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Hilbert's Philosophy of Mathematics

The lens of history tends to diminish losers and magnify winners: because Hilbert's viewpoint seems to have been refuted as decisively as any philosophical view can be, there is a tendency to see his formalist programme and its philosophical basis as a relatively unimportant cul-de-sac in the history of the philosophy of mathematics. Against this, the contention of this paper is that Hilbert's contribution was of central importance in the development of mathematical philosophy and a paradigm of progressive philosophical practice. It is not here argued that Hilbert's programme survives G6del's incompleteness theorems.2 On the contrary, attempts to save the programme by watering down the notion of finitist reasoning do not seem to be philosophically helpful-which is not to deny the interest of seeing how far into the transfinite we must go in order to prove the consistency of given systems. The major claims of this paper are these: