Tortoise wins (Zeno)
The Achilles paradox is an ancient paradox attributed to Zeno, but re-visited from time-to-time. The paradox goes 'approximately' that Achilles and the tortoise are set to have a race. Suppose that Achilles runs ten times as fast as the tortoise and gives him a hundred yard head-start. In order to win the race, Achilles must make-up for his initial disadvantage of a hundred yards to reach the tortoise; but when he has done this and has reached the point, where the tortoise started, the animal would have moved on ten yards. While Achilles runs these ten yards, the tortoise gets one yard ahead. When Achilles runs the one yard, the tortoise is a tenth-of-a-yard ahead. And so on, without end. Achilles never catches the tortoise, because the tortoise always holds the lead, however small.
The paradox seems to highlight a difference in reality between mathematics and physics, where infinite and finite quantities are concerned. There is a degree of overlap, but at one point the premise of the paradox is undermined, when you consider that in practice a mathematically infinite number of actions cannot be carried-out physically. So there might be a point that the paradox breaks-down as you move from physics to maths. An infinite geometric series in maths is inapplicable to a physically real distance.
Even so, do paradoxes such as Zeno's not encourage us to see infinite singularities in physics, for example, as having potential for furthering our understanding? These are not necessarily 'end-points', but can be resolved given the right approach.
Despite successive attempts to resolve this paradox, it seems as if the tortoise still edges-out Achilles.
The paradox seems to highlight a difference in reality between mathematics and physics, where infinite and finite quantities are concerned. There is a degree of overlap, but at one point the premise of the paradox is undermined, when you consider that in practice a mathematically infinite number of actions cannot be carried-out physically. So there might be a point that the paradox breaks-down as you move from physics to maths. An infinite geometric series in maths is inapplicable to a physically real distance.
Even so, do paradoxes such as Zeno's not encourage us to see infinite singularities in physics, for example, as having potential for furthering our understanding? These are not necessarily 'end-points', but can be resolved given the right approach.
Despite successive attempts to resolve this paradox, it seems as if the tortoise still edges-out Achilles.
Comments (150)
I think what this indicates is that this way of looking at movement, as proceeding from a start point to an end point, is somewhat incorrect. We ought to remove those points, those beginnings and ends, from the representation of the movement of the thing itself, and model the movement as moving past the designated points. Then we show Achilles as moving past the tortoise, and as moving past the finish line, instead of modeling the movement as ending at the specified point. I believe that this would resolve all such paradoxes.
Yes, this paradox highlights a problem we may have when it comes to perceiving movement, conceptualising it etcetera. Physics is replete with quandaries such as this, but it is difficult to easily find common ground, some scientists may allude to the paradox of the black hole, but that's a bit far-out.
Sticking to the paradox, I don't think that Achilles can ever reach the tortoise, unless it reaches some sort of Planckian limit in distance and suddenly quantum leaps to become 'the winner'. That suggests that space-time is discretised, that you do reach a limit in physics that does not exist in mathematics.
In the end, quantum leaps aside, although the tortoise moves at an imperceptibly and almost infinitely small pace, it still keeps moving and eventually will cross the line, given that there is no time limit. This seems to accord to what we perceive in reality, we are somewhat subordinated to nature's ultimatum.
Maybe there are true, real limits within time and space, real quanta or discrete units of these, but use of the current way of modeling, which imposes an artificial limit or boundary, and uses calculus to show the approach to this limit, will leave us unable to find these real boundaries. And since relativity theory, which is the most common tool for physicists, assumes the fundamental premise that there is no such thing as absolute rest, modeling an object as reaching, or being at a fixed position in space, is inconsistent with relativity. Simply put, rest frames are imposed according to the purpose. By relativity theory, an object is always moving, and cannot actually be at a fixed position.
To be consistent then, if we employ relativity theory we cannot use the calculus which assumes a fixed position, the boundary or limit. If we quit using these artificially imposed limits, and model moving objects as truly continuous, instead of modeling them as approaching these fixed limits, then the issues and problems which emerge from employing principles of true continuity to the physical world, will reveal whether or not there are true boundaries to space and time. The point being that employing artificially (purposely) created boundaries, which do not correspond with true boundaries will just create confusion and unintelligibility, if we seek the true boundaries.
Quoting Nemo2124
No, the tortoise will never cross the line if there is no time limit. Time will keep going forever, and the tortoise will always have more space to cover before it reaches the line. Therefore the tortoise will never cross the line. this is very similar to the way that Achilles will never reach the tortoise. The latter is a more complex presentation, the complexity designed to create more confusion when looking at the same problem.
This starts to point at a discontinuity that seems to recur between quantum physics and relativity. Relativity says one thing, qunatum physics says another, and the recently nascent field of 'quantum gravity' steps-in with strings and all-manner of additional dimensions and contrivances to reconcile the two divierging worldviews. The fact is that we, perhaps as a result, encounter singularities (infinites) that allude to paradoxes in our perception of the world, regions where space-time breaks down.
Quoting Metaphysician Undercover
We need a starting point here. Do we first take relativity to be valid or the absolute quantisation of space-time? Does the Planck constant suggest that there is a real fabric to space-time at the vacuum level? What is the nature of this fabric? These are questions that start to arise when we have a starting point, that is the discretisation of a space-time. In other words relativity has to make itself compatiable to quantum theory and not vice-versa. We just have to accept that the tortoise wins.
Quoting Metaphysician Undercover
Given an eternity and the fact that the tortoise keeps moving, I think that it will eventually cross any line that is set at a finite distance in the race. If there is a point of 'quantum leap', where either Achilles is within a Planck length (6*10^-34m) of the tortoise or the tortoise is within a Planck length of the finishing line, still the tortoise will get there first, since perhaps it has had the initiative all along. As you put it, this is predicated on validating the Planck-scale first and applying it to the relativistic geometry of space-time.
With the paradox of the tortoise who cannot actually reach the finish line you are only looking at times before the tortoise reaches the end point. The time limit that you are using is the time that it would actually take the tortoise to reach the end point. So it should be no surprise that it appears as if the tortoise never reaches the end point.
In the real world we don't normally look at time as approaching a limit, therefore the tortoise does reach the end point in a finite amount of time.
Mathematically, the infinite sum of the series in question is 1.
Quoting T Clark
Precisely, by mathematical summation the series gives unity, but in practice - physically - it's impossible.
Quoting Metaphysician Undercover
Physical space is not "infinitely" divisible like abstract space. Like most paradoxes, this one is merely apparent it's derived from confusing the physical and abstract.
Yep. The distance covered is finite. The flip side of that is that the time taken for each step is zero at infinity, so while there are a mooted infinity of steps in the process the distance covered and the time taken are both finite.
The apparent paradox is no more than a failure to apply the relevant maths appropriately. It is not a "difference in reality" between physics and mathematics.
So yes, it does help us understand infinity somewhat. For those able to grasp it.
It is my understanding that the appropriate mathematics didnt exist in Zenos time.
But others here do not have that excuse.
