Why Zeno Argued That Achilles Cannot Win

Zeno of Elea did not write to prove some independent thesis of his own so much as to defend his teacher Parmenides by showing that the ordinary, commonsense alternative, a world of real motion and real plurality, leads to conclusions even harder to accept than Parmenides' claim that true being is single and changeless. His best known argument imagines a race between the swift Achilles and a slow tortoise given a head start, and argues that Achilles can never actually catch the tortoise, because by the time he reaches the tortoise's starting point it has moved a little further on, and by the time he reaches that new point it has moved further still, generating an endless series of gaps that Zeno claimed can never be closed. A companion argument, sometimes called the dichotomy, holds that before an object can travel any distance at all it must first travel half that distance, and before that a quarter, and so on without end, so that motion could never even begin. Ancient and modern readers alike have debated whether these paradoxes reveal a genuine flaw in the concept of infinite divisibility or merely a confusion about how infinite series can sum to a finite result, a question that was not given a fully satisfying mathematical answer until the development of calculus and the formal theory of limits many centuries later. Whatever their ultimate resolution, the paradoxes forced Greek philosophy to take seriously the gap between what reason concludes and what the senses report, which was exactly Parmenides' original point.