Taming the Infinite

It has always been a large part of the task of philosophers of mathematics to explain our grasp of the infinite. In the end every credible attempt to do this involves a reduction, an account which shows our understanding of the infinite to be explicable in finite terms. Russell, for example, thought when he wrote The Principles of Mathematics that propositions about infinite classes contain as constituents not the classes themselves but concepts which denote them. 'This,' he said, 'is the inmost secret of our power to deal with infinity. An infinitely complex concept, though there may be such, can certainly not be manipulated by the human intelligence; but infinite collections, owing to the notion of denoting, can be manipulated without introducing any concepts of infinite complexity.' But Russell's appeal to the notion of denoting is plainly question-begging, as he later realized, since it leaves the relation between the denoting concept and its object unexplained. A second, and more promising, route to an account of the infinite in terms of the finite was Hilbert's: treat symbols which appear to designate infinite objects as meaningless and show by a finitistic study of the combinatorial properties of the symbols why we find such meaningless symbols useful. But Hilbert's programme was also flawed, as Godel proved. Shaughan Lavine's book, Understanding the Infinite, as well as containing a wide-ranging historical account of previous attempts to understand the infinite, uses recent work of Jan Mycielski to interpret mathematics within theories of what Lavine calls 'finite mathematics'. This interpretation bears a certain similarity to Hilbert's approach. The extent of the similarity is what I want to discuss in this review. Let us start with some facts from model theory. One version of the Lowenheim-Skolem theorem tells us that every consistent first-order