‘Mathematische Probleme.’ Two offprints from Archiv der Mathematik und Physik, 3. Reihe, 1. Band, 1901.

Leipzig: Teubner, 1901.

First complete publication, extremely rare offprint issue, of Hilbert’s famous and enormously influential address to the International Congress of Mathematicians at Paris in 1900 in which he set forth a list of twenty-three problems that he predicted would be of central importance to the advance of mathematics in the twentieth century. Hilbert’s paper was first published in Nachrichten der Königliche Gesellschaft zur Wissenschaften zu Göttingen, Mathematische-physikalischen Klasse 3 (1900), pp. 253-97, and reprinted with additions as the offered work. Hilbert’s problems came in four groups. In the first group were six foundational ones, starting with an analysis of the real numbers using Cantorian set theory, and including a call for axioms for arithmetic, and the challenge to axiomatise physics. The next six drew on his study of (algebraic) number theory, and culminated with his revival of Kronecker's Jugendtraum, and the third set of six were a mixed bag of algebraic and geometric problems covering a variety of topics. In the last group were five problems in analysis – the direction that Hilbert’s own interests were going. He asked for a proof that suitably smooth elliptic partial differential equations have the type of solutions that physical intuition (and many a German physics textbook) suggest, even though it had been known since the 1870s that the general problem of that kind does not. He made a specific proposal for advancing the general theory of the calculus of variations” (Gray).

“As the nineteenth century drew to its close, David Hilbert (1862–1943), then regarded as a leading mathematician of his generation, presented a list of twenty- three problems, which he urged upon the attention of his contemporaries. They have entered the folklore of professional mathematicians; even a partial solution of one of them has given its author(s) much prestige…

“The motivation was the Second International Congress of Mathematicians, held in Paris early in August 1900, which Hilbert was invited to address. He seems to have thought of the topic by December 1899, for he sought then the opinion of his close friend Hermann Minkowski (1864–1909), and again in March of another ally, Adolf Hurwitz (1859–1919). But apparently he delayed writing the paper until May or June, so that the lecture was left out of the Congress programme. However, by mid-July he must have sent it for publication by the Göttingen Academy of Sciences, of which he was a member, for Minkowski was then reading the proofs; very likely no refereeing had occurred. Hilbert spoke in the Sorbonne on the morning of 8 August 1900, not in a plenary lecture but in the section of the Congress on bibliography and history; he proposed “the future problems of mathematics,” working from a French translation of his text that was distributed to the members of the audience. A summary of it soon appeared in the recently founded Swiss journal L’Enseignement Mathématique; the original seems not to have been published. For reasons of time he described there only ten problems. The full story was soon out with the Göttingen Academy; next year it was published again, with three additions, in the Archiv der Mathematik und Physik … The Archiv version was translated in full into French for the Congress proceedings by the French mathematician and former diplomat Léonce Laugel, who added a few footnotes of his own. His translation appeared both there and as a separate undated pamphlet under the title Mathematical Problems

“The few pages of preamble appraised problems in general and the development of mathematical knowledge as Hilbert saw it; near the end he expressed his optimism with a slogan that he would repeat in later life: “for in mathematics there is no ignorabimus! ”. The modernistic flavour of the problems lay not only in their unresolved status but also in the high status given to axiomatisation in solving or even forming several of them” (Grattan-Guinness).

“The first half-dozen problems pertained to the foundations of mathematics and had been suggested by what he considered the great achievements of the century just past: the discovery of the non-euclidean geometry and the clarification of the concept of the arithmetic continuum, or real number system. These problems showed strongly the influence of [Hilbert’s] recent work on the foundations of geometry and his enthusiasm for the power of the axiomatic method. The other problems were special and individual, some old and well known, some new, all chosen, however, from fields of Hilbert’s own past, present and future interest” (Reid, Hilbert (1970), p. 70). In the second of these problems, Hilbert called for a mathematical proof of the consistency of the arithmetic axioms – a question that, in a later incarnation, turned out to have great bearing on the development of both mathematical logic and computer science when the problem was addressed by Gödel and Turing.

 “Several main branches of mathematics were impressively covered or at least exemplified by problems: number theory and higher and abstract algebra (Hilbert’s two main research specialities up to that time), most of real- and complex-variable analysis, and the still emerging branch of topology. Geometry was more patchily handled; in particular, the achievements of the Italian geometers largely eluded him. Apparently untalented in languages, he had trouble reading even technical Italian. Among problems directly inspired by Hilbert’s own work, the Fourteenth Problem grew out of his proofs in the early 1890s that systems of algebraic invariants always possess finite bases. However, he forgot to cite Hurwitz’s recent contribution [‘Über die Erzeugung der Invarianten durch Integration,’ Nachrichten Königlichen Gesellschaft Wissenschaften Göttingen, math.-physik. Klasse (1897), 71–90]; he apologised to his friend in November 1900 and added a paragraph to the Archiv version.

