Infinity

1. Early Greek thought

The Greek word peras is usually translated as ‘limit’ or ‘bound’. To apeiron denotes that which has no peras, the unlimited or unbounded: the infinite. To apeiron made its first significant appearance in early Greek thought with Anaximander of Miletus in the sixth century bc (see Anaximander §2). He thought of it as the boundless, imperishable, ultimate source of everything that is. He also thought of it as that to which all things must eventually return in order to atone for the injustices and disharmony which result from their transitory existence.

Anaximander was something of an exception, however. On the whole, the Greeks abhorred the infinite (as the old adage has it). More typical of that era were the Pythagoreans, a religious society founded by Pythagoras (see Pythagoreanism §2). They believed in two basic cosmological principles, Peras and Apeiron, the former subsuming all that was good, the latter all that was bad. They held further that the whole of creation was to be understood in terms of, and indeed was ultimately constituted by, the positive integers 1, 2, 3,…; and that this was made possible by the fact that Peras was continuously subjugating Apeiron (the integers themselves, of course, are each finite). The Pythagoreans were followed to some extent in these beliefs by Plato, who also held that it was the imposition of limits on the unlimited that accounted for all the numerically definable phenomena that surround us.

However, the Pythagoreans soon learned to their dismay that they could not simply relegate the infinite to the role of cosmic villain. This was because of Pythagoras’ own discovery that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. Given this theorem, the ratio of a square’s diagonal to each side is $\sqrt{2}:1$ . There are some good approximations to this ratio: it is a little more than 7:5, for example, and a little less than 17:12. Indeed there are approximations of any desired degree of accuracy. Nevertheless, given the basic tenets of Pythagoreanism, it ought to be exactly p:q, for some pair of positive integers p and q. The problem was that they discovered a proof that it is not, which they regarded as nothing short of catastrophic. According to legend, one of them was shipwrecked at sea for revealing the discovery to their enemies. The Pythagoreans had stumbled across the ‘irrational’ within mathematics. They had seen the limitations of the positive integers, and had thereby been forced to acknowledge the infinite in their very midst.

At around the same time, Zeno of Elea was formulating various celebrated paradoxes connected with the infinite (see Zeno of Elea §6). Best known of these is the paradox of Achilles and the tortoise: Achilles, who runs much faster than the tortoise, cannot overtake it in a race if he lets it start a certain distance ahead of him. For in order to do so he must first reach the point from which the tortoise started, by which time the tortoise will have advanced a fraction of the distance initially separating them; he must then make up this new distance, by which time the tortoise will have advanced again; and so on ad infinitum. Such paradoxes, as well as having a profound impact on the history of thought about infinity, did much to reinforce early Greek hostility to the concept.