Online Problem #2

Achilles-and the Tortoise

Heraclitus claimed that the world is always changing, full of fire and flux. Parmenides, on the other hand, held that change is only an illusion, and that all that is really real is the "Changeless One." Parmenides' student, Zeno, defending his teacher's point of view, offered the following argument, called "The Race Course" (see Fig. 1):

Figure 1

To get from point A to point B, the runner must first get to the midpoint, C; but to get to C he must first get to midpoint D but to get to D he must first get to midpoint E; and so on, ad infinitum. So, to get to B the runner must first complete an infinite series of tasks. But it is impossible to complete an infinite series of tasks (that's what infinite means--there's always one more). And therefore it is impossible for the runner to get from A to B. And, since all change is a kind of motion from one place or state to another, change is impossible!

Therefore the world around us, which seems to be so full of motion and change, is actually a great illusion. And if you resolve the above paradox, Zeno had three more, the most famous of which is called "Achilles and the Tortoise." Achilles, renouned among ancient Greeks for his swiftness, agrees to race a lowly tortoise. He agrees to give the tortoise an hour's head start (see Fig. 2):

Figure 2

Now suppose Achilles runs just twice as fast as the tortoise. The tortoise goes one mile per hour, let's say, and so Achilles runs two miles per hour. The tortoise runs for one hour, then, and gets to B, one mile ahead of Achilles. Then Achilles starts. He runs that mile, to B, in 1/2 hour. But in that 1/2 hour, the tortoise has run 1/2 mile ahead, to C. So Achilles runs to C, which takes him 1/4 hour. But in that 1/4 hour, the tortoise has run 1/4 mile further, to D. And so on, ad infinitum.

So, no matter how close Achilles gets to the tortoise, he can never overtake him, because he always has just a bit left to go. In order to overtake the tortoise, Achilles would have to complete an infinite series of tasks, which, as we have seen, is impossible. Therefore--although we seem to see such things happening all the time--a faster runner can never overtake a slower one. Which proves again that the visible world, which appears to be full of motion and change, is really deceptive and illusory.

What's wrong with Zeno's arguments? Or is change only an illusion? There appear to be many things in the world which are changing all the time: but perhaps there is only the Changeless One ...?


Cornford, F.M., Before and After Socrates, Cambridge University Press, New York, 1960.

Kirk, G.S. and J.E. Raven: The Presocratic Philosophers, Cambridge University Press, New York, 1960.