Hamburg: Verlag des mathematischen Seminars, 1922.
First edition of the first major paper in the development of the ‘Hilbert programme,’ in which Hilbert put forward his proposal for a foundation for all of mathematics based on axiomatics and logic. Hilbert’s program was a proposed solution to the foundational crisis of mathematics, caused by the discovery of paradoxes and inconsistencies in the existing foundations..
First edition, journal issue in the original printed wrappers, and a copy with excellent provenance, of the first major paper in the development of the ‘Hilbert programme,’ in which Hilbert put forward his proposal for a foundation for all of mathematics based on axiomatics and logic. Hilbert’s programme “was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic” (Wikipedia). “According to Reid [Hilbert-Courant (1970), p. 154), Hilbert was becoming, in the early 1920s, ‘increasingly alarmed by the gains that Brouwer’s conception of mathematics was making among the younger mathematicians. To him, the program of the Intuitionists represented quite simply a clear and present danger to mathematics’. Hilbert interpreted intuitionism as requiring that all pure existence proofs, a large part of analysis and Cantor’s theory of infinite sets would have to be given up. In particular, this would rebut some of Hilbert’s own important contributions to pure mathematics. Hilbert was especially disturbed by the fact that Weyl, who was his most distinguished former student, accepted the radical views of Brouwer, who aroused in Hilbert the memory of Kronecker. In Reid’s words, ‘At a meeting in Hamburg in 1922 he came roaring back to the defence of mathematics’ (Reid, p. 155). This was the first public presentation of Hilbert’s program” (Raatikainen, p. 158). Hilbert’s Hamburg lecture was published as the present paper. In it, “Hilbert sets out the basic ideas of his proof theory, describes a simple formal axiom system for a fragment of arithmetic, and proves its consistency; he also lays the groundwork for his later investigations of the foundations of set theory and real analysis ... Hilbert makes no attempt to supply a consistency proof for the transfinite part of his theory” (Ewald, p. 1116).
Provenance: Heinrich Behmann (1891-1970) (pencil signature on upper wrapper). Behmann studied mathematics in Tübingen, Leipzig and Göttingen. Hilbert supervised the preparation of his doctoral thesis at Göttingen, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead (1918). This dissertation was primarily intended to give a clear exposition of the solution to the antinomies found in Whitehead & Russell’s Principia Mathematica. Behmann continued to perform research in the field of set theory and predicate logic, proving in 1922 that the monadic predicate calculus is decidable. In 1938 he obtained a professorial chair in mathematics at Halle. See Mancosu, ‘Between Russell and Hilbert: Behmann on the foundations of mathematics,’ Bulletin of Symbolic Logic 5 (1999), 303-330.
Hilbert’s work on the foundations of mathematics has its roots in his work on geometry of the 1890s, culminating in his influential Grundlagen der Geometrie (1899). Hilbert believed that the proper way to develop any scientific subject rigorously required an axiomatic approach, which would enable the theory to be developed independently of any need for intuition, and would facilitate an analysis of the logical relationships between the basic concepts and the axioms. Hilbert realized that the most important questions are the independence and the consistency of the axioms. For the axioms of geometry, consistency can be proved by providing an interpretation of the system in the real plane, and thus the consistency of geometry is reduced to the consistency of analysis. The foundation of analysis, of course, itself requires an axiomatization and a consistency proof. Hilbert provided such an axiomatization in Über den Zahlbegriff (1900), but it became clear very quickly that the consistency of analysis faced significant difficulties, in particular because the favoured way of providing a foundation for analysis in Dedekind’s work relied on dubious assumptions akin to those that lead to the paradoxes of set theory. Hilbert thus realized that a direct consistency proof of analysis, i.e., one not based on reduction to another theory, was needed. He proposed the problem of finding such a proof as the second of his 23 mathematical problems in his address to the International Congress of Mathematicians in 1900 and presented a sketch of such a proof in his Heidelberg talk Über die Grundlagen der Logik und der Arithmetik in 1904, but further progress was delayed because of the lack of a properly worked-out logical formalism.
