Mathematics, foundations of

Conceived of philosophically, the foundations of mathematics concern various metaphysical and epistemological problems raised by mathematical practice, its results and applications. Most of these problems are of ancient vintage; two, in particular, have been of perennial concern. These are its richness of content and its necessity. Important too, though not so prominent in the history of the subject, is the problem of application, or how to account for the fact that mathematics has given rise to such an extensive, important and varied body of applications in other disciplines.

The Greeks struggled with these questions. So, too, did various medieval and modern thinkers. The ideas of many of these continue to influence foundational thinking to the present day.

During the nineteenth and twentieth centuries, however, the most influential ideas have been those of Kant. In one way or another and to a greater or lesser extent, the main currents of foundational thinking during this period – the most active and fertile period in the entire history of the subject – are nearly all attempts to reconcile Kant’s foundational ideas with various later developments in mathematics and logic.

These developments include, chiefly, the nineteenth-century discovery of non-Euclidean geometries, the vigorous development of mathematical logic, the development of rigorous axiomatizations of geometry, the arithmetization of analysis and the discovery (by Dedekind and Peano) of an axiomatization of arithmetic. The first is perhaps the most important. It led to widespread acceptance of the idea that space was not merely a Kantian ‘form’ of intuition, but had an independence from our intellect that made it different in kind from arithmetic. This asymmetry between geometry and arithmetic became a major premise of more than one of the main ‘isms’ of twentieth-century philosophy of mathematics. The intuitionists retained Kant’s conception of arithmetic and took the same view of that part of geometry which could be reduced to arithmetic. The logicists maintained arithmetic to be ‘analytic’ but differed over their view of geometry. Hilbert’s formalist view endorsed a greater part of Kant’s conception.

The second development carried logic to a point well beyond where it had been in Kant’s day and suggested that his views on the nature of mathematics were in part due to the relatively impoverished state of his logic. The third indicated that geometry could be completely formalized and that intuition was therefore not needed for the sake of conducting inferences within proofs. The fourth and fifth, finally, provided for the codification of a large part of classical mathematics – namely analysis and its neighbours – within a single axiomatic system – namely (second-order) arithmetic. This confirmed the views of those (for example, the intuitionists and the logicists) who believed that arithmetic had a special centrality within human thinking. It also provided a clear reductive target for such later anti-Kantian enterprises as Russell’s logicism.

The major movements in the philosophy of mathematics during this period all drew strength from post-Kantian developments in mathematics and logic. Each, however, also encountered serious difficulties soon after gaining initial momentum. Frege’s logicism was defeated by Russell’s paradox; Russell’s logicism, in turn, made use of such questionable (from a logicist standpoint) items as the axioms of infinity and reduction. Both logicism and Hilbert’s formalist programme came under heavy attack from Gödel’s incompleteness theorems. And finally, intuitionism suffered from its inability to produce a body of mathematics comparable in richness to classical mathematics.

Despite the failure of these non-Kantian programmes, however, movement away from Kant continued in the mid- and late twentieth century. From the 1930s on this has been driven mainly by a revival of empiricist and naturalist ideas in philosophy, prominent in the writings of both the logical empiricists and the later influential work of Quine, Putnam and Benacerraf. This continues as perhaps the major force shaping work in the philosophy of mathematics.