It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert's formal axiomatic method from the early 1920s with his existential axiomatic approach from the 1890s. Such a contrast illuminates the circuitous beginnings of the finitist consistency program and connects the complex emergence of existential axiomatics with transformations in mathematics and philosophy during the 19th century; the sheer complexity and methodological difficulties of the latter development are partially reflected in the well known, but not well understood correspondence between Frege and Hilbert. Taking seriously the goal of formalizing mathematics in an effective logical framework leads also to contemporary tasks, not just historical and systematic insights; those are briefly described as “one direction” for fascinating work.
This paper directly builds on my Hilbert's Proof Theory, but focuses on the dramatically different perspectives on the axiomatic method during the 1890s, culminating in the Paris address of 1900, and the early 1920s, when the finitist consistency program developed in a methodologically coherent way; that program was presented for the first time in Hilbert's talk in Leipzig in the fall of 1922. The lectures of Hilbert and Bernays from around 1920 were described in Sieg 1999 and will, finally, be published in Ewald and Sieg 2013.