The problem with this proposal is that there is too much relativity already baked into the procedural methods of quantum physics. Our understanding of energy and how electromagnetic radiation relates to massive objects is relativity based. So it's not a matter of making relativity compatible with quantum theory, it's a matter of falsifying relativity and starting from whatever that falsification reveals. This requires the appropriate attitude, as falsification requires application (experimentation) designed for that purpose.
Quoting Nemo2124
It cannot, by the premises of the example.
Quoting 180 Proof
What would you say that "physical space" is made out of? The divisibility of anything is dependent on what the thing is composed of. If you assert that physical space is not infinitely divisible you need to justify this with some principles, say what space is composed of, and how this limits its capacity to be divided. We tend to think of space as nothing, but then it's just an abstraction, and infinitely divisible. But if it's not nothing, then what is it made of?
The ancient Greek atomists limited the capacity to divide physical substance by positing fundament particles, atoms. The atoms would be indivisible. But Aristotle demonstrated the logical problems with this perspective. Each atom would have to be the same because internal differences would provide for different ways of dividing. And if all atoms are the same, then the differences between different objects could only be a matter of quantity, unless we assume something else to allow for qualitative differences between things. This is why the dualism of matter and form was required.
.
The dualism of two less than physical concepts like prime matter and form to get out of the paradoxes of the physical by marrying the fairy concepts together as a substitute for the physical was a mistake for Aristotle. In the hands of latter thinkers it was a disaster.
You have two runners unable to move but Achilles has more force going in him so he wins by force
:up: :up:
Ergo physical divisibility is finite.
Clearly, Aristotle did not understand that Democritus' atoms are physical and not just abstract (i.e. not formal/metaphysical "platonic").
Take the zero-point energy, for example. In relativity it corresponds to the cosmological constant (lambda term) or 'dark energy' of the Universe. Besides the fact that measurement for the 'dark energy' does not match the theoretical predictions for the zero-point (the cosmological constant problem), we here have grounds for challenging relativity, based on the lambda term, given we affirm the validity of quantum.
The tortoise moves harmoniously even by infinitisimals, at the end, taking an eternity to reach the finishing line, but reaching it in the end (because of the summation of geometric series). Achilles' best hope of reaching the tortoise is that somehow he will quantum leap it at the finishing line. On the other hand I can see the argument that the tortoise might initially stall at the beginning, being unable to move.
You can compare the discrepency between physics and maths in the Achilles paradox and the cosmological constant problem in physics (zero-point energy). Off-the-cuff, if we had quantum field equations for the lambda term in general relativity that might resolve the problem. Finally, the paradox also touches upon a conflict over infinitesimal small distances between quantum and chaos theories.
It's true that the race is a chaotic mix of stops-and-starts, but overall the tortoise moves by infinitesimals, slowly, in what seems like an eternity, to reach a finite distance. Achilles keeps stopping everytime he reaches the tortoise, convinced he is simply faster, and in the end tires-out before even finishing.
I believe "zero-point energy" is the consequence of relativity type thinking. Since relativity denies absolute rest, anything which appears like it ought to be rest, or is assigned "rest" (rest frame etc.), cannot actually be rest, to be consistent with the principle of relativity. Therefore assigning "rest" to something is actually a matter of assigning some form of unknowable motion to it if we adhere to relativity. This manifests in the Heisenberg uncertainty principle.
Quoting Nemo2124
That the "summation of geometric series" will bring one to "the end", is actually demonstrated to be false by the need to assume "zero-point energy". The zero point cannot ever actually be reached in this way, and the practise of the summation of geometric series', is just a rounding off which does not represent physical reality. Since there cannot be any correspondence between the artificial end and any possible real end, due to the relativity premise which dictates that there is no end, then the end produced by summation is simply fiction. It's just a convenient way to avoid the problems created by relativity type thinking, but since it's fiction it produces useless metaphysics.
This is the problem approached by. If the abstract (ideal) is not representative of true reality, we need to understand and respect how this difference may mislead us. In this case, the physical reality of zero-point energy is evidence that the boundary applied by abstract thinking is not consistent with physical reality. So the series summation reaches the boundary (zero), but this is not representative of physical reality, and we are left with something real, called zero-point energy. The physical reality of what is actually represented by that name "zero-point energy" cannot be understood by this way of thinking because it gets swallowed up into the uncertainty principle, as an aspect of reality which cannot be understood.
Quoting Metaphysician Undercover
I'm not sure about this, because this energy of the vacuum represents an intersection between relativity and quantum. Yes, it is the seemingly static reference frame, the inert background against which all motion occurs. This has to be the case in relativity, as well, ever since the introduction of the lambda term, where a static Universe was introduced 'for the time being'. So I do not understand relativity in the same way as you do here, because I think the cosmological constant provides the absolute space-time.
Quoting Metaphysician Undercover
Again, I think that the cosmological constant term in GR places a limit on relativisitic thinking. With the tortoise's movement we have a natural alignment of quantum physics and GR, there is a conformity between physics and mathematics that we cannot imagine. With Achilles' movements, his actions are discretised and limited by his perception that his speed is so much immeasureably greater than the tortoise's that he need only catch-up that eventually he imagines that he will approximately win.
I'm not aware of a mathematical definition of an alternative continuum that resolves all of the logical puzzles posed by Zeno.
Zeno's paradoxes when interpreted mathematically, pose fundamental questions concerning the relationship between mathematics and logic, and in particular the question as to the logical foundation of calculus. The existence and utility of the classical continuum is also called into question.
I dont think that machines (AI) can do this. It represents a limit of reasoning, one that AI is yet to recognise. Its also very difficult, if not impossible, to programme a sense of humour, for example. Here I think its apparent that the paradox highlights the gap between physics and maths.
For no other reason than that the tortoise starts ahead. But the quicker ability of Achilles must count for something? Ultimately Zeno keeps taking the movement backwards a step when it's constantly going forward. Calculus doesn't solve this
Ye but that's assuming there is no motive power to begin with
From a different perspective, the tortoises predominant anima, instinct to move in harmony with nature may give it that advantage already. The paradox maybe also highlights human complacency. We still cannot imagine losing a running-race with an animal as slow as a tortoise!
How can nature have anything infinite within it?
Without end? Sure, it's an infinite series, but it ends when Achilles has run 111 1/9 yards. That's a finite time and a finite distance, simply expressed as a limit of an infinite series. So where is the paradox identified.
There's all sorts of interesting ways to make it more fun, really giving Zeno's argument a run for its money.
You have a FIFO queue, a pipeline of sorts. At 1 second, you put in numbered balls 1 & 2, and take out the next ball, which is '1'. The next half second, you put in 3 & 4 and take out the 2. Each iteration puts in two balls and takes out 1. They're numbered and put in and taken out in order. After the series completes in 2 seconds, how many balls are in the queue? Answer: None since there is no ball that doesn't have a defined time at which it was inserted and another time at which it was removed.
Quoting Nemo2124
The physical has not been shown to be any different than the mathematical model in this scenario, especially since it's a mathematical mind-experiment, not a physical one.
