Zeno’s paradoxes four paradoxes relating to space and motion attributed to Zeno of Elea (fifth century B.C.): the racetrack, Achilles and the tortoise, the stadium, and the arrow. Zeno’s work is known to us through secondary sources, in particular Aristotle. The racetrack paradox. If a runner is to reach the end of the track, he must first complete an infinite number of different journeys: getting to the midpoint, then to the point midway between the midpoint and the end, then to the point midway between this one and the end, and so on. But it is logically impossible for someone to complete an infinite series of journeys. Therefore the runner cannot reach the end of the track. Since it is irrelevant to the argument how far the end of the track is – it could be a foot or an inch or a micron away – this argument, if sound, shows that all motion is impossible. Moving to any point will involve an infinite number of journeys, and an infinite number of journeys cannot be completed. The paradox of Achilles and the tortoise. Achilles can run much faster than the tortoise, so when a race is arranged between them the tortoise is given a lead. Zeno argued that Achilles can never catch up with the tortoise no matter how fast he runs and no matter how long the race goes on. For the first thing Achilles has to do is to get to the place from which the tortoise started. But the tortoise, though slow, is unflagging: while Achilles was occupied in making up his handicap, the tortoise has advanced a little farther. So the next thing Achilles has to do is to get to the new place the tortoise occupies. While he is doing this, the tortoise will have gone a little farther still. However small the gap that remains, it will take Achilles some time to cross it, and in that time the tortoise will have created another gap. So however fast Achilles runs, all that the tortoise has to do, in order not to be beaten, is not to stop.
The stadium paradox. Imagine three equal cubes, A, B, and C, with sides all of length l, arranged in a line stretching away from one. A is moved perpendicularly out of line to the right by a distance equal to l. At the same time, and at the same rate, C is moved perpendicularly out of line to the left by a distance equal to l. The time it takes A to travel l/2 (relative to B) equals the time it takes A to travel to l (relative to C). So, in Aristotle’s words, ‘it follows, he [Zeno] thinks, that half the time equals its double’ (Physics 259b35).
The arrow paradox. At any instant of time, the flying arrow ‘occupies a space equal to itself.’ That is, the arrow at an instant cannot be moving, for motion takes a period of time, and a temporal instant is conceived as a point, not itself having duration. It follows that the arrow is at rest at every instant, and so does not move. What goes for arrows goes for everything: nothing moves. Scholars disagree about what Zeno himself took his paradoxes to show. There is no evidence that he offered any ‘solutions’ to them. One view is that they were part of a program to establish that multiplicity is an illusion, and that reality is a seamless whole. The argument could be reconstructed like this: if you allow that reality can be successively divided into parts, you find yourself with these insupportable paradoxes; so you must think of reality as a single indivisible One.
See also PARADOX, PRE-SOCRATICS, TIME.
R.M.S.