Achilles and the tortoise

One of the paradoxes of the philosopher Zeno argues that Achilles can never catch up to a tortoise if the latter is given a head start.

Written by Josho Brouwers on 23 August 2017

Zeno of Elea was a Greek philospher who lived in the fifth century (perhaps ca. 490–ca. 430 BC). None of his works survive, but among later authors he was renowned for his so-called “paradoxes”. Perhaps the most famous of these is one of his paradoxes of motion, namely that of Achilles and the tortoise.

This is how Aristotle summarizes that particular paradox (Physics VI:9, 239b15), as per the Internet Classics Archive:

The second is the so-called “Achilles”, and it amounts to this: that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

Achilles is picked as an example as he was the fastest of all the warriors assembled before Troy. Zeno’s argument is that at the point where Achilles reaches where the tortoise was, the tortoise will have moved on. Achilles must then reach the new point where the tortoise was before, but once he’s arrived there, the tortoise will again have moved further, and so on. Essentially, Zeno’s argument is that, logically speaking, not even the fastest runner could overtake another once the latter had a headstart.

Naturally, we know this isn’t correct: a tortoise moves at a much slower pace than a human, and regardless of the headst