Quoting Metaphysician Undercover
The two are admittedly modeled as points, which works if you consider say their centers of gravity or their most-forward point. But by your assertion, do you mean that the tortoise is never at these intermediate points, only, the regions between?
Quoting Nemo2124
You think that space being continuous is disproven by this story then. Quantum theory AFAIK has never suggested quantizing spacetime.
Quoting Metaphysician Undercover
Sorry to find a nit in everything, even stuff irrelevant to the OP, but relativity theory doesn't say this. In the frame of Earth, Earth is stationary. There's noting invalid about this frame.
Quoting sime
:100:
Quoting Gregory
I don't see why it would be a problem. For instance, there doesn't seem to be a bound to space or time, making both infinite. Nothing stops working due to that model.
https://www.newadvent.org/summa/1007.htm#article3
I tend to agree with Aquinas but not because of the Aristotle stuff. I have a lot of thinking and mulling to do over the idea of "infinity" before i could say more
I'm not sure what you mean by "alternative continuum."
Quoting sime
The mathematical interpretation of Zeno's paradox seems straightforward to me. Evaluating limits makes the so-called paradox disappear. What is illogical about that? And what does this have to do with calculus. Representing a continuum as an infinite series of infinitesimals seems like a good model of how the universe works, simple and intuitive.
It doesn't seem intuitive to me at all that space divides to infinity and yet has a finite limit. To my mind that is a direct contradiction, like a round triangle
If I remember correctly, Pythagoras or one of those other Greek math guys calculated pi by dividing up a circle into uniform triangular pie slices. It seems intuitive to me, and apparently did to him, that as the size of the slices increase in number and decrease in size, the sum of the area of the triangles approaches the area of the circle with as much precision as we want. Carrying that one step further into calculus using the limit at infinity seems - intuitively - natural and logical.
I mean that if the tortoise is moving it is never at a point. This is because time is continuously passing, therefore motion is continuous too. So the closest thing we could truthfully say is that it is passing a point. To be at a point would require a stoppage in time. There is no time when a moving thing is at a point because that would a stoppage of time, which is a matter of removing the thing from time. That's the point of Zeno's arrow paradox.
Quoting noAxioms
I didn't say it's not valid I said that it's not true. Obviously the earth is not stationary. So that frame in which the earth is stationary is not true, it's an arbitrary (untrue) assumption, made for some purpose.
When you "imagine" infinite points on a segment you are not really imagining an infinity. I realize that the infinity gets smaller and smaller, but it still never ends and hence should have no finite boundary. Each digit of pi corresponds to a slice of space, so infinite space makes finite object, a contradiction, so says the Eleatics. What is intuitive for me is to say there are discrete steps, but it's impossible to explain that geometrically. Infinity seems necessary as a tool, not as a truth
Internet Encyclopedia of Philosophy
"Loop Quantum Gravity (LQG) is a theory that attempts to reconcile general relativity and quantum mechanics by proposing that spacetime is not continuous but rather composed of discrete, fundamental units at the Planck scale, forming a 'fabric' of spacetime loops". Google AI
This is a reinterpretation of General Relativity, which traditionally had continuums in its maths. It's just hard to image space that can't be divided. There is something missing it seems when we try to reason about it. God knows. Food for thought. Weren't the Greek Atomists a reaction to Zeno?
Isn't this an example where the false premises lead to wrong conclusions? Even if the argument appears valid in the form, it cannot reflect the true reality of the world.
Who are you to tell me what I am or am not imagining. Just because you can't imagine something infinitely large or infinitesimally small doesn't mean the rest of us can't. We can imagine things that don't or even can't actually exist.
Quoting Gregory
Points are infinitesimally small in three dimensions, lines in two dimensions, and planes in one dimension, yet we use them all the time in mathematics and physics. They can be used as effective, accurate models of the behavior of actual observable phenomena. Why are the infinitesimals we are discussing any different?
Because as Berkeley said they are ghosts of departed quantities. You are adding up nothing and arriving at something. Why do you post on this forum at all if we could all have magical powers in the mind you don't have?
Zeno's dichotomy paradox corresponds to the mathematical fact that every pair of rational numbers is separated by a countably infinite number of other rational numbers. Because of this, a limit in mathematics stating that f(x) tends to L as x tends to p, cannot be interpreted in terms of the variable x assuming the value of each and every point in turn between its current position and p. Hence calculus does not say that f(x) moves towards L as x moves towards p.
I don't understand how "...every pair of rational numbers is separated by a countably infinite number of other rational numbers." implies "a limit in mathematics stating that f(x) tends to L as x tends to p, cannot be interpreted in terms of the variable x assuming the value of each and every point in turn between its current position and p." It's a model for goodness sake.
I'm going to leave it at that. I'm not a mathematician and I've carried this as far as I can. You can have the last word.
Achilles takes one step. Thats a physical event.
You need to add concepts to this picture to say whether he has moved a fraction of some other step, approached some other limit infinitely, or already won the race. You can not add other physical steps (like infinitely smaller fractional actual steps) to the step that was already the subject of inquiry without denying the existence of the step in the first place. So is there a step, a motion to discuss, or not?
If Achilles cant catch the tortoise, the tortoise cant move either, and there is no paradox, because there is no race.
Theres something like a category issue going on here to fabricate the paradox, and mess with the betting odds at the racetrack.
Measured distances, fractions thereof, and infinity, are concepts. Mental things. We grasp physical things with our hands. We dont grasp infinity like that, ever. Achilles stride and the tortoises pace need have nothing to do with any of those concepts. Strides and pacing are physical things.
If I move ten centimeters, I can be said to have moved one-tenth of a meter. Or I can be said to have moved one whole decimeter. So was this a fraction or whole motion? Does that motion have infinite parts or no parts?
These are concepts, mental constructs, we can only assert apply to physical things. Only by first positing a conceptual scheme in which one meter is equal to ten decimeters can I then name something one tenth of a meter. And only by positing a whole meter (or going the whole distance conceptually) do I fix the denominator that names the decimeter 1/10th meter (10 here meaning whole one). You dont get fractions before wholes; you take wholes and divide them, to conceptualize fractions.
So the race had to be over before anyone could tell you at what point Achilles moved one tenth of a distance, or any fractional distance.
There is no fraction of a physical thing - it is only made a fraction conceptually by relating that whole thing to some other whole thing and seeing the relation is fractional according to your conceptual relational scheme.
You cut an apple in half. You can say you only have a fraction of a whole apple. But lets say you never saw fruit before, and someone hands you a single half-apple - you would have one whole thing in your hand and no means to determine it relates to some other half. You would have a whole thing. And physically, thats all there ever is. The determination of whole versus half of that whole requires concepts, not physical steps or physical processes.
So to reasses who conceptually wins and loses a non-conceptual physical race, one would have to wait until it is physically over before one could properly conceptualize the fractions and partial movements that can be said to make up that whole race.
All of these measurements are post-hoc measurements asserted of some external thing. And for this paradox, they are post-hoc concepts turning a physical thing into a conundrum for those concepts, not for the spectators of the race.
No one ever actually moves one-tenth of any distance. They move an actual, finite, whole distances. In every move.
