember the Church of the Cathode Ray from the movie Videodrome?
Quoting keystone
No problem, take your time. I hope you and your guest have a lovely visit.
I'll address your other comments later, but for now, let's concentrate on one particular issue. It seems that you're either unable or unwilling to acknowledge even the most basic points I've raised. I apologize if this appears to diverge from your interests, but focusing on the image below, can you see how the instructions on the left relate to the image on the right? (This is not a trick question)
I'm unable to understand their point.
Quoting keystone
McDonalds, Sushi, Wok and Roll. Now I'm hungry.
Once again you leave me utterly baffled as to why you posted this.
I've been trying to build towards a more important point but I feel like I have to keep going simpler and simpler to find a common ground with you. I'm hoping interpreting a map is the common ground where we can start from. If you acknowledge that you understand how directions and maps work then I will advance with my point.
Please start any time. I simply have no idea what your overall point is, nor have I understood any of your examples. Start from the top. "I wish to reform the entire corpus of modern mathematics." Then tell me what are numbers, sets, functions, relations, etc.
I just can't figure out what you are doing.
Explain to me as you would if I were standing in front of you, what point you are making with the map.
I'm going to go through a few iterations of modifications to the Google Maps directions I screen captured. Please tell me where I lose you. I will make some simplifications along the way, but the essence of the screenshot remains.
Iteration 1
1) Start at point A
2) Travel the road between point A and McDonalds
3) Arrive at intermediate destination: McDonalds
4) Travel the road between McDonalds and point B
5) Arrive at destination: Point B
Iteration 2
1) Start at point 0
2) Travel the interval between point 0 and point 0.5
3) Arrive at intermediate destination: point 0.5
4) Travel the interval between point 0.5 and point 1
5) Arrive at destination: Point 1
Iteration 3
1) Start at (0,0)
2) Travel the interval (0,0.5)
3) Arrive at intermediate destination: (0.5,0.5)
4) Travel the interval (0.5,1)
5) Arrive at destination: (1,1)
Iteration 4
(0,0) --> (0,0.5) --> (0.5,0.5) --> (0.5,1) --> (1,1)
I don't even get a mention now? That's the only way I know when someone's talking to me.
I don't know what you are talking about. I swear to God, I do not understand what you are doing, what you're talking about, why you're doing this. I am totally lost. I went back over the thread, i simply don't understand what you are talking about. And you flat out refuse to tell me. At one point you were talking about Achilles, so is this something to do with one of Zeno's paradoxes?
Why can't you just give me the top-line summary of what you are doing? A while back we were talking about metric spaces and topological spaces, that at least made some sense. The rest of this, the map, the grid, I just don't know what you are doing. I don't know what is the overall point being made, what I'm supposed to be getting from this. It's very frustrating.
I'll stipulate that you can traverse a grid. Or a line. Your coordinates have two components yet appear on a straight line. That's a little odd. What is your point?
I do have one specific question. Why do your points on a straight line have two coordinates? What does that denote?
I'm taking the Google Maps directions/map and making them more 'mathematical'. Let me try iteration 0 and tell me if this is clear:
Iteration 0
1) Start at 6445-6451 Peel Regional Rd 1
2) Travel the road Erin Mills Pkwy/Peel Regional Rd 1 N towards McDonalds
3) Arrive at intermediate destination: McDonalds
4) Travel the road Millcreek Dr towards 6335-6361 Millcreek Dr
5) Arrive at destination: 6335-6361 Millcreek Dr
Do you honestly not see how this relates to the Google Maps screenshot I sent a few posts back?
Quoting fishfry
I'm developing a framework that applies topological metric spaces to describe continua with arbitrarily fine precision. This might seem esoteric, but achieving this involves turning everything upside downwithout dismissing any past mathematical progress. This approach offers a powerful new perspective on mathematics.
It begins with this map example because I want to (1) describe the continuous journey using intervals and (2) show how those intervals can be described by a topological metric space. However, you're not even letting me do step (1).
-------
Please tell me which iteration you are tripping up on: 0, 1, 2, 3, or 4?
I'm frustrated too, but I know we can make it past this first hurdle. Thanks for persisting!
I'm using interval notation. It's an interval.
I find this incredibly annoying. Can't you get to the point?
Quoting keystone
I don't see the point.
Quoting keystone
It's not esoteric, it's basic high school math. The real number line.
Quoting keystone
You aren't making a case for that.
Quoting keystone
I"ll stipulate to the real number line.
I'm still confused by your describing points on the real line with two coordinates.
Quoting keystone
Quoting keystone
I get that you start at 0, land at .5, and end up at 1. Is that sufficient for your purposes?
I still don't see why you use two coordinates to describe a point on the real line.
Quoting keystone
It's an interval?? What? You are labeling locations on the real line as intervals? That makes little sense. Google maps doesn't do that.
No, that's not right. I referred back to your picture. You described the origin on the line as (0,0). What is the meaning of that?
You described the point commonly notated as .5 as (.5, .5). What am I supposed to take from that?
In fact the notation (.5, .5) is a degenerate open interval. It denotes the empty set. If I take (.5, .5) as interval notation, there are no points at all in it. Do you see that?
So we have two specific questions on the table.
1) What does (.5, .5) represent? In standard mathematical notation, it's the empty set. At best, [.5, .5] would simply be the point .5. But why do that?
2) Don't the standard real numbers already "describe continua with arbitrarily fine precision?" [
I appreciate you asking a specific question about my explanation instead of dismissing it outright. I believe this has helped us move forward.
Quoting fishfry
Yes, it represents the point we would conventionally label 0.5.
Quoting fishfry
Step one involves defining the journey through the use of intervals. Step two entails describing these intervals within the framework of a topological metric space. To successfully carry out step two, it's crucial that all elements involved are of the same type. For instance, I assume that defining a metric on a set that includes both points and intervals is not straightforward. As mentioned earlier, rather than defining continua in terms of points, I am defining points in terms of continua, utilizing intervals (at least in the 1D case).
Quoting fishfry
Before I answer your question, I want to ensure we are on the same page. Do you understand how each of the five steps along the journey from 0 to 1 is represented by intervals, and that the union of these five intervals describes a continuous journey from 0 to 1?
You have used this expression frequently. Do you know what you are talking about? Just curious.
I always engage directly with anything I understand. I don't dismiss the rest, I clearly say I don't understand. This has been true all along.
Quoting keystone
And why do you do that, he asked ...
Quoting keystone
Journey through the use of intervals, I don't understand. Not a dismissal. Direct statement that you said something I can't understand. Can't relate it to anything in my experience or knowledge.
Quoting keystone
It's just a metric space. It's like saying that I petted my cat mammal.
Secondly, "describing these intervals within" etc? First, they're not intervals. The notation (0,0) denotes an empty interval. So you are not communicating.
Quoting keystone
You haven't defined any intervals, you have (0,0) and (.5, .5) and say these are intervals. But as intervals, they are both empty. They denote the empty set. You haven't got any intervals.
Secondly, to define a metric, you need a distance function that satisfies some properties. You haven't done that here. And there's already a perfectly good metric on the real numbers, the usual one.
The real numbers include points and intervals, and the usual metric is a perfectly good metric, and it's very easy to define. distance(x, y) = |x - y|.
Have you read this?
https://en.wikipedia.org/wiki/Metric_space
Quoting keystone
All your intervals so far are degenerate, denoting the empty set. Do you understand that?
Quoting keystone
No, because all of your interval notations denote the empty set and I can't figure out what you are doing. The usual metric on the real numbers seems perfectly satisfactory and you are somehow obfuscating it. You are being unclear. That's not a dismissal. I'm telling you that you are not saying anything clearly that I can figure out.
I can define a continuous "journey," whatever that means, using the identity function on the real numbers f(x) = x.
I'm brand new to topological metric spaces, so I might make some mistakes along the way. As I mentioned at the beginning, my idea is still in an informal stage. Nonetheless, I believe there's some value in these initial thoughts. I'm not claiming to have all the answersI'm here to learn just as much as I am to share my concept. Since you're joining in on the conversation, can you tell me if anything I'm saying makes sense to you?
First off, I want to reiterate my appreciation for your patience and for sticking with me through this. I won't repeat this in every post, but please know that I'm am always thinking it.
[EDIT: IN A LATER MESSAGE I CONCEDE THAT POINTS SHOULD BE DESCRIBED USING CLOSED INTERVALS. INSTEAD OF HAVING YOU RESPOND TO AN INCORRECT MESSAGE, I'M GOING TO EDIT THIS MESSAGE.]
Quoting fishfry
I'm defining the journey from 0 to 1 using the following 5 intervals:
1,3, and 5 correspond to points.
2 and 4 correspond to continua.
To me, it's obvious that the union of the above 5 intervals completely describes the journey from 0 to 1. Do you agree?
I'm using intervals to describe all 5 parts of the journey because I want to use intervals in my topological metric space. Let me go ahead and do this...
Set M is has following ordered pairs (not intervals) as elements:
[EDITED FOLLOWING ORDERED PAIRS VARIABLE LETTERS FROM (X,Y) TO (A,B) ACCORDING TO JGILL'S LATER FEEDBACK]
As I mentioned before, the metric between ordered pairs (a1,b1) and (a2,b2) is defined as follows:
d((a1,b1),(a2,b2)) = | (a1+b1)/2 - (a2+b2)/2 |
This metric essentially measures the distance between the midpoints of two intervals. Hopefully this clarifies why I chose to represent points as intervals.
Quoting fishfry
Partially. To fully claim that one has read and understood the material would mean exploring all the hyperlinks and the nested links within them until everything is perfectly clear. I havent done that, but I believe the first few sections sufficiently address our immediate needs.
Quoting fishfry
I'll address the real numbers once we've clarified the topics above. It's not feasible for me to provide a satisfactory response if we're not in agreement on these preliminary matters.
Paradoxically, Empirical Science "facts" are believed to be true to the extent that they are reducible to mathematical ratios, or other incorporeal abstractions. According to some interpretations of irrational Infinity though, an infinite-sided die is not impossible, only supernatural, in the sense that you can imagine it, as an ideal concept --- e.g. a perfect multidimensional sphere --- but never reach-out and grasp it, in the real world. In what sense does that set of one "imaginary die" exist? :joke:
Math Magic :
Mathematics has a similar structure to certain conceptions of magic. It requires years of studying something entirely incorporeal, it seems to exist independent of the physical realm, its very powerful and has the ability to predict and influence the world around us, and its practitioners are BIZARRE.
https://www.quora.com/What-are-some-examples-of-math-being-magical
Imaginary Dice :
A ten-sided die of Fibonacci, imaginary, and irrational numbers used to abolish intellectual property
https://rollthedice.online/en/cdice/imaginary-dice
Not much, I fear. But I stand aside and try not to impede the ongoing discussion between you and @fishfry, who is more familiar with modern math topics that I am. I am curious about the role of elasticity you describe regarding one of Zeno's paradoxes. I assume this is somehow basic to where you are headed.
Quoting keystone
Is (x1,y1) a point in the Euclidean plane? That's a standard designation. Then (x1+y1)/2 requires some kind of a projection of the y1 down upon the x-axis in order for this expression to represent a midpoint of a line segment. Or, is (x1,y1) an interval on the real line? Perhaps use (a1,b1) instead to keep the level of confusion minimized. Maybe my colleague sees something here I don't. Please carry on.
What is "irrational infinity"? Infinite sided die seems like a sphere in 3D.
Quoting Gnomon
Does magic influence the world around us? Wow, what bizarre powers I wield! :cool:
Okay, I've thought about this further and I think you're right! Do the following 5 intervals make more sense? None of them are empty anymore. For points, let's use closed intervals.
Interval 1: [0,0]
Interval 2: (0,0.5)
Interval 3: [0.5,0.5]
Interval 4: (0.5,1)
Interval 5: [1,1]
d([2,3],[1,4])=0 ? [2,3] not equal to [1,4].
Thanks. I've made this change to my earlier post and noted that it has been updated.
Quoting jgill
In a previous post I defined the elements of the enclosing set to be sets which I called 'continuous sets'. Rewriting what I mean by continuous sets, those are sets whose elements are ordered pairs (a,b) and have the following characteristics:
1) Given any pair of elements (a,b) and (b,c), where a
Returning to your example, (2,3) and (1,4) cannot both be elements of a continuous set so the set you are considering is not included in the enclosing set.
I don't know what you are talking about. You and @fishfry can sort all of this out if he is willing. Good luck.
Sorry, I misspoke. According to the opinion below, Infinity is not a natural or real number, hence the rational vs irrational labels do not apply. Does that agree with your understanding?
Since infinity is un-real and non-dimensional, I assume the infinite die would be a spheroid in all non-real non-dimensions. :joke:
Infinity neither rational not irrational :
Infinity can be expressed as any fraction a0 where a is a natural number. Due to the denominator being zero, it is not rational. An irrational number is a real number that is not rational. As infinity does not exist in the real number system, it is not irrational.
https://www.quora.com/Is-infinity-a-rational-or-irrational-number
Quoting jgill
I think the quote was merely making an analogy between Magic & Math --- not to be taken literally. However, perhaps you can apply your bizarre mathematical powers in a "possible world". :nerd:
What is the relationship between mathematics and reality?
Answer: When plugged into a possible world, mathematics gives us the tools to analyze the logically possible outcomes. Therefore, when a possible world that is expressed mathematically sufficiently aligns with reality, mathematics becomes effective at expressing relationships and outcomes.
https://brainly.ph/question/31124265
Ok, that at least makes sense. But why denote the point 0 as [0,0]? Isn't that obfuscatory and confusing?
Quoting keystone
The union covers the closed unit interval. I don't know what a journey is. If you mean what's mathematically called a path, I'm fine with that. If you mean that you can get from 0 to 1 by first going from 0 to 0, then from 0 to not quite .5, then jumping to .5, then jumping from just above .5 to just before 1 ...
How does all this jumping take place? To get from (0, .5) to 1 involves taking a limit. How do you do that in your journey-mobile?
Quoting keystone
Please stop calling it a topological metric space just as I don't call my cat my cat mammal.
Quoting keystone
How do you accomplish those limit jumps? As a set-theoretic union they're fine, but as a "journey" you have a problem. How do you get from an open interval to its limit?
Quoting keystone
Quoting keystone
Just wondering about those jumps to the limit.
Quoting keystone
How do you get from (0,0.5) to [0.5,0.5]?
Do you understand that any point in [0.5,0.5] is a nonzero distance from .5? How do you jump that gap?
Mathematically, you take a limit.
But how does your "journey" take a limit?
What should I call it?
Quoting fishfry
I did that to facilitate the straightforward definition of the metric. If you permit me to work within a metric space without necessitating an explicit definition of the metric, then I will designate it as point 0.
Quoting fishfry
I'm talking about a top-down analogue to (bottom-up) paths. By this I mean that (bottom-up) paths are defined using points (real numbers) whereas I'm defining the (top-down) 'path' using continua. I would like to use the term 'path' if you permit me to use it without implying the existence of R.
Quoting fishfry
Let me describe both the bottom-up view and the top-down view.
Bottom-up view
The journey from point 0 to point 0.5 can be constructed as follows:
Along this journey there is no finite step where we arrive at precisely 0.5. This approach requires something like a 'step omega' and to get to 0.5 requires a limit to 'jump' the gap.
Top-down view
We begin with the completed journey from point 0 to point 0.5. Some versions of how the journey can be decomposed are as follows:
The various versions correspond to how we might chose to make cuts.
For example, the journey in decomposition version 2 is [0,0] U (0,1/4) U [1/4,1/4] U (1/4,1/2) U [1/2,1/2].
Regardless of how many cuts we make (i.e. regardless of what version we're looking at), the journey is always complete. No limits are required. Limits are only required to make the top-down view equivalent to the bottom-up view (i.e. decomposition version omega = step omega).
The confusion seems to stem from you viewing the interval (0, 0.5) as an infinite collection of points (naturally, since that is a bottom-up perspective of an interval). However, from a top-down perspective, the interval (0, 0.5) represents a single object - a continuum (perhaps I should return to calling it a k-interval to avoid confusion). While this continuum indeed has the potential to be subdivided infinitely (much like an object can potentially have holes), until actual cuts are made, we cannot assert the existence of actually infinite discrete points.
Going back to the set {0 , (0,0.5) , 0.5 , (0.5,1) , 1} , all that exists are 3 points and 2 continua and for a continuous journey we advance through them in this order proceeding from one step to another without taking limits:
Step 1: Start at point 0.
Step 2: Travel the continuum (0,0.5)
Step 3: Arrive at point 0.5.
Step 4: Travel the continuum (0.5,1)
Step 5: Arrive at point 1.
[math][\pi ,\pi ]\cup (\pi ,\pi +1)\cup [\pi +1,\pi +1]\cup (\pi +1,5)\cup [5,5][/math]
exist in your system? Or are you assuming rational numbers only?
Is
[math][5,5]\cup (5,5+\varepsilon )\cup [5+\varepsilon ,5+\varepsilon ]\cup (5+\varepsilon ,5+2\varepsilon )\cup [5+2\varepsilon ,5+2\varepsilon ]\approx 5[/math]
for small epsilon?
Quoting keystone ?
Pi is just as important in the top-down view as it is in the bottom-up view. However, as with many other things, it just needs a little reinterpretation to fit into the top-down picture. As a number, pi, is inseparably tied to actual infinity, so it will need to be elevated to a higher status in the top-down view to break this connection. I hope the conversation continues long enough where we'll be ready to elaborate on this, but for now let's just say that the upper/lower bound of intervals must be (rational) numbers. No doubt, such a restriction has consequences but I hope we will eventually agree that these consequences are features, not flaws. Anyway, why do you ask about pi?
Quoting jgill
The upper and lower bounds of intervals need to be (rational) numbers. It seems you're employing epsilon in its traditional role as an infinitesimal, which does not qualify as a (rational) number. Considering epsilon's role in calculus, let me just say that with some reinterpretation, calculus can be elegantly integrated into the top-down perspective without the need for infinitesimals. This is another topic I hoe we will explore more deeply once we've addressed some of the initial considerations. But again, why do you ask about epsilon?
Quoting jgill
You're right, I meant to say point 1. Thanks for the catch. I've now fixed that post.
Quoting fishfry
Wow, it's one of my favourite films. To each their own, I suppose. It seems we view things quite differently in several respects. That's exactly why I find this conversation so valuable.
Quoting fishfry
JGill noted that using x and y for my upper/lower bounds was confusing. I think that's why you were confused with my earlier post. Hopefully using a and b is less misleading.
Quoting fishfry
I was suggesting that our discussion around topological metric spaces has warmed me up to the idea of sets being fundamental. I now believe that, if there is merit to a top-down view of mathematics, that is will be described using sets. I certainly didn't hold that view at the beginning of our conversation. I didn't watch Videodrome, it was a little before my time.
A metric space. A metric space is already a topological space, as a cat is already a mammal.
Quoting keystone
Calling 0 by the name 0 would be far less confusing. Who was it that said that when discussing transcendental matters, be transcendentally clear. Looked it up, Descartes. Smart guy. You have the burden of being as clear as you can possibly be.
Quoting keystone
If you deny the real numbers then I have no idea what 0 and .5 are, since they are real numbers. What do those symbols mean?
How do you get from (0,0.5) to [0.5,0.5]? Mathematically, you take a limit.
fishfry
Let me describe both the bottom-up view and the top-down view.
Quoting keystone
Quoting keystone
I apologize. All this makes my eyes glaze. It makes no sense to me.
And since you denied believing in the real numbers, I don't know what those symbols mean. Perhaps you can start there.
But look. I asked you this last time.
"Step 2: Travel the continuum (0,0.5)
Step 3: Arrive at point 0.5."
How do you get to .5 fom (0, .5)? Don't you have to take a limit? This is an important question. You seem to be implicitly willing to take limits, while denying the real numbers. I see that as a problem.
Even taken at face value, I fail to understand how posting stills from the movie relates to anything we're discussing. And like I said, humans make lousy batteries. So the premise of the film is wrong. I agree with it as a metaphor for media and government treating us all as tax cattle.
Quoting keystone
a and b is less confusing than x and y? I better go back and re-read the thread.
Quoting keystone
Great flick though the plot gets a little muddle in the second half. Classic Cronenberg.
Quoting keystone
Pi is a computable real number and only encodes a finite amount of information.
Quoting fishfry
0 and 0.5 have distinct positions on the Stern-Brocot tree. If we cut (0,0.5) we'll introduce a new point whose value will lie in the yellow zone. We don't need anything more than the structure of the Stern-Brocot tree to give points and intervals their expected meaning. From the bottom-up view I agree that 0 and 0.5 are real numbers, but from the top-down view 0 and 0.5 are rational numbers.
Quoting fishfry
Let me try again to explain the top-down view.
Consider the path: 0 U (0,1) U 1
This path is traversed in 3 (not infinite) steps. You start at 0, then you step to (0,1), then you step to 1. No limits needed, just 3 simples steps. You don't have to step through all the points within (0,1) because no points exist on (0,1). If you want a point between 0 and 1 then you have to cut (0,1) which will introduce a point having a value between 0 and 1.
Let me try another analogy. Hopefully you like football more than the Matrix.
Imagine a field with no hash lines, only yard lines every 5 yards.
Let's say the on the play the running back catches the ball between the 10 and 15 yard lines and advances it to the 20th yard line. As a commentator we want to be as precise as possible without giving any false information. He would say "He catches the ball between the 10 and 15 yard lines, now he's at the 15 year line, now he's between the 15 and 20 yard lines, oh and he gets tackled on the 20 yard line."
His continuous journey is described as (10,15) U 15 U (15,20) U 20.
Attempting to provide more precision than what the yard markings allow would be incorrect. Furthermore, commentating an exact play-by-play using real numbers is impossible, as real numbers do not have adjacent values and cannot be listed, which complicates precise location reporting in this context.
You're taking that as fundamental?
I like football but these picture posts aren't doing much for me. We were at least having the same conversation about getting from 0 to 1 on the real line. Then you said you don't believe in the real numbers, and then you declined to respond when I asked you twice how you get from (0, .5) to .5 without invoking a limiting process. And now you're changing the subject.
I take it you're not a fan of analogies.
Quoting fishfry
People once mocked movie scenes where detectives would enhance blurry security camera footage with a simple "refine" button, magically clarifying a suspect's face. Now, AI technology has turned that fiction into reality. Indeed, suspension of disbelief has its virtues.
Quoting fishfry
Okay, I'll watch it.
I like analogies fine. I don't understand any of yours. I thought we were making progress on at least having the same conversation when we were traversing the unit interval. Instead of engaging you're changing the subject.
I'm sure there are other ways to define the ordering of rational numbers, that's just my favorite.
Quoting fishfry
I thought I twice answered your question. Let me try again. What you don't seem to appreciate is that with the top-down view we begin with the journey already complete so halving the journey is no problem. If we already got to 1, then getting to 0.5 is no problem. You can't seem to get your mind out of the bottom-up view where we construct the journey from points, which indeed requires limits.
I think we're making progress. 2 steps forward, 1 step back.
So you believe in the rational numbers? But then the reals are easily constructed from the rationals as Dedekind cuts or equivalence classes of Cauchy sequences. If you believe in the rationals you have to believe in the reals.
Quoting keystone
You are the one who started at 0, then got to (0, .5), and then magically completed a limiting process to get to .5. I ask again, how is that accomplished?
You are the one who started at 0, remember?
I'm not quite sure what you mean by "believe in the rational numbers." From a top-down perspective, there's no need to assert the existence of either R or Q, especially since all the subsets within the enclosing 'set' are finite. If you suggest that this enclosing 'set' is infinite, then we must rethink our definition of what an 'enclosing set' actually is in this context. I was hoping to put this particular discussion aside for now, as it will likely divert attention from our main focus.
Regarding Dedekind cuts, they involve splitting the infinite set of rational numbers into two subsets. This presupposes both the existence of an infinite entity (Q) and the completion of an infinite process (the split). If one rejects the concept of actual infinity, then it's questionable whether real numbers necessarily follow from rational numbers.
However, the discussion about actual infinity and the nature of real numbers could go on endlessly. I acknowledge that these concepts are crucial for a bottom-up approach, but can we instead focus on seeing how far a top-down perspectivedevoid of actual infinities and traditional real numberscan lead us? In the top-down view, reals hold a special role, just not as conventional numbers.
Quoting fishfry
I believe the confusion arises from the dual meanings of "start" due to there being two timelines: (1) my timeline as the creator of the story and (2) the timeline of the man running from 0 to 1 within the story.
On my timeline, I start by constructing the entire narrative of him running from 0 to 1. The journey is complete from the start. I can make additional cuts to, for example, see him at 0.5. Regardless of what I do, the journey is always complete.
On the running man's timeline, he experiences himself starting at 0, travelling towards 1, and later arriving at 1.
I think you're trying to build his journey on his timeline, one point at a time. The runner would indeed believe that limits are required for him to advance to 0.5. I want you to look at it from my timeline (outside of his world), where the journey is already complete. If I want to see where he is at 0.5 I just cut his complete journey in half. Does that clarify things?
Unlike supertasks, no magic is required to complete the journey with the top-down view. Assuming you accept the Peano Axioms as a conventional framework, you're familiar with the concept of succession, which defines progression from 1 to 2 to 3, and so on. This is essentially what I'm applying as well; on the runner's timeline he progresses in succession from 0 to (0,0.5) to 0.5, and so on. Please take note, this particular succession from 0 to 0.5 involves only 2 steps. No limit is required, just as no limits are employed with the Peano Axioms.
Quoting keystone
You speak of a metric space. Precisely what are the "points" in such a space? Then explain the metric you have created giving "distances" between these points.
I'm not suggesting that a single metric space can represent all continuous 1D systems. Rather, for each continuous 1D system we generate using a top-down approach, we can define a finite set and a corresponding metric to describe it.
The now familiar example is the unit line with points at 0, 0.5, and 1. This system is composed of the following 5 intervals:
Interval 1: [0,0]
Interval 2: (0,0.5)
Interval 3: [0.5,0.5]
Interval 4: (0.5,1)
Interval 5: [1,1]
For this system, there are 3 points: 0, 0.5, and 1.
As for the corresponding metric space, the set consists of the following 5 'points':
Ordered pair 1: (0,0)
Ordered pair 2: (0,0.5)
Ordered pair 3: (0.5,0.5)
Ordered pair 4: (0.5,1)
Ordered pair 5: (1,1)
The metric is d((a1,b1),(a2,b2)) = | (a1+b1)/2 - (a2+b2)/2 |
This metric corresponds to the distance between the midpoint of two intervals.
I've mentioned this before though...
You confused me a while back. You said you don't believe in the real numbers [or some similar wording].
So I asked you, what are those symbols 0, .5, 1, and so forth? If they're not real numbers, what are they?
That's why I asked you if you believe in the rational numbers. If you do, then you have to also believe in the reals, since the reals are constructed from the rationals. If you don't, then again I ask you what are 0, .5, and 1?
Quoting keystone
You have been freely using the symbols 0, .5, and 1. If they are not real, and they are not rational, then I don't know what those symbols mean. Can you define them?
"... all the subsets within the enclosing 'set' are finite"???? Means what? Lost me there.
Quoting keystone
You're the one with some notion of enclosing set. A metric space is a set with a distance function. If it lives in a larger ambient set, then you have to say what that is. You started a long time ago saying something like "the metric doesn't apply outside the metric space." Ok that's true, but what is outside? You have to say what that is.
Quoting keystone
Ok fine. You reject the real numbers. You already said that.
So I asked you, do you believe in the rational numbers. And you asked me what I mean by that!
If you use symbols like 0, .5, and 1, you have to say what they are.
So, do you believe in the rational numbers? Is that the number system we're working in?
In which case I have to echo @jgill's excellent question as to whether you accept intervals like [pi, pi + 1], and if not, why not.
Quoting keystone
You could bring it to a quick conclusion by saying, "Yes, we are working in the rational numbers."
But you won't even say that! Leaving me totally confused.
Quoting keystone
Sure. Then what are these funny symbols 0, .5, and 1 that you keep on using? What do your interval notations denote?
If you're working in the rationals that's fine, but when I asked you about it, you asked me what I meant by the question.
Quoting keystone
The stories are very unhelpful to me. As are timelines.
Quoting keystone
What is this '0'? What is this '1'? Define your terms.
Quoting keystone
You are using these funny symbols. I know the usual standard mathematical meaning of those symbols, but you have rejected them in favor of your "top down" idea. So what are these symbols? What if we called them "fish" and "bazooka?" Then nothing at all would be clear, but your logic error would be more obvious
You want to reject standard mathematics but freely use symbols like 0, .5, and 1, without defining them.
Do you see the problem?
Quoting keystone
No, since I don't know what 0.5 and "half" mean, in the absence of standard bottom-up math.
Do you see your circularity problem? You want to start by rejecting standard math, but then you won't tell me what these symbols mean in your system.
Quoting keystone
Ah. That's quite a lot already, for someone claiming to reject infinite processes and standard bottom-up math.
So you are willing to start with the Peano axioms? Is that your starting place? Then I know what 0 and 1 are, but I'm still not sure about this 0.5 thing.
Quoting keystone
0.5 is not defined by the Peano axioms. What is it?
Quoting keystone
No idea what 0.5 is. But at least after all this you agreed to stipulate the Peano axioms. That's a start. A start from classical, bottom-up math.
I'll save you some trouble and show you how to build out the rational numbers from the Peano axioms. You extend the natural numbers to the integers, then you do a construction called the field of fractions of an integral domain.
I'm not entirely sure if that construction is legit in Peano without the axiom of infinity, but I can live with it.
So after all this, I think you are working in the rational numbers, and 0.5 has its usual meaning. Is that right?
I can live with that. Although the rational numbers are tragically deficient as a continuum. You know that, right? They're full of holes. They're not continuous in the intuitive sense.
You may not realize it but you are asking a loaded question. I believe in 'rational numbers' but not 'the rational numbers'. The difference is that 'the rational numbers' corresponds to Q, the complete set of rational numbers. With the top-down view, such completeness isn't essential (rather, consistency is the aim of the top-down approach). When constructing my metric spaces, I find that I only need to traverse a certain depth in the Stern-Brocot tree to encompass all the rational numbers I require.
To clarify, I don't believe in the existence of a complete Stern-Brocot tree. Instead, I believe in the existence of the algorithm capable of generating the tree to any arbitrary depth, although not infinitely. No one has ever encountered the entire tree; rather, we've only interacted with the algorithm and finite trees that it creates. Henceforth, let's refer to it as the Stern-Brocot Algorithm to eliminate ambiguity.
