Zeno of Elea

Zeno was Parmenides' pupil and adopted son and visited Athens with his teacher; Plato's Parmenides describes their meeting with the young Socrates. Legend says he later joined a conspiracy against the tyrant of Elea and under torture bit off his tongue and spat it in the tyrant's face.

He wrote one book, aimed at those who mocked Parmenides. Its method was reduction to absurdity: assume plurality and motion, then derive conclusions more absurd than 'Being is one'. The four paradoxes of motion are the most famous. The Dichotomy: to cross a distance you must first cross half, then half of the remainder, and so on without end, so motion cannot begin. Achilles: the fastest runner cannot overtake a tortoise, because whenever he reaches where it was, it has moved a little further. The Arrow: at every instant the arrow occupies a space equal to itself and is therefore at rest at every instant.

The paradoxes touch the nature of infinity, continuity and the instant. Aristotle, Leibniz, Kant and Russell all took them seriously, and the mathematical resolution waited for the nineteenth-century theory of limits. Zeno left no school, but the form of argument he left became a basic tool of philosophy.