Abstract

The relation between mathematical intuition and formal representation of mathematical knowledge is considered in the framework of D. Hilbert’s program in the foundations of mathematics. The notion of sign is connected with both categories: on the one hand, it is an object of intuitive comprehension, thereby represented immediately to mind, on the other hand – the part of formal structure. A double nature of sign plays the special place in D. Hilbert’s finitism. It is shown that D. Hilbert relies heavily on some epistemological characteristics of sign in order to make elementary mathematical structures as the base of mathematical knowledge. Thus Hilbert’s views of intuitive contents of proofs by means of mathematical induction assume that concrete examples of mathematical induction are captured by intuition provided that predicates entering into the proof are in a sense elementary, namely, primitive recursive. According to W. Tait, Hilbert’s finitism, which consists in an assumption of intuitive comprehension of basic mathematical operations and objects, is reduced to acceptance of primitively recursive reasoning. Hilbert relied on the existence of some valid philosophical consensus in what can be considered as correctness of mathematical reasoning. Hilbert had admitted that there is some philosophical ‘minimum’ which is acceptable by all who are ready to accept mathematics.