So, unless Achilles brakes his heel and drops out, the tortoise always loses the physical race. That has nothing to do with any math nor provides any more information about the paradox.
The paradox is really just the irony that it is impossible for the smartest people in the universe to explain a simple motion )which it truly is). Or, it takes sheer genius to prove through concepts, that motion right before your eyes cant happen.
Here, wait a second, I'm going to imagine infinity... There, satisfied? Want me to do it again? It's not a magic power, it's just imagination.
Nuff said.
I'll go along with that.
Rubbish. :roll:
More or less in the case of Zeno. Mathematics is often said to resolve the paradox in terms of the topological continuity of the continuum, by treating the open sets of the real line as solid lines and by forgetting the fact that continuum has points, meaning that the paradox resurfaces when the continuum is deconstructed in terms of points.
In my view, Zeno's arguments pointed towards position and motion being incompatible properties, but the continuum which presumes both to coexist doesn't permit this semantic interpretation.
Mathematical limits are proved in two steps using mathematical induction - which obviously does not involve a literal traversal of each and every rational number in order, which leads nowhere. (The proof of a limit is intensional, whereas the empirical concept of motion is extensional).
You must think finitism is repressive or something. Anyway, explain how a two meter segment placed parallel to a one meter segment is longer when you have a one-to-one correspondance between points, but when you angle the two meter segment to form a triangle they line up the same. Thank you
Continuous means infinitely dense, which in turn would be either then discrete or an infinite series of steps
No, it doesn't.
A continuous path is not reducible to a mere sequence of points; rather, it is a unified whole in which limits make sense without requiring traversal of individual points.
Treat it as points, or as a continuum, but not both.
Why can't you just divide the "unified whole" if it's not discrete?
A general rule: if the description you give of something that happens says that it can't happen, you are using the wrong description.
So does this subdividing result in a series of discrete steps, or goes on forever? Since it's continuous, it must sink down in there forever, which doesn't make any sense since it is a finite segment. See?
I see you haven't understood. I doubt I can help. If it is continuous, the by that very fact it is not discreet.
I didn't say the continuous was discrete. I said the continuous doesn't make sense because spatial infinity squished into a finite size makes no sense.
Like when "experts" say the universe is infinite and expanding. That's called mental masturbation. A bad habit
Heuristics may not be precise, but its value can be substantial in an introductory course. I've never come across this sort of philosophical detail in elementary calculus, which includes lots of motion. "f(x) approaches . . . as x approaches . . ." is common language in math. But, whatever you say.
While the "experts" might say something like that, the experts don't. Space is expanding, but saying the universe is expanding implies that it has a size, which it doesn't if it isn't bounded.
Quoting GregoryZeno did not describe infinite space squished into finite something. It was never spatial infinity.
These comments will also not help you Infinity isn't a hard concept to grasp, but giving it a bound when by definition there isn't one is always going to run into trouble.
There are many physicists who say the infinite-in all-directions-universe is expanding into hyper-dimensional multiverses. I would say Aristotle was right in writing that the spiritual (Heaven) surrounds the finite universe and that's that. He was wrong though about Zeno's arguments and the Greek atomists were on to something
Quoting noAxioms
Zeno said the apparent finite distance was really a series of infinite steps; hence infinite inside finite. It's not that hard to grasp. Infinity as an idea is sound only when it is used to refute itself
The koch snowflake is just sending a finite boundary into infinity. It can't exist. How do you prove that all the even numbers are equal to the whole numbers? Well they line up a few numbers and send it to infinity, not realizing that you can do the same trick with an uncountable infinity. Take all the points on a segment and line each one up one at a time to the whole numbers. Walla they both get sent off into hhe infinite universe. Equal! (The problem i put to T Clark illustrates how geometrically this all doesn't make sense.)Take an orange and cut it in half. Then cut one of the halves in half and do this forever, lining them up largest to smallest. What is the smallest? Obviously something discrete otherwise the orange's partsv would go out infinitely into an infinite universe. Mathematical nfinity swallows itself and there's nothing that save it
Quoting Gregory
A circle has infinite amount of tangents, yet in every place of it's circumference it has just one.
In Mathematics, infinity exists. Clear and simple. We just don't understand everything about it. Hence we have things like the Continuum Hypothesis. Yet our ignorance doesn't make it illogical and false. In Mathematics, it's as real as a finite number or the circle is. Or a Koch snowflake etc.
Quoting Gregory
You can't. Between any two points you select, there are infinitely many more points. There are only countably many whole numbers, but far more points in a segment...
Then why does mathematics combine the two? Real numbers are points representing a continuous number line.
There is the same problem with the odd vs the whole numbers. With these, imagine them going off into the horizon. In how they explain this, they just pull all the odd numbers towards number 1 and say woopy! equal! Why is that a legit move? Why can't you do this move with the uncountable as well?
Where would you start?
First of all, do you know what's the difference between countable and uncountable here?
Basically, the "legit move" is that you can make a bijection with the set of natural numbers (1,2,3,4,5,...) and the set you are thinking about. This means that you basically can write the numbers you are talking about in a way that you get every of them and don't miss any number (if you would have infinite time a so on...). If you can't do this, then it's uncountable.
If I don't make my point clear, just go and look at this site: Countable and Uncountable Sets Remember to look at the proofs.
With any point bijected to 1, 2, and 3.
Quoting ssu
Again, if you can start with 1, 2, and 3 and move the 2 to one and the 3 to 2 ect. you could also take a segment parallel to the whole numbers and move each point down to the left like you did before and assume it's all good at the other infinite end, like you did trying to prove the even numbers are equal to the whole numbers. Also, doesn't this violate the principle that the whole is greater than the part? Infinity doesn't make any sense. What am I missing?
Quoting Gregory
Uh, nope.
OK, let's try another way. I assume (from the above) you know the idea of the Hilbert Hotel works. Please watch this video (only six minutes!), it sums up perfectly the uncountable infinite and Cantor's diagonal argument. And just why sometimes the Hilbert Hotel cannot accomodate every possibility of guests.
Basically this is what in the earlier link I gave you was told in theorem 1.20 of the uncountability of the reals. But for me the above video is more easier to understand.
The very first step of the video i question. If all the rooms are filled you can't move 1 to 2 and 3 to room 4 because all the infinite rooms are already filled. The problem with the diagonal argument for me is this first part about the odd numbers equalling the whole numbers: You are not using the same logic for the two types of infinities.
What do you say?
Quoting Gregory
If you were right then you could specify who does not get a room. In the first case, each individual is assigned to the room one more than the room they are in, and so every individual gets a new room. The person who was in room two is now in room three; the person who was in room three is now in room four; and so on. In the second case, each individual is assigned to the room twice the number of the room they are in. Again, each individual gets a room. In the third case, in which and infinity of new guests arrives, and the spreadsheet is used, each individual is still assigned a room. But for the party bus, the diagonal argument shows that there will always be an individual who does not get a room.
This is what happens when you try to assign a whole number to every point on a segment of the continuum. There are too many points on the segment to be counted.
