Zeno’s paradoxes

In the 5th century BCE, Zeno of Elea devised a number of paradoxes designed to show that reality is single (there is only one thing) and motionless, as his friend Parmenides had claimed. The paradoxes take the form of arguments in which the assumption of plurality (the existence of more than one thing) or motion are shown to lead to contradictions or absurdity. Here are two of the arguments: Against plurality: (A) Suppose that reality is plural. Then the number of things there are is only as many as the number of things there are (the number of things there are is neither more nor less than the number of things there are). If the number of things there are is only as many as the number of things there are, then the number of things there are is finite. (B) Suppose that reality is plural. Then there are at least two distinct things. Two things can be distinct only if there is a third thing between them (even if it is only air). It follows that there is a third thing that is distinct from the other two. But if the third thing is distinct, then there must be a fourth thing between it and the second (or first) thing. And so on to infinity. (C) Therefore, if reality is plural, it is finite and not finite, infinite and not infinite, a contradiction. Against motion: Suppose that there is motion. Suppose in particular that Achilles and a tortoise are moving around a track in a foot race, in which the tortoise has been given a modest lead. Naturally, Achilles is running faster than the tortoise. If Achilles is at point A and the tortoise at point B, then in order to catch the tortoise Achilles will have to traverse the interval AB. But in the time it takes Achilles to arrive at point B, the tortoise will have moved on (however slowly) to point C. Then in order to catch the tortoise, Achilles will have to traverse the interval BC. But in the time it takes him to arrive at point C, the tortoise will have moved on to point D, and so on for an infinite number of intervals. It follows that Achilles can never catch the tortoise, which is absurd. Zeno’s paradoxes have posed a serious challenge to theories of space, time, and infinity for more than 2,400 years, and for many of them there is still no general agreement about how they should be solved.