Zeno of Elea · Philosophy
Zeno’s most famous paradox: give a tortoise a head start, and swift-footed Achilles can never overtake it - for whenever he reaches where it was, it has crawled a little further, and this repeats without end. The paradox seems to prove, against all sense, that the faster runner can never pass the slower.
Of all Zeno’s paradoxes, the most celebrated is the race between Achilles - the swiftest of the Greek heroes, ‘swift-footed Achilles’ of Homer - and a tortoise, the proverbially slow creature. The setup is simple and the conclusion outrageous. Achilles, being far faster, generously gives the tortoise a head start. The race begins. Now, Zeno argues, before Achilles can overtake the tortoise, he must first reach the point where the tortoise started. This takes some time - and during that time, the tortoise, though slow, has crawled a little further ahead. So now Achilles must reach that new point where the tortoise has got to. But again, during the time it takes him to get there, the tortoise has crawled a little further still. And again Achilles must reach the tortoise’s new position - and again the tortoise has moved on. Every time Achilles reaches where the tortoise was, the tortoise has advanced to a new position, however slightly. There are infinitely many such positions to reach, one after another, without end - and so, Zeno concludes, Achilles can never catch the tortoise. The faster runner can never overtake the slower.
The conclusion is plainly false: we know perfectly well that Achilles overtakes the tortoise almost at once. And that is exactly what makes the paradox so unsettling and so important. We are absolutely certain the conclusion is wrong - yet the reasoning seems impeccable at every step. Each individual claim seems true: yes, Achilles must reach where the tortoise was; yes, by then the tortoise has moved a little; yes, this repeats endlessly; yes, there are infinitely many such stages. How can a chain of true-seeming steps lead to a false conclusion? Somewhere in the argument there must be a hidden error - but where? Finding that error turns out to require thinking very carefully about a deep and treacherous notion: the idea of completing an infinite number of tasks. The Achilles is not a trick; it is a probe into the nature of infinity, and answering it took the human mind more than two thousand years.
The flaw in the Achilles paradox lies in a hidden assumption so natural that for two thousand years it was hard even to notice: the assumption that completing infinitely many tasks must take an infinite amount of time (or be impossible). This assumption seems obvious - surely an endless list of tasks can never be finished! - but it is false, and seeing why dissolves the paradox. The crucial point is that the infinitely many stages of Achilles’ pursuit are not all the same size; they get smaller and smaller, without limit. The first stage (reaching the tortoise’s start) might take Achilles ten seconds; the second (closing the smaller gap the tortoise has since made) a fraction of that; the third a smaller fraction still; and so on, each stage taking far less time than the one before. The question is not ‘can you complete infinitely many tasks?’ in the abstract, but ‘can the sum of infinitely many ever-shorter times be finite?’ - and the answer, remarkably, is yes.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned Zeno’s Achilles paradox: a tortoise gets a head start; before Achilles can overtake it he must reach the point where it started, but by then the tortoise has moved a little further; Achilles must reach that point, but the tortoise has again moved on; and so on without end - so Achilles seemingly never catc…
Leads to Aristotle.
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