But this is a different story to the one we started with. Achilles starts behind the tortoise, which has a small head start. By the time he reaches the tortoises starting point, the tortoise has moved forward a shorter distance. Achilles then reaches this new position, but the tortoise has moved again. This process repeats infinitely, but the distances form a geometric series that converges to a finite sum. The total time taken also converges to a finite limit. Achilles reaches the tortoises position in a finite time and then surpasses it. The paradox arises only if one mistakenly assumes that infinite steps must require infinite time, which they do not.
Your statement explains the paradox in an accurate way.
Another simple way to look at this paradox is to see that:
"if you only look at times BEFORE Achilles reaches the tortoise, then it will appear as if Achilles never reaches the tortoise".
There are an infinity of intervals before Achilles passes the tortoise, each one half the time of the previous, and so with a finite sum. The process of Achilles passing the tortoise therefore takes a finite time.
End of story, really.
That would be true if there would be a finite number of rooms. Then the person in the last room would find there's no room for him or her. But it's an infinite hotel. There is no last room.
If there would be a last room, guess what? The hotel would have finite amount of rooms.
Your problem is that you simply don't understand the concept of infinity, so it's quite futile for me to show that these are mathematical proofs, not just my opinions. You see, calculus exists. Infinity is a very useful and logical concept and is used a lot in mathematics. Modern set theory has infinity as an axiom.
I totally understand it's really puzzling. A lot of the brightest minds in the history of math found this very puzzling. Galileo Galilei was one of the first people to point this. (See Galileo's Paradox)
Yeah, but I guess everyone should understand the connection that infinity and an infinitesimal has. (Or limits)
This is just repeating the video. What infinity are we talking about with the first hotel situation. We haven't established what infinities are so maybe you can't move guy 3 to room 4 because theybare all filled. Imagine planks going infinitely into the horizon. Two sets. With the odd vs the wholes you are pulling the odd numbers back to line up with the whole numbers and that is geometrically crazy. Just as it is to say all the points on the edge of a cube are the same as the points in the cube. None of it makes sense
Quoting Banno
False. It's infinite and finite at the same time in the exact same respect
Quoting ssu
More than that, there seems also to be a resistance to learning about infinity - hence flimsy response "If all the rooms are filled you can't move 1 to 2 and 3 to room 4 because all the infinite rooms are already filled". Notice how the OP, which has a relatively simple answer, was exploded into quantum nonsense and "dimensions and contrivances" with such glee, within a few posts of the OP?
Folk don't want an answer...
So what is it they want?
Then answer my arguments
For you to honestly address the issue
Here it is again: A refusal to recognise the answer when it is set out before them.
It reeks of some sort of anti-intellectualism, or at least anti-expertise.
Explains a lot of recent politics, too.
Why couldn't its foundations be wrong?
Then state my argument, or at least ONE of them, in your own words
So are saying that there is more to the continuous number line than the points which are the real numbers? Can you explain that?
Yep. It is also connected and complete; it has a topological structure. Of course, not all the issues are ironed out and answered. If you want more you will need to talk to a mathematician.
I can't, becasue they are incoherent. Take
Quoting Gregory
Yeah, we have, at least enough to be getting on with. For every number there is a next number.
Quoting Gregory
They cannot be wrong, any more than 4+4+2=10 can be wrong. But it can be misunderstood.
I did answer your arguments. Just in the last response I wrote you. Infinite is different from the finite. If you start from a finite situation, now wonder you have problems to understand the infinite.
Quoting Gregory
Hopefully you do notice that calculus is very, very useful in physics or economics etc. It does answer correctly to many real world problems, that can be calculated by using calculus.
For me, one way is to look at the history of mathematics, how new ideas have been responded to, how from one thing we have gotten to another. This way, the theorems aren't taken just as a given.
If you look at the history of calculus, you see the obvious foundational problems it has had. From Newton himself and Leibniz. Yet if your argument is that infinity doesn't exist, then basically calculus wouldn't exist.
No you don't know how a countable infinity relates to uncountable and their qualities before the argument starts.
Quoting Banno
You couldn't try? You're dishonest
Quoting Banno
But you say:
Quoting Banno
So you have only probable assurance that Zeno's paradoxes have an answer?
Exactly. It's useful, not true. Like Gabriel's horn. Obviously false. I've presented at least 5 cogent arguments against infinity on this thread and you didn't indicate that you understood any of them
Are you saying that topology adds something to the line, which is more than just the real numbers? What more could there be?
Yeah, we do. We learn how to count, then notice that whatever number we chose, there is a bigger number. Or most of us do, around the age of seven or eight. Then some see Hilbert's Hotel and the diagonal argument and go "Holly shite! there are numbers that cannot be counted..."
Quoting Gregory
But I did try, in the post to which you are responding. You can't seem to recognise that the responses you are receiving actually answer your questions. It's odd. But it's not about maths, it's about you.
Same with the odd vs whole. Oh it's *different* with infinities of infinities? This is not established in that video
Quoting Gregory
Well, no, since for every whole there is an odd, as has been shown.
That you misunderstand something does not make it wrong.
Not true, but useful?
Ok, that really doesn't make any sense. Calculus is a part of mathematics and totally accepted. Please don't start to argue that Calculus is not true.
Quoting Gregory
No, you haven't been at all convincing. I'm afraid that you don't simply get it.
Quoting Banno
I think I have to agree here with @Banno. Don't want to be harsh here.
It's very first principles are wrong. Like in history when guys started questioning Euclidean postulates? Just because you misunderstood my arguments do not make them wrong. Have a good day
How are the very first principles of calculus wrong? What are you talking about?
Actually you give a perfect example of something not being wrong, but simply limited. It's not that Euclid was wrong, it simply was the case that not everything fell into his understanding of geometry. Root cause was that geometry on a plane and on a sphere are simply different. And you might have to think of geometry of a sphere. That's it. Yet the geometry on a plane is still correct. Hence the error is if one thinks that all geometry happens on a plane. Thus there is Euclidian geometry and non-euclidian geometry (spherical or hyperbolic etc).
Actually, what example would really be false was the Greek idea that "All numbers are rational". And the idea why people believed it was so was because... math is so beautiful. Well, there are irrational numbers. The Greeks found them, and they weren't happy about it. Yet that idea really was a genuine error.
And I think that the idea that "there is no infinity in mathematics" is simply wrong. Similar to the latter example "all numbers are rational". That you only stick to finite mathematics is another thing. Ok, do that. But then what you can do in mathematics is limited.
In this exercise you are imagining the state of the tortoise/hare at a time closer and closer to the time that the hare catches up. But never reaching that time.
You can use this method to approximate this meeting time. No one does, since obviously it can be exactly solved. But if you perform enough iterations of the hare catching up and the tortoise moving on, you will arrive at the effectively exact time and distance that they meet.
The hare never reaches the tortoise, because time, in the thought experiment, never reaches the moment that the hare does pass. As soon as you imagine time proceeding beyond this meeting time, you must imagine the hare passing, for your thought experiment to tension consistent.
