Was Zeno unfair toward Achilles in his paradox?
Last week I was reading the Croatian edition of Bryan Magee’s “The Story of Philosophy” and he reminded me of Zeno’s famous “Achilles and the tortoise” paradox.
Here is how the paradox goes (taken from Wikipedia):
“In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters. If we suppose that each racer starts running at some constant speed (here instead of ‘one very fast and one very slow’ I would stick to the original: Achilles is twice time faster than the Tortoise), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, 50 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever...
Dear Robert, You are right. The key to understanding the paradox is that although Achilles must complete an infinite number of tasks in order to catch up to the Tortoise, he can do so in a finite amount of time, since each successive task takes much less time than its predecessor (as you noted). Of course, today we understand how to add an infinite sequence of terms that converge to a finite quantity. But this wasn't well understood until millenia after Zeno -- and the logical foundations for doing so required Cauchy and Weierstrass in the nineteenth century. So we shouldn't be too hard on old Zeno. By the way, you might find it amusing to consider some more recent Zeno-like puzzles, such as the "New Zeno" discussed by Stephen Yablo in the journal ANALYSIS, vol 60 (April 2000).