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According to Wikipedia's discussion of Zeno's Dichotomy paradox (emphasis mine),

According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions (see solvitur ambulando). To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes.

Why is this argument insufficient? It would seem to be a prime example of a counterexample proof. If Zeno's argument is that there exists a finite distance journey that cannot be completed due to requiring infinite steps, then I can see that a single counterexample is insufficient, but Zeno seems to be claiming that completion is impossible for all finite journeys. If there exists at least one finite journey that can (somehow) be completed in finite time, then it is impossible that all finite journeys are impossible and so Zeno's argument crumbles.

Analogously, if I claim that at least one black cat exists, showing me an orange cat does nothing. However, if, like Zeno, I claim that all cats are black, proof that there exists one orange cat in a box somewhere completely annihilates my argument.

To be clear, I'm not asking whether all finite distance journeys of the type mentioned by Zeno are impossible. I believe that they are possible. I'm asking what is wrong with Diogenes's argument.

I do know that in many models of physics, escaping the event horizon of a black hole is considered impossible as it would require exceeding the speed of light. Quantum physics and the Planck length are potentially relevant to a quantum-based solution, but clearly Zeno was not considering "move this electron to a point halfway between these two orbitals" as a plausible journey. I am not really talking about these advanced physics scenarios, but the "walk across the room" or "cha-cha across the dance floor" scenarios that Zeno was discussing. Alternately, "cat walks to food bowl" or "chicken crosses the road" could be acceptable scenarios that we can not only envision, but actually perform experimentally to verify that the creature is able to complete the journey before the heat death of the universe.

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    Basically, the gist of paradoxes, like Zenos' ones, is not to prove that something does not exist: it is clear that time is real, that speed is real, that the world outside us is real. They are aimed at showing that our current ideas and "theories" have some unsolved puzzles or inconsistencies. Dec 21, 2022 at 12:39
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    What is wrong with it is that all it does is point to the side of the paradox that everyone already knew was there. It added nothing to the discussion. Dec 22, 2022 at 3:42
  • Zeno's making an empirical claim from mathematical assumptions/"truths". Diogenes demonstrates traversing distances is possible by performing the act. Zeno replies "motion is an illusion". It gets weirder and weirder after that as far as I can tell.
    – Hudjefa
    Dec 22, 2022 at 9:09
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    Wikipedia misrepresents the problem. Zeno did not argue that motion does not happen, his conclusion rather was that it cannot be thought coherently. As Diogenes's "response" does not touch that, it is not "insufficient", it is irrelevant. Moreover, there is nothing wrong neither with Zeno's arguments nor with his conclusion. The thought of motion is incoherent on his (or rather Parmenides's) conception of thinking. Many other "responses" fail for the same reason, see Papa-Grimaldi, Why Mathematical Solutions of Zeno’s Paradoxes Miss The Point.
    – Conifold
    Dec 22, 2022 at 13:43

3 Answers 3

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My two cents.

Diogenes solution was a practical démonstration that refuted Zeno's claim. Diogenes demonstrated in practice that Zeno's theoretical construction is not a correct description of this world. And that is it.

Now, theoretically Zeno's argument fails because it assumes infinite divisibility of parts (all in the same sense), thus begging the question of what he wanted to claim. Infinite divisibility is obviously false since, for example, when walking a step size cannot be subdivided arbitrarily but there is a certain minimum step size (depended on the size of our foot) that cannot be subdivided further into smaller step sizes. Assuming infinite divisibility is equivalent to assuming non-termination of the process. In other words, it is equivalent to assuming the conclusion.

To see it from another perspective, Zeno, in a sense, assumes every interval dx however small is always travelled at the same time dt, but this directly contradicts the definition of processes which travel at constant speed, in Newton's law ie v=dx/dt, so for given dx, dt must be adjusted in order for v to be constant along any interval dx. So as dx tends to zero, so does dt and this refutes the claim

In other words, there is no argument, it is simply claimed as truth (and Diogenes simply demonstrated otherwise).

I agree with you that Diogenes' response is in various ways similar to constructive proofs through counterexamples.

One may argue that although Diogenes refuted the claim successfully, he did not elucidate the reason the argument is wrong, he simply demonstrated it wrong, did not explain where it fails in its chain of reasoning. So one may argue that Diogenes provided a refutation but not an explanation.

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    Excellent post!
    – Hudjefa
    Dec 22, 2022 at 9:05
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The Wikipedia quote you cite actually contains the answer to your question.

Suppose you develop a spreadsheet to calculate how many tiles you need to cover the roof of your house. You order that many tiles, and you find that they cover only half of the roof. I would be able to tell you at once that there was a fault in your spreadsheet, because the result it gave clearly did not match reality. What that would not do, however, is to pinpoint the errors in the spreadsheet that led to the incorrect result.

In an analogous way, Diogenes proved that Zeno's paradox was nonsense, but did not shed any light on where the fault in Zeno's reasoning lay. Since the whole point of a paradox is that one cannot see why it yields confusing results, explaining the cause of the paradox is the essence of its resolution.

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Since "reality" is either a "blurry" concept (not for Diogenes obviously), I can help with argument:

The paradox arises from confusing now with infinity, but not consequently applying it:

That which is in locomotion must arrive at the half-way stage before it arrives at the goal.

Seems a rational rule, which (eternally valid) describes the relation of:

  • "locomotion"
  • "arrival"
  • "before"
  • and (half-)way

(now(+past+future)==eternity) ..without even mentioning time.

Makes Zeno maintain:

This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.

  1. Why requires? A linguistic fallacy! To make an (finite) "object" adhere to an infinite rule, which states "it must", doesn't require the object to complete any tasks at all. (Only an observer/subject can(not) verify it accordingly)

  2. why impossible?? now is eternity!!! subject can complete infinite number of tasks (already done) ...

How to solve Achilles vs. Turtle:

  • Time is not "rational" ...it is "scalar" (as numbers), "real", (composed of (natural +) rational + irrational + ???)
  • Same, don't confuse infinity with now.
  • don't confuse subject with object! (...pole with hole)
  • (don't confuse before with after)

What means (Zeno's) arrow paradox

  • time is not (only) rational (has no/infinite "instants")

  • now/eternity confusion:

    that which is in locomotion is always occupying such a space at any moment

    ...when "there is" occupation of space and "locomotion", where is the "liberation" of space?? ;)

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