Only if you are a Platonic realist. Metaphysically, that's an issue with set theory in general, Platonism is presupposed.
And when abstractions such as numbers, are assumed to have independent existence just like physical objects, with no principles to differentiate between the abstract and the physical, we have the problem 180 mentioned:Quoting 180 Proof
This is why the law of identity was imposed, as a principle of differentiation between physical objects and abstract objects. A physical object has an identity unique to itself, an abstract object has no such identity. Therefore all those assumed numbers which cannot be counted, have no identity.
The paradox between discrete and continuous seems a good example, to me on the outside, as something to treat Zeno's paradoxes as genuine paradoxes.
The way I put it to make it make sense to me: I can say there are "more" rational numbers than there are even numbers. Both sets are infinite, but it seems to me that the Rational Numbers > the Even Rational Numbers, as I understand the notions.
But that there can be "larger" infinites is a paradox to my mind.
Is this in any way motivated by the uncertainty principle?
Actual measurements fail beyond Planck's constants. These paradoxes are all hypothetical involving motions of dimensionless points along rational number scales.
Actual measurements fail far above Planck's constants :)
One of the concepts I've found hard to teach is the difference scientists attach to "accuracy" vs "precision"
Normally we'd interchange these words, which is the reason it's hard to differentiate. But they both deal with measurement, in reality, so are needed.
What you read on your fuel gauge on your car is a measure of how much fuel you have in your tank. The accuracy and precision of that gauge can be described as such -- suppose you have a particularly imprecise but mostly accurate fuel gauge, as I suspect most of them are. Then when it reads "1/2 tank" you know it's about, in terms of 16'ths, about 6/16's to 10/16's. Precision is saying "looks, you don't know between these numbers what it actually is" and accuracy is saying "it's definitely within this range that the precision says, and the "real" number is there but this is what you get"
****
What I'm asking is more about Heisenberg's Uncertainty Principle, which as he interpreted it meant that reality itself doesn't allow for a precision of both, but rather demands a[s]precision, or position,[/s]* of any one particle. But due to cuz that's how nature works, not cuz how we measure it.
*Blah. Speaking from memory makes me say wrong things. The precision of position and momentum are proportional to eachother such that a greater precision of position results in a lesser precision of momentum. Einstein interpreted this in mechanical terms, but the quantum scientists, at least of the time, interpreted this in real terms -- it wasn't the apparatus measuring but rather the behavior of the quantum particles which differed from the old billiard ball model.
See, for more, Introduction to Error and Uncertainty.
There's that, and then there's the philosophically more interesting view expressed here:
...the presumption that there is a true value; that given infinite precision we could set out the actual value as a real number. There is no reason to supose this to be true.
Yup.
The "accuracy" part of the distinction is what I consider to be a noble fib. Speaking to a person who believes that the gauge they've always used says exactly what's in there it's time to note a difference between accuracy and precision.
It's only a half-fib, because accuracy still ends up mattering. Using the fuel gauge example if you've used that fuel gauge so many times and know that when it says "a hair up from 1/4 tank" you can easily get from A to B and back to the Gas Station in time then it's accurate, if not precise. So accuracy is important -- it just has more to do with the reason things jump around. If a gauge jumps around over the usual precision limits you might have a problem with accuracy (i.e., the gauge is busted, most of the time -- or occasionally, from the history of science, you actually figure something new out)
But it's not a "fib" at all; the tank really is a quarter full, ±5%. It's a truth.
It's due to the way that time exists, in conjunction with the limitations of our capacity to measure. We are limited in our ability to measure time by physical constraints. If we had a non-physical way to measure time we wouldn't be limited in that way.
You're right it's not a "fib" because measurement requires both, so as I understand it at least.
It's a half-fib because I know the person who thinks in terms of accuracy without precision will most likely not understand the difference. They'll understand that things can be uncertain, of course -- who doesn't? -- but probably doesn't understand that the reason this is uncertain is different from why the other things were uncertain.
Heh.
Well, give it some time. Perhaps we'll figure out the non-physical way to measure time :D
I can't say I agree with your first statement because "the way that time exists" and "the limitations of our capacity to measure" are both things I think about with uncertainty all the time.
We're limited in terms of measuring -- but I want to say that Zeno's paradoxes are not problems of measurement at all. They are logical problems (which is why they evoke the difference between physics and logic and math, as the OP stated already)
Good, becasue it is nonsense. A "non- physical" measurement of a physical quantity... what would be your non-physical units for the fuel left in the tank - not litres, since they are physical.
Physical measurements are not infinitely precise, nor is such precision needed.
Oh, definitely.
Or accurately? Precisely? :D
I think this lays out a good difference between truth and measurement -- we have to be able to say that the fuel gauge is precise, or accurate, in such and such a way in order to do the things we do. Thereby accuracy and precision get relegated to truth -- as the philosopher should want -- but then the truth of truth becomes wildly different from what the philosopher wanted.
No. The measurement is true. Specifying the degree of error does not render the measurement untrue. The tank really does contain 25±1 litres.
Truth, so I'd put it, is a predicate which applies to sentences.
Measurements can be true, but it's not the same as "true" above.
I can be a true friend, and being a true friend is not the same as having a true sentence.
"The tank really does contain 25±1 liters" is true
Accuracy is the "25" and precision is the "±1 liters"
"The tank really does contain 25 liters", in this case, is true
"The tank contains '±1 liters' of what it reads" is also true
Does that confuse, or help, or do I need to say something else?
The logical problems are the result of not having an adequate way of measuring. We are reduced to logical possibility. If we had the proper way we wouldn't have to entertain those possibilities.
So for example, the true divisibility of every physical thing is determined by its physical composition. But if we do not know how it is composed we just assume the logical possibility of infinite divisibility. This is what happens with space and time, and before the atomists, matter itself. We do not know how these things are composed so we just assume the logical possibility of infinite divisibility.
Quoting Banno
I was talking about the problems with the measurement of time (basis of the uncertainty principle), not the measurement of fuel. Show me how time is a physical quantity.
I took a class in psychological measurement in college lo these many decades. This was how precision and accuracy were described. They are both statistical properties. The analogy that was used was to archery. If all the arrows are clustered close together in the bullseye, they are precise and accurate. If they are clustered close together but offset from the bullseye, they are precise but not accurate. If they are centered on the bullseye but are not clustered close together, they are accurate but not precise.
And that's the name of that tune.
Umm... as the set of rational numbers is countably infinite, I would say there's as many rational numbers as there are natural numbers or "even rational numbers".
Because you can go through all rational numbers in the way Cantor showed:
Yep.
The way it was explained to me was the difference between the rationals and the reals -- my thought was to extend that to the rational sets "All Rational Numbers" and "All Rational Even numbers", and note how, intuitively at least, that the first seems to contain about twice as much as the second, even though both are infinite.
That's a paradox to me.
Zeno's paradoxes are very clearly problems of measurement. Like I explain above, if we had the appropriate way of measuring things like time and space, we wouldn't have to entertain the logical possibility of infinite divisibility. Then there would be no such paradoxes. The paradoxes are due to a deficiency in measurement capacity.
