The Beginning of Infinity: Explanations That Transform the World

by David Deutsch

Only the laws of physics determine what is finite in nature. Failure to realize this has often caused confusion. The paradoxes of Zeno of Elea, such as that of Achilles and the tortoise, were early examples. Zeno managed to conclude that, in a race against a tortoise, Achilles will never overtake the tortoise if it has a head start – because, by the time Achilles reaches the point where the tortoise began, the tortoise will have moved on a little. By the time he reaches that new point, it will have moved a little further, and so on ad infinitum. Thus the ‘catching-up’ procedure requires Achilles to perform an infinite number of catching-up steps in a finite time, which as a finite being he presumably cannot do. Do you see what Zeno did there? He just presumed that the mathematical notion that happens to be called ‘infinity’ faithfully captures the distinction between finite and infinite that is relevant to that physical situation. That is simply false. If he is complaining that the mathematical notion of infinity does not make sense, then we can refer him to Cantor, who showed that it does. If he is complaining that the physical event of Achilles overtaking the tortoise does not make sense, then he is claiming that the laws of physics are inconsistent – but they are not. But if he is complaining that there is something inconsistent about motion because one could not experience each point along a continuous path, then he is simply confusing two different things that both ha...