Zeno’s paradoxes

Zeno of Elea (c. 490–430 BCE) was Parmenides’ pupil and, in Plato’s Parmenides (128c–d), his defender. His method is reductio ad absurdum: assume your opponent’s view, then derive a contradiction from it, which refutes the view.

The Dichotomy. Before a runner can cross a track, she must first reach the halfway point. But before that, the quarter point; before that, the eighth — and so on without end. To move at all she must complete infinitely many sub-journeys. Since (Zeno assumes) you cannot finish an infinite to-do list, she can never even start. Motion is impossible.

Achilles and the tortoise. Swift Achilles races the tortoise, which gets a head start. To catch it, Achilles must first reach the spot where the tortoise began. But by then the tortoise has crawled a little farther. Achilles reaches that new spot — the tortoise has moved again. At every stage he must first arrive where it just was, and it is always a bit ahead. So the swiftest runner never overtakes the slowest.

The Arrow. Consider a flying arrow at a single instant. In that frozen instant it occupies a space exactly equal to itself — it is not moving to anywhere or from anywhere; at an instant, it is indistinguishable from an arrow at rest. But the arrow’s whole flight is just a sum of such instants. If it is at rest at every instant, when does it ever move? Motion, Zeno concludes, is a sum of rests — which is no motion at all.

Zeno also aimed paradoxes of plurality at the “many.” If there are many things, each must have some size. If that size is divisible without limit, each thing turns out to contain infinitely many parts — so it must be infinitely large; yet parts with no size add up to nothing — so it must have no size at all. A plurality, he argues, would have to be both infinitely big and vanishingly small. Contradiction — so there is no true plurality.