ABSTRACT
Hilbert was a giant among mathematicians. It is hard to overestimate his influence over the character of twentieth century mathematics; so many of the great names in mathematics studied under him or worked with him in Gottingen. Modern standards of mathematical rigour owe more to Hilbert than to either Frege or Russell, both of whom exerted more influence on philosophy than on mathematics. Hilbert's position was, in a sense, the inverse of Frege's. With his axiomatization of geometry he effectively removed the impulse to treat these axioms as self-evident truths validated by appeal to geometric intuition, or to an intuition of space, for, as he emphasized by saying 'It must be possible to replace in all geometric statements the words point, line, plane by table, chair, mug' (quoted in Weyl 1970 p. 264) there is nothing peculiarly spatial in the conditions laid down by these axioms; they might be satisfied in any domain of objects. Indeed Frege wrote to Hilbert 'It seems to me that you want to divorce geometry completely from our intuition of space and make it a purely logical discipline, like arithmetic' (Frege 1971 p. 14). Yet Hilbert would not agree with Frege's assumption that arithmetic is a purely logical discipline. He did not think that arithmetic could be reduced to logic or to anything more fundamental. The truths of finitary arithmetic are grounded in an intuitive grasp of notions of
unit and of discrete succession as exemplified in the writing of the numerals.
