One of the Parmenides’ pupils was a clever young man called Zeno (known as Zeno of Elea to distinguish him from the founder of Stoicism, Zeno of Citium). This Zeno was brilliant at producing paradoxes, some of wich have puzzled people ever since.
Among these is the story of Achilles and the tortoise. Achilles and the tortoise decide to have a race. Because Achilles can run twice as fast as the tortoise he gives her a long start. Now, says Zeno, by the time Achilles reaches the tortoise’s starting point she will have moved ahead by half the distance of her lead. And by the time Achilles reaches that point she will have moved on by half of that distance. And so on, and so forth, ad infinitum.
Achilles is never able to catch up with the tortoise, because, at each point, by the time he has covered the distance between them she will always have moved on further by half of that distance. So Achilles never overtakes the tortoise.
“Hang on!,” you may cry: “But Achilles does overtake the tortoise. Of course he does. This is all nonsense.” If you say that you will be missing the point – and it is important to be clear what the point of the story is. It is not to convince you that Achilles never actually overtakes the tortoise. He does, and you know perfectly well that he does, and so does Zeno. The point is that here is an impeccably logical argument that leads to a false conclusion. And what are we to say about that?
If that is possible for us to start from unobjectionable premises, and then proceed by logical steps, each of wich is without fault, to a conclusion wich is manifestly untrue, this threatens with chaos all our attempts to reason about the world around us. People have found it terribly disconcerting. There must be a fault in the logic, they have said. But no one has yet been wholly successful in demonstrating what it is.
For this reason, one of the well-known philosophers of the 20th century, Gilbert Ryle, has written of the parable os Achilles and the tortoise: “In Many ways it deserves to rank as the paradigm of a philosophical puzzle.” Perhaps one day it will be solved, as someone has recently solved the problem of Fermat’s Last Theorem.
Bryan Magee, in The Story of Philosophy