I can't. All I can do is lead the donkey to the water. I can't make him drink.
Can you show me a physics text that does not use time?
'cause, you see, as has been mentioned before, your grasp of physics is, shall we say, eccentric?
So better to pay it no attention.
Quoting Metaphysician Undercover
Here I would side with @Moliere. It is a logical problem. Or basically that the measurement problem is a logical problem, hence you cannot just suppose there to be "an adequate way of measurement".
The problem is infinity itself. And that is a logical problem for us.
Quoting Moliere
Ok, it's can be difficult to understand, but I'll try to explain.
Let's say you have a set of numbers, let's call them Moliere-numbers. As they are numbers, you can always create larger and larger Moliere-numbers. Hence we say there's an infinite amount of these numbers. The opposite of this would be a finite number system that perhaps an animal could use: (nothing, 1, 2, 3, many) as that has five primitive "numbers".
If we then say that these Moliere-numbers are countably infinite, then it means that there's a way to put them into a line:
Moliere-1, Moliere-2, Moliere-3,.... and so on, that you can be definitely sure that you would with infinite time and infinite paper write them down without missing any.
If Moliere-numbers are uncountably infinite, then we can show that any possible attempted list of Moliere numbers doesn't have all Moliere-numbers.
Quoting Moliere
Ok.
If you think so, then wouldn't there be more natural numbers (1,2,3,...) than numbers that are millions? Isn't there 999 999 between every million?
No, similar amount, because
(1,2,3,....) can be all multiplied by million
(1000 000, 2 000 000, 3 000 000,...)
And because you can make a list of all rational numbers (as above), the you can fit that line with the (1,2,3,...) line in similar fashion. That's the bijection, 1-to-1 correspondence.
@Banno explains science and math.
Yes, thats just what I was thinking.
How's that relevant? Physics uses mathematics, but that doesn't mean mathematics is physical.
Quoting ssu
Whether or not it is possible to devise an adequate way is irrelevant. The problem is that we do not have an adequate way. And the lack of an adequate way produces the use of an inadequate way. Therefore the problem is not a logical problem, it is a problem in the application of logic. Principles are applied where they are not suitable for the task which they are applied to. That is a measurement problem.
Quoting ssu
Infinity itself is not the problem. The problem is how the concept of infinity is developed and employed. In its basic form "infinity" allows that principles of measurement such as numbers, can be extended indefinitely so that in principle anything and everything can be measured. That is beneficial, it is not a problem. The problem is that there are many misuses of infinity, such as the idea that there is some type of thing which can be infinitely divided. That is not a problem with infinity, but a problem with its application, a problem of applying the wrong principles to the task, a measurement problem.
Is this a "misuse" in mathematics? We are talking about mathematics.
Pick two real numbers, and it can be shown that there are real numbers between them. Pick even two rational numbers, and you have rational numbers between them.
You would wander to the illogical, if you would to start to argue that it isn't so, that it's misuse or something.
That's not misuse, nor is it a problem.
The problem is in Zeno's application, when things like distance, and time, are assumed to be infinitely divisible. It is a measurement problem because instead of determining the natural constraints on such divisions (these constraints are unknown), it is simply assumed that divisibility is infinite.
An obvious wrong assumption?
But it does puzzle us still. Because if you think that we know everything about mathematical infinity, then I guess there should be an answer to the Continuum Hypothesis.
Quoting Metaphysician Undercover
I do get that point, sure. But do actually notice that Zeno belonged to the Eleatic School. Platonists were on the camp of infinite divisibility. The Eleatic School was different.
Quoting ssu
I'm not sure, actually...
That makes sense by what I said, but then your objection also hits home -- if there's a 1-to-1 mapping then really all I'm doing is performing some operation on one set to get to the other set, which is pretty much all a function is (by my understanding).
But then the set "Even numbers" is defined by whether or not they are divisible by 2. So if we have the set of all natural numbers and the set of all even natural numbers then, if there's a 1-to-1 correspondence, we ought be able to lay out a function like the above -- such as f(x) = x * 1 million
But suppose the number 3 -- does it yield 2 or does it yield 4? Once we decide that then we can say there's a function which maps, but before that it seems to me we have to make a decision.
Now that doesn't seem to make them uncountable, and perhaps the sizes of the sets are still the same -- the whole idea of infinite sets having different sizes is the thing that is confusing me. I'm just responding here in my own words and thinking out loud.
Quoting ssu
Thanks for indulging my curiosity. If what I said above is entirely whack then feel free to just point me to a text ;)
But if you're willing to continue....
How could we show that Moliere-numbers are uncountably infinite?
That assumption does create a measurement problem. So unless we think that measurement problems are good, then I'd say it's a wrong assumption.
Quoting ssu
Again, "continuum" assumes something being divided. Simply saying that there is a number between any two numbers does not assume anything being divided, just like assuming that there is always a higher number does not assume anything being counted. These are simply pure mathematical axioms. But when we say things like "there is a continuum", "numbers are objects", then we introduce ontological premises into the mathematical axioms, which may or may not be true.
Correct. Wrong assumptions lead to invalid conclusions. End of the story. I think I wrote this point a while back in the thread.
Quoting Moliere
And there's a 1-to-1 correspondence:
(1, 2, 3, 4, 5, 6, 7,...)
(2, 4, 6, 8,10,12,14,...)
Quoting Moliere
OK, basically how Cantor showed that real numbers are uncountable is the way to do this.
Basically if you have a list where all the Moliere-numbers would be and then you show that there's a Moliere that differs from the first Moliere-number on the list, differs from the second Moliere-number on the list and so on. This way you show that there's a Moliere-number that isn't on the list. Hence there cannot be a list of all Moliere-numbers. The conclusion is a Reductio ad absurdum proof.
How does this enter into a discussion of these Zeno type paradoxes? Define the momentum of a point as it progresses to zero. Does the tortoise have momentum? Too much of a stretch for me.
I think it's because quantum stuff is based on the notion that reality is discrete vs the continuous reality of the block universe; but, yes, if it's just a logical puzzle then the science is irrelevant.
OK that makes sense seeing it like that rather than the muddle I wrote. (While it's interesting to me I claim little knowledge here)
Quoting ssu
That helps me work through the wikipedia page on Cantor's diagonal argument. Thanks!
It's still a concept that confuses the hell out of me, but this gives direction if ever I'm tempted to talk on it again ;)
Yep, it's an easy way to understand the whole thing.
Quoting Moliere
If your smart and observing, it should!!! It is confusing.
Because then you get to the really awesome question: OK, if we have countable and uncountable infinities, what is the relationship between these two infinite sets?
Cantor himself gave us the Continuum Hypothesis as an answer. But what does it mean? And what actually the whole idea of there being larger and larger infinities mean, because the only thing we have shown is this "uncountability" of the reals.
(Oh btw, I think Moliere-numbers sound very cool. You might really think that there are Moliere numbers)
It's better to say that those conclusions are unsound rather than invalid.
Take the rationals in [0,1] and form the union with the irrationals in [0,1] and you get a continuum. They are complementary in the complete interval - which itself is a complete metric space with the usual metric.
