Parmenides had argued that change and plurality are incoherent, which is hard to believe. Zeno of Elea, his student, wrote the counter-attack: if you think motion is obvious, look what follows from it.

The arguments survive mostly in Aristotle's reports, and they are far better than their reputation as schoolboy tricks.

The dichotomy

To cross a room you must first reach the halfway point. To reach that you must reach the quarter point, and before that the eighth, and so on without end.

So before you can make any movement at all, you must complete an infinite sequence of prior movements — and an infinite sequence has no first member. There is no first step to take. Motion cannot begin.

Achilles and the tortoise

Achilles races a tortoise which starts ahead. By the time he reaches where it started, it has moved a little way on. By the time he reaches that point, it has moved again.

Every time he arrives where it was, it is somewhere else. The gap shrinks without limit and there is no moment in this description at which he is level. The fastest runner in Greece cannot pass a tortoise.

The arrow

Consider a flying arrow at any single instant. In that instant it occupies a space exactly its own size and it is not moving — an instant has no duration, and there is no movement in zero time.

But time is composed of instants, and the arrow is motionless in every one of them. A thing that is motionless at every instant of its flight never moves.

What the usual dismissal misses

The standard reply is that the series converges: the distances sum to a finite total, the times sum to a finite total, and calculus handles it.

That is true and it answers less than it appears to. The convergence shows that the sum is finite, which nobody doubted — we can all see Achilles pass the tortoise. What Zeno presses is whether an infinite sequence of tasks can be completed, each one requiring the previous, and a convergent sum does not address the completion. The mathematics tells you where the runner ends up; it does not explain how a sequence with no last member gets finished.

The arrow is worse, because it is not about summing anything. It asks how motion can be composed of instants at none of which there is any motion. The modern answer — that velocity at an instant is defined as a limit, so motion is a relation between instants rather than something occurring within one — is a real answer, and it took the invention of the derivative to state it. It is a reasonable measure of how good the argument is that the reply required that.

What he actually established

Zeno was not trying to persuade anyone that Achilles is slow. The paradoxes are reductio arguments in service of his teacher, and the structure is: you find Parmenides absurd for denying motion, but your own position generates these, so the absurdity is on both sides and you should look again at the premises.

What they really isolate is that the continuum is strange. Any account of space and time as infinitely divisible owes an explanation of how a finite interval contains infinitely many parts, how a sequence with no first member begins, and how motion arises from instants that contain none.

Those questions did not get satisfactory answers until Cauchy and Weierstrass gave a rigorous account of limits in the nineteenth century, and the philosophical residue — supertasks, whether infinite divisibility is physically real, whether spacetime is discrete — is still discussed. Two and a half thousand years is a long run for a trick.