Hilbert's programs: 1917--1922

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 towards finitism; the progression has to be seen against the background of die 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 broad revision of Hilbert's and Bemays'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 modem 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 die title Natur und mathematisches Erkennen, Hilbert emphasized that some physical paradoxes had directed his discussion away from the methods of physics to the general philosophical problem, "whether and how it is possible to understand our thinking by thinking itself and to free it from any paradoxes". Hilbert saw this problem also at the basis of his work in mathematical logic. One might ask polemically, whether there is more to Hilbert's contribution to that problem than the narrow and technical consistency program pursued in Gottingen during the twenties? A critical reader of the relevant historical and philosophical literature, and even of some of Hilbert's own writings, almost certainly would be inclined to give a negative answer. During the last ten or fifteen years, a more positive and also more accurate perspective on the work of the Hilbert School has been emerging, for example, in papers by Feferman, Hallett, Sieg, and Stein; this has been achieved mainly by bringing out the rich context in which the work is embedded. Important connections have been established, on the one hand, to foundational work of the 19th century (that had been viewed as largely irrelevant) and, on the other hand, to a general reductive program (that evolved out of Hilbert's Program and underlies implicitly most modern proof theoretic investigations). However, it remains crucial to gain a better understanding of the development of Hilbert's thought on the foundations of arithmetic, where arithmetic is understood in a broad sense that includes elementary number theory and reaches all the way to set theory. This is admittedly but one aspect of Hilberfs work on the foundations of mathematics, as it disregards the complex interactions with his work on the foundations of geometry and of the natural sciences; and yet it is a most 2 This quotation is found on page 117 of (Hilbert 1919). significant aspect, as it reveals a surprising internal dialectic progression (in the attempt to address broad philosophical issues) and throws a distinctive new light on the development of modern mathematical logic. Standard wisdom partitions Hilbert's work on the foundations of arithmetic, with some justification, into two periods. The first period is taken to extend from 1900 to 1905, the second from 1922 to 1931. The periods are marked by dates of outstanding publications. Hilbert published in 1900 and 1905 respectively iXber den Zahlbegriff and Uber die Grundlagen der Logik und Arithmetik. According to the standard view, the considerations of the latter paper were taken up around 1921, were developed further into the proof theoretic program, and were exposed first in 1922 through Hilbert's Neubegrundung der Mathematik and Bernays's liber Hilberts Gedanken zur Grundlegung der Arithmetik. This "continuity" is pointed out by both Hilbert and Bemays, without emphasizing their early mathematical logical work or the exploration of alternative foundational perspectives. Finally, it is argued that the pursuit of die program was halted in 1931 by Godel's paper Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I. This partition of Hilbert's work does not include, or accommodate easily, the programmatic paper Axiomatisches Denken published in 1918. The paper had been presented already in September 1917 to the Swiss Mathematical Society in Zurich and advocates a logicist reduction of mathematics. In sharp contrast, the 1922 papers by Hilbert and Bemays set out the philosophical and mathematical-logical goals of the Hilbert Program. This remarkable progression is not at all elucidated by publications, but it can be analyzed by reference to notes for courses Hilbert gave during that period in Gottingen. The lectures were prepared with the assistance of Bernays who wrote all the notes, except that Schonfinkel helped prepare the notes for the summer term 1920. I will discuss this development, after sketching in Part A connections to foundational investigations of the 19th century; Part B describes die strikingly novel treatment of general logical and metamathematical issues, whereas Part C is devoted to the emergence of specifically proof theoretic investigations. Thus, here is a first attempt to bridge the gap in the published record between Hilbert's Zurich Lecture and the proof theoretic papas from 1922; the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable indeed. 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. PART A. BEFORE 1917: axiomatic method and consistency. Hilbert viewed the axiomatic method as holding the key to a systematic organization of any sufficiently developed subject; he also saw it as providing the basis for metamathematical investigations of independence and completeness issues and for philosophical reflections. However, consistency was Hilberfs central concern ever since he turned his attention to the foundations of analysis in the late nineties of the last century. For analysis, Dedekind and Kronecker had put forward two radically different kinds of arithmetizations in response to Dirichlet's demand, that any theorem of algebra and higher analysis be formulated as a theorem about natural numbers. The Hilbert Program can be seen properly and fruitfully as an attempt to mediate between the opposing foundational tendencies represented by these two eminent mathematicians. Ai. ARTTHMETIZAT1ON strict and logical. Kronecker admitted as objects of analysis only natural numbers and constructed from them, in now well-known ways, integers, rationals, and even algebraic reals. The general notion of irrational number was rejected, however, because of two restrictive methodological requirements: concepts must be decidable, and existence proofs must be carried out in such a way that they present objects of the appropriate kind. For Kronecker there could be no infinite mathematical objects, and geometry was banned from analysis even as a motivating factor. (Hilberfs critical, but also appreciative discussion in his lectures during the summer term 1920 emphasize these broad methodological points.) Clearly, this procedure is strictly arithmetic, and Kronecker believed that analysis could be re-obtained by following it. It is difficult for me to judge to what extent Kronecker pursued a program of developing (parts of) analysis in an elementary, constructive way. Such a program is not chimerical, as mathematical work during the last two decades has established that a good deal of analysis and algebra can be done in conservative extensions of primitive recursive t:. arithmetic. In contrast to Kronecker, Dedekind defined a general notion of real numbers, motivated cuts explicitly in geometric terms, and used infinite sets of natural numbers as respectable mathematical objects. The principles underlying the definition of cuts were for Dedekind logical ones which allowed the "creation" of new numbers, such that their system has "the same completeness or ... the same continuity as the straight line". Dedekind emphasized in a letter to lipschitz that this continuous completeness is essential for a scientific foundation of the arithmetic of real numbers, as it relieves us in analysis of the necessity to assume existences without sufficient proof. Indeed, it provides the answer to Dedekind's rhetorical question: How shall we recognize the admissible existence assumptions and distinguish them from the countless inadmissible ones...? Is this to depend only on the success, on the accidental discovery of an internal contradiction? 3 Dedekind is considering here assumptions about the existence of individual real numbers. Such assumptions are not needed, when a complete system is investigated: the question concerning the existence of particular reals is shifted to the question concerning the existence of their complete system. 3 Letter to Lipschitz of July 27,1876; in (Dedekind 1932), p.477.