In the Greek colony of Elea (Velia) in southern Italy, Parmenides was born around 515 BCE; Zeno was his student and, in tradition, perhaps his favourite. The only work of Parmenides we have is fragments of a long poem, *On Nature*, in which a goddess sets a mortal before two roads: 'it is and cannot not-be' and 'it is not and must not-be.'

Only the first road is the road of reason. *What is not* cannot be thought; what cannot be thought cannot be spoken; what cannot be spoken does not exist. From this short argument Parmenides draws enormous consequences: true Being is one, indivisible, ungenerated and imperishable, motionless and unchanging. The plurality, birth, motion and death we seem to perceive are mere deceptions of the senses. This is the first great line drawn in Western philosophy between language and logic on one side and the senses on the other.

Zeno defended this paradoxical doctrine with paradoxes of his own. Achilles can never overtake the tortoise, because by the time he reaches the tortoise's previous point it has moved a little farther on. The flying arrow at every instant occupies a place equal to itself — and therefore at every instant is at rest. A moving body must first cover half the distance, then a quarter, then an eighth — and since the divisions are infinite, motion never even begins. These are not weak arguments but a mathematical assault in support of Parmenides: assume motion and plurality and you will fall into contradiction.

The importance of the Parmenidean line is methodological more than doctrinal. Philosophy has moved from 'what do we say about nature?' to 'what does logic force us to say?'. Plato returns to Parmenides in the dialogue named after him; Aristotle calls Zeno the inventor of dialectic. The motion paradoxes themselves were only resolved more than two thousand years later, through the modern mathematics of limits and infinite series — but the problem they posed lies at the root of every subsequent argument about the infinite.