To: Marcus S.
From: Geoffrey Klempner
Subject: Zeno's paradox of Achilles and the tortoise
Date: 1 July 2005 09:59

Dear Marcus,

Thank you for your email of 20 June, with your third essay for the Ancient Philosophy program, in response to the question, 'Analyze in detail any one of Zeno's paradoxes.' You have chosen to analyze the paradox of Achilles and the tortoise.

Achilles does catch the tortoise - you are in no doubt about that. So the question for you is, What is the most plausible reading of Zeno's argument? how can we explain why the argument is felt to be gripping, rather than a trivial sophism? This could be described as an example of the application of the 'principle of charity' in philosophical interpretation. An interpretation is more believable if it makes the philosopher out to be someone worth reckoning with, rather than a fool.

However, I must stop you there. Because Zeno believed - if Plato's account in the Parmenides is correct - that Achilles does not catch the tortoise. No-one catches anything, because nothing moves, nothing changes in Parmenides' One. That is because any attempt to describe a differentiated or changing world leads to self-contradiction. Which is precisely the proposition which Zeno sets out to prove in his paradoxes.

That is not such an absurd claim. We are aware that language is a rather vague, imprecise tool. Perhaps it is true that although we get along with words for practical purposes, any attempt to describe the world with logical rigour will break down. It is not inconceivable (at least, prior to looking at the detailed arguments) that the philosopher who is concerned with the question, 'What is Reality?' in the absolute sense which demands logical rigour might be led to the conclusion that the world we 'know', the world of 'practical purposes' is not Reality, but merely a dreamlike half-world where 'mortals wander, two-headed, knowing nothing'.

You make a robust attempt to explain the assumptions which lead Zeno to this conclusion. If Achilles and the tortoise are conceived as mathematical points, and if Achilles stops at every half-way point (or if the godlike observer who has set up this experiment stops him) then Achilles cannot catch the tortoise. - But they are not, and he doesn't, so he can.

In your last paragraph, you develop this idea by adding a more general explanation, in terms of Parmenides' conception of the 'oneness' of the world: 'Such a world has absolute boundaries for reality.' The implication is that Parmenides was wrong, because in reality there are no absolute boundaries.

There is a paradox which has attracted the attention of contemporary philosophers which you may have heard of: the paradox of the heap. Or, as I prefer, the elephant paradox. A one ounce elephant is a small elephant. Agree? OK. If you add one ounce to an elephant which is not a large elephant, then the resulting elephant cannot be a large elephant. Agree?

It logically follows, that - according to you - there are no large elephants! (The argument is by an application of the method of mathematical induction. If you can prove that 1 has mathematical property X, and prove that *if* n has X *then* n+1 has X', then it logically follows that all numbers have X.)

One plausible response to this paradox is to claim that Reality cannot be vague, only our descriptions of it. It follows that there must be some way to describe any elephant which does not involve any vague predicates. - But are you so sure that such a description can be found? And what happens to the 'elephant' meanwhile? (The problem of vagueness is one of the major themes in the early and later philosophies of Wittgenstein, which we will be looking at in the Pathways Philosophy of Language program.)

I did feel that in your (not implausible) explanation of how motion occurs in an 'infinite world' (which I take to mean a world where there are no fixed boundaries) you glossed over an aspect of the 'grippingness' of Zeno's paradox. It is in fact the case, that in the process of catching up with and overtaking the tortoise, Achilles does an infinite number of things. He does actually get half-way, and half-way again, and again... to infinity. That is not such an easy proposition to accept. If I said to you, 'Do you know, I have the power to do an infinite number of things!' your first reaction ought to be pretty sceptical. But, in fact, by just reaching to my CD recorder to put in the next blank disc I have done just that.

Motion, which we all take for granted, is a pretty scary concept because it implies infinity, and infinity is scary. Zeno was the first thinker to realize, and really appreciate, this fact.

All the best,

Geoffrey