Overtaking Zeno

Shared Article from Slate Magazine

Zeno’s Paradox Is a Trick—But a Very Interesting Trick

The Greek philosopher Zeno wrote a book of paradoxes nearly 2,500 years ago. “Achilles and the Tortoise” is the easiest to understand, but it’s …

David Plotz @ slate.com

O.K., so, briefly: If you think that the point of Zeno’s Paradoxes of motion is to prove that the arrow never will reach its target, or that Achilles never does pass the tortoise, &c. — then I think that you are mistaken about the point of raising the paradox in the first place. Of course, it’s hard to be confident about the motives of dead philosophers who have no surviving books. But what we do know is that Zeno was a student of Parmenides; and Plato tells us that his books were written to defend Parmenides’s doctrines, by negative means,[1] showing that the views of his opponents led to contradictions.

So the most charitable understanding of Zeno’s aims is not that he’s trying to show you that Achilles can never catch the tortoise. Of course he does; just watch them race and you’ll see it happen. His point is to ask, given that Achilles passes the tortoise, well, how is that possible? And, for good or for ill, to argue from the paradox that you can only make sense of Achilles passing the tortoise if you reject presentism, and accept eternalist and Parmenidean conclusions about the nature of time and being.

Maybe he’s right about that, and maybe he’s wrong. (I’m inclined to think he’s wrong.) But note that if your solution is to try and settle the issue by introducing a lot of mathematical notation and conceptual apparatus from modern calculus — for example infinitesimal limit processes, convergent and divergent series, etc. — as is done in the Slate article here, and as is probably the overwhelmingly most common first response to Zeno’s paradoxes by mathematically-trained writers — then probably you are doing a better job than any pre-classical Greek philosopher could do in elaborating the precise nature of the problem.[2] But you’re not obviously refuting Zeno’s claims in any way, at least not yet. At the most you’re kicking the can down the road, and really you’re sort of strengthening Zeno’s own position. After all, naive formulations of mathematical notation are more or less always going to involve you in all kinds of specifically eternalist language, for example about moments in past and future time actually existing, instantiating the value of functions, etc. You cannot normally take the limit of ΔS(t) over values of t that don’t exist (no longer exist, do not yet exist).[3]

Or perhaps you can. But if you can, then doing so, and explaining what you’re doing when you do it, will take some very non-naive reinterpretation of ordinary mathematical language — and some nice metaphysics, too, to justify your reinterpretation. In any case the solution is going to have to be deeply philosophical, not just a matter of applying a technical innovation in maths.

  1. [1]In the Parmenides: I see, Parmenides, said Socrates, that Zeno would like to be not only one with you in friendship but your second self in his writings too; he puts what you say in another way . . . You affirm unity, he denies plurality. . . . Yes, Socrates, said Zeno. . . . The truth is, that these writings of mine were meant to protect the arguments of Parmenides against those who make fun of him and seek to show the many ridiculous and contradictory results which they suppose to follow from the affirmation of the one. My answer is addressed to the partisans of the many, whose attack I return with interest by retorting upon them that their hypothesis of the being of many, if carried out, appears to be still more ridiculous than the hypothesis of the being of one.
  2. [2]Since the 19th century, we’ve done a lot to really nicely rigorize the mathematics of infinites and infinitesimals, in ways that sometimes anticipated by but never fully available to ancient mathematicians.
  3. [3]If anything, this is even more true of late-modern mathematics than it was of classical mathematics. Contemporary mathematics constantly helps itself to a lot of the language of existence, actuality, etc., for mathematical objects, in areas where Euclid and other classical mathematicians were typically much more circumspect about making existence claims for mathematical objects that hadn’t yet been constructed.

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  1. Roderick T. Long

    Plato tells us that his books were written to defend Parmenides’s doctrines

    I think Plato’s interpretation is unlikely. By having Socrates present his interpretation as a hypothesis which Zeno then confirms (in a conversation Plato invented), Plato makes clear that it’s not obvious from Zeno’s book that its goal is to defend Parmenides. And the fact that the real-life Zeno, when asked why he didn’t write about the Parmenidean One, replied that he was waiting for someone to explain to him what the heck it was (Testimonium A16), doesn’t make him sound like a faithful disciple of his teacher. I suspect Zeno’s goals were closer to those of Gorgias in On Nature or What Is Not.

· April 2014 ·

  1. Gabriel

    I’m not sure I agree with either you or the Slate article. You can resolve the matter mathematically, just not with the irrelevant blather about convergent series in the Slate article. The solution I learned in school is that Achilles never reaches the goal because you’re taking shorter and shorter intervals of time, not just distance. That’s a separate matter from deciding on how to measure an infinite sum.

    • Rad Geek

      Well, if the mathematical solutions straightforwardly solve the problem, then I don’t think that the discussion of convergent series in the Slate article is irrelevant, although they do kind of a poor job of explaining why it’s relevant. The reason I think they think it’s relevant is because it is their technical apparatus for talking about the shorter and shorter intervals of time that you mention. So, more or less, something like this: let c₀ represent the interval that it takes Achilles to catch up to the tortoise’s initial position at t=0, c₁ the interval that it takes Achilles to catch up to the position the tortoise had been at after c₀, c₂ the interval it takes him to catch up to the position the tortoise had been after c₁, etc. etc. etc. For the reasons that Zeno lays out, {cₙ} will form an infinite sequence. But, for the reasons that you mention, the sequence of the catch-up intervals is going to be a diminishing sequence (the interval cₙ keeps getting smaller and smaller as n approaches ∞, with a limit of 0). But then if Achilles is traveling fast enough that he could eventually overtake the tortoise’s head start, then the sequence will produce a convergent series, meaning that you can sum up all the catch-up intervals as n goes to ∞, and what you’ll get is the total, finite time that it will take Achilles to overtake the tortoise. So the reason they’re talking about convergent series is because it’s a way of more rigorously expressing what it means that the infinite intervals are getting shorter and shorter.

      Now whether or not this is helpful as a response to Zeno depends on what you take Zeno to be trying to do. And that’s not easy to piece together. But on the interpretation I was sketching out above, it seems to me like this doesn’t offer a prima facie refutation of Zeno’s point, because when you start bringing out the mathematical apparatus (for example, to measure continuous intervals of time and space) you’re already doing something that seems likely to give you some prima facie commitments to the actual existence of things like past moments of time, which on a naive interpretation of mathematical language would seem to commit you to rejecting presentism in favor of eternalism, which would get him to the kind of Parmenidean conclusion that he would (on this, admittedly speculative, interpretation).

      I don’t think that mathematical language actually needs to commit you to eternalism — there are ways of interpreting that away. But then that requires doing some substantial philosophy with some tricky problems involved in it, and the attempt to get out of the philosophical problem by a technical solution using mathematical notation just ended up kicking the can down the road.

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