Mathematics, foundations of

1. Kant’s views; reactions

The ‘Problematik’ that Kant established for the epistemology of pure mathematics focused on the reconciliation of two apparently incompatible features of pure mathematics: (1) the problem of necessity, or how to explain the apparent fact that mathematical statements (for example, statements such as that 1+1=2 or that the sum of the interior angles of a Euclidean triangle is equal to two right angles) should appear to be not only true but necessarily true and independent of empirical evidence; and (2) the problem of cognitive richness, or how to account for the fact that pure mathematics should yield subjects as rich and deep in content and method, as robust in growth and as replete with surprising discoveries as the history of mathematics demonstrates.

In mathematics, Kant said, we find a ‘great and established branch of knowledge’ – a cognitive domain so ‘wonderfully large’ and with promise of such ‘unlimited future extension’ that it would appear to arise from sources other than those of pure unaided (human) reason (1783: §§6, 7). At the same time, it carries with it a certainty or necessity that is typical of judgments of pure reason. The problem, then, is to explain these apparently conflicting characteristics. Kant’s explanation was that mathematical knowledge arises from certain standing conditions or ‘forms’ which shape our experience of space and time – forms which, though they are part of the innate cognitive apparatus that we bring to experience, none the less shape our experience in a way that goes beyond mere logical processing.

To elaborate this hypothesis, Kant sorted judgments/propositions in two different ways: first, according to whether they required appeal to sensory experience for their justification; and, second, according to whether their predicate concepts were ‘contained in’ their subject concepts. A judgment or proposition was ‘a priori’ if it could be justified without appeal to sensory content. If not, it was ‘a posteriori’. It was ‘analytic’ if the very act of thinking the subject concept contained, as a constituent part, the thinking of the predicate concept. If not, it was either false or ‘synthetic’. In synthetic a priori judgment – the type of judgment Kant regarded as characteristic of mathematics – the predicate concept was thought not through the mere thinking of the subject concept, but through its ‘construction in intuition’. He took a similar view of mathematical inference, believing it to involve an intuition that goes beyond the mere logical connection of premises and conclusions (1781/1787: A713–19/B741–7).

Kant erected his mathematical epistemology upon these distinctions and, famously, maintained that mathematical judgment and inference is synthetic a priori in character. In this way, he intended to account for both the necessity and cognitive richness of mathematics, its necessity reflecting its a priority, its cognitive richness its syntheticity.

Kant applied this basic outlook to both arithmetic and geometry (and also to pure mechanics). He did not regard them as entirely identical, however, since he saw them as resting on different a priori intuitions. Neither did he see them as possessing precisely the same universality (1781/1787: A163–5, 170–1, 717, 734/B204–6, 212, 745, 762). None the less, he regarded their similarities as more important than their differences and therefore took them to be of essentially the same epistemic type – namely, synthetic a priori. In the end, it was this inclusion of geometry and arithmetic within the same basic epistemic type rather than his more central claim concerning the existence of synthetic a priori knowledge that gave rise to the sternest challenges to his views.

In the decades following the publication of the first Critique (1781/1787), the principal source of concern regarding its views was the growing evidence for and eventual discovery of non-Euclidean geometries. This led many to question whether geometry and arithmetic are of the same basic epistemic character.

The serious possibility of non-Euclidean geometries went back to the work of Lambert and others in the eighteenth century. Building on this work, some – in particular, Gauss (1817, 1829) – stated their opposition to Kant’s views even before the actual discovery of non-Euclidean geometries by Bolyai and Lobachevskii in the 1820s. Gauss’ reasoning was essentially this: number seems to be purely a product of the intellect and, so, something of which we can have purely a priori knowledge. Space, on the other hand, seems to have a reality external to our minds that prohibits a purely a priori knowledge of it. Arithmetic and geometry are therefore not on an epistemological par with one another.

This reasoning became a potent force shaping nineteenth- and twentieth-century foundational thinking. Another such force was the dramatic development of logic and the axiomatic method in the mid- to late nineteenth century and early twentieth century. This included the introduction of algebraic methods by Boole and De Morgan, the improved treatment of relations by Peirce, Schröder and Peano, the replacement of the subject–predicate conception of propositional form with Frege’s more fecund functional conception, and the advances in axiomatization and formalization brought about by the work of Frege, Pasch, Peano, Hilbert and (especially) Whitehead and Russell.

Certain developments in mathematics proper also exerted an influence. Chief among these were the arithmetization of analysis by Weierstrass, Dedekind and others, and the axiomatization of arithmetic by Peano and Dedekind. Of somewhat lesser importance, though still significant for their effects on Hilbert’s thinking, were Einstein’s relativistic ideas in physics.