Gödel, Realism and Mathematical ‘Intuition’

Abstract

Godel is perhaps the most notable modern mathematician whose writings on the philosophy of mathematics appeal both to a notion of mathematical intuition and also to the Critical Philosophy of Kant. (The others are Hilbert and Poincare, whose views will not be treated here.) The aim of the present programmatic remarks is to approach answers to two questions: the first is to establish more clearly what role, for Godel, ‘intuition’ is meant to play; the second is to establish, and perhaps delimit, the connection to Kant’s philosophy, about which Godel is at best enigmatic. Part of the point of the present paper is to make clear what Godel’s notion of intuition is not. In particular, it is important to say at the outset that, despite his use of the term ‘mathematical intuition’, what Godel is appealing to is not Kantian sensible intuition or an extension of it; neither is Godel (not to speak of this essay) attempting to provide a ‘reading’ of Kant’s Critical Philosophy. In particular, the important and interesting question of how precisely what Godel says fits in with Kant’s theory of mathematics or knowledge in general will not be addressed. Various remarks Godel makes (especially in his Dialectica paper of 1958 and its revision of 1972) show clearly that he regards Kantian sensible intuition (for example, in the way it was employed by Hilbert) as much too restrictive to be of service in attempting to understand the full scope of knowledge of modern mathematics. Furthermore, Godel can hardly be described as a transcendental idealist in Kant’s sense, but is rather a straightforward realist about mathemat-