Infinity

3. The rationalists and the empiricists

For over two thousand years Aristotle’s conception of the infinite was regarded as orthodoxy. Often this conception was motivated by a kind of empiricism: the actual infinite was spurned on the grounds that we can never encounter it in experience. But does the potential infinite fare any better in this respect? Is experience of an infinitude that is given over time any less problematic than experience of an infinitude that is given all at once? The more extreme of the British empiricists were hostile to the infinite in all its guises. Where Aristotle had felt able to accept that space and time were infinitely divisible, Berkeley and Hume denied even that. They thought that the concept of the infinite was one that we could, and should, do without (see Empiricism).

This was partly a backlash against their rationalist predecessors. The rationalists had argued that we could form an idea of the infinite, even though we could neither experience it nor imagine it. They thought that this idea was an innate one, and that it constituted, or helped to constitute, a vital insight into reality. They did not see any difficulty in this view. As Descartes put it, the fact that we cannot grasp the infinite does not preclude our touching it with our thoughts, any more than the fact that we cannot grasp a mountain precludes our touching it (see Rationalism §2).

Descartes believed that our idea of the infinite had been implanted in our minds by God (see Descartes, R. §6). Indeed this was the basis of one of his proofs of God’s existence. Only a truly infinite being, Descartes argued, could have implanted such an idea in our minds. Note here the assimilation of the infinite to the divine: this was a legacy of medieval thought which is nowadays quite commonplace. But when the assimilation was first made, at the end of antiquity – most famously, by the Neo-Platonist Plotinus–it marked something of a turning point in the history of thought about the infinite (see Plotinus §§3, 4). Until then there had been a tendency to hear ‘infinite’ as a derogatory term. Henceforth, it was quite the opposite.

The empiricists, meanwhile, needed to defend their rejection of the infinite against the charge that it invalidated contemporary mathematics. They had more or less sophisticated ways of doing this, though in the case of geometry, where the problem was at its most acute, Hume took the rather cavalier step of simply denying certain crucial principles which mathematicians took for granted. (Berkeley’s chief concern was with the use of infinitesimals in the recently invented calculus. In fact, his reservations were perfectly justified: it was a century before they were properly addressed.)