Equipped with the Stern-Brocot Algorithm, the mathematical symbols of rational numbers retain their conventional meanings. If we could execute the Stern-Brocot Algorithm to its limiting conclusion and produce the entire tree, there would theoretically exist a 'row-omega' containing the real numbers. This implies that, theoretically, real numbers necessarily follow from the rational numbers and the Stern-Brocot Algorithm. However, it's evident that running the Stern-Brocot Algorithm to completion is impossible. Consequently, the existence of real numbers doesn't necessarily follow from the existence of rational numbers.
Again, I have a strong affinity for the Stern-Brocot Algorithm, but I don't assert that it's the exclusive method to assign meaning to rationals.
Quoting fishfry
The difference lies in our perspectives on the existence of mathematical objects. I assume you are with the bottom-up majority who adhere to the belief that all mathematical entities actually exist, accessible when required, and that these objects fit neatly into sets. In contrast, my perspective maintains that no mathematical object inherently exists; it only manifests when a mind conceives of it. Therefore, if no mind currently contemplates the number 42, it does not exist in actuality; it merely holds the potential for existence.
Regarding the enclosing set, I don't subscribe to the notion of its inherent existence. Instead, I endorse an algorithm capable of generating sets to have arbitrarily many elements, albeit not infinite. If you run this algorithm long enough, it will generate the set we're looking for to define our metric space.
Quoting fishfry
I refuse to regard pi as a boundary for my intervals because it cannot be generated using the Stern-Brocot Algorithm. Pi does hold significance in my perspective, but I think it's more appropriate to delve into that explanation if/once we move on to two dimensions.
Quoting fishfry
I only referred to the Peano Axioms to point out the concept of succession. When viewed from the top-down perspective, numbers are not constructed from the naturals (I agree, that would imply a classical, bottom-up math start). Natural numbers are only distinctive in that they are positioned on the right-most branch of any tree created with Stern-Brocot Algorithm, which indeed makes them quite unique.
Quoting fishfry
I agree that rational numbers alone cannot model a continuum. With the top-down view, this is equivalent to saying that points alone cannot model a continuum. And that's why I'm starting with a continuum (i.e. using intervals rather than numbers). It's much easier to get points from a continuum than it is to get a continuum from points.
This is perhaps the second time this oddity from number theory has cropped up on this forum. I knew a tiny bit about it since it can involve elementary continued fraction theory. How did this become so important to you?
Is this a legitimate "path" ?. A linear arrangement of points and intervals.
Ordered pair 1: (3,3)
Ordered pair 2: (3,3.7)
Ordered pair 3: (3.7,3.7)
Ordered pair 4: (3.7,4.2)
Ordered pair 5: (4.2,4.2)
Ordered pair 6: (4.2,5.1)
Ordered pair 7: (5.1,5.1)
So, a "metric space" for this path consists of "points" (a,b) within this structure. For example, d((3,3.7),(4.2,5.1))=|(3+3.7)/2 - (4.2+5.1)/2| = 1.3. So you do not compare "points" from one path to another. Altering the path, even slightly, places it in another metric space. But a ms could be a subspace of a bigger ms. Just talking to myself, here.
You spoke earlier of an "elastic band". where does that come into the picture? Especially with regard to metric spaces? Can a path be circular?
I learned of it and, as I delved deeper, my amazement grew. For instance, from a paper by Niqui, I learned that the Stern-Brocot tree can serve as a basis for performing arithmetic on rational numbers. Although this method is far too inefficient and complex to be taught to children, the mere possibility of its application reinforces the idea that the Stern-Brocot tree has capabilities that are often underestimated.
Quoting jgill
I'll write the legitimate path from point 3 to point 5.1 with interval notation:
[3,3] U (3,3.7) U [3.7,3.7] U (3.7,4.2) U [4.2,4.2] U (4.2,5.1) U [5.1,5.1]
Quoting jgill
Exactly.
Quoting jgill
I initially used non-technical language due to a lack of knowledge. Through this discussion, I've learned that my actual interest lies in metric spaces. To illustrate, imagine a point as a pin that anchors a 1D continuum in place. From a bottom-up perspective, the continuum, being pinned throughout, appears rigid and static. However, from a top-down perspective, there are always gaps between the pins where the continuum can deform freely. In this way, the metric space is topological and resembles what I initially described as elastic.
Here's another perspective. Consider the following question: Where is the point 0.25 in the image below? Is it at position a, b, c, or d? As I've been suggesting, until a cut is made, there is no actual point 0.25. This isn't a rigid continuum where one could say point 0.25 definitely sits at position c, the middle 0 and 0.5 in the visual representation. All that can be stated is that the interval (0,0.5) contains the potential for point 0.25. I would even be open to saying that that interval contains the potential for aleph-1 points.
Quoting jgill
Certainly. I think that every object or concept in the bottom-up view has a counterpart in the top-down approach. It typically just needs some reimagining, often involving the transformation of an actually infinite object into a potentially infinite process.
Um ... ok ... I think ...
Quoting keystone
Ok whatever. You accept some rational numbers. Not much of a continuum you have there. You understand that, right?
Quoting keystone
There's no difference between an algorithm and the number it generates. 1/3 = .3333..., an infinite decimal, but 1/3 has a finite representation, namely 1/3
Pi has a finite representation. All the computable real numbers do.
Quoting keystone
I already understand that your mathematical ontology includes some but not all rational numbers. Can we move past this please?
Quoting keystone
Fine whatever. Enough already.
Quoting keystone
Can we move on please?
Quoting keystone
A view that has near universal mindshare, but ok, I'm a brainwashed mathematical sheep if you like.
Quoting keystone
Ah. Perhaps you would enjoy intuitionism.
[quote=Wiki]
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality.[/url]
Quoting keystone
I think you are an intuitionist. I think that's what you're getting at. Can you read the Wiki link and tell me if that's what you're getting at?
Intuitionism is closely related to constructivism, the idea that mathematical objects only exist if there's an algorithm or procedure to construct them. Intuitionism is like constructivism with an extra bit of mysticism that I can never quite grasp.
Quoting keystone
You reject the algorithm given by the Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...?
Quoting keystone
Ok. My eyes glaze a little more every time you mention the S-B tree, I have no idea why this idea has such a hold on you.
Quoting keystone
If you have a continuum but disbelieve even in the set of rationals, the burden is on you to construct o define a continuum.
Ditto.
Quoting keystone
So, a continuous deformation takes path A to path B, but inside the ms of path A? Or a new ms of path B? You might illustrate this. I'm curious about these continuous deformations in the contexts of your ideas. A topology, on the other hand . . .
Wikipedia
On those very rare occasions in which the subject arises I have felt the two to be more or less alike. But, here is what Wiki has to say:
I concur that rational numbers alone, represented as points, are insufficient for constructing a continuum. That's not the argument I'm making. You keep thinking I'm trying to build a continuum. No, I'm starting with a continuum, defined by the interval notation we have discussed, and working my way down to create points.
Quoting fishfry
Oh no, the classic debate about whether 0.9=1. I know you dislike the S-B tree but it makes the top-down and bottom-up views very clear. Maybe use some eyedrops? :P
Top-down view: 0.9 corresponds to a journey down the tree, symbolized by the string "LR". In contrast, 1 corresponds to a specific node at the top of the tree, symbolized simply by "[]". These are fundamentally different concepts: one is a potentially infinite journey with no final destination (since there is no bottom of the tree), and the other is a definitive destination. From the top-down perspective, 0.9 does not equal 1.
Bottom-up view: Using a supertask, the creation of the tree is completed and we are able to go the limit to observe the bottom row of the tree: 'row-omega'. In this case, the journey "LR" does indeed arrive at a destination (at row-omega), and that destination is precisely 1. When working with row-omega (i.e. real numbers), the journey is indistinguishable from the destination. From the bottom-up perspective, 0.9 equals 1. (Or does it equal a pumpkin?)
I'm pretty sure that you won't like my depiction of the bottom-up view as I frame it in a way that make's it clearly problematic. I'm fine with not investing further on this specific topic at this time as it really will just be a distraction from the main topic.
Quoting fishfry
I'm not questioning the mathematics itself, but rather the philosophical underpinnings of the mathematics. For instance, I recognize Cantor's remarkable contributions to math, even though I personally do not subscribe to the concept of infinite sets. His contributions have a valuable top-down interpretation.
Quoting fishfry
You make a good point. However, I'm not sure about the details of the constructivist approach - my impression is that a typical intuitionist would say that the number 42 permanently exists once we've intuited it. So while I'm hesitant to label myself hastily, I do think that broadly speaking I fit into this camp.
Quoting fishfry
I totally accept and am in awe with the algorithm. I just don't think the algorithm can be run to completion to return a number. I also don't think it has to be run to completion to be valuable.
Quoting fishfry
I agree, but isn't that what I've been doing all along? Doesn't [0,0] U (0,0.5) U [0.5,0.5] U (0.5,1) U [1,1] define a continuum? Maybe it would be valuable if you detail what you think a continuum must be. For example, will you only accept the definition if it is composed solely of points (and no intervals)?
Quoting fishfry
I'd like to move forward since we haven't yet reached the most interesting topics, but if you believe that I'm not defining a continuum, then there's no point in proceeding further.
Continuous deformations cannot alter topology (i.e. convert the path to a different path). Let's illustrate with an example: imagine a class of 100 students tasked with 2D graphing y = x^2 within the domain of [-10,10] and range of [-10, 10], including grid lines every 1 unit. Suppose you collect all the completed drawings and stack them. What are the chances that every graph aligns perfectly? Essentially zero. Some graphs might be larger, others smaller, some with lines drawn with rulers, others not. However, assuming there are no errors, each graph conveys the same underlying information. You could, theoretically, continuously deform one student's graph to match another's. It is in this sense that I mean they are all topologically equivalent. What would mess things up is if someone added an extra line, e.g. y=x. Then it is a different graph with a different topology described with a different metric space.
Do you think I'm using the term topological incorrectly?
Also, as an aside, think about what the students actually did - they started off with a blank piece of paper (a continuum) and drew lines on them (cuts). They didn't use pointillism (points) to create the graph (continuum). I believe that, in many senses, we've been approaching things top-down all along.
By the way, I just want to say that I truly appreciate our interactions on this thread. You are genuinely trying to understand what I'm saying and giving good feedback. Thanks!
Well, if you were to avoid both metric spaces and variations of the word "topology" it might mitigate what seems to be a questionable attempt to employ legitimate mathematical notions within a somewhat murky mix of ideas. However, I applaud your enthusiasm. I used to teach point set topology and metric spaces, so I am biased toward their traditional interpretations. In any event at some point you must present a clear and detailed description of your ideas that mathematicians might have reservations about but can follow the logic.
What would be a homeomorphism of [0,0]U(0,.3)U[.3,.3]U(.3,.5)U[.5,.5] ?
The late George Simmons of Colorado College wrote a marvelous book many years ago, perhaps the finest introduction to modern analysis and topology ever written: Introduction to Topology and Modern Analysis. Slowly work your way through this book and you will see why we ask so many questions. And don't mix philosophy of mathematics with the real deal. Just an I idea.
I'm not saying that [0,0]U(0,.3)U[.3,.3]U(.3,.5)U[.5,.5] is homeomorphic to anything. Rather, that interval description describes paths which can be transformed into each other via stretching and compressing, such as the following 3 paths:
If you are suggesting that perhaps I shouldn't say they're homeomorphic because I haven't formally defined how they are 'essentially the same' then technically you're right, but might using that term help convey the general idea in this informal discussion? After all, it seems easier to say homeomorphic than it is to repeatedly say 'paths that can be transformed into each other via bending, stretching, and compressing, without any cutting or gluing.'
Quoting jgill
I acknowledge that I might be using the term "topology" somewhat loosely, but based on the above comment, it seems to fit what I'm trying to describe, doesn't it? Also, am I not using metric spaces correctly?
Quoting jgill
Agreed. At this point I'm presenting a very informal idea and you and fryfish are having a really tough time following. This is the least desirable scenario for me. Here's how I see this possibly turning out:
I know you guys don't owe me anything, but I'll be lucky if you guys stick with me long enough for this to conclude in one of the first two scenarios. I have a history of working in isolation for extended of periods of time before getting feedback so this conversation is refreshing.
Quoting jgill
If you leave this thread soon, I'll still appreciate your suggestion. However, if you stay longer and gain a deeper understanding of my needs, I'll be even more grateful. Perhaps by the end, you might even suggest that I need a mathematics degree, not just a single textbook.
Quoting jgill
I don't understand why you would say this. Some of the greatest mathematicians were philosophers and vice versa. There's a lot of overlap.
It looks like you simply move the point [.3,.3] down the line segment to different (faulty) positions.
How does this affect your metric?
Quoting keystone
:cool:
Quoting keystone
Very very few contemporary mathematicians give a fig leaf about Platonic vs non-Platonic arguments or similar discussions about whether math is embedded in nature or in the mind.
Right. Constructivism is purely technical. Intuitionism is constructivism plus some kind of psychological motive or mystic woo. That's my understanding.
But you haven't got a continuum if your intervals contain only rational numbers
How can you say you exclude the real numbers, then write down an interval and call it a continuum?
Quoting keystone
No that is not what I said at all and it has nothing to do with that.
Quoting keystone
Just gonna skip it. Can't relate, don't see its relevance. I'm more focussed on what you just said: that you are "starting with a continuum" that does not include the real numbers.
I'm afraid I can't comprehend that at all.
Quoting keystone
Time is not an aspect of the tree, there is no supertask.
Quoting keystone
I don't even dislike it. I don't get the relevance of the entire subject. Tell me more about your continuum made up of only rational numbers. If we could get to the bottom of just one thing ...
Quoting keystone
You should renew your subscription :-)
Quoting keystone
I don't understand intuitionism, but you said mathematical objects come into existence via the imagination or acts of will of mathematicians (paraphrasing what you said earlier, sorry if I mis-stated it) and that reminded me of intuitionism.
Quoting keystone
Do you believe in the number 1/3 then?
Quoting keystone
Not if there are only rational numbers in the intervals. Do you understand this point? The rationals are full of holes. More holes than points in fact. Swiss cheese continuum.
Quoting keystone
Me? The continuum is the real number line. Totally workable definition. Avoids all the philosophical overhead. But the nature of a continuum is pretty deep, way beyond my knowledge of philosophy.
Quoting keystone
That's because you refuse to get there.
Quoting keystone
You said you don't have any real numbers in your intervals, only rationals, and not even all the rationals.
Do you understand that your rational continuum is full of holes? At least tell me if you understand what I mean by that. In other words the rationals contain the points 1, 1.4, 1.41, 1.4142, ... but they don't contain the square root of 2. There's a hole there.
The reals are the completion of the rationals. The reals plug up all the holes in the rationals. That's why the reals are a continuum and the rationals aren't.
You are using interval notation but you are not including the reals. Moths ate your continuum.
Perhaps you can explain to me how an interval of rationals can be a continuum in your mind.
Bottom line: Define [0, 0.5]. Because I have no idea what you mean by that notation.
Consider one of your rational intervals [0,1]. What is its length?
Well, if it were an interval of reals, it would have length 1. That's how length is defined in the real numbers.
But if it's an interval containing only rationals, then its length is 0. Why? Because of my very first post in this thread. It's countable additivity again. The length of a point is zero; and the length of an interval composed of countably many points is still 0.
That's another problem with your rational continuum. I said it's full of holes. And the problem is that all the length is stored in the holes!
Why is it faulty? That's the thing - such continuous deformations don't affect the metric. They don't affect anything at all. That's why I say those three paths are homeomorphic.
Quoting jgill
I believe that's where the issue lies. In physics, some who dismiss the quest to grasp the meaning of quantum mechanics simply say, 'shut up and calculate.' In my opinion, that approach is misguided.
Ok, this was an excellent post! I better understand your criticism. It lies in the fact that I'm using the term 'interval' in an unorthodox manner. I use the term interval to describe the objects (whatever they may be) lying between the upper and lower bounds.
Let's consider the interval (0,0.5).
From a bottom-up perspective, the objects within the interval are aleph-1 actual points.
From a top-down perspective, the object within the interval is a single continua. It doesn't contain the rational points between 0 and 0.5, it contains no points. However it holds the potential for rational number points between 0 and 0.5.
Quoting fishfry
It's only deep from a bottom-up perspective. From the top-down perspective it is elementary.
Quoting fishfry
I believe that I could use the Stern-Brocot algorithm to generate a 3 layer tree whose third layer will contain a node described by LL and having all the properties that we generally attribute to 1/3.
Quoting fishfry
The length of continuum (a,b) is b-a. So consider the continuum defined by interval (0,0.3). It's length is 0.3 for all 3 paths depicted below because all 3 are homeomorphic.
Unfortunately, your approach is a muddled mixture of traditional ideas and speculative continua. I think you need to go back to the beginnings of your efforts and truly start with continua and develop an original approach to math in which points arise from these continua, avoiding the real line entirely at first. MU has written a similar notion about continua and points. Perhaps you can put some meat on the table.
Define a continuum as an abstract entity and not in terms of the real line. As a matter of fact, use another word for your creation. State the properties of the continuum, again not referencing the real line or numbers. This is a tall order. Metric spaces and topological functions are perhaps inappropriate in this regard. I don't know. You will be going into unexplored territory.
Please stop talking about the S-B algorithm. Both my colleague and I would rather not contemplate this thing. Leave the realm of real numbers at first.
Or, do your thing and persist until the thread dries up and vanishes. Good luck.
I find myself agreeing with very little of what I've read from MU.
Quoting jgill
Quoting jgill
Quoting jgill
Quoting jgill
Seriously, am I not supposed to mention numbers? Or lines? Or relevant mathematical concepts like the S-B algorithm? I'm not sure if this is meant as advice, or if it's just a polite way of telling me to be quiet.
Quoting jgill
I'm not in the realm of real numbers. I'm not working with the real line. Could it be that your understanding of my view is more influenced by your preconceived notions than by what I'm communicating?
Quoting jgill
You dodged my question about this. Why is it faulty?
Quoting jgill
...and yet you give advice as if you do. I think your advice might be a bit premature. It seems like you haven't fully grasped what I'm trying to say, and I haven't even reached the most important parts yet. Perhaps you're ready to move on from this discussion, and those were your final thoughts. If that's the case, that's perfectly fine. Thanks and I wish you all the best.
Yay! Thanks.
Quoting keystone
I have no problem with intervals. But intervals of rationals make terrible continua.
Quoting keystone
By this I take you to mean the set of some but not all rational numbers between those values, inclusive. Yes? I say some but not all because you have said that yourself.
Quoting keystone
Not if the interval contains only rationals. It depends on how you define your notation.
As a side remark, there are [math]2^{\aleph_0}[/math] real numbers, and the question of whether that happens to be equal to [math]\aleph_1[/math] is the continuum hypothesis.
Quoting keystone
How can it be if it contains only rationals? I have challenged you on this point several times already without your providing satisfactory explanation.
Quoting keystone
No points. It's empty?
Quoting keystone
Even if I accept that, its length would be zero and it would be full of holes, hardly a continuum.
Quoting keystone
So far you have expressed strange and unjustified ideas about continua. Such as that they are empty or have length 0.
Quoting keystone
Um ... ok I guess that's a yes ...
Quoting keystone
But that interval contains only rational numbers in your notation. Its length is zero by the countable additivity of measures.
You have not grappled with this problem yet.
In fact all the length of an interval is carried by the irrationals. There aren't enough rationals to have any length at all.
I fully understand your criticism. The problem is that you are missing my point (or perhaps I should say you are missing my 'continua').
Let's continue to work with the path defined as [0,0] U (0,0.5) U [0.5,0.5] U (0.5,1) U [1,1] as depicted below.
I say that (0,0.5) and (0.5,1) contain no points so you think I'm only working with three objects - the points as depicted below. The length of all points within my system is 0 so you think the objects I'm working with have zero length.
I say that (0,0.5) and (0.5,1) describe continua so I say I'm working with 5 objects as depicted below. The length of all points within my system is indeed 0 but the length of the continua within my system add up to 1.
I prefer working with such simple paths as described above but let's do the impossible and say that somehow I could cut my unit line aleph-0 times such that there is a point for each rational number between 0 and 1.
You say that the length of all these rational points adds up to 0. I agree.
You say that there are gaps between these points. I disagree. In between each pair of neighbouring points would lie an infinitesimally small continua. If I add up the lengths of all of these tiny continua it would add up to 1. These infinitesimally small continua are indivisible.
I'm not fond of discussing impossible scenarios as they tend to lead to incorrect conclusions. Indeed, rational points do not have neighbors, and continua are inherently divisible (unless we're treating points as 0D continua, in which case they are indivisible). Therefore, we shouldn't lend too much credence to this example, but I thought it was necessary to address your points more directly.
The problem is that you're not allowing continua to be valid objects in themselves. It is as if you are only allowing points to be valid objects.
Quoting fishfry
I accept this correction.
So I figured out a better way to talk about this instead of using metric spaces. Instead, it is better to use Graph Theory.
Consider the following path:
It can be described using the following graph:
where:
vertex 0 = [0,0]
vertex 1 = (0,0.5)
vertex 2 = [0.5,0.5]
vertex 3 = (0.5,1)
vertex 4 = [1,1]
To travel from vertex 0 to vertex 4 we simply walk the connected path. One nice thing about this view is that it's clear that no limits are required to walk these graphs.
I'm not missing your point, I'm challenging it. The length of your rational intervals is zero. That causes a problem for your argument.
Quoting keystone
The length of that union is zero, if the intervals are restricted to rationals. Do you agree with that point?
Quoting keystone
No points. So they're all the empty set? I'm not supposed to push back on this?
Quoting keystone
No countable additivity? What then? How do your segments add up to 1 if they only contain rationals.
I hate to be so pedantic about this but I don't understand how an interval of rationals can have a nonzero length.
Quoting keystone
The length would still be zero. Countable additivity again.
And it's not a "unit line" if it only contains rationals. This is where your intuition is failing you.
Quoting keystone
Ok good.
Quoting keystone
There are no "neighboring points." Bad intuition again. Between any two rational numbers are infinitely many more distinct rationals.
Quoting keystone
There are no "infinitely small continua." You're just making all this up out of bad intuitions about the nature of the real numbers.
Quoting keystone
Why on earth do you troll me into arguing with your points, then admitting that you agree with me in the first place?
Quoting keystone
Of course I do. I very much believe in the continuum, which is pretty well modeled by the standard real numbers. I say pretty well because there are other models such as the constructive real line and the hyperreal line, but those lines are not Cauchy-complete. [The constructivists wave their hands at this with some technicalities].
Quoting keystone
As opposed to what? You keep hypothesizing things that do not exist, like empty continua.
[==== your second post ====]
Quoting keystone
Ok.
I am at an utter loss as to what you have been getting at all this time. Can you get to the bottom line on all this? So far I get that your "continua" are either empty or have length 0. Or that they somehow have length 1, despite being composed of only rationals.
Quoting fishfry
If an interval corresponds to a set of points (and nothing else) then I agree that an interval containing only rationals has no length.
Quoting fishfry
Our problem is that you are only allowing points in your sets. Suppose I introduce a new concept called 'k-interval' to define the set of ANY objects located between an upper and lower boundary. Would you then consider allowing objects other than points into the set?
Quoting fishfry
I wanted to show you that even if I cut my unit line to contain all rational points between 0 and 1 that there would still be stuff in between the points -- continua. Perhaps I used the wrong tactic by talking about an idea which I don't support. I did say at the start of the paragraph that it was impossible but maybe I could have been clearer.
Quoting fishfry
Yes, you believe in continua, but not as 'objects in and of themselves'. You believe that continua can't exist in the absence of points. Please confirm.
Quoting fishfry
My preference is that you accept non-points into sets, however, if you're unwilling to do that then here's an alternate approach. To move this conversation forward, let's say that when I say 'a line', you can think to yourself that I'm referring to 2^aleph_0 points (which somehow assemble to form a line), and I'll think to myself that I'm simply referring to a line (which cannot be constructed from points). In other words, you can go on thinking that points are fundamental and I'll go on thinking that lines are fundamental. How does that sound to you? All I need from you really is to allow me to restrict my intervals to those whose bounds are rational (or +/- infinity). Could you let that fly? ...Just to see how far my position can go in the absence of the explicit use real numbers (I'm fine if in your eyes their use is implied but I just won't ever mention them)...
So for example, can you allow me to say that there are 5 objects depicted below? You can go on thinking that 2 of the objects are composite objects and I'll go on thinking that all 5 objects are fundamental (they're either 0D or 1D continua).
Likewise.
Quoting keystone
But you are the one saying that you only have rationals.
Quoting keystone
In standard set theory, the only thing that sets can contain is other sets. We can call them points but that's only a word used to convey geometric intuition. Actually sets don't contain points, they contain other sets.
Quoting keystone
I don't know anything about set theory with urlements.
Quoting keystone
Only adds to my annoyance level. But that's a low bar so no worries.
Quoting keystone
Too deep for me. I don't even know what that means.
Quoting keystone
So set theory with urelements? I don't know much about that subject past the definition.
Quoting keystone
You only believe in rationals. Where are you getting these things?
Quoting keystone
If you have a line and you have the rationals, you will get the real numbers by Cauchy-completing the line.
Quoting keystone
Your idea is not coherent. If you start with a line (a thing you have declined to define) and say it's populated by the standard rational numbers by cuts, then you can construct the standard real numbers.
Quoting keystone
By bounds you mean endpoints? So now you are backing off entirely from your last half dozen points, and saying that your ontology consists of intervals with rational endpoints. But the real numbers are indeed present inside the intervals after all? Is that what you are saying?
What are these lines of yours, anyway?
Quoting keystone
You can't get anywhere as far as I can see.
Quoting keystone
You haven't given a coherent definition of these objects. All along you've been saying they are intervals of rationals. That's at least coherent, even if such intervals lack all properties of being continua.
But now only the endpoints are rational, leaving me baffled as to what those objects are.
ps -- A forum member once pointed me to the ideas of Charles Sanders Peirce (correct spelling) who said that the mathematical idea of a continuum could not be right, since a true continuum could not be broken up into individual points as the real numbers can.
Perhaps you are getting at some idea like that. Here's one link, you can Google around to find others if this interests you.
https://en.wikipedia.org/wiki/Charles_Sanders_Peirce
His ideas on continuity:
https://plato.stanford.edu/entries/peirce/#syn
No, I'm saying that within an open interval, such as (0,0.5), lies a single objects: a line. Absolutely no points exist with that interval. If you say that 0.25 lies in the middle of that interval, I will say no, 0.25 lies between (0,0.25) and (0.25, 0.5). And what this amounts to is cutting (0,0.5) such that it no longer exists anymore. In its place I have (0,0.25) U [0.25] U (0.25,0.5).
Quoting fishfry
Let's move away from directly using sets to describe the path. Instead, we'll describe the path using a graph, and then define the graph with a set.
Quoting fishfry
Urelements are indivisible 'atoms'. The lines that I'm working with are divisible.
Quoting fishfry
That is not what I believe. I can define a line using no rationals: (-inf,+inf). I see this line as a single object (a line). It is not populated by rational points. It is not populated by any points for that matter. I've drawn it for you below in between points at -inf and +inf. To walk this path from -inf to +inf you don't need limits, you just walk the corresponding graph from vertex 0 to vertex 1 to vertex 2.
You would call this green line the 'real number line'. You see this as 2^aleph_0 points. You believe that to walk any length on this green line you need limits. I understand what you're saying. We're just starting from different starting points. You're starting from the bottom and I'm starting from the top.
Quoting fishfry
Yes, I mean endpoints. I used the term 'bounds' because it is a more general term that applies to higher dimensional analogues. I'm searching for a way to keep this conversation going so it doesn't end prematurely out of frustration. Currently, I don't believe I can persuade you that a continuum can exist without points. However, I've come to realize that convincing you of this isn't necessary. Heres my new approach:
1) Start at the bottom
2) Build up to the top
3) 'Start' at the top
4) Approach the 'bottom' from the top
I see this equivalent to starting at the bottom of the S-B tree -> working our way to the top of the tree -> then proceeding back down to approach the bottom. I know you won't see it that way, which is fine. But to be clear, I still believe things are very ugly at the bottom filled with pumpkins. Nevertheless I do understand how mathematicians think things work at the bottom and if starting at the bottom is the only way you'll allow me to get to the top then I'll go with it. I understand your criticisms of starting at the top, I just don't accept them. Once you allow me to get to (3) I endeavor to show you that (3) and (4) alone fully satisfy our needs and if I'm careful (e.g. by only allowing for rational endpoints) that (1) and (2) are not only superfluous but undesirable. Is that a fair approach?
Quoting fishfry
Yes, the endpoints are rational, and the object between any pair of endpoints is simply a line. It doesn't go deeper than that. I understand you see that line as a composite object consisting of 2^aleph_0 points. However, I view the line as a primitive object. Clearly, our starting points differ. To move the discussion forward, could we agree to a compromise where we refer to a line as a "composite" object? This way, by including composite it's evident that I recognize your perspective, but the quotes indicate that my viewpoint doesn't necessitate this classification.
Quoting fishfry
I agree with this point. The issue has been the lack of viable alternatives. I see that Peirce was suggesting the use of infinitesimals, and you're aware of my stance on thosethe one from the comment where you thought I was just trolling.
I'm afraid I don't know what a line is, absent the real numbers, unless you mean the original line of Euclid, "A line is breadthless length." I'm not a scholar of Euclid so I really can't say.
I mentioned Peirce to you because it seems to me that you are interested in the "top down" definition of a continuum. I'm deeply unqualified to discuss the matter. I can only give you the standard mathematical interpretation, which is unsatisfying to both of us. I don't know enough about the philosophy of the continuum to comment.
Quoting keystone
Sigh. Your pictures don't help. What is a line? What does the notation [0, 0.5] mean?
Quoting keystone
You said that your sets contains things other than sets. You just keep making up your own terminology. I don't think we are making any progress, and I no longer know what we are discussing.
Quoting keystone
That directly contradicts what you said earlier. And I don't know what your notation means.
Quoting keystone
What is a line?
Quoting keystone
It's empty? We're going in circles.
Quoting keystone
Ok.
Quoting keystone
Ok. We're going in circles. I have no idea what you're talking about.
Quoting keystone
I'm not qualified to discuss the philosophy of the continuum with you, except as it relates to the standard mathematical real numbers.
Quoting keystone
I'm perfectly willing to believe it, I just don't know anything about it.
