ACHILLES AND THE TURTLE

So, here, as promised, is the famous paradox of Zeno Eleaticus, known as Achilles and the Turtle. First we have the testimony of Aristotle, in Physics (3):

---The Achilles: “In a race, the quickest runner can never overtake the slowest, since the pursuer has first to reach the point whence the pursued started, so that the slower must always hold a lead.”

Here is how this paradox is explained to young readers in schoolchildren’s textbooks:

“Suppose, Achilles is in a race with a tortoise. Achilles runs 10 times faster than the tortoise, but starts at point A, 100 yards behind the tortoise, at point T1. To overtake the tortoise, Achilles must first reach the point T1. However, when Achilles arrives at T1, the tortoise has moved 10 yards in front at point T2. Again Achilles runs to T2. But as before, as soon as he has covered the 10 yards, the tortoise is now a yard ahead of him, at point T3, and so on. Therefore, Achilles can never overtake the tortoise.”

In his Philosophy in the Tragic Age of the Greeks, Nietzsche talks about this infinitely more philosophically and elegantly as well:

Nothing infinite can exist, for to assume it, would yield the contradictory concept of a perfect infinity. Now since our reality, our world, everywhere bears the stamp of just such perfect infinity, the word signifies in its very nature a contradiction to logic and hence to the real and is therefore an illusion, a lie, a phantasm. Zeno especially makes use of indirect proof. He says, for example, “There can be no movement from one place to another, for if there were, we would have had a perfect infinity, but this is an impossibility. Achilles cannot catch up with the tortoise which has a small start over him, for in order to reach even the starting point of the tortoise, Achilles must have traversed innumerable, infinitely many spaces--- first, half of the interval, then, a fourth of it, an eighth, a sixteenth, and so on, ad infinitum. If he, in reality, does catch up with the tortoise, this is an illogical phenomenon, and not a real one. This is not true Being, but merely an illusion. For it is never possible to finish the infinite.”

As far as I am concerned, I remember thinking about the Achilles paradox at one time myself, and coming up with the following answer: In order to solve this puzzle we must divide the race track into a number of stages, equal or non-equal making no difference. As long as the number of these stages is finite, it is easy to see how Achilles will overtake the turtle on the way (unless the race handicap is so skewed that there is no physical chance for Achilles to run the whole course in less time than it takes the turtle to finish a few inches allotted to it by the organizers of the contest). If however the number of stages is infinite (assuming that each stage is measurable above zero, as otherwise, the division into stages should make no sense at all), we are embarking on an infinite journey, which cannot be finished in any time-limited, or even time-unlimited fashion. Please notice that in my argument I substitute Zeno’s infinitely small spaces by my own infinitely long stretch of a journey, without a bit altering the conditions set by Zeno himself. Indeed, when we deal with infinity, infinitely large and infinitely small are basically indistinguishable by the very nature of the concept.