Über das Unendliche. Offprint from: Mathematische Annalen 95. Bd., 2. Heft.

[Berlin: Springer, 1925].

First edition, very rare offprint, of this famous lecture which contains Hilbert’s most detailed exposition of his proposal for the foundation of classical mathematics, which became known as Hilbert’s Programme. “No one shall expel us from the paradise which Cantor has created for us,” Hilbert famously declared in this lecture (p. 170). This paper further contains his attempted proof of the ‘continuum hypothesis’, that no set can have both ‘fewer’ elements than the set of real numbers and ‘more’ than the set of integers – the terms ‘more’ and ‘fewer’ referring to the (infinite) numbers of elements in these sets, understood in terms of Georg Cantor’s theory of ‘cardinal numbers’ of infinite sets. “Hilbert devoted to the foundations of mathematics a series of papers … Among all these contributions the 1925 paper [the offered item] stands out as the most comprehensive presentation of Hilbert’s ideas. It is the text of an address delivered in Münster on 4 June 1925 at a meeting organized by the Westphalian Mathematical Society to honor the memory of Weierstrass … The paper consists of two quite distinct parts. The first is a clear and forceful presentation of Hilbert’s ideas at the time on the foundations of mathematics. It starts by recalling how Weiserstrass eliminated references to ‘infinity’ in analysis (Hilbert could have mentioned Cauchy and D’Alembert) and reviews the role played by the infinite in physics, set theory, and, when we deal with general propositions, logic. Leaning on the example of arithmetic, it introduces the distinction between finitary and ideal propositions, and it undertakes a simultaneous formalization of logic and arithmetic … The second part of the paper is a sketch of an attempted proof of the continuum hypothesis. Stated for the first time by Cantor at the very beginning of the development of set theory (1878), presented by Hilbert as Problem no. 1 in his famous list of unsolved mathematical problems (1900), the continuum problem for years resisted the efforts of mathematicians … The continuum hypothesis was proved by Gödel (1938) to be consistent with the customary axioms of set theory and by Cohen (1963) to be independent of these axioms … In his review of Gödel’s second paper on the consistency of the continuum (1939), Bernays wrote: ‘The whole Gödel reasoning may also be considered as a way of modifying the Hilbert project for a proof of the Cantor continuum hypothesis, as described in [Über das Unendliche, 1925], so as to make it practical and at the same time generalizable to higher powers’” (Van Heijenoort, From Frege to Gödel, pp. 367-92). OCLC locates just two copies, in Germany and the Netherlands, but we have located three others: one in the Paul Hertz collection of the University of Pittsburgh, and two in a private collection.

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, but progress was delayed because of the lack of a properly worked-out logical formalism. It was the publication of Russell and Whitehead’s Principia Mathematica (1910-13) that provided the required logical basis for a renewed attack on foundational issues. A little later Brouwer began to develop his intuitionistic mathematics, to which Hilbert’s gifted former pupil, Hermann Weyl, was drawn.

Contrary to some other contemporary mathematicians, including Leopold Kronecker and Henri Poincaré, Hilbert was greatly impressed by Cantor’s set theory. This he made clear in a semi-popular lecture course he gave in Göttingen during the winter semester 1924-1925 and in which he dealt at length with the infinite in mathematics, physics, and astronomy. A few months later he repeated the message in a wide-ranging lecture on the infinite given in Münster on 4 June 1925 [published as the offered paper]. The occasion was a session organized by the Westphalian Mathematical Society to celebrate the mathematical work of Karl Weierstrass. Hilbert’s Münster address drew extensively on his previous lecture course, except that it was more technical and omitted many examples” (Kragh, p. 4).

Hilbert agreed with the long tradition according to which there is no such thing as an actual or completed infinity. In the present paper, he wrote (p. 190):

The infinity is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought – a remarkable harmony between being and thought ... The role that remains for the infinite is solely that of an idea – if one means by an idea, in Kant’s terminology, a concept of reason which transcends all experience and which completes the concrete as a totality – that of an idea which we may unhesitatingly trust within the framework erected by our [mathematical] theory.’

But, unlike Kronecker and Brouwer, he did not therefore want to rebut all the achievements of infinistic mathematics. What Hilbert rather aimed at was to bring together the safety, or reliability, of critical constructive mathematics and the liberty and power of infinistic set theoretical mathematics – to justify the use of the latter by the uncontroversial methods of the former. More exactly, Hilbert’s program for a ‘new grounding of mathematics’ was planned to proceed as follows:

“First, all of mathematics so far developed should be rigorously formalized. In Hilbert’s own words:

‘All the propositions that constitute mathematics are converted into formulas, so that mathematics proper becomes an inventory of formulas … The axioms and provable propositions, that is, the formulas resulting from this procedure, are copies of the thoughts constituting customary mathematics as it has developed till now.’

“Next, Hilbert intended to isolate what he viewed as an unproblematic and necessary part of mathematics, an elementary part of arithmetic he called ‘finitistic mathematics’, which would certainly be accepted by all parties in the foundational debate, even the most radical skeptics such as Kronecker, Brouwer and Weyl. Indeed, Hilbert stated explicitly that Kronecker’s view ‘essentially coincided with our finitist standpoint’. Further, Hilbert divided mathematical statements into ideal and real statements. The latter are finitistically meaningful, or contentual, but the former are strictly speaking just meaningless strings of symbols that complete and simplify the formalism, and make the application of classical logic possible. Finally, the consistency of the comprehensive formalized system is to be proved by using only restricted, uncontroversial and contentual finitistic mathematics. Such a finitistic consistency proof would entail that the infinistic mathematics could never prove a meaningful real statement that would be refutable in finitistic mathematics, and hence that infinitistic mathematics is reliable” (Raatikainen, p. 160).

“Hilbert’s addresses in Göttingen and Münster also merit attention because of their references to microphysics and cosmology. On the one hand, he said in Münster, the new atomic and quantum physics had demonstrated that the infinite divisibility of space and matter was nothing but an abstract idea inapplicable to the real world studied by the physicists and chemists. On the other hand, the infinite seemed also to be contradicted on the largest possible scale, the universe as a whole. ‘We need to investigate the expansion of the universe [Ausdehnung der Welt], to establish if it results into something infinitely large.’ Of course, Hilbert’s words should not be mistaken for an anticipation of what a few years later became known as the expanding universe. Well aware of the new curved-spaced cosmology based on Einstein’s theory of general relativity, he sketched its essence to the mathematicians gathered in Münster (p. 165):

‘The abolition of Euclidean geometry is today not just a purely mathematical or philosophical speculation, but we have also arrived at it from quite a different perspective that originally had nothing to do with the question of the finitude of the world. Einstein has proved the necessity of deviating from Euclidean geometry. Based on his theory of gravitation he attacks the cosmological questions and demonstrates the possibility of a finite world. Moreover, all the results found by the astronomers are also in full agreement with the assumption of the elliptic world’” (Kragh, p. 5).

Although many offprints of Hilbert’s papers in the Mathematische Annalen were issued in printed wrappers, we have seen several copies of the present offprint and all were bound in the same way as this copy, with a green paper spine strip and no wrappers. An exactly similar copy was offered by Quaritch in 2005 (Cat. 1335, no. 94, £8,000).

8vo (226 x 157 mm), pp. [161], 162-190. Bound with a green paper backstrip as issued, a fine copy.

Item #3181

Price: $2,500.00