Hilbert's Programs: 1917-1922

Hilbert’s finitist programwasnot created at the beginningof the twenties solely to counteract Brouwer’s intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert’s considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert’s and Bernays’s contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert’s Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work. Introduction. At the very end of a sequence of lectures he gave in 1919 under the titleNatur und mathematisches Erkennen, Hilbert emphasized that some physical paradoxes had directed his discussion away from the methods Received May 15, 1997; revised October 19, 1998. Versions of this paper were presented at the workshopModern Mathematical Thought in Pittsburgh (September 21–24, 1995), at the conference Philosophy of Mathematics, Logic, and Wittgenstein in Chicago (May 31–June 2, 1996), at Gödel ’96 in Brno (August 25–29, 1996), and at the Annual Meeting of the ASL in Boston (March 22–25, 1997). Critical questions and constructive suggestions from all audiences helped me to clarify some issues; I am particularly grateful to W. A. Howard, G. H. Müller, H. Stein, and A. Urquhart. I want to acknowledge the continuing, fruitful discussions with myHilbert-Edition-colleagues W. Ewald, M. Hallett, R. Haubrich, and U. Majer, but also with my students J. Byrnes and M. Ravaglia. Finally, I profited from remarks by Jeremy Avigad, Andreas Blass, Charles Parsons, and an anonymous referee; the latter pointed me to work reported in [2] and [80].— A summary of the main findings will appear in the Proceedings of the Roskilde Workshop on Proof Theory under the title Toward finitist proof theory. c © 1999, Association for Symbolic Logic 1079-8986/99/0501-0001/$5.40