Mathematics, foundations of

2. Intuitionism

A variety of views concerning the asymmetry of geometry and arithmetic emerged in the late nineteenth and early twentieth centuries. That of the early intuitionists Brouwer and Weyl retained Kant’s synthetic a priori conception of arithmetic.

They responded to the discovery of non-Euclidean geometries, however, by denying the a priori status of that part of geometry that could not be reduced to arithmetic by such means as Descartes’ calculus of coordinates. They retained, none the less, a type of a priori intuition of time as the basis for arithmetical knowledge (see Brouwer 1913: 127–8). They also emphasized the synthetic character of arithmetical judgment and inference, and sharply distinguished them from logical judgment and inference.

Brouwer described his intuition of time as consciousness of change per se – the human subject’s primordial inner awareness of the ‘falling apart’ of a life-moment into a part that is passing away and a part that is becoming. He believed that, via a process of abstraction, one could pass from this basal intuition of time to a concept of ‘bare two-oneness’, and from this concept to, first, the finite ordinals, then the transfinite ordinals and, finally, the linear continuum. In this way, parts of classical arithmetic, analysis and set theory could be recaptured intuitionistically. (See Brouwer 1907: 61, 97; 1913: 127, 131–2.)

Brouwer thus modified Kant’s intuitional basis for mathematics. He also modified his conception of knowledge of existence. Kant believed that humans could obtain knowledge of existence only through sensible intuition. Only this, he believed, had the type of involuntariness and objectivity that assures us that belief in an object is not a mere compulsion or idiosyncrasy of our subjective selves. Like the post-Kantian romantic idealists, however, Brouwer (and Weyl, too) believed as well in knowledge of existence via a kind of ‘intellectual intuition’ – an intuition carried by a purely internal type of mental construction (1907: 96–7).

The early intuitionists (especially Brouwer and Poincaré) remained Kantian in their conception of mathematical reasoning and took it to be essentially different in character from ‘discursive’ or logical reasoning. Brouwer believed logical reasoning to mark not patterns in mathematical thinking itself but only patterns in its linguistic representation. It was therefore not indicative of the inferential structure of mathematical thinking itself and had no place within genuine mathematical reasoning per se. This was essentially the idea expressed in Brouwer’s so-called ‘First Act of Intuitionism’ (1905: 2, 1981: 4–5).

Thus the early intuitionists (especially Brouwer and Weyl and, to some extent, Poincaré) discarded Kant’s view of geometry, revised his conception of arithmetic and existence claims, and preserved his basic stance on the nature of mathematical reasoning and its relationship to logical reasoning. Later intuitionists (for example, Heyting and Dummett) did not keep to this plan. They rejected Brouwer’s view of the divide between logical and mathematical reasoning and made a significant place for logic in their accounts of mathematical reasoning. Some of them (Dummett and his ‘anti-realist’ followers) even went so far as to make the question ‘What is the logic of mathematical reasoning?’ central to their philosophy of mathematics (see §5 below).