I don't think the tortoise actually wins. In a similar way to the way that Achilles cannot catch up to the tortoise, the tortoise also cannot actually reach the finish line. To reach the finish line, the tortoise must first cross half the distance to it, then half the remaining distance, then half of the remaining distance again, and onward infinitely.

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.

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. — Metaphysician Undercover

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.

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. — Nemo2124

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.

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. — 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.

Simply put, rest frames are imposed according to the purpose. — Metaphysician Undercover

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.

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. — 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.

Time will keep going forever, and the tortoise will always have more space to cover before it reaches the line. — 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.

Both the Achilles paradox, and the paradox of the tortoise who cannot actually reach the finish line, are illusions created by looking at time as if it was approaching a limit.

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.

Time's arrow, time's cycle... Already differing, almost contradictory perspectives on the passage of time. Even if it would take an eternity (infinite time) for the tortoise to cross the finishing-line, it would get there one-way-or-another. The trouble with Achilles, is that he tries too hard. His movements are limited, he only moves when he realises that the tortoise is ahead and yet he is certain that he will always catch-up with it and will eventually win the race. The tortoise's strategem is to stay one slight step ahead.

Mathematically, the infinite sum of the series in question is 1. — T Clark

Precisely, by mathematical summation the series gives unity, but in practice - physically - it's impossible.

In a similar way to the way that Achilles cannot catch up to the tortoise, the tortoise also cannot actually reach the finish line. To reach the finish line, the tortoise must first cross half the distance to it, then half the remaining distance, then half of the remaining distance again, and onward infinitely. — Metaphysician Undercover

Not really a contribution to the discussion, but I think this expression of MU's is simple, elegant, and final.

If gentle reader buys Zeno's setup, he must then buy MU's conclusion. Of course Achilleus passes the tortoise and both achieve the finish line. MU, then, illuminating that the difficulty is not so much in the problem, as with some particular, but with the problem, as a whole.

To reach the finish line, the tortoise must first cross half the distance to it, then half the remaining distance, then half of the remaining distance again, and onward infinitely. — 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.

the infinite sum of the series in question is 1. — T Clark

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.

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. — Banno

It is my understanding that the appropriate mathematics didn’t exist in Zeno’s time.

Yep.

But others here do not have that excuse.

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. — Nemo2124

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.

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. — Nemo2124

It cannot, by the premises of the example.

Physical space is not "infinitely" divisible like abstract space. Like most, this paradox is merely apparent – in this case it's derived from confusing the physical and abstract. — 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.

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else to allow for qualitative differences between things. This is why the dualism of matter and form was required — Metaphysician Undercover

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

What would you say that "physical space" is made out of? [ ... ] The ancient Greek atomists limited the capacity to divide physical substance by positing fundament particles, atoms. The atoms would be indivisible. — Metaphysician Undercover

:up: :up:

Ergo physical divisibility is finite.

But Aristotle demonstrated ... why the dualism of matter and form was required.

Clearly, Aristotle did not understand that Democritus' atoms are physical and not just abstract (i.e. not formal/metaphysical – "platonic").

The problem with this proposal is that there is too much relativity already baked into the procedural methods of quantum physics. — Metaphysician Undercover

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.

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. — Nemo2124

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.

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). — 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.

The zero-point energy of the Universe (or null-point energy) represents the energy of the vacuum according to quantum physics. This has been proven to exist via the Casimir effect, as far as I am aware. Nonetheless, we also know that the minimal physical distance is the Planck length, a discrete almost infinitessimal distance for Achilles and the tortoise to move. Below this distance-scale, nothing physical can exist, except the zero-point energy (equivalent to the lambda-term in GR).

I believe "zero-point energy" is the consequence of relativity type thinking. — 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.

It's just a convenient way to avoid the problems created by relativity type thinking, but since it's fiction it produces useless metaphysics. — 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.

It is my understanding that the appropriate mathematics didn’t exist in Zeno’s time. — T Clark

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.

A feature of Zeno’s paradoxes is the counterintuitive conclusions that the thinker is forced into. This seems to go against the faculty for reason, but you supersede the apparent contradiction by recognising an alternative route to explanation, thereby making sense of the paradox.

I don’t think that machines (AI) can do this. It represents a limit of reasoning, one that AI is yet to recognise. It’s also very difficult, if not impossible, to programme a sense of humour, for example. Here I think it’s apparent that the paradox highlights the gap between physics and maths.

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 — Nemo2124

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

Achilles gives the tortoise a head-start because he is so convinced of his superior speed. As it turns out, even if he gives the tortoise as much as a hair’s breadth advantage, the wise animal would probably still win…

Ye but that's assuming there is no motive power to begin with

From a different perspective, the tortoise’s 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?

When Achilles runs the one yard, the tortoise is a tenth-of-a-yard ahead. And so on, without end. — Nemo2124

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.

Precisely, by mathematical summation the series gives unity, but in practice - physically - it's impossible. — 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.

We ought to remove those points, those beginnings and ends, from the representation of the movement of the thing itself — 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?

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' — Nemo2124

You think that space being continuous is disproven by this story then. Quantum theory AFAIK has never suggested quantizing spacetime.

By relativity theory, an object is always moving, and cannot actually be at a fixed position. — 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.

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. — sime

:100:

How can nature have anything infinite within it? — 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.

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 — noAxioms

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 aware of a mathematical definition of an alternative continuum that resolves all of the logical puzzles posed by Zeno. — sime

I'm not sure what you mean by "alternative continuum."

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. — 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.

Representing a continuum as an infinite series of infinitesimals seems like a good model of how the universe works, simple and intuitive — T Clark

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