Quoting keystone
Was this supposed to be helpful?
Quoting keystone
Jeez man ...
Quoting keystone
Two seconds ago you denied this.
Quoting keystone
What is a line?
Quoting keystone
I did not say that, and there are other characteristics that a line must have. I am perfectly willing to adopt your ontology, if only you will state it clearly.
What is a line?
Quoting keystone
Ok. Euclid again?
Quoting keystone
You could move this forward by telling me what a line is. But I don't think I'm helping anything by sniping at your ideas in frustration.
Quoting keystone
Just giving a reference to what you seem to be getting at. A continuum that can't be divided into points.
Yes!!! I agree with Euclid's definition of lines and points. I appreciate that he provides foundational definitions of both as separate, fundamental entities. Thanks for pointing this out.
Quoting fishfry
Euclid also said that "The ends of a line are points." When I describe a path as 0 U (0,1) U 1:
(0,1) corresponds to the object of breadthless length and
0 and 1 correspond to the points at the end.
It seems that some people intepret Euclid as saying that a line without endpoints extends to infinity. I do not think this is necessarily the case. While (-inf,+inf) is a valid line, I believe (0,1) is also a valid line in and of itself.
---------------------
Please give the following figure a chance as it captures a lot of what I'm trying to say:
1) In the first row, we have a line with two endpoints, totaling three objects.
2) I can represent this path as a graph composed of three connected vertices. Notice that the lines and points are all represented as vertices in the graph.
3) I want to put the information from all vertices into a set. That's 3 objects in one set. Not just the end points.
4) When the runner travels from 0 to 1, they don't run a path composed of infinite points. They walk the graph, which in this case is the journey from vertex 1 to vertex 2 to vertex 3.
5) If you cut the line, you'll end up with five objects: the two endpoints, a middle point, and the two line segments in between. This is what we have in the second row.
6) The length of the points is zero. In fact, no matter how many times we cut the line, the total length of the points will always be zero.
7) The total length of the line segments is one. In fact, no matter how many times we cut the line, the total length of the segments will always be one.
8) Notice that in the second row, the interval (0,1) is not present because it has been cut.
9) We can continue cutting the line indefinitely, and one particular sequence of cuts is depicted across the subsequent rows.
10) Notice the pattern in the columns labeled 'length of lines' and 'path length'. As we progress downward, we approach the familiar geometric series.
11) Unlike the bottom-up approach, which requires a limit to make the summation total one, the top-down approach results in a summation that totals one at every stage.
Quoting fishfry
I believe that someone even as intelligent and knowledgeable as yourself is not qualified to discuss the bottom-up philosophy of a continuum because it is flawed. I'm 100% certain you have the capacity to understand, discuss, and criticize the top-down philosophy.
Quoting fishfry
You're right, I did say that the endpoints were necessarily rational numbers. (-inf, +inf) has no endpoints. While there are scenarios where it is useful to include points at infinity, for this discussion, let's agree that the points at -inf and +inf are not real points. I'm only using infinity as a shorthand. I should have been clearer.
Didn't I ask you about this several posts ago? Ok, Euclid's line.
Quoting keystone
Ok so you are doing classical Euclidean geometry (not modern Euclidean geometry, please note).
Quoting keystone
Euclid would not recognize that notation; and at this point in our conversation, neither do I. You have variously stated that (0,1) contains only rationals, or that it may even be empty.
In view of my new understanding that by line, you mean Euclid's line, what does the notation (0,1) mean? Euclid did not have numbers as we know them.
Quoting keystone
Utterly baffled. Utterly. Baffled. No idea what it means. 0, 0 + 0, 0 + 0 + 0, no idea what I am supposed to glean from that. And by the way, what is this "+" symbol? Have you defined it? Is this the standard + of the rational numbers?
I feel terrible ignoring these diagrams that you put so much work into, and that hold so much meaning for you. I wish I could be more helpful. I don't mean to just continue to snipe at you. It pains me. I just don't know what you are saying and have no idea how to respond.
Quoting keystone
I never claimed to be able to discuss the philosophy of the continuum. On the contrary, I've admitted that I can't. Except, that I know a bit about the real numbers, and they are the standard mathematical model of the continuum. And that counts for something.
Quoting keystone
Possibly, but not the inclination. If I could dispatch a clone, I'd have him read Peirce. I'm not a philosopher of the continuum. I'm not a philosopher at all. I only come to this forum to clarify people's mathematical misunderstandings. And it's a full time job :-)
Quoting keystone
Ok. So far, your line is Euclid's original line. Leaving undefined, your notation (0,1), which you have repeatedly pointed out is NOT the open unit interval of real numbers.
ps -- Ok I took another look at your picture. You correctly note that the sum of the lengths of the points is 0. But then you say that the sum of the lengths is 1, and I'm not sure how that follows.
Since your intervals are entirely made up of rationals, the total length must be 0.
Where is the extra length coming from?
I'm willing to let you say that the length of the interval (0,1) is 1 even though it's only made of rationals. I'll stipulate that for sake of discussion, even though it's hard to understand how it works.
But what does it all mean? I'm lost and dispirited. It's not my role in life to feel bad about myself for endlessly sniping at your heartfelt ideas.
Sorry, I didn't appreciate the point when you first mentioned it. Yes, I'm starting from classical Euclidean geometry.
Quoting fishfry
Yes. Formally the arithmetic is performed as described here (https://www.sciencedirect.com/science/article/pii/S1570866706000311) but informally it's performed using the standard method we teach kids. The formal and informal results are equivalent.
Quoting fishfry
It describes the line's potential. I'm going to provide a shorthand answer involving real numbers that I don't want you to take literally. If this explanation lands, great, otherwise forget it.
Quoting fishfry
The length of a line comes from its potential.
Quoting fishfry
Sometimes its a bit frustrating when my explanations dont connect, but this conversation is exactly what I need right now, so please dont feel bad. I'm very appreciative that you've stuck around.
Path Length = Length of Lines + Length of Points
Path Length = Length of Lines + 0
Path Length = Length of Lines
So referring to row 3 of that figure...
Path Length = Length of Lines
1 = 1/2 + 1/4 + 1/4
Ok. I wanted to say that I'll stipulate to your non-rigorous conception of a continuum of being made of tiny little continua "all the way down," with no need for actual points, if that's your idea. I think this is what Peirce is getting at.
In any event to save us some time, I'll stipulate to your vision, even if it's a bit contradictory and not totally clear.
So then what?
Quoting keystone
LOL You are committed to that idea, I'll give you that.
Quoting keystone
I'll stipulate to arithmetic on the rationals, I think we can agree on that.
Quoting keystone
Ok. With my earlier stipulation, I could say:
(a) What? That makes no sense; or
(b) I sort of get it. The line contains a frothing sea of tiny little micro-continua that are not points. Is that about right?
Quoting keystone
Well here you are in trouble. If you allow "cuts" then à la Dedekind we have the real numbers. But you don't want to go there so ok. There are cuts but not so many as to allow the reals.
Quoting keystone
Oh my. You are now allowing the reals? Ok. Maybe that's good.
Quoting keystone
I sort of get your thinking. Not sure where you're going but I'll stipulate to all this, even with the vagueness.
Of course all mathematical entities are fictional, so I can't see what the difference is between and actual and a fictional point. Once you stipulate to fictional points, they become actual by virtue of being used and accepted. Just as negative numbers and imaginary numbers once did.
The life cycle of a mathematical idea is is:
Regarded as impossible ---> Fictional but useful ---> Normal everyday stuff.
Same path taken by non-Euclidean geometry. Impossible, then Riemann's curiosity, then Einstein's platform for general relativity
So once you admit a "fictional" entity you might as well grant it actual status, since you will eventually.
Quoting keystone
But here's the thing. I said earlier that in the standard unit interval, the length is stored in the irrationals.
You are saying the exact same thing, but changing the name of irrationals to "fictionals." I don't see how that changes anything. You just changed their name but they're the same irrationals.
Quoting keystone
Ok. I'm adopting a less rigorous and more intuitive sense of what you are saying.
But you have also met me more than halfway. You have agreed finally that there ARE irrationals on the line, and that they carry, or store, the length. You just call them fictionals instead of reals. But they are the same thing.
[==== next post ====]
Quoting keystone
I'll stipulate that the length is stored in the fictionals, which I'll continue to think of as the reals till you claim otherwise.
I also think that's what Peirce was getting but that's definitely not what I'm getting at. Remember when I "trolled" you by introducing a scenario involving infinitesmals? I believe that approach aligns with Peirce's thinking and I believe it's wrong.
Quoting fishfry
You keep trying to concieve of my line as something built from smaller more fundamental elements (before points, now infinitesimals). It is not built from anything. (0,1) is one object - a line. The smaller elements emerge from the line, not the other way around.
Quoting fishfry
I'm not allowing a single real number. We can partition the S-B tree at a rational node (e.g. 1/2), but we cannot partition it at a real node (because real nodes don't exist).
Quoting fishfry
Just as you don't grant infinity actual status as a natural number, I don't grant irrational points actual status as points. After all, infinity and irrational points are inseparably linked in the S-B tree, since irrational points become actual points at row infinity. If there is no actual row infinity, there are no actual irrational points.
Quoting fishfry
The difference is that you believe individual irrationals can be isolated, whereas I think we can only access irrationals as continuous bundles of [math]2^{\aleph_0}[/math] fictional points. A mathematical 'quanta' if you will. In a 1D context, I refer to this continuous bundle as a line. And if we cut a line, we have two lines (i.e. two bundles of [math]2^{\aleph_0}[/math] fictional points). No matter how many times we cut it, we will never reduce a bundle down into individual points. Since we can only ever interact with these bundles, it is meaningless to discuss individual irrationals - they are fictions. The bundles are not. Do you see the distinction?
How do you propose to pass from a finite line to a circle, say? If you are considering topological transformations, how can you express them? Sorry for butting in, but I remain curious.
Welcome back. I've transitioned from topology to graph theory, which (in this context) maintains similar concepts but is much simpler. To convert a path graph into a cycle graph, I would use vertex identification. Not sure what you're getting at. And really this is beyond the scope of what I'm covering. Right now, I'm just focused on reinterpreting the Cartesian coordinate system.
It's the only way I can make sense of what you're saying.
Quoting keystone
[0,1] is standard mathematical notation for a particular set of real numbers. You can't fault me for bringing my preconceptions about that notation. You should use less suggestive notation.
How about "L". If you say, "I have a line, I call it L," then I can't come back and challenge you about that notation.
You want it both ways. You think you can traverse the line from 0 to 0.5 to 1, freely borrowing our high school intuitions about the real number line. And when I bite on that bait, you say, "Oh no it's not the real line!"
You call your line [0,1], you treat it as if it's the usual unit interval, and then you object when I believe you!
Quoting keystone
Ok back to no real numbers.
Quoting keystone
It's not a natural number. It's not 0 and it's not the successor of any number. I can PROVE "infinity", whatever you mean by that, is not a natural number.
Quoting keystone
Last post you started believing in the real numbers, but you called them fictitious. Now you're backing off even that.
Quoting keystone
Whatever that means.
Quoting keystone
Ok. Can't agree, can't disagree.
Quoting keystone
That's true of the real numbers as well. You know, I think you are just coming to understand the nature of the standard mathematical real numbers.
If you start with the real line and cut it any number of times, you can never isolate a point that way. Do you agree?
Quoting keystone
No, because the real numbers have the same property. I cut [0,1] in half, I get [0, 1/2) and [1/2, 1]. I'm arbitrarily placing 1/2 in the second segment.
If I cut [0, 1/2) in half, I get [0, 1/4) and [1/4, 1/2), and so forth.
No number of cuts will ever isolate a point.
So I think what is happening here is that you are coming to a better intuition of the standard real numbers. Because you can keep cutting the unit interval in half and you will never isolate a point.
Throughout our conversation, my perspective and how I express it have greatly developed, leading me to believe it's best to reformulate and clarify my position. I'll be on a short holiday for the next few days, and I'd also like to take the necessary time to gather my thoughts before responding. For now, let me make two points:
I'll reach out again in a few days. I look forward to continuing this discussion. Enjoy your weekend!
No prob, I'll be here.
Just remember: Once you stipulate to the rationals, you get the reals for free. I thought at one point that those were your fictional numbers, but then you said they weren't.
Please allow me to refine and restate my position on reals.
Grandi's series has no sum but it should be 1/2.
Analogously, I believe a line is not made of points but it should be made of [math]2^{\aleph_0}[/math] points.
Analogously, I believe a line is not modeled by numbers but it should be modeled by the real numbers.
Just as Grandi's series only sums to 1/2 in a very particular light, my view amounts to the belief that there is great mathematical value in irrationals, but that they only make sense in a very particular light - when considered collectively as bundles, rather than individual, isolated points. This is the essence of the top-down view where we start with such a bundle of 2^aleph_0 points - a line in this case - and then we make cuts to selectively isolate segments of this line. I refer to any point nested within such a bundle, as opposed to being isolated, as a potential point.
Revisiting the analogy above, when I utilize an interval to describe a range, I am referring to the underlying and singular continuous line between the endpoints, which should correspond to the set of real numbered points contained within these endpoints.
Let's look at two examples:
0D cut example - a cut of line (0,2) at 1.5:
[math](0, 2) = (0, 1.5) \cup [1.5, 1.5] \cup (1.5, 2)[/math]
Length of continua (a,b) = b-a
1D cut example - a cut of line (0,2) around [math]\varphi[/math]:
[math](0,2) = (0, \varphi - \epsilon_1) \cup [\varphi - \epsilon_1, \varphi + \epsilon_2] \cup (\varphi + \epsilon_2, 2)[/math]
Where
Potential points in [math](0, \varphi - \epsilon_1) = 2^{\aleph_0}[/math]
Potential points in [math][\varphi - \epsilon_1, \varphi + \epsilon_2] = 2^{\aleph_0}[/math]
Potential points in [math](\varphi + \epsilon_2, 2) = 2^{\aleph_0}[/math]
Length of [math](0, \varphi - \epsilon_1) = \varphi - \epsilon_1[/math]
Length of [math][\varphi - \epsilon_1, \varphi + \epsilon_2] = \epsilon_1 + \epsilon_2[/math]
Length of [math](\varphi + \epsilon_2, 2) = 2 - \varphi - \epsilon_2[/math]
I believe performing an arbitrarily small 1D cut around [math]\varphi[/math] instead of a 0D cut is more true to the Cauchy definition of [math]\varphi[/math]. With this approach, [math]\varphi[/math] is never isolated as an actual point.
What do you think?
Why on earth would you think that? It clearly has no sum, since the sequence of partial sums has no limit.
Quoting keystone
It's not made of points but it's made of points? How am I supposed to understand that?
Quoting keystone
It's not but it is?
Quoting keystone
I can't imagine what that light is. The Wiki page is misleading on that point.
Quoting keystone
Needs explanation.
Quoting keystone
Cuts as in Dedekind cuts? If you already have continuum-many points, why do you need cuts?
Quoting keystone
But that is exactly the standard view.
Quoting keystone
I don't know what an "arbitrarily small cut] means. It conflicts with your previous use of cut.
Quoting keystone
There are no arbitrarily small real numbers.
The golden ratio?
This isn't working for me.
I agree that it lacks a sum, but do you think that terms like Cesàro summation and Ramanujan summation are completely misnomers? Do you truly think that there's no meaningful way to assign a value of 1/2 to that divergent series? I'm taken aback by this, though perhaps debating Grandi's series is merely a distraction.
Quoting fishfry
I think there's a bit of confusion around what I mean by "bundle." Let me explain using an analogy. GULP. Consider a fitness membership that includes access to cardio equipment, swimming pools, sauna rooms, group classes, and more. When you join the club, you pay a single price for this all-inclusive membership bundle. This means one price covers numerous amenities. There isnt a separate charge for the sauna or the swimming pools. However, there should ideally be underlying individual prices, right? Like, when setting the bundle price, the gym owner should have calculated costs for each component. But what should have been done doesn't necessarily reflect what isa single price for the entire bundle.
Similarly, in my scenario, the bundle of interest (a line) is represented simply as (0,2). Just as there's no itemized pricing for each gym amenity, there's no infinite set detailing every coordinate on the line.
Quoting fishfry
Dedekind cuts have perfect precision. I claim that the best we can do is plan to cut an arbitrarily narrow line surrounding an irrational number. Cuts are used to decompose the bundle. Initially, the bundle price for the membership is established, and it's only afterwards that we attempt to deconstruct it into an itemized price list. Itemizing a membership can become an endless endeavor, breaking the price down into increasingly smaller fragmentsfrom the cost of each toilet to each square of toilet paper, and even down to the cost of each atom in that toilet paper. Attempting to detail a gym membership to such minute components is a fool's errand. The same goes for breaking a line into individual points.
Quoting fishfry
The process of making cuts involves two distinct phases: (1) planning the cut and (2) executing the cut.
(1) We can devise a perfect plan. During the planning phase, we dont commit to specific values for epsilon; we only recognize that it can be arbitrarily small. This stage is the realm of mathematicians.
(2) Conversely, executing the cut requires selecting specific values for epsilon, which inevitably introduces some imprecision. Applied mathematicians handle the execution, often employing approximate values for irrationals like pi, such as 3.14. While this approach might seem dirty, it's also quick, and this has allowed applied mathematicians to significantly improve the world.
Youre correct that previously, I was focused on the execution, but I've realized that the planning phase is indeed more critical for this discussion.
I think they are chainsaws, not to be trifled with by the untrained masses.
Quoting keystone
I do. But I assume it's possible that there's some clever Ramanujan insight to get some deep mathematical context that makes the equation come out. And it might even mean something. But as a factoid to be dropped into casual mathematical conversation, no. Not any more than the execrable 1 + 2 + 3 + ... = -1/12. That's also "False as stated, and true only in rarified mathematical contexts of no relevance to a general audience."
Quoting keystone
That I think there's no sensible way to assign a value to the series, regardless of whether there's some deep mathematical context? Why?
Quoting keystone
Yes.
Quoting keystone
There are fiber bundles in math. A hairbrush with bristles sticking out is a fiber bundle. Off topic but reminded me of the name.
Quoting keystone
Ok, it's an aggregate price where the components haven't necessarily been priced. So you have aggregate lengths, but no individual ones. Something like that?
Quoting keystone
You know, you can recover the real line just from its open intervals. Is that what you mean? In fact the rational intervals are good enough, the cover all the reals anyway.
I honestly think that what you are doing is coming to understand, in your own way, the nature of the real numbers. Which in my opinion is one of the most worthy endeavors a person can do. Clarifies so much formerly bad thinking about infinite processes.
Quoting keystone
Well sure, every irrational can be identified with a descending sequence of open intervals. I can locate pi in the sequence (3, 4), (3.1, 3.2), (3.14, 3.15), (3.141, 3.142), ...
Does that idea resonate with you?
What I mean "identified with" is this. I don't mean that my descending sequence of nested intervals "traps" the number pi or closes down on it.
No. I mean that the sequence itself IS the number pi. If pi didn't already exist, we'd just define it to be this sequence; or more precisely, the equivalence class of all such sequences. That's how mathematicians think. Pi doesn't have to be any particular thing. If you can define it as something that behaves the way pi is supposed to behave, then you might as well just consider it to be pi.
Does this make sense to you?
Which is why I say that once you grant me the rationals, you get the reals for free. If you believe in the rationals there's no point in trying to make the irrationals second class citizens in number country. Once you have the rationals the reals are already right there on equal footing. They're first class numbers.
You just have an ... ahem ... irrational prejudice against irrational numbers.
Quoting keystone
You do appear, to my humble and untrained eye, to be recapitulating the notion of descending down to points on the line by means of a sequence of downward nested open intervals; which is exactly the method of defining a real number as an equivalence class of Cauchy sequences. That's an alternative construction to Dedekind cuts but it gives you the same set of real numbers, since there is only one set of real numbers up to isomorphism. That's a handy thing to know. No matter how you conceive them, there is essentially only one set of real numbers.
Quoting keystone
This bit about planning and execution is a little off the mark. I'm with you descending down to points via sequences of open intervals. In math when we conceive a thing it's automatically done. Would that the rest of the world were so simple!
Quoting keystone
I see this as making an infinite sequence of descending intervals as above.
pi = {(3, 4), (3.1, 3.2), (3.14, 3.15), (3.141, 3.142), ...}
Do you relate to this at all?
Quoting keystone
Does the world seem improved to you? I'm afraid that one of my bad habits is following the news, so I can't share your enthusiasm. And a lot of the trouble I see comes directly from the data mungers trying to eat us all. Hope you're glad you asked my opinion about that :-)
Quoting keystone
In math you can do it all at once. You can decree that the number pi be located via a decreasing sequence of nested open sets converging on pi. And if you don't believe pi is really there, then no problem. You just define pi as the sequence of nested open intervals and you've got an object that, if it's not the "real" pi, is just as good. That's how they build the reals from the rationals.
This sounds a lot like gatekeeping. Cesàro summations are very simple. Nevertheless, let's set aside Grandi's series for now. It doesn't have any relevance to my position...until further on.
Quoting fishfry
Wow, I feel like a generation alpha kid trying to come up with an email address - all the good names are already taken! Anyway, I'd actually rather call the bundles 'quanta', but to avoid QM-washing this discussion I'll keep calling them bundles.
Quoting fishfry
Further. bundles have interval descriptions but individual points (within a bundle) do not. In other words, the bundle is the fundamental unit. Sure, we can perform a cut actualize a 0D bundle, such as [0.5,0.5], but that point is emergent.
Quoting fishfry
I dont believe thats the case. It seems there are three factors involved here. (1) I'm refining my ideasthank you for your assistance with this. (2) I'm improving how I communicate my ideasagain, thanks for your help. (3) You are starting to understand that my perspective doesnt undermine any established mathematics; it mainly reinterprets it (making bundles fundamental).
Quoting fishfry
ABSOLUTELY (except for the underlined part). I did note this 2 days ago when I said that 1D cuts around ? are more true to the Cauchy definition of ? than 0D cuts. I also chose the golden ratio in that message because it has a beautiful description using the SB-algorithm. Look at the figure below. Instead of focusing on the yellow tree, consider the blue intervals with the red labels. The golden ratio is not any particular interval, but rather the descending sequence of open intervals captured by the sequence ([], R, RR, RRL, RRLR, RRLRL, ...) simply written as [math]R\overline{RL}[/math].
Quoting fishfry
I've never denied the significance of irrationals. My view is simply that because irrationals are always encompassed within bundles, or rather are the bundles themselves, they differ distinctly from isolated points/rational numbers.
Quoting fishfry
Wait, I'm not proposing that an irrational is a descent down to a point. Rather, I'm proposing that irrationals are infinite descents involving arbitrarily smaller intervals. The interval never has a length of 0 whereby a single irrational point is isolated.
Quoting fishfry
Ah, okay, so you don't require a point at pi. Awesome. It seems like we're making progress. Once we're completely aligned, I'd like to explore what I believe are the unseen and surprising consequences of this perspective with you.
Quoting fishfry
Distinguishing between planning and execution is paramount. The inability to differentiate between them is precisely why there are so many infinity cranks. Cranks reject the concept of completing a supertask. On the other hand, mathematicians refuse to reject supertasks (or ideas implicitly associated with them) because they carry profound aesthetic and practical value. I find myself in the middle ground. What I suggest is that mathematicians would find complete satisfaction in merely planning the supertask, without concerning themselves with the imperfections of its incomplete execution.
Quoting fishfry
Theres a lot wrong with the world today, but would you really want to live an Amish or Mennonite lifestyle? Personally, I appreciate living in the most interesting of times, despite the uncertainty of our future.
I won't argue this point with you. I stand by it.
Quoting keystone
I can't see that you have a position. I see you as struggling to understand the nature of the mathematical continuum.
Quoting keystone
Funny.
Quoting keystone
Bundles is is. Should I think of them as tiny little wriggly micro-continua?
Quoting keystone
Ok, You have all the intervals, but no individual points. Is that right I can live with that. But I already showed that if you have all the intervals, you can derive the points as downward nested sequences of intervals.
Quoting keystone
I am not the lord high defender of the mathematical realm. I don't care if you overturn mathematics. I'm not defending mathematics. I'm trying to clarify ideas about mathematics, and trying to frame your ideas in the context of what's already known about mathematics.
Quoting keystone
Nothing showed up underlined so I don't know what you are referring to. But if you agree that a descending stack of intervals can be taken as the definition of a point, that's a major agreement between us.
Quoting keystone
It's irrational. I've joined your church. I no longer believe in irraionals.
What, now you believe in irrationals? You know the S-B tree is not the only kind of tree structure that represents the real numbers. I don't know why you are fixated on it.
Quoting keystone
I don't relate. Sorry.
Quoting keystone
Well, irrationals are downward nested stacks of intervals. That's the next best thing. Can we agree on that?
But ... so are the rationals! Right?
Quoting keystone
That's exactly what I said. There is no point. All there is is the downward nested stack. We identify the stack with the point that it represents, even if the point isn't "there."
Quoting keystone
No real number is isolated. Any interval that contains it necessarily also contains infinitely many other real numbers on either side of it. But if you mean that a point has length 0, and an interval has a positive length, the unsigned difference of its endpoints, we agree.
Quoting keystone
Yes. I don't require a point at pi. I simply define pi to be the nested stack "above" where it should be. Or more accurately, the equivalence class of all the nested stacks above where it should be.
Quoting keystone
Ok! Very much making progress. But do note that what I'm describing is actually the standard mathematical way of thinking about the real numbers. The method of nested stacks is a small variation on the method of Cauchy sequences.
Quoting keystone
Forget supertasks please. There's been enough silly talk about them.
Quoting keystone
Mathematicians in general have no interest in supertasks. They're mainly a curiosity for the computer scientists as I understand it.
Quoting keystone
I don't know why you think that supertasks are an interest of mathematicians.
Quoting keystone
It's a thought. I don't believe I'd take well to getting up at 5am to milk the bull.
Quoting keystone
I like the modern world, but I don't think that applied mathematicians are universally engaged in creating good.
You persist in searching for an infinite set made up of tiny fundamental building blocks to assemble, like a mosaic. This bottom-up approach colors your perspective, but it's not feasible to represent the interval (0,1) as a union of such micro-continuait simply doesn't work. I am seeking a construction that, at least theoretically, could be explicitly written down.
In contrast, a top-down approach begins with a singleton set that includes a large fundamental bundle to trim, like a sculpture. Each cut can split it into finitely more, smaller bundles. Although we can continue making cuts indefinitely, there is no necessity to complete a supertask and produce an infinite set of tiny micro-continua. Our strategy only needs to involve a finite number of cuts to produce the necessary elements for the computation at hand. Why do you believe it's necessary to have all the intervals?
Quoting fishfry
And I greatly appreciate this.
Quoting fishfry
Oops, I meant to edit the quote as follows with the underlined part being the part I disagree with: "I mean that the sequence itself IS the number pi".
Quoting fishfry
In our discussion, I've always acknowledged the value and beauty of irrationals. However, I believe they don't share the same status as rational numbers. Rationals correspond to singleton intervals and represent specific points, whereas irrationals correspond to non-degenerate intervals and represent lines, albeit arbitrarily small ones. You're correct, the S-B tree isn't fundamental to my perspective.
Quoting fishfry
Yes, but let me qualify my position as I think we will disagree on some details. We can execute cuts to isolate rationals within singleton intervals. We can plan to isolate computable irrationals within arbitrarily small intervals. However, even that plan alone is not feasible for non-computable irrationals. The best we could plan for is to isolate non-computable irrationals within a finite length interval. I hold this view because any plan we devise must, at least theoretically, be expressible in a finite number of characters. Besides, why would we even need to isolate non-computables? They're social creatures that like to live in large communities.
Quoting fishfry
No, I view rationals as singleton intervals. I can devise a plan to target an irrational whose midpoint is arbitrarily close to a rational, but when I actually execute the cut, I must choose a positive epsilon value, and the resulting distance between the point and the resulting line segment's midpoint will necessarily be non-zero. It is for reasons like this why it is critical to distinguish between the plan and the execution of the plan. With the top-down view, there is an inherent approximation in the act of executing a plan - a principle analogous to the Uncertainty Principle in QM.
Quoting fishfry
Excellent. This is a crucial point that I will revisit as we continue our discussion.
Quoting fishfry
I acknowledge that most mathematicians are Platonists and therefore see no necessity for supertasks. However, constructivist (and people like me) needs supertasks to arrive at the objects that Platonists consider to exist. And my stance is that supertasks are impossible, so while maintaining my perspective, I can never fully align with yours. Instead, I am interested in understanding why you think my position falls short.
Quoting fishfry
You know you've made the wrong life choices if you're trying to milk a bull.
Quoting fishfry
I don't think anyone is universally engaged in anything. What's concerning is that within the next decade, it might only take one rogue individual to cause problems for everyone else.
The nested interval construction can be explicitly written down. I perhaps am not sharing your vision here.
Quoting keystone
Finitely many cuts won't get you enough of the points. Your continuum will be full of holes. The set of real numbers approximable by finite sequences is only countably infinite.
Quoting keystone
The sequence is defined as pi. And thereafter, it might as well be taken for pi since, by suitably defining arithmetic on the set of sequence stacks, it will have all the required properties of pi.
Quoting keystone
The reals are logically constructed from the rationals. If you have the rationals you get the reals for free.
Quoting keystone
Glad to hear that!
Quoting keystone
I don't see why. You haven't mentioned algorithms, and it's the existence of certain algorithms that distinguishes computable from noncomputable reals.
Quoting keystone
Ok. But now you're saying that just because you can't express something, it doesn't exist. Maybe things exist that are beyond our ability to express.
Quoting keystone
Well, some of them can be isolated, if by that you mean defined. Most can't.
Quoting keystone
Yes. Agreed. But they can ALSO be taken to be nested stacks. And then there is no difference in status between the rationals and the irrationals.
Quoting keystone
I don't follow all this but maybe it's ok.
Quoting keystone
Ok good.
Quoting keystone
No. Supertasks are simply not part of math. Not because anyone's a Platonist or a realist or a formalist or whatever, but because the subject never comes up. It's like you're learning to drive, and you ask, well, how do I peel the apple? The answer is, peeling apples is not part of learning to drive. And you say, "Oh, is that because most drivers are Platonists?"
Makes no sense. Apples have nothing do do with driving, supertasks have nothing to do with math.
Quoting keystone
If that's true, then you are saying that supertasks are a formalism or a concept that let you reproduce standard math, while pretending that you reject parts of standard math.
I can write (0,1) as the union of arbitrarily many intervals. However, I cannot write (0,1) as the union of infinitely many intervals.
For example, consider describing (0,1) as the union of N equal-length non-degenerate intervals (plus a bunch of points).
Length of (0,1) = Length of each interval * number of intervals
Length of (0,1) = (1/N) * N
This equation is valid when N is any positive natural number, but it is not valid when N is infinity. Therefore, it is not sensible to define the interval (0,1) as the union of infinitely many intervals. And what I'm saying is that since we have to pick a finite number, why not just pick N=1?
Quoting fishfry
I don't have to cut (0,1) at all to give it length. All [math]2^{\aleph_0}[/math] points that you are looking for are there from the start, albeit bundled together in one single object. Cuts don't create length, all they do is divide length.
Quoting fishfry
Quoting fishfry
Nothing is for free, not even the rationals. When I start with path [math](-\infty, +\infty)[/math] I have no numbers. Instead I have [math]2^{\aleph_0}[/math] points bundled together in a single object. Again, with the fitness membership bundle, there's not a price for every atom (or rather every point) in the gym. There's just a price for the bundle. We don't get the price per point for free. What would that price even be - seriously? $0/point? A bottom-up pricing model is absolute nonsense.
Quoting fishfry
What is a plan if not a form of algorithm?
Quoting fishfry
I'm not suggesting that non-computable points don't exist within the (0,1) bundle; rather, I'm saying it's impossible to isolate such points. Humans have never isolated a non-computable point and never willit's simply unfeasible to come up with a plan to do so. It seems almost as if you're adopting a stance based on faith. But why? What is the need to isolate non-computable points?
Quoting fishfry
I'm not referring to a verbal description, but rather to isolating it using cuts. For instance, you might attempt to pinpoint Chaitin's number within the interval (0.007, 0.008) for a specific Turing machine. However, this interval has a finite length of 0.001. It is not feasible to devise a plan to successively refine these intervals to confine Chaitin's number within an arbitrarily narrow range.
Quoting fishfry
A point can never be perfectly represented using a line, no matter how small that line is.
Quoting fishfry
I see (0,1) as a bundle of [math]2^{\aleph_0}[/math] points. You see (0,1) as [math]2^{\aleph_0}[/math] isolated points. In order for me to get to where you are I need to make [math]2^{\aleph_0}[/math] cuts. I need to absolutely mince that line until it's made of individual objects that have no length. This requires a supertask. It's not possible. It's not sensical.
(0,1) is the union of (1/n, 1 - 1/n) as n goes to infinity. I just wrote (0,1) as the union of infinitely many open intervals.
Quoting keystone
Infinity is not a natural number.
Quoting keystone
I just showed you exactly how to do that.
Quoting keystone
Lost me entirely with this line of thought.
Quoting keystone
Ok ...
Quoting keystone
I just don't follow your idea of assigning status to various classes of numbers. That can be done historically, but not mathematically. The rationals can be extended to the reals by a logical process. If you believe in the rationals you must believe in the reals, since the reals are what you get when you plug all the holes in the rationals.
Quoting keystone
Lost me again.
Quoting keystone
Plans are far more general than algorithms. Not every plan is computable. Chaitin's Omega is "planned" if you like, but not computable. Chaitin's number illustrates the subtle distinction between definable and computable.
https://en.wikipedia.org/wiki/Chaitin%27s_constant
Quoting keystone
Chaitin's Omega is one such noncomputable number that can be specifically defined.
You're right that most of the noncomputables have no unique definition and can't be "isolated," but so what?
Quoting keystone
Chaitin again. You're wrong.
Quoting keystone
Please read the Wiki page on Chaitin's Omega. It's a noncomputable number that is perfectly understandable and specific enough, depending only on the chosen encoding of Turing machines.
Quoting keystone
Ok whatever. Not sure where this is going. All real numbers are defined by Dedekind cuts.
Quoting keystone
Did I say the contrary? I don't recall doing that.
Quoting keystone
No real numbers are isolated.
Quoting keystone
Whatever. This is depressing me a bit. I no longer know what we're talking about.
Based on your description:
I don't follow. What exactly are you combining into a union?
-----------------------------
I've summarized key aspects of my argument in the following table, where I provide two analogous examples (a fitness gym vs. a path). Can you please tell me which cells in the table you disagree with or do not understand? This will help us identify the confusion and hopefully advance the conversation. You don't have to read the column for the fitness gym. I've only included it to ensure that our thinking is grounded in reality.
Quoting fishfry
Okay, forget about 'plans'. The notion of algorithms much better captures what I'm talking about.
Quoting fishfry
My argument is that the top-down perspective lays equal claim to the irrational points. You can't claim that there are gaps in my intervals just because the non-computable irrational points in my view do not have corresponding numbers. Both of our views of a line involve the exact same [math]2^{\aleph_0}[/math] points. The difference is that my points are bundled together as a single package (thus not needing numbers) whereas your points are independent (thus needing numbers).
Quoting fishfry
You mentioned that a rational number, which is a point, can also be considered as nested stacks of intervals, essentially an arbitrarily small line. However, no matter how small that line is, it can never truly be a point.
Quoting fishfry
Interesting. It sounds like you might agree that the best one can do is isolate a small bundle around the real number. What is the length of such a bundle - positive rational? Zero? Ever shrinking?
Quoting fishfry
We don't need to discuss supertasks. They're not relevant to either of our positions.
Yes you're right, I should say that n goes from 3 to infinity here, forget 1 and 2.
So I'm unioning (1/3, 2/3), (1/4, 3/4), (1/5, 4/5), ...
You can see that if x is in (0,1), then x is in a least one (actually all but finitely many) of the sets (1/n, 1 - 1/n).
Quoting keystone
I would have to give this some thought. Would it make progress if I stipulate to your metaphysics? I don't know what to say anymore.
Quoting keystone
Yes ok, so if you have an alternate way of getting to the same real numbers, what does it matter? The real numbers are categorical. That means that any two models of the second-order real numbers are isomorphic. Second order basically means that you can express the least upper bound property.
Quoting keystone
Not a line, a nested collection of lines. The point zero is (-1, 1), (-1/2, 1/2), (-1/3, 1/3), etc.
Quoting keystone
Every interval containing a given real number, necessarily contains other real numbers. That's the definition of (not) being isolated.
Quoting keystone
That was my only point.
I'm currently feeling unwell and will reply shortly. Cheers, and thank you for continuing this dialogue.
No worries, as they say. Get well soon.
Thanks! Mostly better now.
Quoting fishfry
Allow me to further clarify my position. I can write (0,1) as the union of arbitrarily many disjoint intervals. However, I cannot write (0,1) as the union of infinitely many disjoint intervals.
Quoting fishfry
Do you think you understand my position so far (and perhaps don't agree with it) or do you have no clue what I'm proposing?
Quoting fishfry
I don't have an alternate way of getting to the real numbers. What I lay claim to is the real points, not the real numbers. Consider the ruler depicted below. It features 96 tick marks, to which we can assign 96 numbers. Yet, between each tick mark, there exists a bundle of [math]2^{\aleph_0}[/math] points to which we can assign an interval. Each bundle has a length 1/16 in. When measuring an object that falls between adjacent tick marks, the best I can do with this ruler is to assign the corresponding interval to the object. My goal is to move beyond the Cartesian coordinate system by separating the concepts of numbers from points. I see no need to assert the existence of [math]2^{\aleph_0}[/math] numbers.
Quoting fishfry
The fact that the length of each line in your sequence is getting shorter is a red herring. Every single line in your sequence is composed of exactly [math]2^{\aleph_0}[/math] points. The point count isn't converging to 1. What you've exhibited is not actually a nested collection of lines but an algorithm for generating such a collection (or at least the essence of an algorithm). This distinction is crucial because the algorithm, if executed, doesn't halt. If you chose to execute the algorithm, the best you can do is wait for a long time and interrupt it when the last line produced is sufficiently small. In other words, the output of the algorithm is an arbitrarily small line, not a point.
Quoting fishfry
Do you believe individual rational numbers can be isolated? I believe they can. I'm going to use the SB tree to illustrate my view, not because it's essential but because it's familiar. I can cut this tree such that left of the cut is (0,1/2) and right of the cut is (1/2,inf). With this cut, I've isolated 1/2. I cannot do the same for irrational numbers.
Hello Sime, thank you for your comment. It prompted me to explore point-free geometry a bit online. I found this paper that adopts intervals instead of points in its framework, which is quite relevant. I appreciate Whitehead and others' rationale behind their approach, but I must admit, as someone not deeply versed in pure mathematics, I find their concepts a bit challenging to grasp just by skimming. It seems like a thorough reading might be required to truly understand these ideas, something I'm not quite ready to dive into, especially in terms of applying it to something like the SB tree anytime soon.
Not bird flu I hope. Jeez the medical propaganda is everywhere these days. Are we all doomed? Like not eventually, but as soon as next week?
Quoting keystone
Correct, but why does that matter? (0,1) is already the disjoint union of open intervals, namely itself.
Quoting keystone
I literally have no idea what we've been talking about the past several weeks. Which makes me feel foolish sniping at it.
Quoting keystone
But points and numbers are entirely synonymous in this context. The "real line" is just the set of mathematical real numbers.
Quoting keystone
Your diagrams make my eyes glaze. I appreciate that they're meaningful to you. They are not helpful in terms of communicating with me.
Quoting keystone
Eyes glazed. I'm sorry. I feel terrible because you put so much work into these diagrams.
You actually lost me when you distinguished between the real numbers and points; since as far as I know, they are the same thing.
Quoting keystone
Actually algorithms are not relevant, because I need not have a finite description for the set of open intervals that descend to a point. It's not a computational process.
Quoting keystone
Perhaps you are using a different definition of the word. Mathematically, no rational number is isolated, because every interval containing a given rational necessarily contains others.
But if you have some other definition, you might convince yourself otherwise.
Quoting keystone
Eyes glazing like a ham on Easter.
Quoting keystone
This example is useful to you but it is not helpful to me.
Quoting keystone
If that's what you mean, then so be it.
It also demonstrates the difficulty of trying to do something original and noteworthy in mathematics. It's a very complicated game requiring perseverance and dedication. Are you in it for the long haul?
Why do you want to know?
As a sickly child, when I felt ill, I would imagine myself as heroically fighting severe illnesses, attributing my survival to extraordinary strength. Turns out, I'm just wimp. I was probably just dealing with a common cold last week. Fortunately I wasn't in tune with any of the bird flu news...anxiety doesn't usually help...
Quoting fishfry
The issue revolves around whether the part or the whole is primary.
If parts precede the whole, then logically, I should be able to union such parts to create the whole, which you acknowledge is not feasible.
Conversely, if the whole precedes the parts, then I should be capable of bisecting the whole into smaller sections, continuing to do so until I have arbitrarily small parts. This approach is feasible.
Quoting fishfry
Yes, individual points are entirely synonmymous with numbers. However, continuous bundles of points are synonymous with intervals. And what I'm saying is that it's these continuous bundles of points (described using intervals) that are fundamental, not the individual points (described using numbers). We start with a continuous bundle of points (described using an interval) and when we cut it (ie. bisect this interval), we not only create smaller continuous bundles of points but also isolate an individual point in between (described using a number). Hence, the individual points and their associated numbers emerge from the bisecting process; they do not exist as independent objects before it. Individual numbers and points are emergent.
Quoting fishfry
When I presented that table and you wrote 'I would have to give this some thought' but didn't follow up on it, is it that you don't want to consider an alternate view? The common theme throughout all of my posts (including this one) and summarized in that table is that I believe we must start with the whole and manipulate it to produce the parts. Building (or defining) the whole from the parts is hopeless. Do you understand what I mean by this?
That's why the Amish communities weren't hit hard by covid. They don't watch tv :-)
Quoting keystone
How is that issue resolved by the question of whether you can partition the open unit interval into countably infinite many pairwise disjoint open intervals? You can't, but it certainly is the pairwise disjoint union of ONE interval, and so what?
Quoting keystone
I already showed you a non-disjoint countably infinite union of open intervals that equals the open unit interval. And the interval is the disjoint union of its uncountably many points. How many decompositions do you want? Why do you insist on the one decomposition that we can't do?
Quoting keystone
You surely can't do that with countably many cuts.
Quoting keystone
I asked you earlier: Suppose that rather than snipe line by line at this paragraph, I just accept it for sake of discussion. Can we move forward? I just don't see how any of this matters. And your bisection idea doesn't work, you can't get any irrationals that way. But I believe you've already agreed with that.
Quoting keystone
I found
The common theme throughout all of my posts (including this one) and summarized in that table is that I believe we must start with the whole and manipulate it to produce the parts. Building (or defining) the whole from the parts is hopeless. Do you understand what I mean by this?[/quote]
I found the fitness gym analogy confusing and pointless. But of course your whole approach is pointless (that's a pun) so maybe I'm getting it.
When it comes to the real numbers, I do think building the parts from the whole is difficult, because you'd need uncountably many cuts. But Dedekind has already built the reals from cuts of rationals, so it can be done. But there are uncountably many cuts.
Quoting keystone
What you are doing seems to me to be more metaphysics than mathematics. And that's OK. But without studying what is accepted mathematics you have a rough road ahead if you wish to contribute to that discipline. However, there have been amateurs within the last half century who have made significant discoveries. Marjorie Rice. Thankfully, is there to help guide you.
LoL.
Quoting fishfry
I want to make it clear that a line cannot be constructed from/defined by infinite isolated points (numbers) or micro-lines (intervals). If that's clear then what is a line -->
Quoting fishfry
Yes! Forget about declaring that the line is infinite individual things and instead call it ONE thing, ONE bundle, described by ONE interval. This is an important distinction because it frees us from actual infinity allowing for a stronger foundation. We don't need [math]2^{\aleph_0}[/math] individual numbers to describe a line because we have ONE interval to describe the entire bundle.
Quoting fishfry
I can't do what? How small do you want the bundles to be? I assure you, I can divide them as small as you wish. Of course, I can never cut a line down to indivisible bundles, but I never claimed I could. Why would we even need that?
Quoting fishfry
You're right that I can't execute a cut to isolate an irrational point. However, what I can do is develop an algorithm that defines an endless cutting of the line such that the line segment containing the desired irrational point gets arbitrarily small. As we've agreed, that algorithm is the irrational. There's no need to declare that the algorithm can be run to completion to output an irrational number. The algorithm is sufficient in and of itself. And if I need a number, I can interrupt the algorithm to deliver an arbitrarily narrow interval with rational end-points and I can pick a suitably close rational number within that interval.
Now, I can't isolate a non-computable this way, but that's not a problem. The non-computable points are not missing from my view. They are included - my line is continuous. The non-computable points just cannot be isolated. But we don't need to isolate them. They fulfill their job being constrained to bundles. Do they not?
Quoting fishfry
You're right that IF I needed to completely cut my line to isolate all [math]2^{\aleph_0}[/math] points/numbers then it would require uncountably many cuts. My argument is that we don't need completeness. Let's embrace our inability to fully execute a non-halting program. Our inability to isolate everything is a feature of my view, not a flaw. After all, why do you need [math]2^{\aleph_0}[/math] isolated numbers?
Quoting fishfry
Let's lay out all countably infinite rationals in an ordered line. How many gaps are there - countably many? What is the difference between a gap and a Dedekind cut? If they are the same, how do we arrive at uncountably many cuts? The answer is that Dedekind doesn't ever execute the cut. Dedekind Cuts only make sense if they correspond to non-halting algorithms which by definition cannot be executed completely.
Quoting fishfry
I had asked whether you understood what I was saying and you said you literally have no idea. It's hard to move forward if nothing I'm saying is coming through.
Quoting fishfry
Ha. My view has points, they're just not fundamental. Points emerge when a cut is made, but the line doesn't come precut and nobody could ever completely cut a line.
Without seeing where I'm going with this, amidst all of my non-technical dialogue, it is admirable that he has stuck around for so long. Suffice it to say that I'm VERY appreciative of fishfry.
The real line is the set of real numbers. That's the modern view. Or as Euclid said, "A line is breadthless length."
Quoting keystone
Ok. One thing.
Quoting keystone
I don't know. Why is any of this important? You claim you can get to every real number via cuts, but only finitely or countably many of them. And that's not true. And the reason we need it is so we can have the real numbers.
Quoting keystone
You could not do that, for the reason that there are only countably many algorithms, and uncountably many irrationals. You haven't got enough algorithms.
Quoting keystone
There aren't enough algorithms to define all the reals. Most reals are noncomputable.
Quoting keystone
Actually that is untrue. There aren't enough algorithms.
Quoting keystone
Well, if you want to see it that way, I suppose so. But without the noncomputables, the real line is not continuous. So they are there but they're not there?
I ask again: Suppose instead of endlessly sniping at your ideas, I just agree. What then? What is the point of all this?
Quoting keystone
Well then the intermediate value theorem is false. Calculus would collapse.
Quoting keystone
They'e not isolated, I've explained that several times. But we need them for completeness! The completeness of the real numbers is the defining property of the real numbers. Without completeness they're not the real numbers.
Quoting keystone
Uncountably many. Each gap represents an irrational.
Quoting keystone
A Dedekind cut is a pair of sets of rationals. A gap is a non-rational point. They amount to the same thing, expressed differently.
Quoting keystone
Because there are uncountably many reals and only countably many rationals.
Quoting keystone
That's just nonsense. It's wrong. But again, what is the end game of all this? Suppose I stop objecting and ask you, what is the point? Actually I have asked you several times recently.
Quoting keystone
Well you've made progress. You did say something I understand, namely that Dedekind cuts must correspond to non-halting algorithms. The problem is that it's utterly false. It's just a massive misunderstanding on your part.
Quoting keystone
Ok. Fine. Now what?
No good deed goes unpunished :-)
Quoting fishfry
In later posts, I aim to demonstrate that calculus not only remains intact with my perspective, but is actually built on firmer foundations. However, before we advance I'm going summarize the essentials so far. If you understand what I'm saying (even if you don't agree) we'll be ready to proceed.
1) Initial Composition: My line consists of the same points and numbers as the real line. However, initially, the continuous points bundle together to form a line, and the continuous numbers bundle together to form an interval. Thus, we begin with a single object (a line) described by a single interval.
2) Isolation Through Cuts: A point/number can only be isolated from the line through a cut. Until the cut is executed, it is meaningless to refer to the point/number as an independent entity.
3) Rational Cuts: A rational cut corresponds to isolating a rational by bisecting the line.
4) Irrational Computable Cuts: An irrational computable cut corresponds to a non-halting algorithm that isolates an irrational computable within an arbitrarily small interval. This cut cannot be executed completely.
5) Irrational Non-Computable Cuts: These cuts don't exist. Irrational non-computables cannot be isolated.
6) Completeness: All the points are there from the start (bundled in the line) so in a sense the line is complete. However, it is impossible to fully cut the line such that all points/numbers are isolated so in a sense the isolated points/numbers are incomplete.
PLEASE try to understand the following example (including the figures!). This is essential for me to make any progress explaining why calculus doesn't collapse with my view. Notice that in these 1D examples the figures contain the same information as the unions. It contains no additional information, but when we move to 2D, the figures become much more significant.
1) I start with a line (-inf,+inf)
2) I execute a rational cut at 0 such that it's now:
(-inf,0) U 0 U (0,+inf)
3) I then plan an irrational computable cut corresponding to ? such that it's now:
(-inf,0) U 0 U (0,?-?1) U ?-?1 U (?-?1,?+?2) U ?+?2 U (?+?2,+inf)
where:
? is the familiar irrational number and ?1 and ?2 are arbitrarily small positive numbers. Their independent values are not important as they are never used in isolation. What's important is that ?-?1 and ?+?2 are rational numbers and ? lies within the arbitrarily narrow interval (?-?1,?+?2).
Do you follow what I'm saying?
This isn't going anywhere. And as I've mentioned, the real numbers are [url=https://en.wikipedia.org/wiki/Categorical_theory]categorical[/math]. That means that up to isomorphism, there is only one model of the real numbers. If you have a construction of the real numbers, it produces the same real numbers as anyone else's construction.
So I will stipulate that you have a construction of the real numbers (though I don't think you do). But then, so what? I keep asking you that. You have an alternate construction of the real numbers. But whatever you have constructed is exactly the same thing as the regular old real numbers.
Just like you can use Dedekind cuts or Cauchy sequences. You can have a preference for one or the other, but it makes no difference.
Quoting keystone
Well if you have the intermediate value theorem and the least upper bound property -- ie, completeness -- then what you have, whatever it is, is isomorphic to the standard real numbers. Perhaps you have better philosophy, or a story you find more satisfying. But logicians have shown that the real numbers are categorical. There aren't any alternate models.
Quoting keystone
No, I don't need to. If calculus works, then you have the standard real numbers.
Perhaps you are building the constructive or intuitionistic real line. Did I already mention those earlier? If so, they have a theory of computable completeness that lets them finesse the issue. Perhaps you mean that.
Quoting keystone
Skipping the next bits ...
Quoting keystone
No such thing as an arbitrarily small positive real number. Not in the standard reals, not in the constructive reals, and not in the hypereals. That's because if epsilon is a positive real number, then
0 < epsilon/2 < epsilon.
It's essential that you understand that. The real numbers are a field. You can ALWAYS divide by 2.
Quoting keystone
There is no such thing as an "arbitrarily narrow interval." I believe I've identified the exact flaw in your thinking.
Quoting keystone
I think I've refuted it. Two points to sum up:
1) The real numbers are categorical. Any two models are isomorphic; and
2) There is no such thing as an arbitrarily small positive real. That's because the real numbers are a field, in which you can always divide by 2.
I don't have an alternate construction of the complete set of isolated real numbers.
Quoting fishfry
I don't have the intermediate value theorem or the least upper bound property.
Quoting fishfry
I acknowledge that for the bottom-up view, calculus requires the complete set of isolated real numbers, the intermediate value theorem, and the least upper bound property to "work"...I use quotes because it also requires some mental gymnastics. However, that's just not the case for the top-down view. It works perfectly in absence of all of the above...including the mental gymnastics.
Quoting fishfry
Consider the following Python function:
If executed, this function will print 1, 1/2, 1/4, 1/8, 1/16, and so on to no end. By saying that there is no smallest positive number you are essentially acknowledging that this program does not halt. I agree with that. What I'm saying is that for any positive number you provide, x, I can run the code in a finite amount of time to print out a number smaller than x. In other words, it has the potential to print out a number as small as you want but it cannot actually print out the smallest positive number, any more than it can halt. Do you understand this distinction?
Moreover, without ever executing the program I can describe the function's potential. Assuming it will run for at least a little while, at any time the last number it will have actually printed, ?, necessarily cuts the line (0,1) as depicted below:
Again, until we execute the function ? doesn't hold an actual value. In this illustration, ? is simply a placeholder. The fact that I drew it approximately 1/3 between 0 and 1 is inconsequential. All that can be said is that if executed, ? will correspond to a point somewhere between 0 and 1. That's how you should interpret the drawing.
In this light, I ask that you revisit the example from my last post and see if you understand step (3) where I plan an irrational computable cut at ?. I specifically wrote plan there instead of execute because I wanted to focus on the potential of the cut, as I have done for the program illustrated above.
Quoting fishfry
I keep trying to advance forward but your responses continue to either directly or indirectly show that you're not following. If you don't understand what I'm illustrating when I plan an irrational computable cut at ? then you won't understand my 2D illustrations that demonstrate that the IVT and the LUB property are not required for the top-down view.
Then what on earth are you doing?
Quoting keystone
You claimed completeness. Do you now retract that? Or have a private definition?
Quoting keystone
How does it work? What does it do?
Quoting keystone
If executed, such a program will eventually output the same number over and over, until its computing resources run out. You are factually wrong and I hope you can see why. Even so, there is no smallest positive real number, and you have not provided an argument.
Quoting keystone
What? There is no smallest positive real number.
Quoting keystone
Not even wrong. You are talking nonsense.
Quoting keystone
There is no smallest positive real number. I see that you are confused about this.
Quoting keystone
There is no smallest positive real number. You have convinced me that you are simply confused about this point. The real numbers are a field. You can always divide by 2.
Quoting keystone
Your most recent exposition postulated a smallest positive real number. There is no such thing. There is nothing for me to follow.
But even so. I have repeatedly asked you to give me the big picture. Give me something.
What is an isolated real number?
Show us elementary calculus from the top down. I am curious.
As I said earlier, I've got the points bundled into a continuous line, but not all of the points can be isolated. So if by 'completeness' you mean a line without gaps then my line is complete. However, if by 'completeness' you mean a line that can be described as the disjoint union of infinite points/numbers then my line is incomplete.
Quoting fishfry
What you are essentially saying is that a turing machine cannot operate on an infinite memory tape since such a tape cannot exist in a finite world. Ok, you're right.
Quoting fishfry
I largely agree but I would phrase it as 'there is no smallest possible positive number'. This distinction is important if numbers are emergent but it's not worth discussing at this time.
Let me rephrase my argument to address these points you've made.
Is that more clear now?
Quoting fishfry
As always, I'm grateful for this discussion and I'm certainly not complaining, but I hope you see that I have to walk a very thin line with you. I can't talk too high level as you will ask for the beef, I can't show figures as they will make your eyes glaze over, I can't use analogies because my analogies don't stick, and when I try to talk technical you often skip over or misunderstand my ideas. Of course, it doesn't help that I'm not a trained mathematician. Again, I'm extremely grateful for this discussion, just trying to put things in perspective.
Here's the next 4 steps of my plan:
Hopefully this plan will at least give you confidence that I'm heading somewhere with all of this...
With the top-down view we start with a continuous bundle of real numbers forming a line. In other words, we have one object (an interval), not [math]2^{\aleph_0}[/math] objects (isolated real numbers). We can isolate numbers as an afterthought by cutting the line (something like a Dedekind Cut). For example, I can isolate 1 by cutting (0,2) to produce (0,1) U 1 U (1,2). Rational numbers can be isolated in this fashion. One can develop an algorithm to isolate irrational computable numbers (akin to a Cauchy sequence of intervals) but the cut cannot be executed. Non-computable numbers cannot be isolated in any sense. In other words, they will forever be stuck interior to an interval.
Quoting jgill
I hope to show you...just need to get to step 4 of the plan I outlined to fryfish.
If there are no gaps, it's complete. That's the informal definition. The official definition is that every Cauchy sequence converges. You haven't defined Cauchy sequences so I don't know.
Quoting keystone
Right, all physical computers have bounded resources.
Quoting keystone
Hmmm, murky.
Quoting keystone
I believe you are doing computer math. Well known. I believe you are describing IEEE-754 floating point numbers. There is a smallest and largest possible value. It's actually not even hardware dependent, it's in the standard.
https://en.wikipedia.org/wiki/Fixed-point_arithmetic
Or you might be describing fixed point numbers.
https://en.wikipedia.org/wiki/Fixed-point_arithmetic
Either way you're doing computer arithmetic, where there is a minimum smallest interval and even a largest possible number.
Way way short of being the real numbers. So you can do approximate calculus or discrete calculus that way if you like. I think this is what you are getting at.
Am I close?
Quoting keystone
It's ok, I have my second wind I think. Especially now that I know you're just doing computer arithmetic, fixed or floating point.
Quoting keystone
Yes! Now we're communicating :-)
Quoting keystone
If you're doing computer arithmetic or some variant, we're on the same page.
Quoting keystone
I'll stipulate that you can do discrete calculus on a computer.
https://en.wikipedia.org/wiki/Discrete_calculus
Quoting keystone
Well if we're doing computer arithmetic and some variant of discrete calculus, that's interesting to know. What do you think?
A point of clarity. Thanks. Calculus started with discrete, then moved to infinitesimal, then with technology back to discrete in some sense.
Discrete calculus is certainly important to my view but it's not what I'm talking about.
There are two steps to a cut:
Step 1: Planning the cut with an algorithm
Step 2: Executing the plan by completing the algorithm
EXAMPLE: Let's cut (0,4) by ?.
Step 1: My plan is illustrated as follows.
Step 2: It is impossible to execute the plan to infinite precision. This is where discrete calculus comes into play. On one computer, the execution of the plan might look as follows.
On a different computer step 2 might look a little different. However, regardless of the computer, the structure of the execution always corresponds to the structure of the plan as captured by the following graph (after all, what good is a plan if the execution looks nothing like it):
where 0|2|4|6 are points and 1|3|5 are lines.
This graph doesn't seem to add much value in 1D systems but when we get to 2D systems it becomes much more useful.
Quoting keystone
I am one of those and I doubt your claim, but there may be others who find it of interest. I don't see anything of substance here so far, but I may be missing the point.
Yes. This is the whole idea behind discrete calculus, right? The analyst discretizes their model such that their machine can manage the computation.
Quoting jgill
To be fair, the 1D case isn't particularly exciting. Things get much more interesting in 2D. I've concentrated on the 1D case because it provides a simpler framework to establish (though I say "simpler" with some irony, as it's taken longer than expected to reach this point).
I recommend you move on to 2D. Just a thought.
Is that right? I'm not sure if I have studied that, the pre-Newton and pre-Leibniz developments. You could be right.
What about computer arithmetic, fixed and floating point representations, smallest and largest possible values?
Quoting keystone
All that followed went way over my head.
Quoting keystone
So you are doing normal math except within the limits of a finite computational space. If not fixed/floating point, something else. But computer arithmetic regardless.
Quoting keystone
You cannot telescope down to pi on computer-limited representations of numbers. If you mean that your number pi is actually a little interval around pi with approximation bounds given by the limitations of your computer representation, I'm fine with that.
Don't see the point though.
Quoting keystone
I didn't follow much of this. Ok check that. I re-read it twice and I do not understand any of it. I'm genuinely sorry I can't be of more help. BUT ... I have an idea:
If you reject the noncomputable reals, what you have is the constructive real numbers, and the calculus based on them is called constructive real analysis.
All the explanations on the web immediately get into technicalities involving intuitionist logic, and seem like heavy reading.
My understanding is that they are just doing calculus using the computable numbers. There is one trick I know in the subject, which is that when you want to see if a sequence converges, you have to have a modulus of convergence which is a function that lets you know how well the sequence is converging.
I'm not explaining that very well, because I don't understand it very well. In fact I believe I understood it for a small while a few years ago, but I seem to have lost it. There were some advocates of constructive analysis and intuitionist logic on this site a few years ago, but they seem to have drifted away. Cauchy sequences that never converged, as it were.
But I wonder if this is what you are getting at. The numbers you define consist of a sequence and a function (or plan, or algorithm) saying how the function converges. Or something.
