Hilbert's Programs: 1917–1922

Abstract

Hilbert's finitist program was not created at the beginning of 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.

Hilbert's Program Revisited

“…1 See e.g., Kitcher (1976), Giaquinto (1983), Detlefsen (1993a, b), Sieg (1984, 1990a, 1999. 2 Detlefsen (1986Detlefsen ( , 1990; cf.…”

“…There is a widely accepted, rather sophisticated logical explication of Hilbert's program that is essentially due to Kreisel (1958Kreisel ( , 1968Kreisel ( , 1976) and endorsed by various distinguished logicians, e.g., by Feferman (1988Feferman ( , 1994, Sieg (1984Sieg ( , 1988Sieg ( , 1990aSieg ( , 1999, Prawitz (1981), Smorynski (1977Smorynski ( ,1985Smorynski ( , 1988, and Simpson (1988), as well as by philosophers such as e.g., Kitcher (1976) and Giaquinto (1983). This interpretation relates Hilbert's requirement of consistency proof closely to the soundness and conservativity properties.…”

Hilbert's Program Then and Now

“…Hilbert at the time considered the problem as essentially solved by Whitehead and Russell's work in Principia. Nevertheless, other fundamental problems of axiomatics remained unsolved, including the problem of the "decidability of every mathematical question," which also traces back to Hilbert's 1900 address. 1 See [Sieg, 1999;Sieg, 2002], [Stein, 1988], [Hallett, 1990;Hallett, 1994], [Mancosu, 1998b], and [Avigad and Reck, 2001] for further discussion of the conceptual framework and historical background of Hilbert's thought, and [Resnik, 1974a] on the Frege-Hilbert correspondence. On Hilbert's foundational interests before 1917, and his engagement for Husserl, Zermelo, and Nelson in Göttingen, see [Peckhaus, 1990].…”

“…The two-volume Grundlagen der Mathematik [Hilbert and Bernays, 1934;Hilbert and Bernays, 1939] was prepared by Bernays alone, and included new re-5 See [Avigad and Zach, 2002] for a general introduction to the ε-calculus and [Zach, 2003b] and [2004] on the history of the ε-calculus and the substitution method. [Sieg, 1999] presents a detailed and perceptive analysis of the development of the program and its influence as a whole.…”

Book Review: The search for mathematical roots, 1870-1940: Logics, set theories, and the foundations of mathematics from Cantor through Russell to Gödel

“…The work of Bernays, Gentzen, Herbrand, Gödel, and Tarski, depending as it does on the concepts of metamathematics and of formal system, would not have been possible without this radical rethinking. For a sophisticated description and analysis of these lectures and of the development of Hilbert's views in 1917-22, see [7].…”