The publication of Russell and Whitehead’s Principia Mathematica provided the required logical basis for a renewed attack on foundational issues. “In the interval since [Über die Grundlagen der Logik und der Arithmetik], numerous important developments had taken place in the foundations of mathematics. Hilbert’s younger Gottingen colleague Ernst Zermelo had proved his well-ordering theorem. The paradoxes of set theory had become widely known, and had led to spirited discussions between Russell, Poincare, Richard, Konig, Zermelo, and Peano. Hilbert’s axiomatic method, first employed in his Grundlagen der Geometrie (1899), had been applied in numerous investigations in geometry, algebra, and mathematical physics. Poincare published his criticisms of [Über die Grundlagen der Logik und der Arithmetik] and of the logicists; Zermelo supplied axioms for the set theory of Dedekind and Cantor; Whitehead and Russell published Principia Mathematica; Brouwer began to develop his intuitionistic mathematics; Hilbert’s gifted former pupil, Hermann Weyl, was drawn to Brouwer’s ideas” (Ewald, pp. 1105-1106).
In September 1917, Hilbert delivered an address to the Swiss Mathematical Society entitled Axiomatische Denken, his first published contribution to mathematical foundations since 1905. In it, he again emphasizes the requirement of consistency proofs for axiomatic systems, and states his belief that this had been achieved by 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.
Within the next few years, however, Hilbert came to reject Russell’s logicistic solution to the consistency problem for arithmetic. At the same time, Brouwer’s intuitionistic mathematics gained currency. In particular, Hilbert’s former student Hermann Weyl converted to intuitionism. Weyl’s paper Über die neue Grundlagenkrise der Mathematik (1921) was answered by Hilbert in three talks in Hamburg delivered on July 25-27, 1921, published as Neubegründung der Mathematik.
“The first period [of Hilbert’s work on the foundations of mathematics] is taken to extend from 1900 to 1905, the second from 1922 to 1931. The periods are marked by the dates of outstanding publications. Hilbert published in 1900 and 1905 respectively Über den Zahlbegriff and Über die Grundlagen der Logik und der Arithmetik … the considerations of the latter paper were taken up around 1921, were quickly expanded into the proof theoretic program, and were exposed first in 1922 through Hilbert’s Neubegründung der Mathematik” (Sieg, p. 3).
In the Neubegründung Hilbert attempts to rebut the critiques of classical mathematics by Brouwer and Weyl and to clear away the doubts engendered by the paradoxes they raised. “At the level of metamathematics, Hilbert adopted the intuitionistic criticisms of infinitary mathematics, and sought to use only reasoning that was intuitionistically acceptable; indeed, at this level, Hilbert’s finitism went further than that of Brouwer himself. The disagreements stemmed rather from a difference of opinion about what constitutes a foundation for mathematics, and concerned, first, the desirability of formalized mathematics uberhaupt; second, the usefulness, legitimacy, and mathematical interest of the classical, infinitary modes of inference expressed in Hilbert's formal system.
“In contrast to [Über die Grundlagen der Logik und der Arithmetik], the present essay now draws a clear distinction between the logico-mathematical formalism and the inhaltliche metamathematical reasonings about it. This distinction allows Hilbert to answer the charge of circularity raised by Poincare … [who] had charged that Hilbert needed to presuppose the truth of mathematical induction in order to prove its consistency; but Hilbert can now distinguish (as he does in §31) between the strong principle of complete induction expressed in the formal language and the weaker principle used in the metalanguage” (ibid.).
Ewald, From Kant to Hilbert, Vol. II, Oxford, 1996; Raatikainen, ‘Hilbert’s Program revisited,’ Synthese 137 (2003), pp. 157-177. Sieg, ‘Hilbert's Programs: 1917-1922,’ Bulletin of Symbolic Logic 5 (1999), pp. 1-44.
8vo (240 x 162 mm), pp. 99-177, , 4, uncut and mostly unopened. Original printed wrappers, spine slightly sunned.