You know, I do think you have some good intuitions about things ... but you keep rejecting all my examples. Computer math, constructive analysis, etc.
Still ... tell me what it all means. I have had conversations with proponents of constructive analysis, and at one point I put a little effort into trying to learn it, but in the end, I didn't really get the hang of it.
It's popular these days because it's good for doing computer math and automated proof checking. There's a lot of research in related areas.
I'm sure you can appreciate the problem of substituting rational number approximations of irrational numbers too early in a computation. The best approach is to do all the manipulation first and only perform the substitution at the very end when the computation is required. I would rephrase this as follows:
This is analogous to the 2-step cutting process I outlined in my previous post. In both cases, step 2 is crude and done using computer arithmetic. It's the realm of applied mathematicians and not of interest here. I'm solely concerned with step 1.
Quoting fishfry
That's pretty much what I'm saying! But instead of talking about any particular computer (which only becomes relevant to step 2), I want to remain in step 1 and talk in general terms. As such, would you allow me to say that ? is (?-?1,?+?2), and that the value of ?1 and ?2 only need to be determined in step 2? And the following figure is simply saying that ? is somewhere between 0 and 4.
Quoting fishfry
If you say that the above figure makes sense to you, then I can show you a 2D figure, and the benefits and consequences of my perspective will hopefully become clear.
Quoting fishfry
If the noncomputables reals can describe continua it is because below the surface they rest upon a more fundamental scaffolding which can describe continua in and of itself. So no, I'm not interested in constructive real analysis. I'm interested in good ol' real analysis. I just want to place it on a stronger philosophical foundation. I think my perspective will become clearer to you when explained in 2D.
Quoting fishfry
From the outside, it may seem like this conversation isn't progressing, but your reluctance to accept my informal ideas has highlighted areas where I need to strengthen my arguments. So, you are indeed helping me a lot.
I'll bet I've conjectured and proven over a hundred theorems, almost all involving convergence/divergence of sequences and series of one sort or another and have never used this expression.
I go to Wikipedia when I encounter something in math I'm not familiar with to see what the daily average of views is - a very rough idea of how popular the topic is. My own math Wiki site gets about 19 per day, and the topic is way, way off in the margins of mathematics. However, I score higher than the 3 for this topic. But thanks for opening my mind a bit.
Constructive analysis was almost a passing thought until I read about it. I would have called myself something of a constructivist in that I rarely if ever used the excluded middle - if I postulated an entity I constructed it. But reading this description shows how far I am from contemporary mathematics. Once again, I go to Wiki to see how popular this topic is. And I find it scores a 17 - not bad, but still less than my virtually unknown page.
Quoting fishfry
Probably not. I was thinking of the ancient Greeks breaking apart a sold object and measuring the pieces to approximate the object's volume or whatever. But even Archimedes recognized the infinitesimal.
Quoting keystone
And it may be purely philosophical. Given a line segment, points in this object are purely potential, non-existent until a device is used to "isolate" them. Is that about it? If so I doubt any practicing mathematician would be interested. But math philosophers might be. A lot depends upon where you go from here. Just my opinion.
Was my mention of constructive analysis helpful?
Quoting keystone
You don't believe in the real numbers, how can you manipulate them? You think every real number can be arbitrarily approximated by an algorithm. That's false. But if your approximation only needs to be to the minimum distance in a system of computer arithmetic, then you're doing computer arithmetic.
Quoting keystone
Only if you are doing computer arithmetic.
Quoting keystone
What if none of your figures make sense to me? Your latest uses these epsilon quantities, which you've defined as the minimum possible length in a given physical computer. So you are doing computer arithmetic. Not that there's anything wrong with that! But it seems to me that's what you're doing.
Quoting keystone]
You don't even believe in the noncomputable reals. And yes, they do rest on a deeper scaffolding, namely (1) Dedekind cuts; or (2) Equivalence classes of Cauchy sequences; or (3) Forget specific constructions, just write down the axioms for the real numbers, which are categorical.
Quoting keystone
Ah. What do you think is wrong with the current philosophical foundation? And why would a mathematician care?
Quoting keystone
Glad I can help.
You did say something interesting. You are trying to patch up the philosophical basis of the real numbers. Do you understand that this is the first time that you've told me what you're doing? It's a bit of a revelation, for all the times I've said that I don't understand what you're doing.
But the real numbers are categorical. Any two models are isomorphic. So you are not going to be able to produce a "better" model of the real numbers. One representation, construction, or description gives you exactly the same set of real numbers as any other.
Modulus of convergence is a thing in constructive math. I learned about it arguing with a constructivist here a couple of years back.
Quoting jgill
Andrej Bauer wrote a paper called Five Stages of Accepting Constructive Mathematics. It might be of interest.
https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf
Where do you find these scores? Is that a Wiki feature?
Quoting jgill
He was way ahead of his time.
Click on "View History", then "Pageviews". It's a crude estimate of the popularity of a topic. For example, group theory gets 513 views per day and non-standard analysis gets 80.
I believe the following:
1) The following two algorithms (written with a finite number of characters as infinite series) correspond to e and pi:
[math]e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots[/math]
[math]\pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \cdots[/math]
2) It is possible to compute the partial sums to a finite precision (e.g. ? can approximately be represented as 3.14).
3) It is impossible to compute the complete sums to infinite precision (i.e. ? cannot be represented as an infinite decimal number).
4) The algorithm itself does not apply any restrictions on the precision (.e. imprecision is only introduced during computation).
5) To prevent imprecision from being introduced, one should work with the algorithm and delay the computation for as long as possible.
6) There are algorithms for performing arithmetic on infinite series (i.e. algorithms on algorithms).
7) It is possible to compute the partial sum corresponding to ?+e to a finite precision (e.g. ?+e can approximately be represented as
[math]e = \frac{1}{0!} +4 + \frac{1}{1!} - \frac{4}{3} = \frac{14}{3} [/math])
8) It is impossible to compute the complete sum corresponding to ?+e to infinite precision (i.e. ?+e cannot be represented as an infinite decimal number).
9) Such arithmetic algorithms itself do not apply any restrictions on the precision (i.e. imprecision is only introduced during computation).
10) To prevent imprecision from being introduced, one should work with the arithmetic algorithm and delay the computation for as long as possible.
11) One can avoid computation altogether and just speak in terms of algorithms.
Quoting fishfry
I'm taking (11) seriously and avoiding computation. By doing so, I'm not approximating anything; By sticking to the algorithms I'm working with perfect precision. While computers can work with algorithms, I'm not talking about the finite arithmetic you are referring to.
Quoting fishfry
This is false. I think that non-computable real numbers exist but only within intervals. They do not exist as isolated objects. Since numbers are isolated by cuts and cuts are described with algorithm, we cannot even describe how to isolate non-computable real numbers.
Quoting fishfry
Then I'll keep trying until you quit. It may be impossible to convince you to adopt my view, but I'll be fully satisfied if, by the end of this discussion, you can at least argue my position, even if you don't accept it.
Quoting fishfry
As described above, I care about the algorithms, not the numbers - plans, not the computations. The figure with epsilons illustrates the algorithm defining the cut corresponding to ?. As I said earlier, it illustrates the plan, not the execution of the plan. To execute the plan then I need computer arithmetic, but I'm only interested in the plan.
Quoting fishfry
The current philosophical foundation is riddled with actual infinities and paradoxes. Mathematicians have elegant ways of sweeping these paradoxes under the rug (like Russell's Paradox, Riemann's Rearrangement Theorem, the Dartboard Paradox, Zeno's Paradoxes, etc.), but they're still there. However, if you believe there's nothing under the rug, it becomes harder to convince you to care. I see a paradigm shift towards a top-down view having significant consequences across philosophy, especially in the interpretation of quantum mechanics. Such a statement might not seem 'beefy', but let me just say that truth has a history of being useful, even if its utility isn't immediately apparent when it's uncovered.
Quoting fishfry
I said things like...
...but I should have explicitly said that I'm trying to patch up the philosophical basis of the real numbers.
Quoting fishfry
Numbers are the objects of computation, while algorithms are the objects of plans. I aim to shift the concept of reals from numbers to algorithms, from computations to plans. As such, I'm not proposing an alternate number model of the reals. I'm proposing an algorithmic model of reals. This model is structured very different. For one, while the real numbers are used to construct/define the real line, the real algorithms are used to deconstruct/cut the real line. However, I endeavor to show that switching to the top-down view has absolutely no impact on applied mathematicians, even those working with calculus.
At the heart of my view is a simple idea: that infinity is a potential, not an actual. This idea leads to many consequences, one being that points are potential until isolated, as you have noted. However, I hope to show that there are far more interesting consequences to this small idea.
Sometimes a small idea can have huge consequences. For example, at the heart of relativity is the simple idea that the laws of physics are the same for all observers, regardless of their relative motion. The importance of relativity speaks for itself.
And no, I'm not comparing myself to Einstein. I'm just saying that even significant consequences can follow from simple ideas.
That is really cool, thanks! I think I'll be hooked on looking these up now.
For those in the profession who do not deal with transfinitisms and set theory or foundations it's likely they would agree. When I say that a sequence converges to a number as n goes to infinity I simply mean n gets larger without bound. I don't think I have ever spoken of infinity as a number of some sort, although in complex variable theory one does speak of "the point at infinity" in connection with the Riemann sphere. But I am old fashioned.
I have no problem with you identifying the various computable real numbers with any one of the many algorithms that generate their decimal digits. Or, as a mathematician might do, identify the number with the equivalence class of ALL such algorithms, so that the real number does not depend on which algorithm you choose.
However, you can't do the noncomputable numbers that way. And there are a lot of them.
I'm not sure how your idea of approximation works. If your computer has to truncate the series, then you are only defining an interval around pi or e. You are not being exact enough.
Am I following you?
Quoting keystone
None of that adds anything. If you are defining pi by some interval around it, I'm sure you can get it all to work. It's like error bars in engineering calculations.
Wait I think I've got it. You are doing Engineering math. In particular, when you have a number x, you also have error bars, so it's really x plus or minus a little wiggle room. And you are taught how to calculate that way, how to calculate with the error bars.
Is that helpful? Is that what you are doing?
https://en.wikipedia.org/wiki/Engineering_mathematics
Quoting keystone
You seem to be going off in directions. Those lists, they don't tell me much. I get that you are doing arithmetic with error bars. That's good, that's how they build bridges.
Quoting keystone
Ok. So in your ontology, there are:
* Computable numbers, which have algorithms, or are identified with their algorithms, or are found by executing their algorithsm. Not sure which of those you mean but they're all about the same. But each computable number is the number that WOULD be computed if you finished executing the algorithm, but you can't; so each computable number is a number inside a little interval.
Have I got that right? And
* Inside each of these computable intervals, live all the noncomputable numbers.
I think all that's fine. In fact I could go you one better. Consider for each computable number [math]c[/math], the set of open intervals [math](c - \frac{1}[n}, c + \frac{1}[n})[/math], for n = 1, 2, 3, ...
That would give you a countable set of open intervals whose union is the real numbers, including the noncomputables. But you'd never have to "identify" a noncomputable. And in fact each of the endpoints [math](c \pm \frac{1}[n}[/math] are themselves computable.
What do you think of that? Would you say that's a reasonably fair mathematical model of your idea?
Quoting keystone
Why? The diagrams make it harder for me to read your posts. I think the diagrams are important to YOU. But they are not in general doing me much good. What if the overabundance of diagrams was increasing the likelihood I'd quit? You can see that under that hypothesis, you are acting against your own interests by battering me with diagrams.
Quoting keystone
How does battering me with diagrams help? I'm trying to understand your view and your diagrams in general don't reach me and they also make it more difficult for me to read your posts.
I do think the diagrams are very helpful for YOU. So you should do them, just be judicious in how often you include them in posts.
Quoting keystone
So far I get that your system involves little intervals centered at the computable numbers. I think intervals of 1/n on each side of each computable gives the same topology as your idea of using all the truncations of the series given by the algorithms.
Are we on the same page here? I really feel that we are.
Quoting keystone
You have infinitely many truncations and infinitely many little open intervals around each computable.
And you surely aren't going to resolve the standard set-theoretic paradoxes with your intervals. I don't see the connection at all.
Quoting keystone
Viewing the real numbers as the union of a bunch of open intervals centered at the computable numbers isn't going to resolve those.
Please feel free to show me how to resolve any paradoxes with what you've talked about so far.
Quoting keystone
The paradoxes have been resolved mathematically over a century ago. The philosophical paradoxes are in fact of interest to me, but not of high interest.
But I am not saying there are no problems. I'm saying that I can't imagine how your computables-and-intervals idea solves anything. Please give me an example of how this would work.
Quoting keystone
Russell's paradox and QM as well? Please, show me how this is supposed to work.
Quoting keystone
None of those things have to do with the philosophy of math or with problems thereof. Calculus was formulated without the need for infinitesimals in the late nineteenth century. It's a terrific intellectual achievement, but it's a solved problem.
Quoting keystone
Yes it would have been helpful. Glad we've established that, I find it quite helpful to understand what it is you are doing.
Quoting keystone
Ok. So as far as I get this: The real numbers are made up of a bunch of open intervals centered at the computable reals. Is that right? And FWIW I think your truncated algorithm idea will give the same reals as my plus/minus 1/n intervals.
I agree. I keep hoping for an interesting idea to appear, but so far there is nothing novel about the mathematics. If one studies existing mathematics one begins to get a recognition of what has been established. Exploration is the soul the subject, but one does not explore the heart of Africa by strolling around city park. Sorry . Perhaps when you present your ideas in 2D instead of (rather boring)1D (and the mind-numbing SB Tree) something of interest will appear. Philosophically, however, your ideas of potential points may go somewhere, but I don't know what has been done along those lines.
Yes and no.
Yes - The person tasked to execute the cut is an engineer doing engineering math. He knows he'll never be able to cut the line exactly at ? so he cuts an interval containing ? to give him wiggle room - kind of like a safety factor.
No - The person tasked to generalize all engineer actions is a mathematician. Instead of assuming any particular engineer, the mathematician aims to describe the actions of the 'arbitrary engineer'. Instead of saying that the interval width is any particular value, the mathematician just says that the interval width is ?2-?1, where ?1 and ?2 can be any arbitrarily positive number.
The cut of (-?,+?) at ? is generalized as (-?,?-?1) U ?-?1 U (?-?1,?+?2) U ?+?2 U (?+?2,+?)
Quoting fishfry
Computable reals are identified with their algorithms.
Computable rationals are found by executing their algorithms.
Quoting fishfry
Yes. I would like to distinguish between real numbers and real algorithms. A computable real number WOULD be computed if you finished executing the corresponding real algorithm, but you can't; so, the real algorithm only ever defines an interval within which the real number is inside. No real number can ever be isolated.
Quoting fishfry
I want to avoid talk of the existence of an actually infinite set. We need to frame it in terms of the potential to create an arbitrarily large set. It is very important that the endpoints be rational, otherwise nothing is gained by defining ? using intervals.
Quoting fishfry
Point taken. I will be more judicious. SB-tree aside, I will grant that I didn't need to use a single diagram for the discussion so far. Interval notation would have been entirely sufficient. I was just hoping that you would warm up to 1D diagrams because when I go to 2D it will be very hard for me to describe what I'm thinking with words. I suppose I'll cross that bridge when we get there.
Quoting fishfry
It almost sounds like you're suggesting that I'm saying that (-?,+?) is the union of infinite little intervals. It is not. With the top-down view, we don't construct (-?,+?), rather we start with it. Engineer1 may cut (-?,+?) five times. Engineer2 may cut (-?,+?) five million times. What the mathematician would say is that the 'arbitrary engineer' will make N cuts, where N is an arbitrary natural number. The is no 'privaledged engineer' who has a system that has been cut infinitely many times. Rather, each engineer must work within their own finite system.
Quoting fishfry
I do feel like we're very close to being on the same page now!
Quoting fishfry
Let's save the paradox discussion for later. I only mentioned it at this point because you asked why a mathematician would care.
Quoting fishfry
The real number is interior to the interval defined by the corresponding real algorithm. However, it doesn't necessarily have to be at the center. ?1 and ?2 don't have to be equal. I do think your 1/n values for epsilon works, but I'm not sure if we need to constrain the values of epsilon as such. If we're cutting (-?,+?) then it seems to me we should be as general as possible and say that epsilon can be any positive number - even 5 billion.
Well I'd say infinite sets are pretty pervasive in modern mathematics. The reason is that sets are very useful and finite sets are very restrictive - they're not a satisfactory alternative. What I'm proposing is nuanced - arbitrarily big sets. I want to replace 'infinity' with 'arbitrary'...at least most of the time. A sequence converges to a number as n gets arbitrarily big.
I have no problem with the treatment of infinity on the Riemann sphere or in projective geometry.
Quoting jgill
Richard Feynman once said that "the chance is high that the truth lies in the fashionable direction. But, on the off-chance that it is in another direction a direction obvious from an unfashionable view of field theory who will find it? Only someone who has sacrificed himself by teaching himself quantum electrodynamics from a peculiar and unfashionable point of view; one that he may have to invent for himself."
My ideas have no relation to field theory, they might not be true, and even if they're true, they might not be interesting. I accept that chances this goes nowhere is high. But you shouldn't discredit my view just because I choose to stroll through unfashionable parks.
I look forward to a breakthrough in your quest. But I am very old and have multiple medical conditions, so I may not be around. Smooth sailing, fellow explorer.
Thank you! I wish you very many wonderful years ahead.
What? You know, none of this makes any sense. [He's crabby tonight]
Quoting keystone
What? There's no difference with respect to algorithms. Consider 1/3 = .333...
Quoting keystone
Of course, because they are entirely different things, and there are a lot more real numbers than algorithms.
Quoting keystone
I think I am nearing the end here. You just are not making any sense. You've just made all this terminology up.
Quoting keystone
If there is a difference between 1.0 and 1.00000... you are off on your own. I can't hold up my end of this. Nothing you write is correct.
Quoting keystone
Not feelin' it tonight.
Quoting keystone
I don't see where this is going. I might be doing you a disservice by encouraging you.
Quoting keystone
You have said so many times, or at least I have understood you to say that. And besides, it is. Just not pairwise disjoint open intervals.
Quoting keystone
Ok whatever.
Quoting keystone
Ok that's good. Can we turn the page?
Quoting keystone
Then you give me no reason to care. You are not going to "solve QM" with your line of discourse.
Quoting keystone
Fine.
:100: :100: :100: :100: :100:
I was trying to go along with your idea of engineering math. If the mention of an engineer and a mathematician working together doesn't help, then fine - we'll drop the idea of engineering math. But, I disagree with your statement that none of it makes sense.
Quoting fishfry
Consider the follow program which writes the specified fraction in the specified base (NOTE: You can skip over the code):
For 1/3 in base 3, this program returns 0.1.
For 1/3 in base 10, this program returns nothing because the program does not halt. After all, it's trying to compute the sum of an infinite series. Impossible.
Clearly, these are not the same outcomes. They're different because [math]0.1_3[/math] is a rational number, while [math]0.\overline{3}_{10}[/math] is a real algorithm.
Quoting fishfry
Take [math]1.0[/math] and [math]1.\overline{0}[/math] literally. Here's what they mean:
[math]1.0 = 1 + \frac{0}{10}[/math]
[math]1.\overline{0} = 1 + \frac{0}{10} + \frac{0}{100} + \frac{0}{1000} + \ldots[/math]
If I were to write a program for each of those computations, the former would halt and the latter wouldn't, similar to the 1/3 example above. I understand why you and everyone else think they're exactly the same, but, in the purest sense, they are algorithmically different. Do you not see that?
It's not that I'm incorrect. It's that mathematicians have been so sloppy with the distinction between reals and rationals not realizing that this distinction truly matters, especially from a top-down perspective.
Quoting fishfry
Allow me to clarify: I want to distinguish between a real number and it's corresponding real algorithm. A real algorithm corresponding to ? can be written perfectly with finite characters, such as:
[math]\underline{\pi} = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \cdots[/math]
or better yet, let me take that infinite series and derive 'a sequence of intervals' version of ?:
[math]\begin{aligned}P_N &= \text{Partial sum for first } N \text{ terms of the above series} \\\min_a &= \min(P_a, P_{a+1}) = \begin{cases} P_a & \text{if } P_a < P_{a+1}, \\ P_{a+1} & \text{else} \end{cases} \\\max_a &= \max(P_a, P_{a+1}) = \begin{cases} P_a & \text{if } P_a > P_{a+1}, \\ P_{a+1} & \text{else} \end{cases}\end{aligned}[/math]
Here is the algorithm for the infinite sequence of intervals:
[math]\begin{aligned}\underline{\pi} = (\min_0, \max_0), (\min_1, \max_1), (\min_2, \max_2), \ldots\end{aligned}[/math]
The real number ? is the only number that lies in [math]\begin{aligned}(\min_{N}, \max_{N+1})\end{aligned}[/math] regardless of what N you choose. However, since no interval in the series has zero length, the real number ? cannot be isolated. Numerically, ? is the only number that lies in any interval described by ?, and geometrically, ? is the only point that lies in any line described by ?.
From the bottom-up view ? is equivalent to ?.
From the top-down view ? is not equivalent to ? (any more that an algorithm is equivalent to it's output, or a line is equivalent to a point).
Quoting fishfry
That's because when it comes to reals, mathematicians have been so sloppy with their terminology. I'm trying to make things more precise.
Quoting fishfry
First off, I'm only claiming to (at least partially) solve the issue of how to philosophically interpret QM. I'm certainly not claiming to have solved quantum gravity or anything like that. Are you saying you want me to jump right to the implications of the top-down view without even explaining the top-down view? I'm certain that without understanding my view you'll just think I'm injecting quantum woo into the top-down view. If you stick with it, what you'll see is that quantum intuitions follow from the top-down view. It is this way because QM is a top-down view of reality whereas classical mechanics (CM) is a bottom-up view of reality. Whether we're talking about mathematics or physics, the bottom-up view has been undoubtedly and demonstrably useful. It's just not correct at a foundational level. The reason why we struggle to interpret QM is because the mathematical top-down view has been neglected. Zeno was the first canary in the coal mine urging us to consider it.
Quoting fishfry
You've been of great help to me so far and I greatly appreciate that. If you ever want to call it quits I will accept that, thank you for your help, and that will be the end of it. Of course, I reallllly hope that doesn't happen...
Ok. So far, after all this, what I understand of your idea is that the real line consists of a countably infinite set of overlapping open intervals, each containing a computable number. So far so good? But this is not a very deep idea, there are many ways to express the reals as a union of open intervals.
What? There's no difference with respect to algorithms. Consider 1/3 = .333...
fishfry
Quoting keystone
This is a common misunderstanding of what a halting program means.
The number pi is computable. Clearly no computer program can generate all the digits in one go. But that's not what halting means.
Pi is computable because, given a positive integer n, I can run the program for a finite number of steps, and output the n-th decimal digit.
It's absurd to claim that 1/3 is not computable or not algorithmic (or whatever you're claiming) just because its decimal representation is infinite. Given n, the n-th digit is 3. That's a halting program. Therefore 1/3 is computable. Likewise pi.
Quoting keystone
No. I deleted some text but you were trying to convince yourself that 1 and 1.0 and 1.0000... are different numbers. They are not. And again, it's because you're confused about what a halting program means. Now that you understand it (now that I've explained it to you), you are no longer confused.
All an algorithm is required to do is, given n, output the n-th digit in finitely many steps.
Quoting keystone
You are delusional. Could it be that you are the one who's confused, and not mathematicians?
Does my explanation of what's a halting algorithm to compute a real number cause to you reframe your understanding? Input n, output the n-th decimal (or any base) digit in finitely many steps.
Quoting keystone
What are you doing that, when I quote your numeric examples, the quote text comes out in a column?
Quoting keystone
Did you understand my point about halting? Does it help you understand what's going on?
Turing defined computability another way. A real number is computable if, given epsilon > 0, there's an algorithm that generates an approximation to the number within epsilon. You can see that this amounts to outputting the n-th digit. The point is that a number can be computable by an algorithm even if the number has infinitely many decimal digits, like 1/3 or pi.
Quoting keystone
Is that really likely? Or is it more likely that there are some basic things you don't understand, like the definition of halting or the fact that it's trivial to write the real line as a countably infinite union of open intervals?
Quoting keystone
It would help me to understand what you're talking about, and why.
After all, you say your top-down view starts with the real line. But I say, I don't know what the real line is. How do you know there is any such thing unless you construct it from first principles?
Quoting keystone
This sounds very cranky.
Quoting keystone
Cranky. Grandiose claims not backed by anything coherent.
Quoting keystone
I'll slog on a little longer. It would help if you'll engage with my key point tonight, which is that you've been misunderstanding the nature of halting with respect to computable numbers.
Can you see that 1/3 = .333... is computable, because the program "print 3" halts in finitely many steps for an n, giving the n-th decimal digit of 1/3?
Quoting fishfry
Yay!! Thanks a tonne :)
Quoting fishfry
My understanding is that a program halts if it reaches a point where it completes its execution and stops running. Do you actually disagree with this definition of halt?
You are employing a straw man argument. I'm saying that the program that computes 0.333...to infinitely fine precision does not halt, and you are saying that the program that computes 0.333... to an arbitrarily fine precision does halt. I agree with you, but your argument doesn't address my point.
I believe the term 'computable number' applies to a number which can be represented by an algorithm. Am I wrong? If so, it is a very misleading name because the definition makes no mention of computers, finite resources, or anything of the sort. I would much rather call them 'algorithmic numbers' but let's stick with the current terminology.
I believe the term 'halt' applies to the execution of the algorithm. If it cannot be executed to completion then it does not halt.
Your failure to see the above distinction relates to one of my central complaints about the current (bottom-up) view of mathematics: mathematicians too often obfuscate the program (the algorithm) with it's execution (the generation of output by the algorithm). And it doesn't help that we call both the program and it's output the same thing: numbers. This is where I'm trying to bring clarity to the situation by redefining terms (such as what it means to be a rational vs. a real), but it turns out that such efforts just makes you think I don't know what I'm talking about.
Quoting fishfry
Shouldn't the first principles be self-evident? We experience continua and finite numbers all the time in our physical reality. The same cannot be said about points and transfinite numbers. It is the points which must be constructed from first principles. It is infinity which must be derived, not axiomatized into existence.
Euclid's line is so simple -- breadthless length. It's hard for anyone to say that's not self-evident. And I can easily construct a point from that line - I cut it and the midpoint emerges. The bottom-up view is far less self-evident. Somehow combining sufficiently many objects of no length results in an object of length. And even though nobody has a good explanation of how this works we nevertheless proceed by saying that the continuum constructed in this fashion is paradoxically beautiful and only to be seriously discussed by the experts. Really?
You expect a deeper structure to my line, such that, say, when I cut line (0,4) at point 2 that this involves identifying a pre-ordained point and isolating it by means of a cut. That's not what I'm proposing. My line has no deeper structure or additional properties beyond continuity and breadthless length.
I've come to realize that I've been heading down the wrong path by saying that my line is a bundle of [math]2^{\aleph_0}[/math] points. I ended up here because I was defending against your arguments that my line has gaps. It turns out that my defense has just made you expect a structure to these points. It is better for me to just claim as a first principle that the line is continuous. As such, I'd like to discard the 'bundle' argument.
Instead, the structure you are looking for comes from the cutter, not the line. I can cut line (0,4) anywhere I want and label the midpoint that emerges '2'. In labelling that point '2' I am making an agreement with myself that any subsequent cut I make, I will label it to maintain the structure we have come to expect with numbers (as captured by the SB tree). For example, if I subsequently cut the line (2,4) I agree to label it with a number between 2 and 4.
In the top-down view, the cutter/mathematician plays a central and active role in maintaining structure and, moreover, actualizing objects....not unlike the the observer in QM...hmmm....
Quoting fishfry
There's nothing actually infinite about the line. What is infinite is the potential for the cutter to make [math]2^{\aleph_0}[/math] cuts to the line. But since (1) an infinitely precise computable irrational cut requires the completion of a supertask, (2) non-computable irrational cuts cannot be algorithmically defined and (3) the cutter can only ever perform finitely many cuts, this potential can never be completely actualized. When working in 1D, the mathematician will forever be stuck working with a finite set of lines and points. However, because the mathematician can continue to make arbitrarily many more cuts (i.e. any natural number of cuts), that set can grow to be arbitrarily large (i.e. have any natural number of elements).
Quoting fishfry
Of course that's possible (and likely). After all, that's why these ideas are being discussed in this chat forum and not eternalized in the Annals of Mathematics. But since yelling 'I'm not crazy' only makes one sound crazier, I won't challenge this point further and hopefully the ideas will eventually speak for themselves.
Quoting fishfry
I know how it sounds, that's why I'm reluctant to talk about QM and paradoxes at this time. When I communicate the fundamentals, you ask for the implications. When I communicate the implications, you ask for the fundamentals. My only hope is that at some point the fundaments become coherent to you, after which the implications will naturally follow. I admire you for sticking with me for this long given that you think my ideas have so far been incoherent.
Quoting fishfry
I ask ChatGPT to give me the Latex equivalent of an expression and I insert that Latex string with the math tag.
A program P computes a real number x to infinitely fine precision ifff F [where F is a formula whose only free variable is x and that uses only previously defined terminology]
Meanwhile, along the lines mentioned:
(1) For any computable real x, there is a program that lists the digits of x and does not halt so that any step in execution, only finitely many digits have been listed.
(2) For any computable real x, there is a program whose domain is the set of natural numbers, and for any n, the program outputs the nth digit of x and halts.
(3) The set of digits in the decimal expansion of 1/3 is the singleton {3}, thus a computable set.
That's mathematics. For "compute to infinitely fine precision" to be mathematics, it requires a mathematical definition.
/
Mathematicians do not "obfuscate" programs with executions of programs. If one claims that mathematicians do that, then what are specific written examples? If there are no specific written examples, then it's a strawman.
/
Classical mathematics does not call program 'numbers'. (However, programs can be assigned Godel-numbers.)
/
Mathematics uses a specific method for definitions. Alternative definitions would themselves need to adhere to the methodology of definitions which includes the criteria of eliminability and non-creativity, which insure against vagueness, circularity, and infinite regress, most pointedly that the definiens use only primitives or previously defined terminology.
/
This is a non sequitur: We do not physically experience infinitudes therefore the existence of infinite sets must be derived and not axiomatic.