“Some problems were handled with great perspicuity. In particular, in the Fifth Problem on the theory of Sophus Lie (1842–1899) of continuous groups of transformations, not only did he pose a specific problem invoking the differentiability of the pertaining functions, but also a broader one about weakening that property. The latter is still far from a general answer …

“The importance of Hilbert’s lecture was grasped quite soon after the Congress; for example, Laugel’s translation of the full version was published in its proceedings with the plenary lectures although it had not been so delivered. But the reaction after the lecture was “a rather desultory discussion,” to quote from the report on the Congress prepared by Charlotte Angas Scott (1858–1931) for the Bulletin of the American Mathematical Society. Two comments were made. Firstly, the Italian mathematician Giuseppe Peano (1858–1932) remarked that the Second Problem on the consistency of arithmetic was already essentially solved by colleagues working on his project of mathematical logic and that the forthcoming Congress lecture by Alessandro Padoa (1868–1937) was pertinent to it... Secondly, the German mathematician Rudolf Mehmke (1857–1944) made a point about numerical methods that bore upon the Thirteenth Problem on resolving the quintic: it led to a new paragraph in the Archiv version...” (Grattan-Guinness).

“Hilbert’s problems ranged greatly in topic and precision. Some of them are propounded precisely enough to enable a clear affirmative or negative answer, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis). For other problems, such as the 5th, experts have traditionally agreed on a single interpretation, and a solution to the accepted interpretation has been given, but closely related unsolved problems exist. Sometimes Hilbert’s statements were not precise enough to specify a particular problem but were suggestive enough so that certain problems of more contemporary origin seem to apply, e.g. most modern number theorists would probably see the 9th problem as referring to the conjectural Langlands correspondence on representations of the absolute Galois group of a number field. Still other problems, such as the 11th and the 16th, concern what are now flourishing mathematical sub-disciplines, like the theories of quadratic forms and real algebraic curves. There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the axiomatization of physics, a goal that twentieth century developments of physics (including its recognition as a discipline independent from mathematics) seem to render both more remote and less important than in Hilbert’s time. Also, the 4th problem concerns the foundations of geometry, in a manner that is now generally judged to be too vague to enable a definitive answer. The other twenty-one problems have all received significant attention, and late into the twentieth century work on these problems was still considered to be of the greatest importance. Paul Cohen received the Fields Medal during 1966 for his work on the first problem, and the negative solution of the tenth problem during 1970 by Yuri Matiyasevich generated similar acclaim. Aspects of these problems are still of great interest today” (Wikipedia).

Here are the 23 problems enumerated by Hilbert.

  1. The cardinality of the continuum, including well-ordering.
  2. The consistency of the axioms of arithmetic.
  3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.
  4. The straight line as shortest connection between two points.
  5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining a group.
  6. The axioms of physics.
  7. Irrationality and transcendence of certain numbers.
  8. Prime number theorems (including the Riemann hypothesis).
  9. The proof of the most general reciprocity law in arbitrary number fields.
  10. Decision on the solvability of a Diophantine equation.
  11. Quadratic forms with any algebraic coefficients.
  12. The extension of Kronecker's theorem on Abelian fields to arbitrary algebraic fields.
  13. Impossibility of solving the general seventh degree equation by means of functions of only two variables.
  14. Finiteness of systems of relative integral functions.
  15. A rigorous foundation of Schubert’s enumerative calculus.
  16. Topology of real algebraic curves and surfaces.
  17. Representation of definite forms by squares.
  18. The building up of space from congruent polyhedra.
  19. The analytic character of solutions of variation problems.
  20. General boundary value problems.
  21. Linear differential equations with a given monodromy group.
  22. Uniformization of analytic relations by means of automorphic functions.
  23. The further development of the methods of the calculus of variations.

 A 24th problem, on the simplicity of proofs, was omitted by Hilbert but rediscovered in his original manuscript notes by German historian Rüdiger Thiele in 2000.

OOC 320 (Göttingen Nachrichten issue); I. Grattan-Guinness, ‘A sideways look at Hilbert's twenty-three problems of 1900,’ Notices of the American Mathematical Society (2000), pp. 752-7; J. Gray, ‘The Hilbert problems 1900-2000,’ European Mathematical Society Newsletter 36 (2000), p. 10-13.

8vo (236 x 163 mm), pp. [44]-63 & [213]-237. Original printed wrappers.

Item #5482

Price: $6,500.00