David Hilbert's finitistic standpoint is a conception
of elementary number theory designed to answer the intuitionist doubts
regarding the security and certainty of mathematics. Hilbert was
unfortunately not exact in delineating what that viewpoint was, and
Hilbert himself changed his usage of the term through the 1920s and 30s.
The purpose of this paper is to outline what the main problems are in
understanding Hilbert and Bernays on this issue, based on some
publications by them which have so far received little attention, and on
a number of philosophical reconstructions of the viewpoint (in
particular, by Hand, Kitcher, and Tait)