If one proposes a mathematics without infinite sets, then that is fine, but the ordinary mathematics for the sciences uses infinite sets, which are not derivable from the rest of the set theoretic axioms, thus requiring an axiom.
For that matter, we don't physically experience breadthlessness, so breadthlessness is itself an idealization, just as infinitude is an idealization.
/
The notion of length is quite coherent. In context of the reals, length is a property of segments not of points. And we have a perfectly rigorous explanation of how it works: Let x and y be two different points, the length of the segment with x and y as endpoints is |x-y|. This is a notion that is clear not just to mathematicians but to school children.
/
Who said anything equivalent with "the continuum constructed in this fashion is paradoxically beautiful and only to be seriously discussed by the experts"? Specific quotes are called for. Otherwise the claim is a flagrant strawman.
"Really?" No, really, who said anything equivalent with "the continuum constructed in this fashion is paradoxically beautiful and only to be seriously discussed by the experts"?
/
'1.0' not= '1.0...'
1.0 = 1.0...
'1.0' and '1.0...' name the same number, whether is is named as a finite sum or infinite sum.
/
Mathematicians have not been sloppy in distinguishing rationals from reals. Mathematics uses a rigorous method definitions. The definitions of 'is a rational number' and 'is a real number' are completely rigorous. Suggestion to anyone not familiar with that fact: Read any one of a number of books in which the constructions of the rationals and of the reals are shown step by rigorous step.
A sequence is a function. A function has a domain. If the domain is not infinite, then n cannot be arbitrarily large.
One is welcome to work it out in some other way. But then the natural question is: What are your primitives, formation rules, axioms and inference rules? Mathematicians give us the courtesy of stating those rigorously so that it is utterly objective, by machine-verification, whether a purported formal proof is indeed a formal proof. Or, one could say that one doesn't do things formally. That's fine, but then a comparison with mathematics is not apt since mathematics rises to a challenge that informal quasi-mathematical ruminations do not.
The number and the program are different things. The number and the equivalence class of programs are different things.
:up:
Just to be clear:
There is no first order theory of complete ordered fields. That is, there is no first order theory whose models are all and only the complete ordered fields. Yes, in set theory we define 'complete ordered field' (some people call the clauses in the definition 'axioms') and show that all complete ordered fields are isomorphic, but that is different from a first order theory whose models are all and only the complete ordered fields.
There are first order categorial theories of such things as real closed ordered fields, but they do not include the completeness property that is crucial for an adequate account of the real numbers.
It is false that the interval (0 1) is not an infinite union of disjoint intervals.
Ostensively:
(0 1/2)
[1/2 3/4)
[3/4 7/8)
...
Formally:
Let f be the function whose domain is the set of natural numbers such that:
f(0) = (0 1/2)
for n>0, f(n) = [(2^n - 1 )/2^n (2^n+1 - 1)/2^n+1)
The range of f is an infinite partition of (0 1). That is: the range of f is infinite; every member of the range of f is an interval; the range of f is pairwise disjoint, and the union of the range of f is (0 1).
/
It was claimed that Amish communities weren't hit hard by Covid. I take that to mean that among people who lived in Amish communities there were starkly lower rates of infection, painful sickeness, hospitalization, death and long Covid. What is the source for that claim? (Or was the claim merely a fanciful joke premise so that both the setup and punchline are not presumed factual?) The punchline was that Amish communities didn't suffer so much because they avoided media coverage of the pandemic. Even though the context is that of a joke, is it nevertheless being suggested that knowing less about the pandemic has a causal relation in reducing infections, painful sickness, hospitalization, death and long Covid?
'gap' is mentioned but not defined nor is 'execute a cut'.
'Dedekind cut' is not defined in terms of 'gaps', nor 'executions', nor algorithms.
'Dedekind cut' is mathematically defined and we see mathematical proofs that the set of Dedekind cuts is uncountable.
And it is serious misunderstanding to argue that there can't be more Dedekind cuts than there are rationals. A Dedekind cut is a certain kind of pair of subsets of the set of rational numbers. The set of rational numbers is countable, but the set of subsets of the set of rational numbers is uncountable. A person who argues from the fact that there are only countably many rationals to the claim that therefore there can be only countably many Dedekind cuts is a person who does not know what a Dedekind cut is.
Just to be exact: The continuum is the ordered pair
I'm gratified to know I'm being helpful. Thanks.
Quoting keystone
Ok correct to there.
Quoting keystone
"Infinitely fine precision" is imprecise. If you change it to arbitrarily fine precision, we already have well known programs for both.
Perhaps by infinitely fine precision you mean specifying all of its digits at once. Sort of like running the Turing machine for 1, for 2, for 3, ... such that all of them were run and the outputs collected into the full decimal representation of pi.
If so, we agree that no physical computer could ever do that, simply because it would require an infinite amount of computing resources: time, energy, and space. If those are finite, then so is what we can practically compute. We agree.
pi an 1/3 are both computable, so the same argument applies.
By the way, set theorist Joel David Hamkins is investigating [url='https://jdh.hamkins.org/ittms/']Infinite-time Turing machines." Here's the abstract:
[quote=Hamkins]
We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of computation on the reals and concepts of decidability and semi-decidability for sets of reals as well as individual reals.
[/quote]
Is that cool, or what!
@Michael Perhaps this is of interest.
Quoting keystone
I agree, adding the picky detail that it is required to terminate on a Halt state. If it terminates on an error state, it doesn't count. I believe that's one of the rules. With that proviso, we agree.
Quoting keystone
You lost me. You AGREE with me but I didn't address your point.
I am not sure what you agree with and what I didn't address.
I said: No halting algorithm can print out ALL of the digits in one execution. It would take infinitely many steps. That's against the rules for Turing machines. That's obvious.
But to be computable, a number doesn't need an algorithm to print out all its digits at once and then halt. That's impossible.
Rather, all that's needed is a machine that inputs n and outputs the n-th digit.
Now that's pretty clear. And it was very sensible on Turing's part, to realize that this is the right definition, and to avoid all talk of computations that run forever. He avoided all that. All computations must halt after finitely many steps. That is the property that characterizes computation from everything that is not computation. At least in the formal sense. Of course real life programs are designed to keep running, such as web servers and operating system kernels and the like. Still, actual computers can't compute anything that's not already computable by a Turing machine, so it doesn't matter.
So tell me what you agree with and what point I didn't address, and I'll try to address it.
Quoting keystone
Turing invented the abstract computer. That's what a Turing machine is.
There is no accounting for terminology, it's mostly historical accident. Open and closed sets in topology are confusing because a set can be open, closed, neither, or both. Generations of students have been confused about open sets. But the terminology is dug in deep, there is no changing it.
When we say computable, we mean relative to Turing's definition of an abstract computer. A definition that has served computer science well ever since then.
Perhaps you are making the distinction between computer science and computer engineering! Turing machines are not made out of transistors and chips and wires; real computers are.
I hope you are not going to let yourself get hung up on the notion of abstraction. The purpose of the abstract Turing machine is to allow us to reason logically about what computers can and can't do. The abstract theory informs computer engineering, it's just not all of it.
Hope this helps. You do not want to get bogged down in confusion about the realm of the abstract versus the realm of the actual.
Quoting keystone
Yes, correct. But if you are referring to the idea of printing out all of the digits in one shot, you are just denying or refusing to accept the standard usage of the term. It pertains to an abstract computer, not a real one.
Are you getting hung up on that?
Quoting keystone
You seem to be unhappy that abstract math doesn't climb into the wiring cabinet and start patching cables. Do I have that right?
I don't think you don't know what you're talking about. I think I don't know what you're talking about. I'm throwing out guesses. You don't like infinities, ok there's finitism. You want things to be algorithmic, ok there's constructivism. You want there to be a minimal positive real, ok that's computer arithmetic. None of it sticks. Why do you reject the doctrines you espouse?
Quoting keystone
Of course not. That belief was demolished by non-Euclidean geometry leading to General Relativity. Kant said we have a priori knowledge of the Euclidean and Newtonian universe, and that this was true knowledge of the world. He was wrong. We are born with an intuition of Euclidean space, but that turns out to just be a local approximation to the world we live in. Who knew, right?
Quoting keystone
Ok. I get that you feel enthusiastic about this. I'm on your side. I hope you can work out your ideas. I do think they are a little half-baked at the moment. That's an honest assessment.
If you don't like the axiom of infinity you are a finitist, but perhaps not an ultrafinitist. You might be interested in learning about [url=https://en.wikipedia.org/wiki/Finitism]finitism[/ul].
Quoting keystone
If I stipulate that every mathematicians that ever lived is a bad person for doing whatever you think they did ... would it help?
Can you step back a tiny bit and see that if every smart person who ever lived is a dummy acting from bad faith ... well, maybe it's you, and not them. Maybe a lot of people already thought about these issues. Excessive grandiosity is often an indicator of crankitude.
I mean actually, if you feel that everyone else is wrong and you are right ... you should keep this to yourself! You don't want me to know you feel that way.
Quoting keystone
More breathless than breadthless. (Sorry couldn't resist!)
Quoting keystone
When I Quote your posts, those expressions always render as one character per line. Makes it hard to read.
Anyway ok, you claimed that many times, and now you're not sure. That's fine, since your theory is a long way from being able to define [math]2^{\aleph_0}[/math]. That expression is a fairly sophisticated "bottom up" construction.
Quoting keystone
Abandoning the entire bundle argument? So the real line is no longer made of a countably infinite union of overlapping open intervals, each characterized by a particular computable number it contains? I thought that was a pretty good thing to achieve agreement on. You are abandoning this now?
As far as gaps go, they're important. The completeness property, aka the Least Upper Bound property, aka Cauchy-completeness, is the defining characteristic of the real numbers. Accept no substitute! If someone tries to sell you a model of the real numbers, ask them if it's complete! Mathematical shopping advice.
Quoting keystone
Your focus on the Stern-Bricot tree is ... well some adjective anyway. Do you know the infinite complete binary tree? Start with a binary point at the top node. Underneath each node from now on is a left node called 0, and a right node called 1. The infinite tree has countably many nodes, but uncountably many paths through the tree. Each path corresponds to a real number and all the real numbers are represented by some path. Or sometimes two distinct paths, as in .0111111... and .10000... I find it helpful to visualize the real numbers that way sometimes.
But your cut idea, you just can't get to enough of the real numbers that way, with each number represented by finitely many cuts, if that's what you're doing.
Quoting keystone
Ah, well that's like the active intelligence of intuitionism. You should read up a bit on this too.
Here's what Wiki says:
[quote=Wiki]
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality.[1] That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.[/quote]
How did we ever survive before Wikipedia?
Quoting keystone
Well you are a long way from making that many cuts when you start by denying even countable infinity!! Isn't that a little inconsistent?
Quoting keystone
I don't think you should tie in supertasks. There's already enough fuzzy thinking on that topic going around.
Quoting keystone
Very difficult to get a model of the real numbers while denying infinite sets. It's been tried, really. The constructivists have a pretty interesting model. They even have their own version of completeness, even though their model is only countably infinite. Maybe you should look at constructivism.
Quoting keystone
You have seemingly expressed the idea that you are right and thousands of years of the smartest people in the world looking at these issues are wrong. That's always a bad sign. You should note it yourself, as an objective check on your own thought process.
Quoting keystone
Ah, you have discretion. Well actually that's a point in your favor. But QM, really? That's a stretch.
Quoting keystone
And I've understood precious little of either :lol:
Quoting keystone
I'm in shock that you're abandoning the little open intervals. I was clinging to that like a reed in the ocean.
Quoting keystone
Jeezus. That ain't workin'. Maybe some control characters in there. And don't use ChatGPT, it rots your brain. Lot of foolishness floating about in the culture lately.
/
From my non-exhaustive readings on Turing machines, I don't know what an 'error state' is for a Turing machine. Basically, a Turing machine takes an input and either halts with a state and whatever is on its tape when halted or it does not halt.
But there are different formulations of Turing machines, so I would welcome knowing of one that defines an error state.
I'm very keen to respond but there's a lot to reflect on and respond to and I'm completely overloaded with work and family responsibilities so I might not respond until next weekend. As always, I appreciate your comments and this dialogue.
No worries mate.
I've further refined my thinking on numbers and algorithms so please allow me to start from scratch here without us bringing in any of the out-of-date baggage from my past few posts.
Fractions
I would like the objects of computation to be fractions. I have an unorthodox view of fractions. I don't see them being constructed from other number systems (such as integers) or as elements of other number systems (such as real numbers) but rather as numbers in and of themselves. The structure of the fractions are captured by trees, my preference being the Stern-Brocot tree, which places fractions at the nodes and whose basic arithmetic can be performed by navigating the Stern-Brocot tree. But when I mention the SB-tree in the following post, I acknowledge that any equivalent structure (such as the infinite complete binary tree) can be used in its place.
Reals
Continua
In my post directed to both you and TonesInDeepFreeze, I either directly or indirectly addressed a lot of your shared points. Below are my responses to your unshared points. Please let me know if I failed to respond to any of your points.
Quoting fishfry
I'm skeptically curious :)
Quoting fishfry
Quoting fishfry
Ok fair point. Agreed.
Quoting fishfry
We need to make a distinction between the core mathematical idea and language with which it's communicated. For example, the Pythagorean Theorem was known and used in various forms long before the formalization of bottom-up number-based systems. And it will continue to hold value even if we move past bottom-up number-based systems to top-down continuum-based systems. I'm not proposing that any such mathematical idea is wrong.
And even IF I'm right, it doesn't mean that bottom-up number-based systems are useless. They would remain useful in the same sense that Newtonian mechanics remains useful (it just cannot be used to describe our reality at a fundamental level). But yes, I agree that it is likely me who is the dummy. I am likely experiencing the DunningKruger effect. But nevertheless ideas should be challenged on their merit, not on how unlikely it is for an important math idea to originate from an engineer on a chat forum.
Quoting fishfry
I had to abandon the bundle argument, in part because it seems to imply a structure that's not there. For example, I never proposed that the real line was made of a countably infinite union of overlapping open intervals. Rather, I proposed that a computer can begin to cut the line but it will never exhaust cutting such that the line is divided into infinitely many partitions.
Quoting fishfry
I agree that a number-based system that has gaps cannot be used for calculus. However, a continuum-based system begin with a continuous line, and if all we do is make cuts, there will never be a state of the system where gaps are present.
Quoting fishfry
I agree that the SB-tree and the infinite complete binary tree capture the same information. In fact, the binary tree might be preferable since binary is the language of computers. However, I prefer the SB-tree since it places fractions at the nodes. As described in my post to both you and TonesInDeepFreeze, I see binary numbers as algorithms operating on fractions. Since paths down the binary tree can also be seen are algorithms operating on fractions, I feel that the distinction between a node and a path is less clear with the infinite complete binary tree.
Quoting fishfry
Yes, I do think my view falls near the intuitionist camp.
Quoting fishfry
Yeah, you're right. In my reformulated view (where I posted to you and TonesInDeepFreeze) I make no mention of uncountably many cuts.
Quoting fishfry
I think it's impossible to model a continuum using a finite set of indivisibles. However, I'm proposing that we model a continuum using an evolving finite set of divisibles.
Quoting fishfry
Is the problem simply that I'm using Latex? And ChatGPT is imperfect but nevertheless awesome as a co-pilot.
Quoting fishfry
I genuinely appreciate this sentiment. Given that my ideas continue to get reformulated throughout this discussion, I could only agree that they are in the baking process (and to be realistic, they are likely less than half-baked at this point). I also want to acknowledge that when the baking is complete the end product may not be anything anyone wants to eat!
In my post directed to both you and fryfish, I either directly or indirectly addressed a lot of your shared points. Below are my responses to your unshared points. Please let me know if I failed to respond to any of your points.
Quoting TonesInDeepFreeze
I agree that applied mathematicians often formulate their theories with infinite sets, but that is largely because those are the tools provided by pure mathematicians. In practice, applied mathematicians do not actually use infinite sets. I believe that applied mathematicians would welcome a different set of tools which allowed their theories to be reformulated free of infinite sets, while still retaining their usefulness.
Quoting TonesInDeepFreeze
1D continua (i.e. lines) are a simpler version of the 3D continua we experience. Comparatively, infinitude is a more complex version of the finitude we experience. As such, the idealization of lines and the idealization of infinitude are not comparable.
Quoting TonesInDeepFreeze
I agree with this. But if the segment is built entirely from points, where would the length come from if not the points?
Quoting TonesInDeepFreeze
I started writing a defence for that comment but given that fryfish has entertained a long conversation on the topic with me, a non-expert, I think it's better for me to just take that statement back.
Quoting TonesInDeepFreeze
I am proposing that we not work with the infinite sequence itself (which cannot exist in a computer) but rather the algorithm designed to generate the infinite sequence (which is described with finite characters and, as per Turing, halts if executed.) As for the domain, I am proposing that we not work with an infinite set of numbers, but rather a line upon which arbitrarily many fractions can emerge by means of cuts.
Quoting TonesInDeepFreeze
I agree that a formal treatment is the ultimate destination for a math idea. I see this informal forum conversation as the journey which may be slowly taking me toward that destination or may be leading me toward the junkyard to dump my ideas. Either way, this journey is useful to me.
Quoting TonesInDeepFreeze
I think it's best for me to just take this claim back.
Quoting TonesInDeepFreeze
In earlier posts I described 'potential' points which existed only in bundles that can be isolated by means of a cut. I have since scrapped that idea. Instead of proposing that cuts isolate points, I am now proposing that cuts create points.
Quoting TonesInDeepFreeze
Treating your example algorithmically and using the notation described in my recent post I would say the following: (0, 1/2) U [1/2, 3/4) U [3/4, 7/8) U ... !? (0, 1)
In other words, that algorithm can never output the interval (0, 1).
But I think the original discussion was about describing a line as the union of fundamental objects. Defining a line as the union of smaller lines would be a circular definition, which is why I want to take the line as fundamental.
Quoting TonesInDeepFreeze
In the context of computation, this distinction is moot since we can only ever speak of arbitrarily many cuts. As such, I'm willing to withdraw my comment about uncountably many cuts made to countably many rationals.
Instead of that comment, let me reframe my position in the context of cuts as described in my recent post to you and fryfish. If we inspect the outcome of a real cut, we cannot determine what real algorithm was used to make that cut. This is because when executing real cuts we are forced to select a fractional interval at some finite depth in the Cauchy sequence of intervals, and whatever that interval may be could correspond to potentially [math]{\aleph_0}[/math] different real cuts.
If a Dedekind cut does not correspond to the cut described above, it likely corresponds to one that is infinitely precise such that it results in a cut at a point (not a line). This would be akin to the impossible and non-sensical task of selecting the final interval in the Cauchy sequence of intervals, whose interval would contain only a single number. Of course, there is no final term. From the computational perspective, this type of cut is not possible so has no relevance in the top-down view. If it holds together in the bottom-up view then fine.
Let me know when (or if) you have a system with formation rules, axioms and inference rules.
Otherwise, discourse with you is such that what obtains is just what you say obtains, without your interlocuters having access to checking your arguments by the objective reference of mathematical proof.
Quoting keystone
Depends on what you mean by "applied". Ordinary mathematics, even as used for basic applied interests such as speed and acceleration use ordinary calculus, which is premised in infinite sets.
/
In context of ordinary mathematics, a line in 2-space is a certain kind of set of ordered pairs and a line in 3-space is a certain kind of set of ordered triples. If one wishes to propose an alternative, then they can set it up coherently with axioms and definitions. On the other hand, if you wish to limit yourself to ostensive, imagistic musings then I wouldn't begrudge you from entertaining yourself that way; only that count me out, as I have better avenues of intellectual engagement.
/
You ask where does length "come from" if not points. The mathematical definition of 'distance between points' is given by a well known formula in high school Algebra 1. Look it up.
Quoting keystone
Even better would be to figure out why you are prone to such things to begin with.
Quoting keystone
No, the execution does not halt. You have it completely wrong.
Quoting keystone
That's nice. But, by magnitudes, less useful to me than actually reading mathematics and philosophy of mathematics. For that matter, even less useful to me than, say, watching my screen saver. But, meanwhile, I don't mind correcting falsehoods you post about mathematics itself and to comment on the irremediable handwaving in the description of your own musings.
Quoting keystone
See earlier in this post.
Quoting keystone
You seem really not to recognize that it is a potentially infinite game with you, where you pretend to define or explain terminology or concepts in terms of yet more undefined terminology and concepts.
And who can keep up with you? So 'isolate' is now out? And 'create' is now in?
Quoting keystone
I guess I'm supposed to glean that that means something like "as goes to infinity" or something?
Anyway, what you said makes no sense and is not an answer to my correction of your false claim.
What I wrote:
Quoting TonesInDeepFreeze
There is no "algorithm" or "going to infinity" there.
Purely that I proved that, contrary to your false claim, there is an infinite partition of (0 1). But you can't be bothered to actually looking at the proof to understand it (though it is very simple), as instead you're too preoccupied with handwaving and strawmaning that I've invoked an "algorithm" and a "going to infinity", which suggests of you a kind of a narcissism as you wish that other people indulge your undefined and confused musings while you ignore (worse, strawman) clear proofs and exact explanations from your interlocuter.
Quoting keystone
I didn't invoke any circularity. Read the proof.
Quoting keystone
You wave "In the context of computation" here like it's a "Get Out Of Jail Free" card that you can use to evade that you're mischaracterizing what a Dedekind cut is.
Quoting keystone
Again, better not to be prone to making outlandishly false claims.
Quoting keystone
Again, Dedekind cuts do not invoke algorithms.
Quoting keystone
Learn to use mathematical terminology coherently. You suffer the misconception that throwing around jargon in inapposite imagistic clumps is mathematically meaningful.
Etc.
I thought you don't believe in points :-)
Quoting keystone
TMs are arbitrarily large but finite.
Quoting keystone
Murky.
Quoting keystone
Murky. To me, anyway.
Quoting keystone
You've said it repeatedly, haven't you?
Quoting keystone
Suppose I grant that you have some alternative construction of the reals. What of it? All models of the reals are isomorphic to one another.
Quoting keystone
My point exactly. If you have an alternative view of the reals, nothing changes.
Quoting keystone
I don't see what ideas you've challenged.
Quoting keystone
It took weeks for me to understand your bundles, and just when I did, you took them away.
Quoting keystone
Well we agree on that.
Quoting keystone
Ok. Not that I understand.
Quoting keystone
"Binary numbers" aka real numbers can never be algorithms, since there are way too many of them. There are uncountably many reals and only countably many algorithms.
Quoting keystone
You should study intuitionism then.
Quoting keystone
Sigh. I am not getting much from this latest post.
Quoting keystone
ok!
I'm not commenting on your other post, but I did note this:
Quoting keystone
Then you are a constructivist. I don't understand why you disagree.
I understand the previous post was lengthy, and I know you don't owe me anything. However, I wonder if this marks the end of our discussion. I'm unsure how to keep it going since anything more might just be more words to skip over. If you have any advice on how to continue, I'd appreciate it. If you'd prefer to end our conversation here, I accept that and thank you for the discussion!
Didn't say that, just got a little overwhelmed by all the line items. I'll take a look at it.
But what about my point about constructivism? If you reject the noncomputable reals, you're a constructivist.
ps -- I read through your list and I don't get it. Suppose I stipulate all that. Then what? So what? I just don't get it. And I surely don't have the heart to pick apart every line item. So what should I do? I'm impressed at the energy you put into this. I really can't comment on the rest of it, because there would be no end. "I would like the objects of computation to be fractions." What am I supposed to make of that? What is a computation? What is the object of a computation? In what way is 1/2, say, the object of a computation, but pi isn't? I could go on like this for every item but what would be the point? If that list of items encapsulates your philosophy of math, I'm happy for you. I'm happy for everyone's philosophy of math. Bertrand Russell's philosophy of math turned out to be wrong, but presumably he was happy with it. I don't think I'm going to be able to make you happy regarding that particular post.
Oh ok, that's great to hear. Yeah, sorry for the large number of line items...
Quoting fishfry
You're right, I'm likely a constructivist/intuitionist. I say 'likely' because there's a lot of material to go through, and I need more time to fully understand it all. However, my views align with the key principles of constructivism. My main frustration with the material I've found so far is that it doesn't seem to address what I'm talking about...
Quoting fishfry
With my view, reals are constructed one at a time. It is impossible to construct [math]{\aleph_0}[/math] reals, let alone [math]2^{\aleph_0}[/math] reals. Given this, it's pretty clear that I'm not constructing the familiar reals.
Quoting fishfry
I think it is more correct to say that I have an alternate view of continua for which reals only play a supporting role. If mathematics were reformulated to be entirely absent of actual infinities would that be significant?
Quoting fishfry
I'm working towards a foundation free of actual infinities.
Quoting fishfry
Okay, I was too rash to take the bundles away. I think they're a useful way for us to find common ground. One thing I need to make clear though is that when I write (0,4) I'm referring to a bundle between point 0 and point 4. I'm not referring to [math]2^{\aleph_0}[/math] points each having a number associated with them. If I cut that bundle, a midpoint will emerge and I can assign to that point any number between 0 and 4. A number is only assigned when the cut is made. How does that sound?
Quoting fishfry
I comment on this in the long post which you haven't responded to.
Quoting fishfry
Yeah, the other post of mine was more beefy.
I think only at that time would we enjoy talking with each other.
Quoting TonesInDeepFreeze
https://www.youtube.com/watch?v=jreGFfCxXr4
Your line items are helpful to you, and that is the ultimate goal. Technically it doesn't matter whether I ever understand your ideas or not, as long as I am useful as a sounding board. So if you will take the glass half full approach to my not relating to your charts and graphs and lists, then you can feel free to keep posting them and my eyeballs will feel free to be glazed.
Quoting keystone
Check out this guy.
https://en.wikipedia.org/wiki/L._E._J._Brouwer
I must say that in my modest studies of those subjects, constructivism seems more reasonable. I'm perfectly ok with working out the consequences of restricting the real numbers to the computable reals.
But intuitionism, with its active intelligence creating sequences as they go ... that's just a little out there for my taste.
Quoting keystone
You're not constructing the familiar reals? First time I'm hearing this. Maybe you're constructing the computable reals. Is that what you're doing?
Quoting keystone
I'm pretty sure, but have no specific info about this, that people already decided you can't do analysis, that is calculus and the theory of the reals, without the axiom of infinity. But I could be wrong. I think if you could do analysis without the axiom of infinity that would be impressive.
Quoting keystone
But infinities are one of the most fun and interesting part of math! I always liked infinities. I think I just don't understand the psychology of someone who doesn't like the axiom of infinity.
Tell me, what makes you interested in trying to do math without infinite sets?
Quoting keystone
I would interpret that as your intuition that the open intervals with rational endpoints are a basis for the usual topology on the reals. All the open sets are unions (perhaps infinite) of open intervals with rational endpoints. But then again ... do you allow infinite unions and intersections of sets? Do you want to get rid of infinitary operations as well as infinite sets?
Quoting keystone
Did I miss a post? Or do you mean the long list of definitions and principles that glazed my eyes a bit?
By the way my eyes glaze over frequently at many things. It's nothing personal.
Yeah, the other post of mine was more beefy.[/quote]
Sorry I'm still confused. Did you mean the big list?
You give me link to some unidentified video so that I would have to take my time to watch through to find out, or guess, what it is you want me to know about it.
Trying to make my ideas clearer so that your eyes might not glaze over has indeed helped me collect my thoughts. I've also benefitted in other of your recommendations, such as construtivism which I really appreciate. So thanks for the glass half full. But there will come a point where no further progress can be made if I can't produce post that you are able to digest.
Quoting fishfry
I do plan to do a deeper investigation into Constructivism and certainly Brouwer will be a part of that. Thanks.
Quoting fishfry
I, on the other hand, am particularly drawn to intuitionism because I find it to be the least 'out there'. In this perspective, what exists are not infinite, eternal abstract objects in some inaccessible realm, but rather the finite set of objects currently being 'thought' by active computers. In my view, if the number 42 is not presently within the thoughts of any computer, then 42 does not currently exist.
Quoting fishfry
In line with my intuitionistic view, I'm not constructing any infinite set, rather constructing computable reals one cut at a time. More importantly, I can stop at any point and still have a working system. There's no need to complete the impossible task of constructing all the real numbers...after all, computers do math without ever having the complete set of real numbers in memory.
Quoting fishfry
I disagree with this decision. I believe it is possible to perform analysis without relying on the axiom of infinity. While I don't have formal rules or detailed structures yet, I possess concepts that would be found in an introductory calculus textbook, or perhaps an introductory engineering calculus textbook. Admittedly, this is a significant claim that requires substantial support...it's just that your eyes glaze over...
Quoting fishfry
Cantor's proofs are quite fascinating. Many people, often labeled as "infinity cranks," argue that actual infinities are riddled with contradictions. These individuals are in the minority, as most mathematicians do not share this view. I'm intrigued by the idea of a mathematics that does not rely on actual infinity, as I believe this approach is more aligned with true mathematics. It promises to be free of contradictions and brings with it the potential for beauty and advancement.
Quoting fishfry
Why do you talk of everything, such as 'all the open sets'? I can't imagine a computer holding this infinite set in memory. I'd rather talk about what I know is possible, such as a computer which holds a few open intervals with rational endpoints. As for infinitary operations, my long post with many bullets (let's call it the bullet post) addresses my view.
Quoting fishfry
Yes, the bullet post.
Quoting fishfry
Yes, the bullet post.
Such books don't axiomatize the principles used. And those books make use of infinite sets.
I think even constructivist and intuitionist set theories have a version of the axiom of infinity. But the logic of those systems is different from classical logic, so a statement in one system might not mean what it means in another system with a different logic.
I'd like to know whether a "no complete, only potential, infinity" concept has been axiomatized in a way that would be to the satisfaction of cranks if they were ever to actually learn about such things.
Also, keep in mind the amount of complication an alternative axiomatization might be. Already with intuitionistic logic, the semantics is much more complicated than with classical logic. Of course, that price might be worth paying.
That's what I was trying to conveyI don't have axioms and formal rules, but rather concepts relevant to a typical practitioner of calculus. My approach would achieve the same methods as traditional calculus but without relying on actual infinities. I agree that current calculus texts depend on actual infinity, which I believe is both unnecessary and undesirable.
Quoting TonesInDeepFreeze
But isn't it more like a potential infinity?
Quoting TonesInDeepFreeze
I wonder if what is missing isn't the axiomatic systemsof which constructivists offer manybut rather the relevant and accessible intuitions for a typical practitioner of calculus.
Quoting TonesInDeepFreeze
To be honest it's because I didn't feel like continuing the dialogue with you because I find some of your comments offensive.
I don't see why. One of the best ways you can respond to someone who brings a problem to you is to just ask them to explain it all to you in detail. By the time they're halfway done they usually solve it themselves. It's their explaining that does the trick, not my understanding. Smart bosses do that.
Also, the thing is, and I thought of mentioning this to you the other day, I don't actually care about anyone's ideas about what the real numbers are or how math should work. There's nothing at stake for me here. I enjoy trying to relate your ideas with things I know in math, but there's never going to come a point where I "digest" this. All this is a very half-baked brew, to mix metaphors.
Quoting keystone
Welcome. If you think everything's generated by algorithms, constructivism's your math philosophy.
Quoting keystone
Computers don't have thoughts. In fact, there aren't any numbers in computers. There are electrical circuits. It's the humans that interpret certain bit patterns in certain circuits as numbers. The Chinese room does not understand Chinese.
Let me ask you a question. Suppose there is a computer in the world that contains, at this moment, a bit pattern corresponding to the character string "XLII". Then suppose there's another computer somewhere else in the world, and it contains the bit pattern 101010. And in yet a third computer, we find a bit pattern corresponding to the character string "42".
Do these three computers each instantiate the existence of the same number 42? And how would you know?
I am drawing your attention to the problem of representation. There is never a number inside the computer. There are only bit patterns. And depending on the encoding scheme, the same bit pattern may mean different things; and different bit patterns may mean the same thing.
Quoting keystone
You can do engineering like that, but you can't do math. In finitism (rejecting the axiom of infinity) we can do a fair amount of number theory, but not analysis. You can't do calculus, you can't do physics. You can do finite approximations, but the underlying theory is infinitary.
Quoting keystone
You should research that claim rather than just proclaim it. This is one of the reasons I am never going to "digest" your ideas. Many clever people have given these matters considerable thought. You should do a literature search on this idea to clarify your thinking.
Can you see that grandiose claims made without sufficient background come down to untrained feelings and intuitions? Not that there's anything wrong with that. But it supports my belief that there is nothing to digest.
Quoting keystone
There's a mathematician and and engineer joke in there somewhere. And eye glazing is something else. I think you have a bad idea, not in the sense that it's absolutely wrong; but in the sense that you have a very naive understanding of what's involved, so that it seems grandiose.
But that is NOT what makes my eyes glaze. Certain diagrams and definitely long lists make my eyes glaze. Eye glazing is entirely independent of reasonableness. Your claim about calculus is unreasonable. That doesn't make my eyes glaze. The diagrams and lists make my eyes glaze. Hope that's clear. Someone could show me a diagram or list that was 100% correct and brilliant, and my eyes would still glaze. Eye glazing is no measure of the quality of an idea, it's just how I process or have difficulty processing it. But grandiose claims that you personally have figured out how to do analysis in the absence of the axiom of infinity, that's a bad idea. My eyes are perfectly clear about that.
Hope I made this distinction. Some of the best ideas make my eyes glaze.
Quoting keystone
Why? Infinitary set theory is perfectly clear of contradictions. Well, as far as we know. We can't prove its consistency without assuming more powerful systems.
Are you a Cantor crank by any chance? A long time ago (when I was younger and I suppose less harmless looking than I am these days) a traffic cop pulled me over. As he was writing my ticket, he asked me if I'd ever been arrested. "I just want to know who I'm dealing with," he said. That's why I ask if you're a Cantor crank. I just want to know who I'm dealing with.
Quoting keystone
The open sets were your idea. And the standard topology on the reals is generated by the open intervals with rational endpoints. There are only countably many of those.
Nobody's saying you can't approximate things with computers. You're imagining some kind of mathematical metaphysics that isn't really there. But the constructivists have this all worked out. Did I post this Andrej Bauer article, Five Stages Of Accepting Constructive Mathematics? Give it a read, tell me if any of it makes sense to you. I read it a while back, don't remember much. I never get very far trying to understand constructivists.
https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf
Quoting keystone
Ok good, thanks.
Agreed. But sometimes the person will not see the flaw in their argument unless explicitly identified by someone else.
Quoting fishfry
By "digest" I didn't mean to suggest that you would accept it. But there's value in being able to entertain a thought without accepting it.
Quoting fishfry
If a human thinks of a duck and somehow in their computations the duck behaves exactly like the mathematical object 42, then (within that person's thoughts) the duck represents an instance of the number 42. As the old saying goes, "If it swims like a 42, and quacks like a 42, then it probably is a 42." I think we both agree that absent of an intentional being giving mathematical meaning to the duck (or to electrical activity within a computer), no mathematics is going on.
But if at a later time the human's thoughts of the duck do not correspond to the number 42, then the duck is no longer an instance of 42. The number 42 is contingent on thought. It's existence is temporary.
If we frame our views within this context, the difference is that you believe in an infinite consciousness whose thoughts eternally encompass all numbers. On the other hand, I believe there is no such preferred consciousness; rather, there are only finite consciousnesses whose thoughts can hold only a finite number of numbers at any given time.
Quoting fishfry
Computers perform calculus, and everything they do is finite. So, you're essentially arguing that there's a disconnect between the theory and the practice. Remember, in the case of calculus, the practice came first, and mathematicians later developed an actual-infinity-based theory to justify the practice. Might it be possible that a potential-infinity-based theory could provide a better justification for the practice? This one-minute video by Joscha Bach, titled "Before Constructive Mathematics, People Were Cheating," eloquently captures my view: https://www.youtube.com/watch?v=jreGFfCxXr4
Quoting fishfry
While I haven't done much research on logic, I have a reasonably strong grasp of basic classical calculus. I understand that continuity is essential for classical calculusmy view starts with continua. I also understand that limits are essential for classical calculusmy view achieves the same ends by using arbitrariness. If you don't want to entertain my ideas simply because clever people weren't able to make calculus work within a finitist framework, that's fine as well. But let's be clearit's not that you can't digest my ideas; it's that you won't entertain them.
Quoting fishfry
I understand how my claims appear. I'd like to support my position but it's quite hard if you don't look at my figures or words. You ask for the beef but the only comments you respond to are the bun.
Quoting fishfry
I believe my view is naïve in the same sense that Naïve Set Theory is naïve (minus the contradictions).
Quoting fishfry
Joscha Bach seems quite confident that classical mathematics is filled with contradictions.
Quoting fishfry
Youve probably heard the story of Penzias and Wilson, who struggled with persistent background noise on their radio receiver, initially attributing it to pigeon droppings. It turned out to be the cosmic microwave background radiation from the Big Bang, earning them a Nobel Prize. I believe Cantor has interpreted his incredible discoveries as mere pigeon droppings.
Quoting fishfry
I don't think I mentioned open sets.
Quoting fishfry
Funny you mention this. I skimmed through it a few days ago and then watched his YouTube lecture by that name yesterday. Now, I'm in the middle of his lecture on LEM. I'm really excited about watching his lectures.
Quoting fishfry
Is your preferred format essay?? How did you become a mathematician and not an english major? But seriously, how am I supposed to communicate my ideas to you? This might not be the best chat forum etiquette, but would you be open to a Google Hangout? ...Please feel no need to even respond to that idea...
I am trying to understand you.
Quoting keystone
I'm trying to understand your argument or thesis or idea.
Quoting keystone
You said that numbers get instantiated when they appear in a computation. I asked you whether one number or several numbers get instantiated when various representations exist. Who determines that they act the same? Where is that process, that brings a number into existence?
Quoting keystone
Is God watching all this and keeping track of everyone's version of each number? This seems like a cumbersome idea.
Quoting keystone
I believe no such thing, what are you talking about? I believe in the axioms of ZF and not much else. They are purely a human artifact.
Quoting keystone
Yes, that's how constructivists think. Thanks for telling me it was a short vid, you got me to watch it. Math doesn't need a justification. It doesn't have to make constructivists happy.
Quoting keystone
You are making a grandiose claim that's likely to be false. But there are plenty of productive finitists, and the constructivists are on the march these days due to computer proof systems. But there is something to be said for infinitary math. Why shouldn't we enjoy having such a lovely theory of the infinite? What is the harm?
Quoting keystone
LOL. It's hard to develop a theory of the reals without the axiom of infinity. The figures and words haven't helped much so far. You say, Start with a line. Make a cut. I don't know what these things are. You're just approximating the reals. I don't see anything to grab on to.
Quoting keystone
Your idea isn't naive. It's grandiose. And let's talk about something else please.
Quoting keystone
You say that like it's a bad thing!!
[quote=Wiki]
Paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating[clarification needed] way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" systems of logic, which reject the principle of explosion.[/quote]
https://en.wikipedia.org/wiki/Paraconsistent_logic
But some guy thinks standard math has contradictions. He could be right. And if it did, the contradictions would be repaired. People wouldn't stop doing infinitary math. If a contradiction were found in ZF, it wouldn't affect group theory or differential geometry .
Quoting keystone
I don't follow the analogy you're making. Cantor has underestimated or overestimated his discoveries?
Quoting keystone
You have been making use of open intervals all along, haven't you?
Quoting keystone
Ok. Well if I turned you on to constructivism I'm happy and I've done some good. But I can't go down that road with you too far, because I have tried to understand constructivism a few times and it just doesn't speak to me. I like infinitary math and I think that if you reject the noncomputable, you are missing a lot.
You know, that's a particular bit of philosophy I can assert. The noncomputable reals are telling us something. Infinitary math is telling us something. The history of math is expansive, never contractive. Nobody says, "Those complex numbers, they were a step too far." But they say that about infinitary math.
Quoting keystone
Well I didn't become a mathematician! I got to grad school and my eyes glazed. Well there were a lot of things going on. I ended up in programming. In math when you're stuck, you're stuck. In programming you can always code something, get something running, solve a bug, do something useful. I don't have an eye glaze factor when I'm coding, but I do when I do math. Interesting, I never thought of it that way but it's true.
Quoting keystone
Are you getting frustrated? I'm sorry, I thought I was helping the best I can. You're doing fine, I've understood a lot. It's a tall order to reformulate analysis without the axiom of infinity. Even constructivists have infinite sets, not not noncomputable ones. It's like if you told me you were going to do brain surgery. You might be able to learn to do it, but you are not there yet as far as I can see.
Quoting keystone
No thanks, this forum's all I can handle and barely at that. Plus I hate Google. They went from Don't be evil to being evil.
I don't know why you are acting as if I'm not attending to what you say. I sense a difference of perspective that I'm not privy too. Everything seems fine at my end.
I was trying to convey that the representation itself is not important; what matters is the behavior. If in my mind x+x=2, then x behaves like 1. Similarly, if y+y=2, then y also behaves like 1. In this scenario, 1 has multiple representations (x and y) in my mind, but that isn't an issue because they both behave the same.
But I must highlight that to conclude that x=1, I dont work through an infinite checklist, considering all possible arithmetic equations involving 1. No, I'm mindful of the consistent and finite set of rules associated with the construction and arithmetic of the SB-tree (or equivalent tree), so all I need to do is declare that x will behave like the node occupied by 1. At that moment, I bring 1 into existence and it is representation in my mind is the character x.
Quoting fishfry
We each are the god of the mathematical systems that inhabit our own minds. If we want to compare my (-inf,1) U 1 U (1,+inf) with your (-inf,42) U 42 U (42,+inf) we need to agree to the SB-tree and compare the nodes where my 1 and your 42 lie. If they correspond to the same node, then our systems are equivalent. While nobody explicitly does this, it's the unspoken agreement we make when comparing systems. I don't see why we would need a third party to arbitrate the comparison.
Quoting fishfry
Since no human artifact can be infinite, is it fair to say that you believe in the axioms but not in the infinite objects they describe? If so, this directly supports my thesisforget about the existence of infinite sets and instead focus on the (Turing) algorithms designed to generate these infinite sets.
Quoting fishfry
I'm receptive to a constructivist approach to the axiom of infinity. If were talking about computable infinite sets in the same way that Turing talked about computable real numbers I have no problems, provided we do not assert the existence of infinite objects.
Quoting fishfry
What harm is there in relying on Newtonian mechanics when it performs admirably for slow-moving objects like ourselves? Similarly, what harm is there in embracing General Relativity when the singularities it predicts are distant from our everyday experience? There's a certain beauty in their simplicity; as a mechanical engineer, I rely on Newtonian mechanics daily and will continue to do so regardless of advances in physics.
Yet, as physicists began pulling the loose threads of classical physics, a more fundamentally robust and aesthetically compelling framework emerged: Quantum Mechanics. There is something to be said for pulling the loose threads.
Quoting fishfry
It would be much easier if you would just roll with the intuitions for a little while so we wouldn't get stuck on the first step. Let's sweep through the whole idea informally and if it has any merit then we can sweep through again and formally define things. Think of a line as a piece of string. Think of a cut as what you do with scissors to partition the string. You're making this more complicated than it needs to be.
Quoting fishfry
No, I'm not. Yes, I'm referring to (Turing) algorithms that produce rational approximations with arbitrary precision, but the algorithm itself is exact. The algorithm perfectly encapsulates the essence of the real. That's why I'm emphasizing the algorithm itself, not its output.
Quoting fishfry
I agree to both sentences! (1) That's what I'm trying to do and (2) I'm just trying to throw a 'potential' in front of the 'infinitary math'.
Quoting fishfry
Cantor has already received considerable acclaim, making it difficult to envision greater recognition for him. What I meant to convey is that Cantor unearthed something monumental, yet his interpretation was poop (actual infinities). I believe that in the future, it will be recognized that his true discovery lay in articulating the potential within continua and mathematics as a whole.
Quoting fishfry
I'm using open interval notation to describe the bundle (line) the lies between its endpoints. This bundle cannot be described as an infinite set of individual points, because as I mentioned before, we can only talk about individual points when the line has been cut. For this reason, I'm reluctant to say that I've been proposing open sets.
Quoting fishfry
I agree with this sentiment. Whether it's noncomputable reals, the halting problem, Gödel's incompleteness theorems, or the liar's paradox, they are all screaming at us that there is a potential in mathematics that cannot be fully actualized. But Classical mathematics aims to actualize everything, much like classical physics. They both suffer the same flaw...and I believe are both addressed with the same resolution: a top-down view.
Quoting fishfry
Yeah, you're going to lose some things with constructive mathematics, be it LEM or the axiom of choice. But by and large I'm proposing a much more beautiful structure. Just as classical physics was a natural stepping stone to QM, actual-infinitary-math is a natural stepping stone to potential-infinitary-math.
Quoting fishfry
I can reframe my examples in Python if that's your preference. The main drawback however is that my posts would get longer.
Quoting fishfry
It's frustrating to think that I'm running out of ways to communicate my ideas so I'm starting to think that the conversation might end prematurely with the least desirable conclusion (that you don't know whether my ideas are right or wrong). But perhaps it's too soon to talk about the end. I'm getting value out of every post you and I write so I'd be grateful if we keep going and just take it one day at a time.
Quoting fishfry
It's just that at some point we'll need to talk beef and I'll need to figure out an alternative way to communicate the bullet post.
Yes but who determines that? You said that numbers come into existence when some computer represents them. If one computer represents "XLII" and other represents "42", does your creation engine instantiate one or two numbers? How does God or whoever decides what numbers are instantiated, immediately know whether those two expressions always behave the same? If you think about it it's a hard problem.
Quoting keystone
"... the consistent and finite set of rules ..."
Oh, you use an axiomatic BOTTOM UP system after all!!!!!
Quoting keystone
If you have a set of rules (a bottom up concept) that let you know when two representations denote the same number, then why do you need the computer? Why not just accept that the rules themselves bring all possible numbers into existence already?
Quoting keystone
I believe in the mathematical existence of the abstract objects they describe. I make no claim that they are physically real. Just like when I read Moby Dick I accept the existence of Ahab and the whale, without making any ontological commitments outside of the context of the novel.
Quoting keystone
There simply aren't enough algorithms to generate all the sets. There are countably many algos and uncountably many subsets of the natural numbers.
Quoting keystone
Ok. You should study constructivism. I can be of no assistance, I don't know much about it.
Quoting keystone
Completely wrong analogy. Infinitary math always works. It just doesn't apply to physics (as far as we know). And if it DOES someday turn out to be important for physics, it will be a good thing if the pure mathematicians have been studying it all along.
Just as Riemann developed non-Euclidean geometry 70 years before Einstein needed it for general relativity.
Or just like Diophantus studied number theory some 2200 years before public key cryptography became the foundation of online commerce.
Quoting keystone
Then you are agreeing with my point. I read something interesting once about sailing. Sailboat technology did not get really good until sail was no longer a useful means of practical transportation. Today's high-tech sailboats were developed by purely recreational sailors, after sailboats became obsolete in commerce.
Quoting keystone
I disagree with the analogy. Constructivism is not a deeper form of conventional math. But I can see the argument being made. Just not by me.
Quoting keystone
I'm fine with cutting strings. You have never explained to me how this serves as a new foundation for math.
Quoting keystone
That's fine, you're making a constructivist argument. I'm the wrong person to have that discussion with, since I have no feel for constructivism, despite making a run at the subject several times.
Quoting keystone
I'm not stopping you. You have a ways to go in terms of developing a comprehensive, logical theory that I can understand. It could just be me.
Quoting keystone
That's a stretch. Please don't be a Cantor crank. I'd be disillusioned. If you are one, it's better to keep it to yourself.
Quoting keystone
So after all this time your interval notation does not not stand for its conventional meaning?
How nice of you to let me know.
Can you see that in terms of communication, you could have either made that clear up front, or defined a different notation, like
Quoting keystone
I would say that the noncomputable reals show us the limitations of algorithms. And, being a skeptic of simulation theory and the trendy thesis that human minds are Turing machines, this is an important plank in my platform. Algorithms are vastly insufficient to express what's important, either about math or reality. This is perhaps why the constructivists annoy me.
Quoting keystone
I have no doubt you have a beautiful structure in mind. Your challenge is to express it. Maybe I'm a poor sounding board.
Quoting keystone
Jeez people's code fragments definitely make my eyes glaze. No code please.
Quoting keystone
No risk of that. I already know your ideas are wrong. Hey you set yourself up for that one :-)
Quoting keystone
That's fine with me. I can't promise to meet your expectations about what you wish I'd write. I've responded sensibly and I think relevantly to each point you've made. I don't have to agree with your overall point of view, or even understand it.
But of course you could start by defining your notation.
Those parenthesis (a,b) don't stand for standard open sets. Can you see that you redefined a notation that's so universal that I had no choice but to be sent into a state of confusion about your meaning? Can you see why I've been confused for weeks?
Quoting keystone
I make no commitment to meet your conversational expectations. What you see is what you get. You have expectations about my state of mind that are unlikely to be met. I'd ask you to accept that rather than continually expressing disappointment with my posts.
According to the Buddhists, unfulfilled expectations are the source of suffering.
Like I say, if you want to communicate, don't redefine extremely well-known standard notations without announcing it clearly.
And also, if I could make this request ... can you write shorter posts? Short and to the point.
In my first semester as a grad math student I was required to take a course called Introduction to Graduate Mathematics. It was basically naive set theory and at the end the professor said, "You should only continue in foundations if this course really appeals to you. How many of you find that to be the case?" I recall out of thirty students one or two hands went up. The rest of us wiped the glaze out of our eyes and went on into other areas of mathematics.
But this was 1962, back in the dark ages. And at a state university, not a top-notch school, like Harvard. Things have changed since then. When were you in a grad math program?
Not true. I published papers when I was active that never assumed infinity was actualized. Fryfish and I, sometime back, argued about the use of transfinite math in analysis, particularly functional analysis. He pointed to the use of Zorn's lemma or the axiom of choice as a required tool to prove the Hahn-Banach theorem, and I replied that that was true, but by altering the hypotheses slightly, they were not required. Hahn-Banach was my only very brief encounter with transfinites in my career. But then I sought interesting theory in classical analysis - a far cry from foundations. So, your statement is not entirely correct.
But I admire your tenacity.
I believe the main issue is that new topics are added more often than old ones are removed, leading to bloated posts. I'll not respond to a few of your comments to address this request.
Quoting fishfry
Label the original string (-inf,+inf). Cut it somewhere. Label the left partition (-inf,42). Label the right partition (42,+inf). Label the small gap between the strings 42. Now you have a new system: (-inf,42) U 42 U (42,+inf). But you seem to get hung up on those intervals number being continuous even though I'm saying that those intervals describe continua - abstract string in this case.
Quoting fishfry
Moving forward, instead of writing "computer+mind", I'm just going to write "computer".
I believe that true mathematical rules exist independently of computers. These rules are necessary truths and finite in number. If one assumes they describe actually existing objects, such objects must exist beyond our comprehension, as no computer could contain them.
However, if we assume that mathematical objects must exist within a computer, then not all mathematical objects can actually exist and it becomes a matter of a computer choosing which objects to actualize.
Quoting fishfry
Please allow me to use the SB-tree as something concrete to talk around. I acknowledge that any infinite complete tree will do.
We outline the rules for constructing the SB-tree and can mentally construct it to an arbitrary depth. Everything we ever actually construct is finite. Why insist on believing in the computationally impossible the existence of the complete SB-tree?
Quoting fishfry
You've said that the reals correspond to unending paths down the infinite complete binary tree, so indeed, there are potentially [math]2^{\aleph_0}[/math] paths that cannot be algorithmically defined. This doesn't mean the rules for constructing the tree are incomplete; it simply means there are paths computers can never traverse. Computers cannot exhaust these rules.
Or here's how I see it. When I see the tree, I do not see paths and nodes. Instead I see a continua at each row, being cut by the numbers at each row. For example, I see the top two rows of the SB-tree as:
Row 1: 0 U (0,1) U 1 U (1,+inf)
Row 2: 0 U (0,1/2) U 1/2 U (1/2,1) U 1 U (1,2) U 2 U (2,+inf)
...
With this view, I would rephrase the conclusion as follows: computers cannot completely cut continua. Computers cannot exhaust cutting. Actually, I would go one step further and assume that computers are all that's available, so I would simply say that continua cannot be completely cut. But we know that already, you'll never cut a string to the point where it vanishes.
Quoting fishfry
Sometimes I push back as a form of defense. Nevertheless I'll try and be more mindful of this. I'm very appreciative of our conversation. Thanks!
I'm putting this comment in a separate post because I wanted my main post to be 'smallish' and I'm not expecting a response from this.
Quoting fishfry
Here are quotes from my earlier posts. You don't have to read all bullets as they all say the same thing. I'm just trying to highlight that the confusion is not for lack of me trying.
Certain areas of mathematics, like combinatorics, are sufficiently distant from foundational issues and actual infinities. These areas transcend the label of 'classical' mathematics.
In classical set theory, we define 'is infinite'. I don't know whether there is a definition of 'is potentially infinite' in any theory. But, it seems reasonable to say that the notion of potential infinity is supposed to be upheld in constructivist and intuitionist set theories, which is the very reason I've brought them to your attention.
Note that Brouwer himself was not interested in formalization. But some constructivists and intuitionists have been.
Quoting keystone
A litte earlier:
Quoting keystone
But of course, people have the right to change their mind, even for the least of reasons.
Nicely done that at the end of a post in which you did continue to comment on my remarks, you say that you don't want to continue. And however you've taken other comments by me, the one I made about the link to a video is not very much offensive, if at all. On the other hand, it is at least a bit rude to link to a video without saying what point you want to make with the video or what part of the video you have in mind, so that I wouldn't have to have to watch all of a video without even context of your point about.
Anyway, I gave a proof that you are incorrect when you claim that the interval (0 1) is not an infinite union of disjoint intervals, whether or not you want to take a minute to understand the proof.
What makes them cranks is not that they don't accept that there are infinite sets nor that they find the notion of infinite sets nonsense or fatally problematic. What makes them cranks is that their arguments about classical mathematics are ignorant, ill-informed, misrepresentational and astoundingly irrational. Sure, it's great that classical mathematics can be critiqued, but the ignorance, misinformation, misrepresentations and irrationality of the cranks is noxious
No Cantor crank would ever have the self-awareness to know that he or she is a crank.
They are included in classical mathematics. They may be developed in set theory, without using the axiom of infinity, but they are still included in set theory. Moreover, with mathematical logic we have the formalizations of primitive recursive arithmetic that can be interpreted in set theory, and PA that can be interpreted in set theory, and set theory except with the negation of the axiom of infinity, which is inter-interpretable with PA.
You clearly have a lot of knowledge, and many of your posts, including your recent ones, are informative and well-intentioned. However, sometimes I feel bad after reading your posts. Even though I have a thick skin for criticism (as seen with fryfishs posts), I still sense that fryfish likes me despite their criticisms. On the other hand, I feel like you dislike me or what I represent. I cant debate classical mathematics to your level of formality, and you don't seem interested in my ideas, so Im not sure what we have left to discuss.
Regarding the hyperlink, I admit I was rude. I was upset and wanted to throw some rudeness back at you. I should go back and switch the link to be a Rickroll for good measure.
Quoting TonesInDeepFreeze
I don't think we'll agree on terms. For example, in another message to you which you ignored I explained that I want to think of the infinite series 9/10 + 9/100 + 9/1000 + ... as a Turing computable algorithm, which can output arbitrarily precise partial sums but never output a 1. I get what you're saying, but in this sense, your function will never output intervals which will union to (0 1).
Quoting TonesInDeepFreeze
I've been very open about my views on Cantor, actual infinities, my informal training, and my motivations. While I don't believe any Cantor crank shares my perspective, if someone wants to label me a Cantor crank, that's their prerogative.
Quoting TonesInDeepFreeze
I need to rephrase my statement as I was using 'classical mathematics' in an unorthodox sense. I should have said bottom-up mathematics instead of classical mathematics. Combinatorics transcends the distinction of bottom-up and top-down mathematics. Again, this distinction wouldn't interest you since it relates to the ideas I'm proposing.
Whatever I say will be flavored by my top-down view which you're not interested in. I think I don't have anything for you.
It doesn't matter whether I like you. For that matter, I can't have any fair opinion of you as a person aside from this extremely narrow context of posting. I have no reason to doubt that you are a decent and likable person away from posting. On the other hand, yes, I very much dislike your posting modus operandi.
It's not only a matter of discussing to a degree of formality. Rather, it's that you say a lot of things that are incorrect. But, yes, you handwave through just about everything.
As to my interest, I took a whole lot of time a while back to go through all the details of your proposal at that time. My participation was indeed generous. But even as I adapted to your many revisions, it ended up in a dead end where your proposal was still hopelessly vague and reliant on sophistical ambiguities. And even if I don't have the time and interest to engage yet again the broad handwaving scope and details of your musings, it is still eminently reasonable to point out particular clear falsehoods and misunderstandings you post. For example, rather than take a minute to understand my refutation of your false claim about (0 1) and to understand my proof, you fuss that I don't seem to like you.
I didn't respond to your notion of an algorithm, since it doesn't vitiate that (0 1) is an infinite disjoint union of intervals. Again, an example of your modus operandi. You make a false claim, but then complain that it's not refuted because of some other red herring about some other notion you have.
What you just said is an utter disconnect. That no finite partial sum is 1 in no way contradicts that (0 1) is an infinite disjoint union of intervals. You argue on the basis of your kinda sorta associations about two different things rather than be responsible to make actual logical connections.
For the record, I did not say you're a 'Cantor crank'.
More childishness from you. From the fact that I'm not interested in going over all your stuff all over again, it is not entailed that I rule out being interested in any ideas you might mention. And again your childish penchant for turning blame around. You admit that you misused 'classical' but still manage to blame me anyway.
Bringing your original comment back...
Quoting TonesInDeepFreeze
What I'm trying to convey is that f(0) U f(1) U f(2) U ... doesn't fit into any computer, just as complete infinite series do not fit on a computer. What makes sense computationally is f(0) U f(1) U f(2) U ... f(N) when N can be an arbitrarily large natural number. Just as Turing focused on computations of partial sums when considering computable reals, I want to focus on computations of partial unions when considering infinite unions. Within the constraints of computation, there is no partial union of intervals from f which corresponds to (0,1).
My knowledge of mathematics, logic and philosophy is quite meager. But I do have a good grasp of certain basics and an intent not to misstate them (at least within reasonable informal explanations).
Of course, we understand that computations are finite.
But the specific mathematical statement you made earlier was incorrect. You'd do yourself a favor by recognizing that fact.
'75 -'77.
Sounds like your professor just didn't like foundations.
Ok thanks. I realize that I myself write long posts.
Quoting keystone
Define your notation, since you already told me this is NOT an open interval in the real numbers.
Did I not complain bitterly enough about this in my last post?
Quoting keystone
I don't know what your notation means and you are not going to tell me.
Quoting keystone
In another thread going on right now, it's been pointed out that there are uncountably many mathematical truths, and that most of them can't even be expressed, let alone proven. See
https://thephilosophyforum.com/discussion/15304/mathematical-truth-is-not-orderly-but-highly-chaotic
Quoting keystone
I don't think anyone believes that, not even the constructivists. Maybe they do. It's pointless to argue constructivism with me, I know nothing about it.
Quoting keystone
I don't know why. It's not helpful to me at all.
Quoting keystone
I don't think it's productive for you to try to talk me out of my mathematical beliefs. I believe in the axiom of infinity and the higher transfinite cardinals.
Quoting keystone
So what? My mathematical ontology is not confined to what's computational. Yours is. So you should study constructivism. It's pointless to try to discuss it with me, since I have already made good-faith efforts to get interested in constructivism, without getting interested.
Quoting keystone
Whatever. I can't argue these points. Computers are woefully inadequate to express mathematical truth.
Quoting keystone
You're welcome. I just can't argue constructivism. There use to be some constructivists on this board. Long gone.
Please use a different notation. The notation (a,b) means something else.
But you immediately have problems. What does "between" mean unless you define an order relation?
Some do.
I've never seen one. Every one of the hundreds and hundreds of cranks I've seen lacks the self-awareness to understand the ways in which they are a crank.
You might think so from what I said, but he was young and pretty enthusiastic about teaching the subject. We had numerous worksheets that eventually led to the construction of the exponential function. So, his comment at the end came as a bit of a surprise. :cool:
Foundations are a bit of a backwater. My grad school had an excellent math faculty but no interest in foundations. They had one professor who was an eminent young set theorist. He didn't get tenure even though he was becoming quite famous. He quit math and went into medicine. Quite a loss for set theory.
Looking back I wish to hell I'd worked harder. I was depressed or having some grad school blues, never had very good study habits, didn't develop any. Oh well. Regrets of the past.
