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.

Josho Brouwers

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 headstart, even the slowest human being will eventually overtake the tortoise. Yet, Zeno’s logic is, on the face of it, sound. This is because Zeno’s argument is based on the assumption that Achilles will always need to catch up the tortoise. Even when the distance that Achilles has to cover is infinitesimal, he still needs to cross that distance, at which point, the tortoise will again have moved ahead, even if just another infinitesimal amount of space.

Of course, the paradox could be easily disproved just by getting up and overtaking a random tortoise wandering around. We overtake slower people or animals all the time. But the problem lay in how to disprove Zeno’s paradox through either logical argument or mathematics. Over the centuries, various philosophers and scientists have attempted to solve it in different ways, including ancient Greek thinkers such as Aristotle and Archimedes, but without much success.

It wasn’t until the late nineteenth century before it could be satisfactorily explained using new developments in mathematics. Most notably, the theory of transfinites, first developed by Georg Cantor (1845–1918), showed that it’s possible to have an infinite number of steps plus one, with the final step being the point at which Achilles is able to overtake the tortoise. Naturally, all those infinitesimal steps required to catch up pass in the blink of an eye, so to speak, and thus Zeno’s paradox can be resolved.