Infinity

2. Aristotle

Aristotle’s understanding of the infinite was an essentially modern one in so far as he defined it as the untraversable or never-ending. But he perceived a basic dilemma. On the one hand Zeno’s paradoxes, along with a host of other considerations, show that the concept of the infinite really does resist a certain kind of application to reality. On the other hand there seems to be no prospect of doing without the concept, as the Pythagoreans had effectively realized. As well as $\sqrt{2}$ , time seems to be infinite, numbers seem to go on ad infinitum, and space, time and matter all seem to be infinitely divisible.

Aristotle’s solution to this dilemma was masterly. It has dominated all subsequent thought on the infinite, and until very recently was adopted by almost everyone who considered the topic. Aristotle distinguished between the ‘actual infinite’ and the ‘potential infinite’. The actual infinite is that whose infinitude exists, or is given, at some point in time. The potential infinite is that whose infinitude exists, or is given, over time. All objections to the infinite, Aristotle insisted, are objections to the actual infinite. The potential infinite is a fundamental feature of reality. It is there to be acknowledged in any process which can never end: in the process of counting, for example, in various processes of division, or in the passage of time itself. The reason why paradoxes such as Zeno’s arise is that we pay insufficient heed to this distinction. Having seen, for example, that there can be no end to the process of dividing a given racecourse, we somehow imagine that all those possible future divisions are already in effect there. We come to view the racecourse as already divided into infinitely many parts, and it is easy then for the paradoxes to take hold.

Even those later thinkers who did not share Aristotle’s animosity towards the actual infinite tended to recognize the importance of his distinction. Often, though, Aristotle’s reference to time was taken as a metaphor for something deeper and more abstract. This in turn usually proved to be something grammatical. Thus certain medieval thinkers distinguished between categorematic and syncategorematic uses of the word ‘infinite’. Putting it very roughly, to use the word categorematically is to say that there is something with a property that surpasses any finite measure; to use the word syncategorematically is to say that, given any finite measure, there is something with a property that surpasses it. In the former case the infinite has to be instantiated ‘all at once’. In the latter case it does not.

The categorematic–syncategorematic distinction heralds another distinction, whose importance to the infinite is hard to exaggerate. This is the distinction between saying that there is something of kind X to which each thing of kind Y stands in relation R, and saying that each thing of kind Y stands in relation R to something of kind X (not necessarily the same thing each time). This is referred to below as the ‘Scope Distinction’ (see Scope).

But Aristotle himself was not thinking in these very abstract terms. He took the references to time in his own account of the actual–potential distinction quite literally, and this gave rise to his most serious difficulty. He held that time (unlike space) is infinite. He also held that time involves constant activity, as exemplified in the revolution of the heavens. When our attention is focused on the future, there is no obvious problem with this. Past revolutions, however, because they are past, seem to have an infinitude which is by now completely given to us, and hence which is actual. This difficulty, in various different guises, has been a continual aggravation for philosophers who have wanted to see the infinite in broadly Aristotelian terms.