This is my statement you're referring to: "I can write (0,1) as the union of arbitrarily many disjoint intervals. However, I cannot write (0,1) as the union of infinitely many disjoint intervals."
In your response, you did not explicitly present the union. Instead, you described a function that takes a natural number as input and outputs an interval. At no point did you actually show a union of infinite intervals in its entirety.
I understand that, classically, your function is interpreted as identifying infinite intervals that exist simultaneously. However, I do not accept that perspective. I believe that you need to construct the objects you are discussing.
In that thread they state that "any set of sentences can be a set of axioms." I want to distinguish between what is (i.e. actual) and what can be (i.e. potential). It is tempting to actualize everything and declare that there are uncountably many mathematical truths. However, I would argue that these truths are contingent on a computer constructing them. When I speak of finite necessary truths I'm referring to the rules of logic itself.
Quoting fishfry
I'm trying to establish a view of calculus which is founded on principles that are restricted to computability (i.e. absent of actual infinities). You don't have to abandon your view of actual infinities to entertain a more restricted view. Perhaps we can set aside the more philosophical topics and return to the beef.
Quoting fishfry
The term 'line' comes loaded with meaning so to start with a clean slate I'll use 'k-line' to refer to objects of continuous breadthless length (in the spirit of Euclid). I'll use
The systems always start with a single k-line described by a single k-interval (e.g. <-?,+?>). A computer can choose to cut the k-line arbitrarily many time to actualize k-points. For example, after one cut at 42, the new system becomes <-?,42> U 42 U <42,+?>.
The order relation comes from the infinite complete trees.
Are we at a place where we can we move forward?
Read the proof to its end. The union of the range of the function is an infinite union of disjoint intervals and that union is (0 1).
The use of 'can' there is merely colloquial. We may state it plainly: Any set of sentences is a set of axioms. More formally: For all S, if S is a set of sentences, then S is a set of axioms.
From my perspective, we can only discuss objects that can be explicitly constructed. Since the complete output of the function cannot be generated all at once, it is meaningless to talk about the range of the complete output. However, I acknowledge that the standard view assumes that you can discuss the range of the function whereby the range unions to (0 1). You don't have to keep repeating your point; I understand it. I'm just viewing this from a constructivist standpoint, and from my perspective, my statement holds: you cannot explicitly write the union of infinite disjoint intervals.
Quoting TonesInDeepFreeze
Again, we're approaching this from different ontological perspectives. It seems you're trying to point out flaws in my viewpoint by identifying how it differs from yours. If you want to challenge my perspective effectively, it would be more impactful to identify actual contradictions or limitations within my own ontology rather than highlighting its differences from yours.
You hadn't said that you understand the point, so the point deserved repeating.
Quoting keystone
But you don't have to keep repeating that point, as it has many times been recognized that the there is no finite listing of an infinite set. Even more simply than that there is no finite listing of the members of the infinite union of intervals, we may observe that even more basically there's no finite listing of the set of natural numbers.
I wish you wouldn't presume to speak for "a constructivist standpoint". You do seem to be along the lines of constructivism (along with a notion of potential infinity) but there's a lot more to constructivism than you know about, so I think it invites error when you speak on behalf of constructivism.
For that matter, it is not a given that my argument is not constructive. I constructed an infinite set in the sense that I used only intuitionistic logic to prove the existence of a particular, named set. If I am not mistaken, constructivism in the broadest sense does not disallow construction of infinite sets. It only requires that an assertion of the existence of a set with a certain property is only allowed when at least one particular, named set is proven to have that property. I claimed that there is a set that has the property of being an infinite disjoint union of intervals such that the union is (0 1). And I proved the existence of a particular, named such set. However, granted, the proof requires the axiom of infinity, which may not be considered constructive, since it merely asserts that there exists a successor inductive set without adducing a particular one. However, the axiom of infinity is equivalent to the claim that there exists a set whose members are all and only the natural numbers, which is a particular, named set.
For example, Cantor's proof that there is no enumeration of the set of real numbers is accepted by constructivism. From any enumeration of a set of real numbers, we construct a real number that is not in the range of that enumeration. Constructive.
It seems that you heard about constructivism and your reaction is "Goody, now I can put my own wonderful alternative framework under the banner of a cool, authoritative school of thought" but without actually understanding what all is involved in constructivism.
That is incorrect. In the instance about "can", I merely provided you the information that mathematics doesn't need to use "can" but rather can use "is".
Quoting keystone
As I've said about four times already, months ago I spent a quite generous amount of time and energy following up the finest details in your proposal, but that ended up in a bust with your continually shifting equivocations, handwaving and contradictions. Now it seems you're proposing yet another revision. I don't have interest in going down another path like that with you.
Meanwhile, when I do mention certain individual misstatements, you repeat your whining that I am not engaging the full glory of your wonderful alternatives. But I don't have to do that merely to correct certain misstatements and provide you with explainations, even though too often you evidence that you lack the maturity and restraint from self-grandiosity to truly think about the explanations.
And I haven't claimed a particular ontology, so "your ontology" is inapposite.
I am getting in halfway into the chat. Is "constructivism" here used in the context as related (not synonymous) to Brouwer's intuitionism, or something else?
Constructivism is broader than intuitionism. Intuitionism is one form of constructivism. I don't opine as to what other poster's notion of constructivism is, except that it would not be correct to claim that arguments such as Cantor's argument that there is no surjection from the naturals to the reals are not constructive. Similarly, not correct to regard my recent argument about a certain union as not constructive.
I think it's fair to say that my top-down view likely fits under the constructivism umbrella, but my view does not represent constructivism as a whole. It is a fair request that I represent my view, not constructivism as a whole.
Quoting TonesInDeepFreeze
Within the context of my view, we can talk about algorithms designed to construct infinite sets (as in your example) but we cannot talk about the complete output of such algorithms. Rather, in the spirit of Turing, we can only talk about the partial output of such algorithms, which necessary is a finite set.
Quoting TonesInDeepFreeze
Cantor's proof holds value within the context of my view.
Quoting TonesInDeepFreeze
So be it.
Classical mathematics itself first formulated that there is no algorithm that prints all the members of an infinite set and halts.
Quoting keystone
How nice. My point is that it is constructive.
Quoting keystone
Let "so be it" be.
Alright, so it was what I was thinking. Just confirming.
Mercifully short. Thank you muchly.
Logic isn't constrained to computability.
"However, I would argue that these truths are contingent on a computer constructing them."
Argue that all you like. I can't engage, since I'm not a constructivist. AND, having tried to learn constructivism from time to time, it just doesn't resonate with me. But nevermind the other thread then.
Quoting keystone
I have answered this several times already.
(1) Constructivism is fine, you should study it.
(2) I'm the wrong person to discuss this with.I have no affinity for constructivism despite trying over the years.
Quoting keystone
Ok. I feel like we're about to go through this same exposition again. At least the notation's less confusing.
Quoting keystone
I certainly hope so.
I'll try to do a better job this time. But first, one other area of confusion has been the distinction between infinite and arbitrary as it relates to an algorithm's design vs. it's execution. A Turing algorithm for constructing N, is designed to output a set of m elements, where m can be arbitrarily large. By this I mean that the algorithm itself sets no limit on the size of its output; rather, the size of the output depends on the execution (i.e., the chosen 'precision' based on available resources). Please note that I'm not saying that m is a particular number, nor is it infinity. Instead, when talking about the algorithm itself, m serves as a placeholder for a value that is determined only upon execution of the algorithm. Upon execution, m is replaced with a natural number and the output is a finite set. In a similar vein, when I speak of ? being arbitrarily small, I am using it as a placeholder to describe an algorithm. Upon executing that algorithm, ? is replaced with a positive rational number that is small, but by no means the smallest. Is this clear?
You know, that's a very interesting point.
One difference is that rationals get arbitrarily close to 0. But I'm not sure it's all that different. Maybe you have a good analogy. I am not 100% sure what I think about this yet.
I believe you're saying this:
You can always find a rational interval small enough to suit the needs of any computation you do, by analogy with always being able to find a suitably large but finite natural number when you need it for a computation.
Is that a fair understanding of your point?
Also yes, that's arbitrary. You can always find as many as you like, but always a finite number.
We call that "finite but unbounded." There are computations that need 1, that need 2, etc. There's no upper limit to how large a number you can use."
To be actually infinite is far stronger. It's like putting out a good but rational approximation to a real number, versus "printing it all out at once" as it were. Having not just as many digits of pi as you need; but rather all of them at once.
That's the magic of the axiom of infinity. The difference between unbounded, or arbitrary, and infinite.
That is a fair understanding of my point but I do want to highlight one thing: it's not always about the computation. If I want to focus on algorithm design (and not execution), I can keep ?'s floating around. The ?'s only need to be replaced when I execute the algorithm and perform the computation. Fair?
Quoting fishfry
Yes, actual infinities are beyond computation.
Quoting fishfry
It does seem to be a bit magical. I'd like to avoid magical thinking if at all possible.
Yes fair. Though I am not entirely sure I ever understood the distinction you're making. Of course I do understand the difference between an algorithm written on paper and an execution of the algorithm in a digital computer; a physical process requiring time, space, and energy; and outputting heat. That's something a guy I worked with said. The only observable output of a cpu is heat.
Anyway. Yes I understand the difference. No I don't understand what POINT you are making about the difference.
Quoting keystone
Yes but indirectly IMO. Once you get a countably infinite infinity, you immediately get from the powerset axiom an uncountable infinity. But here are only countably many ways to talk about things. Leaving most of the world inexpressible.
Now the constructivists are entirely missing that much of the world, and imagining it doesn't matter. I think it matters. So I have a philosophy about his. I think constructivists and computationalists of all kinds: simulation theorists, mind uploading theorists, philosophers who claim mind is computational, etc -- I think all these people are missing something really important about the world. So I think I'm a bit of an anti-constructivist!!
Quoting keystone
The magic is the best part. I love the mathematics of infinity. I would never deny it from my world. I am a great devotee of the axiom of infinity.
I haven't made a point yet. I just wanted to clarify this as I previously found it a stumbling block in our conversation.
Quoting fishfry
I'm a computational fluid dynamics analyst, so I naturally approach things from a simulation perspective.
Quoting fishfry
In the context of a potentially infinite complete tree, to me it makes sense to talk about potentially (countably) infinite nodes and potentially (uncountably) infinite paths. In this sense, the paths have more potential than the nodes. I don't think any beauty is lost in reducing infinity to a potential.
Quoting fishfry
As QM suggests, something funny happens when we're not observing the world. I consider this to be the magic of potential. Anyway, this is fluffy talk about potential...let me get to the beef.
Really? I thought you made a pretty good point. You got me to understand what epsilon is.
Quoting keystone
Ahhhhhhhh ... now I understand your point of view. You use computers to study continuous flows. This makes perfect sense now.
Quoting keystone
I'm on record as disagreeing. The axiom of infinity adds a whole beautiful new world. AND is indispensable to developing the very approximation techniques that you use!
Quoting keystone
ok
I'd like to distinguish between a fraction and a real. The fraction description is finite (e.g. [math]\frac{1}{1}[/math]), whereas the real description is infinite (e.g. [math]1.\overline{0}\[/math] which can be represented as an algorithm that generates the Cauchy sequence of fractional intervals: [math](\frac{9}{10}, \frac{11}{10}), (\frac{99}{100}, \frac{101}{100}), (\frac{999}{1000}, \frac{1001}{1000}), (\frac{9999}{10000}, \frac{10001}{10000}), \ldots[/math]).
Because fraction descriptions are finite, a cut at a fraction can be planned and executed all in one go. A cut of [math](-\infty, +\infty)[/math] at the fraction [math]\frac{1}{1}[/math] results in: [math](-\infty, 1) \cup \{1\} \cup (1, +\infty)[/math].
Because real descriptions are infinite, a cut at a real must be planned and executed separately.
The algorithm to cut [math](-\infty, +\infty)[/math] at the real [math]1.\overline{0}\[/math] is generalized as: [math](-\infty, 1-\epsilon) \cup \{1-\epsilon\} \cup (1-\epsilon, 1+\epsilon) \cup \{1+\epsilon\} \cup (1+\epsilon, +\infty)[/math] where [math]\epsilon[/math] can be an arbitrarily small positive number.
In the spirit of Turing, the execution of the cut of [math](-\infty, +\infty)[/math] at the real [math]1.\overline{0}[/math] could be: [math](-\infty, \frac{9}{10}) \cup \{\frac{9}{10}\} \cup (\frac{9}{10}, \frac{11}{10}) \cup \{\frac{11}{10}\} \cup (\frac{11}{10}, +\infty)[/math], where [math]\epsilon[/math] in this case is replaced by [math]\frac{1}{10}[/math].
eyes glazed?
I'm curious to know what that notation 1/1 means. In abstract algebra class I learned how to construct the rational numbers as the field of quotients of the integers. That's as bottom-up as you can be.
So what is this 1/1 you speak of?
Quoting keystone
If you fixed your notational issues I could quote your markup. Can you figure out why your ChatGPT output is doing that? And as I said before, stop using ChatGPT. The purpose of AI is to make everyone stupid.
Quoting keystone
I can't read any of this when I'm replying to your post, without going back to the original.
Quoting keystone
Do you not know what I'm talking about? Quote one of your own posts.
But just tell me what 1/1 means. Start with "1". What's that?
I've wondered about that. Thanks for illuminating.
Please allow me to respond in the context of the SB-tree. Fractions correspond to nodes. Reals correspond to arbitrarily long paths (well, almost but providing clarifying details would bloat this post). There's no point to introduce natural numbers, integers, or rational numbers as disagreement would ensue. I would say they are all fractions but you would likely say they are all reals.
The cut at fraction 1/1 is fully captured at row 1 of the tree. The algorithm corresponding to the cut at real 1.0 generalizes how the cut would be captured at any arbitrary row beyond row 1 (well, to be precise I should really use ?_left and ?_right instead of just ?). Finally, the execution of the cut at 1.0 happens on a particular row once the computer chooses values for the ?'s. What should be clear is that none of this happens at the bottom of the tree. This is an entirely top-down approach.
Quoting fishfry
I think it's just that Latex does not get used properly in quotes. I'm rewriting my last post in plain text and using the notation I recently proposed.
---------------------------------------------------------------------------
I'd like to distinguish between a fraction and a real. The fraction description is finite (e.g. 1/1), whereas the real description is infinite (e.g. 1.0, which can be represented as an algorithm that generates the Cauchy sequence of fractional k-intervals: <9/10, 11/10>, <99/100, 101/100>, <999/1000, 1001/1000>, <9999/10000, 10001/10000>, ...
Because fraction descriptions are finite, a cut at a fraction can be planned and executed all in one go. A cut of <-?, +?> at the fraction 1 results in: <-?, 1> ? 1 ? <1, +?>.
Because real descriptions are infinite, a cut at a real must be planned and executed separately.
The algorithm to cut <-?, +?> at the real 1.0 is generalized as: <-?, 1-?> ? 1-? ? <1-?, 1+?> ? 1+? ? <1+?, +?> where ? can be an arbitrarily small positive number.
In the spirit of Turing, the execution of the cut of <-?, +?> at the real 1.0 could have us replace ? with 1/10 as follows: <-?, 9/10> ? 9/10 ? <9/10, 11/10> ? 11/10 ? <11/10, +?>
---------------------------------------------------------------------------
One thing that I've failed to get across is that I'm not outlining a procedure which will be used to construct infinite numbers. These systems I'm outlining, such as <-?, 1> ? 1 ? <1, +?>, are valid systems in and of themselves. Finite systems such as these are all we can ever construct in the top-down view.
I'm afraid that in the absence of a bottom-up approach, I have no idea what are fractions or reals.
Perhaps your entire approach is pre-axiomatic, in which case we have to accept a lot of things we can't formalize.
Quoting keystone
You missed the point of my asking you what the notation 1/1 means, in the absence of building up the rationals from the integers, the integers from the naturals, and the naturals from the axioms of set theory. Or even PA if you can do that.
You have to ask me to imagine I know what those things are, hoping that you yourself are not committing errors by declining to define your own notation.
Quoting keystone
I still don't know what 1/1 signifies unless you are secretly assuming all the bottom-up stuff you pretend to reject.
Quoting keystone
I didn't check that but I'm not sure you're right. Maybe you are. I'll check that here.
[math]e^{i \pi} + 1 = 0[/math]
I'll commit my post then quote it and see what I get.
Quoting keystone
Ok. You start by assuming we all know what these symbols like "1" are, while you reject the standard math definitions, but secretly use them anyway. Ok as far as it goes, but I have to suspend disbelief.
As far as the sense of what you're doing, it eludes me. Are you building the constructive real line? Lost on me.
Ok I posted this, then quoted my LaTeX and got
Quoting fishfry
so clearly that's not right, on the one hand, but not a column of 1-character lines as you get.
Each row of the tree involves medians, which require ratios of integers and arithmetic of these ratios. So, your top down approach always involves bottom up procedures. You cannot correlate rational numbers with nodes without using expressions like a/b. Instead of simplifying, you are complicating something you assume. Just my opinion.
Quoting fishfry
Peano arithmetic can be formalized in Coq. Similarly, Niqui arithmetic on the SB tree, which builds on Peano arithmetic, has been proven in Coq. There's an unquestionable structure to natural numbers and fractions that we both agree on. What we disagree on is the ontology related to these necessary truths. You believe that Peano arithmetic applies to infinite natural numbers, whereas I believe it applies to arbitrary natural numbers. By this, I mean that Peano arithmetic corresponds to an algorithm designed to take as input any arbitrary pair of natural numbers and output the expected natural number. My ontology does not require the existence of any number. I only need numbers when I want to execute the algorithm, and I only need two numbers at that, not an infinite set.
Although the above focuses on Peano arithmetic, the same applies to Niqui arithmetic. While the actual computations of Niqui arithmetic involve the manipulation of symbols or electrical signals, an elusive structure emerges in our mind when studying the algorithmthe SB-tree. Nobody has ever envisioned the complete tree, but we have seen the top part, and when I say 1/1 occupies a particular node, that top part is all I need to see. I don't need to assert the existence of an unseen complete tree; after all, it is merely an illusion that helps us understand the underlying algorithm (Niqui arithmetic).
Quoting fishfry
I'm trying to establish parallel ontologies: Actual vs. real. At this point, we have actual numbers (fractions) and real numbers. We have actual points (k-points corresponding to fractions) and real points (k-lines corresponding to real numbers). This distincting is rather bland in 1D but it becomes much more consequential in 2D when establishing a foundational framework for geometry and calculus.
Quoting fishfry
The Philosophy Forum appears to be quirky. I tried quoting this multiple times, sometimes including the spaces surrounding it, sometimes not, and about half the time it puts a column of 1-character lines.
As I was suggesting to fishfry, in the context of Stern-Brocot, what fundamentally exists is the algorithm for Niqui arithmetic. The illusion of the SB-tree emerges from our contemplation of this algorithm. When we examine the illusion, we notice interesting mediant relations between the nodes, but all of these relations stem from Niqui arithmetic. What I'm proposing is that Niqui arithmetic is more fundamental than the SB-tree. Interestingly, the SB-tree was discovered before Niqui arithmetic, but mathematics often reveals its truths in unexpected ways.
And perhaps you'll argue that almost no one views the SB-tree as a result of explicitly considering Niqui arithmetic. I would counter that by saying that when looking out the window no one consciously processes the mathematical properties of the light waves hitting their eyes, yet that is precisely what we are doing subconsciously.
NIQUI ARITHMETIC
Niqui arithmetic doesn't start by populating the nodes of a tree with fractions using the mediant operation. Rather, it doesn't assign any number to any node. It simply takes a pair of arbitrary nodes and outputs a resulting node based on the chosen arithmetic operation. It's entirely location-based. As an afterthought, we can observe the behavior of these nodes and realize that their behavior corresponds perfectly to the familiar arithmetic, provided we label the nodes according to the SB-tree's labeling. And when we label the nodes in this manner, a peculiar property related to mediants becomes apparent. But in no way does Niqui arithmetic rely on the mediant operation.
I will dispatch a clone. Meaning, I will add it to my input queue, which is now very long. If you supply a two-sentence summary I'll read it. In Silicon Valley they call it your "elevator pitch."
Quoting keystone
Oh now I have to converse about proof assistants? You know, if you've been picking up the lingo, that's great. Not of interest to me. It's impressive what they're doing. Just not an interest of mine.
Quoting keystone
No such thing as infinite natural numbers, nor do I hold any such belief, mental state, or psychological disposition towards any such thing as what you wrote.
Quoting keystone
I think that's great. I have no response. I have no function here except as a sounding board. Honestly you sound very crankish about all this. Why not just go learn some math.
Quoting keystone\
I'm glad you find meaning in this.
Quoting keystone
I'm not the guy for this any longer.
Quoting keystone
The Philo forum giveth, and the Philo forum taketh away. I have learned that over the years.
Unknown territory for me. No Wikipedia page I can find (among 26,000+), but perhaps it's under a different heading. You are full of surprises. Are you Niqui? :cool:
Niqui arithmetic: Niqui's method allows you to take as input a symbol and a pair of locations in an unlabelled tree and it returns a corresponding location in the tree. It does not presuppose any mathematics other than Peano arithmetic.
My interpretation: It just so happens that if you label the nodes of the tree according to Stern-Brocot then those symbols correspond to the familiar operators of arithmetic.
Why this is important: If one can informally say that Peano defined the natural numbers according to discrete ordered positions along a line then that is no different than me saying that Niqui defined the fractions according to discrete ordered positions along a tree.
Quoting fishfry
I mentioned this only to suggest that my view may not be pre-axiomatic. I think Peano arithmetic is very important and Niqui took that to the next level. I also don't care to converse about proof assistants.
Quoting fishfry
Sorry, I meant to say "an infinite number of natural numbers" as in "[math]{\aleph_0}[/math] natural numbers". I can see how this was misleading because when I later wrote "arbitrary natural numbers" I was referring to placeholders that can be populated by any natural number you come up with.
Quoting fishfry
I'm just looking at things from the perspective of a computer. A computer doesn't access infinite sets, it always works with the finite set of finite inputs provided to it - so why not only assert the existence of those inputs (and whatever abstract objects it actually manipulates to deliver an output) and see how far this restricted math can go?
Quoting fishfry
What I'm proposing is not entirely philosophical.
Quoting fishfry
I think we got stuck in the weeds because I began to justify how fractions can exist in my view but that justification doesn't interest you.
Quoting fishfry
I think temporary suspension of disbelief is probably the best path forward so that we can jump to the good stuff before you decide to quit...or have you already decided...
No, it's just a paper which perhaps only I believe is important. I'm guessing I'm the only person to have used the term 'Niqui arithmetic'. And it's only important if we are to drill into the ontology of numbers. If I am not asked 'but what is 1/1 at a fundamental level?' then I don't need to mention Niqui arithmetic or the SB-tree. At this point, I think this is the better option because we've been stuck in the weeds of 1D continua far too long...
I know you asked me to proceed a long time ago but weeds get in the way...
I"m happy you find meaning in this.
Quoting Ludwig V
I am happy this para has value for you.
Quoting Ludwig V
That's a relief.
Quoting Ludwig V
Common error.
Quoting Ludwig V
Why not? I haven't told anyone not to do this.
Quoting Ludwig V
It's not math.
Quoting Ludwig V
Correct.
Quoting Ludwig V
I'm within epsilon. I no longer have any idea what we are conversing about.
Does the axiom of identity mean Ludwig V = keystone ?
Just curious. :smile:
LOL I don't think so but I see what you mean.
Did you try to make sense of my 'elevator pitch'? I wasn't communicating nonsense. It wasn't even my work I was talking about...
Quoting fishfry
Epsilon, eh? Will a few figures push you over the edge? That's where I need to go to move this forward...
Perhaps the paper by Milad Niqui. In that case things may get technical and out of the realm of TPF.
viz, Quoting keystone
That's a bridge too far for me.
What did I miss?
Quoting keystone
I've done all I can. We're at a point where our interests have diverged. I need to wrap up my end so perhaps if you have any final questions, or perhaps if you come back now and then with more ideas formed over time, we can chat. So a pause, if nothing else.
Quoting fishfry
Quoting fishfry
If you click on the link to the quotations in your message, you will find yourself here:-
Quoting Ludwig V
That is my message. It is on the "Infinite Staircase" thread, and does not include any of the passages attributed to me in your quotations. So I have no idea who wrote them.
I'm not particularly pleased by this, though I'm flattered at the extent of mathematical knowledge that is attributed to me. It is so far from the truth (as fishfry knows) that it couldn't even count as a forgery or a parody.
In a way, no harm is actually done, but it is annoying and it should at least be taken down and perhaps prevention measures implemented.
I'm hoping that @Baden is enough to draw moderator attention.
It's not clear to me what happened. You're complaining you have been falsely accused of being @keystone?
All I know is that a link to a message that I did post, on the Infinite Staircase thread, has been added to text and quoted by fishfry in this post on the Fall of Man thread. Judging by this post and my exchanges with them on the Fall of Man thread, they did not do this. I don't know who did it and have no way of finding out. The suggestion that it came from keystone came from .
I hope that's clear. I would like the false attribution to be corrected or removed.
You stuck with me for an incredibly long time, and while I wasn't specifically looking for a sounding board, it turns out that's exactly what I needed. However, we've now reached a point where this discussion requires more than just a sounding board; it needs someone who can truly digest and engage with what I'm saying. Given this, it makes sense for you to disconnect now. This isn't due to a lack of topics to discuss, but rather because you're not interested. As such, I don't foresee us picking up this conversation again in the future, but who knows. Thank you so much for staying with me up to this point. I've gained a lot from our discussions, and I wish you all the best!
Well it's been a couple of months but we're finally at the state you predicted. I know you've been on the bleachers largely but now and again you've injected a comment which suggests that you've been following. Do you care play a more active role in the discussion or would you rather leave it at that and let this thread 'dry up and vanish'?
Some time back this thread shifted to the idea of starting math with continua and deriving points, rather than the other way around. MU has spoken of this, but has yet to put any meat on the bones. You, on the other hand, got into the discussion with some sort of ideas, and I was intrigued. I assumed you might begin with something akin to contours in the plane, but you went another direction, and sticking with one dimension I think was very limited, and rather boring I fear.
There are many thousands of ideas, large and very small, floating around in the world of mathematics these days, each one championed by one or more individuals. I enjoy playing with contours in the complex plane, and I hoped what you had to say would somehow involve this concept. But, instead, the discussion moved towards reconstructing the reals, devolving into an obscure approach - interesting I am sure to a few - but not to the, relatively speaking, many.
If you were to return to the beginning and speculate continua that precede points, or something similar, the thread might continue. Just my opinion.
Indeed, I went in another direction and 1D is boring.
Quoting jgill
This is exactly what I want to return to. Here's what I'm thinking:
If we keep going, I'll summarize the 1D story so far and then spend a couple of messages discussing the visual representation (using lines instead of intervals). After that, I'll move on to 2D, exploring higher-dimensional analogues of intervals/lines, namely equations/faces. Then, I'll spend a few messages examining the meaning of roots, derivatives, and integrals from this top-down perspective.
However, I'm not sure if your advice was about how to appeal to you specifically or more about attracting a new audience. If you're not interested, that's totally fine, and I'll move on. If you want to continue but prefer to take it one message at a time, then we want the same thing. I'm not asking for a long-term commitment like the one fishfry gave me. But if we proceed, I want to ensure it's a conversation, not just me 'lecturing' to the bleachers. What do you think?
Why not go directly into 2D. If you stay away from the SB-tree and Niqui arithmetic I might linger a bit longer. Let's see. You might even tempt fishfry back.
Oh I'm terribly sorry. That was for @keystone. I think I know what might have happened. Sometimes when I'm responding to someone, I have their quote tag in my copy buffer. Then I go to a different mention to reply to that, and when I was talking to keystone, I might have had your quote tag there instead.
No harm no foul I hope.
Did I screw up the quoting? Are we having an interesting convo about infinity and time or did I hallucinate that?
I do confess that on this site I process my mentions and don't always know what thread I'm in. It all kinds of blends together. Just for my curiosity, what did I say you said? Like I said earlier, I probably just copy/pasted the wrong quote tag.
You're very welcome. You know you had a profound effect on my life. I had been away from this board for quite some time, a couple of years or more I think. I did read it on occasion; and when you posted your original question in this thread, I jumped in because I happened to know that the answer to your puzzle is that there is no uniform probability on a countably infinite set. Your top post lured me back from my vacation. For what that's worth. The jury's still out or maybe they're measuring the rope.
I enjoyed our chat. Thank you.
OK. No problem. I was just annoyed because I didn't understand what was going on.
Thanks for your message. As you see, it's sorted out now.
Our relationship was mostly one-sided, with me being the main beneficiary of our conversations. I'm glad to hear that you found some benefit in meeting me as well. This is a great way to conclude our conversation. Cheers!
I see no need to mention those topics any time soon.
Quoting jgill
Without sufficient priming, you will almost certainly reject the 2D images I intend to share. I'm going to just spend this message covering the remaining 1D topics and if you're still following and interested I'll follow up with a message on 2D.
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In 1D, figures adds no value as they are equal to the k-interval union. However, this is an ideal situation as it allows us to ease into unorthodox top-down figures.
Here's a k-interval union:
<-?, 1-?> ? 1-? ? <1-?, 1+?> ? 1+? ? <1+?, +?>
Below is that same k-interval union placed above a grey box containing an illustration of k-line geometry (with matching components vertically aligned and color coordinated such that k-lines/k-intervals are green and k-points/fractions are red).
I wrote <1.0> as a k-interval and placed it below <1-?, 1+?> because they both describe the same k-line.
I want to highlight one important difference between k-line systems and Cartesian systems.
All k-line system properties (such as order and length) are preserved under continuous deformations. In other words, all of the 3 systems illustrated below correspond to <-?, 1-?> ? 1-? ? <1-?, 1+?> ? 1+? ? <1+?, +?>. It is for this reason that complete k-line systems can fit on a finite page, since 'k-points' at infinity need not be far off in the distance. From our discussions, I've learned not to call these systems topological as that term comes loaded with meaning that doesn't apply. However, to capture the idea that properties are being preserved under continuous deformations I'm going to say these systems are k-topological. So if in 2D you see a graph that looks weird, I will likely justify the weirdness by saying that it is k-topological.
Also, please note that if ? were allowed to equal 0 then a k-topologically different system would result. As such, ? must never equal 0.
What do you think?
I'm done here. Sorry. Maybe another mathematician will appear.
Sounds good!
Likewise.