Zeno's paradoxes of motion

Zeno of Elea, a pupil of Parmenides, devised a set of arguments around 450 BCE that seem to prove movement impossible. We know them mainly from Aristotle's Physics. In the Dichotomy, a runner crossing a track must first reach the halfway mark, then half the remaining distance, and so on without end, so he must complete infinitely many tasks. In Achilles and the Tortoise, the swift Achilles gives a tortoise a head start, but whenever he reaches where the tortoise was, it has moved a little further, so he never catches it. In the Arrow, a flying arrow at each instant occupies a space equal to itself, so at each instant it is at rest, and a sum of rests is no motion.¹²

Zeno was not denying that people see things move. He was defending his teacher's thesis, in Parmenides' poem, that reality is one and unchanging, by showing that the opposing belief in many moving things produces contradictions. The method of assuming the opponent's claim and deriving an absurdity made him, in the ancient tradition, an inventor of dialectic. Aristotle replied that distance is only potentially divisible without end, so it does not contain infinitely many completed tasks. Mathematicians later answered with infinite series: the steps one half, one quarter, one eighth add up to exactly one. The rigorous theory of limits that makes this precise came only in the nineteenth century, with Cauchy and Weierstrass.¹²

Philosophers still disagree about whether the arrow and the idea of completing infinitely many tasks are fully resolved. Zeno's puzzles bear on the nature of space, time and infinity, and they were taken seriously by Bertrand Russell in his work on the foundations of mathematics. They remain the standard first example of how a plain-looking argument can expose a deep problem.²