Unsound argument means the premise was false, and also invalid reasoning was applied for the conclusion. Here reasoning seems valid, but the premise was false, which led to the false conclusion. Hence the argument is invalid.
To be precise, conclusion is either true or false, but arguments could be either valid or invalid. If the conclusion was true and followed from the premise, then the argument is valid and sound. If the conclusion was false and had false premise, then the argument is invalid and unsound.
Arguments can be valid if it followed from the premise even if the conclusion is false. Argument is invalid, if it didn't follow from the premise even if the premise was true. Do you agree with these points?
The statement P -> Q was false could have been proved via MT.
P -> Q
~Q
~P
Replace "and" with "or" here, and you'll see that if the reasoning is valid and the premise is false, then the argument is valid but unsound. So you should conclude "Hence the argument is unsound', instead of the following:
Quoting Corvus
https://iep.utm.edu/val-snd/
Depending on the type of two points in the distance (which is not clear in the OP), the conclusion can be true, which makes the argument invalid.
Think of the case where the two runners are running around a circle, not a straight line. When A takes over T, he is still behind T in the circle. A must run again to take T over, but when he does, he is still behind T and so on ad infinitum, which makes the conclusion true, and argument invalid.
Doing so without clear evidence or information of the type or nature of the distance in the track would be commiting a fallacy of illicit presumption. Until all is clear and evident, the argument must be judged as invalid.
If you mean the Heisenberg uncertainty principle no - although I'm tempted to think that Zeno was close to discovering a logical precursor to the Heisenberg Uncertainty Principle on the basis of a priori arguments.
The semantic problems of calculus with regards to Zeno's arguments stem from the fact that calculus isn't resource conscious. Sir Isaac Newton and Leibniz had no reason in 17th century to formulate calculus that way, given the use cases of calculus that they had in mind.
A notable feature of resource-conscious logics is how they naturally have "quantum-like" properties, due to the fact their semantic models are state spaces of decisions that are generally irreversible, thereby prohibiting the reuse of resources; indeed, the assumption that resources can be reused, is generally a cause of erroneous counterfactual reasoning, such as when arguing that a moving object must have a position because it might have been stopped.
So in the case of a resource-conscious calculus that avoids mathematical interpretations of Zeno's paradoxes (as in a function having a gradient but also consisting of points), a function must be treated as a mutable object whose topology undergoes a change of state whenever the function is projected onto a basis of functions that "measure" the function's properties -- Thus the uncertainty principle of Fourier analysis has to be part of the foundations of a resource-conscious calculus rather than a theorem derived from real-analysis of the continuum that is the cause of the semantic unsoundness of calculus with respect to the real world.
An obvious candidate for contributing to the foundations of such an alternative calculus is some variant of differential linear logic, which incidentally has many uses in quantum computing applications.
Can you explain a bit more thoroughly what you mean by "resource-conscious"?
But that doesn't answer the questions we have about infinity. If we have countable infinity and then uncountable infinity, Cantor argues there's this hierarchial system of larger and larger infinities (from aleph-0 to aleph-1 and higher). Is that really how it goes? And the Continuum Hypothesis is a hypothesis, it's not a theorem.
Set theory, which can be viewed as the foundations of mathematics, simply takes infinity as an axiom. That's not a proof and that's the problem. The questions that have been around for thousands of year, things like if there is either a potential or an actual infinity are still puzzling. That's why Zeno's paradoxes are talked over and over again, even if we do understand that Achilles brushes past the tortoise in reality and we do have the math to calculate it, the foundational question are still there.
As I have stated before in this thread, we don't have a similar debate about of "Are all numbers rational?". Nobody is saying that they would be so. No, we can prove that there are irrational numbers. Above all, we can understand that there are and have to be irrational numbers. There wouldn't be this kind of over and over repeating debate Zeno's paradoxes, if we fully would understand the infinite or infinity.
Resource conscious logics such as Linear Logic don't automatically assume that the premise of a conditional can be used more than once. They are extensions or refinements of relevance logic. The best article relating resource-sensitivity to the principles of quantum mechanics is probably nlabs description of linear logic
https://ncatlab.org/nlab/show/linear+logic
As for uncertainty principles:
Recall that classical logic has the propositional distributive law, that for all A, B and C
A ? ( B ? C) = (A ? B) ? ( A ? C)
Here, the meaning of "and" is modelled as the Set cartesian product, and the meaning of "or" by set disjunction, neither of which are resource conscious - therefore one always has the same cartesian product, even after taking an element from one of its sets. The negation of this principle is more or less a definition of the uncertainty principle and characterizes the most remarkable aspect of quantum logic, which is in fact a common-sense principle that is used extensively in ordinary life.
The connectives of Linear logic cannot be interpreted in terms of the cartesian product and set disjunction. Instead it has the tautology
A ? ( B ? C) ? ( A ? B ) ? ( A ? C )
If this formula is interpreted to be a true conclusion that needs to be proven with respect to unknown premises , then it has the interpretation "Assume that we are sent an A i.e. an element (a : A), and that we are also sent either (b : B) or (c : C) at our opponent's discretion, neither of which consume the (a : A) (that is to say B and C are independent of A). Then we end up with either (a : A) and (b : B), or (a : A) and a (c : C)".
Likewise, our opponent's side of this interaction is then described by the tautology
¬A ? ( ¬B & ¬C) ? ( ¬A ? ¬B ) & ( ¬A ? ¬C)
"If our sending of (a : A) also implies our sending of either (b : B) or (c : C), where B and C are independent of A , then we either send both (a : A) and (b : B), or we send both (a : A) and (c: C).
But there isn't the theorem
A ? ( B & C) ? ( A ? B ) & ( A ? C )
The inability to derive this theorem is the common-sense uncertainty principle of linear logic: getting an A together with a choice of B or C for which this act of choosing is independent of the existence of A, isn't equivalent to the outcome of the choice being independent of the existence of A.
(Imagine winning a bag of sugar together with a choice between winning either ordinary ice cream or diet ice cream. It might be that the awarders of the prizes use the awarded bag of sugar to produce the chosen ice-cream.)
By analogy, by using a resource-conscious logic as the foundation of an alternative calculus, smoothness and pointedness can be reconciled by defining them to be opposite and incompatible extremes of the state of a mutable function that is affected by the operations that are applied to it. This is also computationally realistic.
Or the limit concept.
Exactly.
And actually the case of the troublesome infinitesimal, which both Newton and Leibniz could quite well describe to be calculated, still was so problematic that we got the concept of limits (even if we have today non-standard analysis). Yet then it's typical that the limit approaches infinity.
I would argue that this is a case of us having the ability to use the math, but not having the philosophy behind it.
"x goes to positive infinity" simply means "positive x increases without bound". No need for the word "infinity". Simply shorthand. But, set theory is another matter and postulates all sorts of things.
I belong to a past generation who didn't venture beyond the Peano axioms. These days searching for relationships between the broad spectrum of math subjects is in fashion. I cheer them on from the sidelines. :cool: