Infinity

4. Kant

Kant played his characteristic role of conciliator in the debate on the infinite (see Kant §§2, 5, 8). He had an empiricist scepticism about the infinite, based on the fact that we cannot directly experience it. Nevertheless, he sided with the rationalists by insisting that there are certain formal or structural features of what we experience, which are accessible a priori and which do involve the infinite. Thus he thought that space and time were infinite (both in the sense of being infinitely extended and in the sense of being infinitely divisible): it is written into the form of whatever we experience that there can also be experience of how things are further out, further in, earlier or later. These, on Kant’s view, were mathematical truths, a priori and unassailable.

But there is a question about how the topology of space and time can be a priori. Kant’s celebrated reply was that space and time are not features of ‘things in themselves’; they are part of an a priori framework which we contribute to our experience of things. What then of the contents of space and time, the physical universe as a whole? This was different. Kant did not think that what was physical was constructed a priori. Nor, on the other hand, did he think that it was ultimately real, that is to say real in a way that transcends any possible access we have to it. It had no features, on Kant’s view, that exceed what we are capable of grasping through experience. So here the concept of the infinite did resist application. It still had what Kant regarded as a legitimate regulative use. That is, we could proceed as if the physical universe as a whole were infinite, thereby encouraging ourselves never to give up in our explorations. But we ultimately had no way of making sense of such infinitude. Kant was forced to take an extreme empiricist line by denying that the physical universe as a whole is infinitely big, that it has infinitely many parts and, going this time beyond Aristotle (thus bypassing the difficulty that had beset Aristotle himself), that it is infinitely old.

However, there was a dilemma. Kant was also forced to deny that the physical universe is finite in each of these three respects. Apart from anything else, to postulate infinite, empty space or time beyond the confines of the physical universe is itself to postulate that which exceeds what we are capable of grasping through experience.

This dilemma looks acute. Kant himself presented it in the form of a pair of ‘antinomies’. These antinomies consisted of the principal arguments against the physical universe’s being infinite in each of the specified respects, and the principal arguments against its being finite. But he believed that the dilemma contained the seeds of its own solution. If what is physical is not ultimately real – if there is no more to it than what we are capable of experiencing – then we are at liberty to deny that there is any such thing as the physical universe as a whole. There are only the finite physical things that are accessible to us through experience. The physical universe as a whole is neither infinite nor finite. It does not exist.

Kant’s solution involved him in a direct application of the Scope Distinction (see §2 above). On the one hand he affirmed that any finite physical thing is contained within something physical. On the other hand he denied that there is something physical within which any finite physical thing is contained. Both of these, the affirmation and the denial, were grounded in the fact that there is nothing we can identify in space and time such that we cannot identify more. This is fundamentally a fact about us: the fact that we are finite. Our identifications are always incomplete. What Kant added, in an idealist vein, was that what we cannot identify does not exist. Here, as in so many other places, we see how deeply involved with human finitude Kant’s philosophy was, and how seriously he took it.