Hilbert's problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. The problems were all unsolved at the time, and several of them turned out to be very influential for 20th century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21 and 22) at the conference, speaking on 8 August in the Sorbonne; the full list was published later.
Nature and influence of the problems
While there have been subsequent attempts to repeat the success of Hilbert's list, no other broadly-based set of problems or conjectures has had a comparable effect on the development of the subject, or attained a fraction of its celebrity. For example the Weil conjectures are famous, but were rather casually announced. André Weil was perhaps temperamentally unlikely to put himself in the position of vying with Hilbert. John von Neumann produced a list, but not to universal acclaim.
At first sight this success might be put down to the eminence of the problems' author. Hilbert was at the height of his powers and reputation at the time, and would go on to lead the outstanding school of mathematics at the University of Göttingen. On closer examination matters are not quite so simple.
The mathematics of the time was still discursive: the tendency to replace words by symbols, and appeals to intuition and concepts by bare axiomatics, was still subdued, though it would come in strongly over the next generation. In 1900 Hilbert could not appeal to axiomatic set theory, the Lebesgue integral, topological spaces or Church's thesis, each of which would change permanently its field. Functional analysis, in one sense founded by Hilbert himself as the central notion of Hilbert space witnesses, had not yet differentiated itself from the calculus of variations; there are two problems on the list about variational mathematics, but nothing, as a naïve assumption might have supposed, about spectral theory (problem 19 does have a connection to hypoellipticity).
In that sense the list was not predictive: it failed to register or anticipate the coming swift rises of topology, group theory, and measure theory in the twentieth century, as it didn't roll with the way mathematical logic would pan out. Therefore its value as a document is as an essay: a partial, personal view. It suggests some programmes of research and open-ended investigations.
In fact many of the questions posed belie the idea of a professional mathematician of the twenty-first century, or even of 1950, that the form of a solution to a good question would take the shape of a paper published in a mathematical learned journal. If that were the case for all twenty three problems, commentary would be simplified down to the point where either a journal reference can be given, or the question can be considered still open. In some cases the language used by Hilbert is still considered somewhat negotiable, as far as what the problem formulation actually means (in the absence, to repeat, of the axiomatic foundations, installed in pure mathematics starting with work of Hilbert himself on Euclidean geometry, through Principia Mathematica, and ending with the Bourbaki group and 'intellectual terrorism' to finish the job). The First and Fifth problems are, perhaps surprisingly, in an unsettled status because of less than full clarity in formulation (see notes). In cases such as the Twelfth, the problem can reasonably be taken as an 'inner', fairly accessible version in which it is quite plausible that we can know what Hilbert was driving at, and an 'outer', speculative penumbra.
With all qualifications, then, the major point is the swift acceptance of the Hilbert list by the mathematical community of the time (less of a conventional form of words than now, in that there were few research leaders and they generally were in a small number of European countries, and personally acquainted). The problems were closely studied; solving one made a reputation.
At least as influential as the problem content was the style. Hilbert asked for clarification. He asked for solutions in principle to algorithmic questions, not practical algorithms. He asked for foundational strength in parts of mathematics that were still guided by intuitions opaque to non-practitioners (Schubert calculus and enumerative geometry).
These attitudes carried over to many followers, though they were also contested, and continue to be. Thirty years later, Hilbert had only sharpened his position: see ignorabimus.
The problems as Hilbert's manifesto
It is quite clear that the problem list, and its manner of discussion, were meant to be influential. Hilbert in no way fell short of the expectations of German academia on empire-building, programmatic verve, and the explicit setting of a direction and claiming of ground for a school. No one now talks of the 'Hilbert school' in quite those terms; nor did the Hilbert problems just have their moment as Felix Klein's Erlangen programme did. Klein was a colleague of Hilbert's, and in comparison the Hilbert list is far less prescriptive. Michael Atiyah has characterised the Erlangen programme as premature. The Hilbert problems, by contrast, showed the good timing of an expert. In pharmaceutical terms they operated both within 20 minutes and by slow release.
If the 'school of Hilbert' means much, it probably refers to operator theory and the style of mathematical physics taking the Hilbert-Courant volumes as canonical. As was noted above, Hilbert did not use the list to pose problems directly about spectral theory. That, one could say, would have been in Klein's style. He also did not give any undue prominence to commutative algebra — ideal theory, as it would then have been known), his major algebraic contribution and preoccupation from his invariant theory days; nor, at least on the surface, did he preach against Leopold Kronecker, Georg Cantor's opponent, from whom he learned much but whose attitudes he almost detested (as is documented in Constance Reid's biography). The reader could draw ample conclusions from the presence of set theory at the head of the list.
The theory of functions of a complex variable, the branch of classical analysis that every pure mathematician would know, is though quite neglected: no Bieberbach conjecture or other neat question, short of the Riemann hypothesis. One of Hilbert's strategic aims was to have commutative algebra and complex function theory on the same level; this would, however, take 50 years (and still has not resulted in a changing of places).
Hilbert had a small peer group: Adolf Hurwitz and Hermann Minkowski were both close friends and intellectual equals. There is a nod to Minkowski's geometry of numbers in problem 18, and to his work on quadratic forms in problem 11. Hurwitz was the great developer of Riemann surface theory. Hilbert used the function field analogy, a guide in algebraic number theory by the use of geometric analogues, in developing class field theory within his own research, and this is reflected in problem 9, to some extent in problem 12, and in problems 21 and 22. Otherwise Hilbert's only rival in 1900 was Henri Poincaré, and the second part of problem 16 is a dynamical systems question in Poincaré's style.
A round two dozen
Hilbert originally included 24 problems on his list, but decided against including one of them in the published list. The "24th problem" (in proof theory, on a criterion for simplicity and general methods) was rediscovered in Hilbert's original manuscript notes by German historian Rüdiger Thiele in 2000.
Of the cleanly-formulated Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19 and 20 have a resolution that is accepted by consensus. On the other hand, problems 1, 2, 5, 9, 15, 18+, 21, and 22 have solutions that have partial acceptance, but where there exists some controversy as to whether it resolves the problem.
The + on 18 denotes that the Kepler problem solution is a computer-assisted proof, a notion anachronistic for a Hilbert problem and also to some extent controversial because of its lack of verifiability by a human reader in a reasonable time.
That leaves 8 (the Riemann hypothesis) and 12 unresolved, both being in number theory. On this classification 4, 6, 16, and 23 are too loose to be ever described as solved. The withdrawn 24 would also fall in this class.
Hilbert's twenty-three problems are:
|1st||The continuum hypothesis (that is, there is no set whose size is strictly between that of the integers and that of the real numbers)||No consensus|
|2nd||Prove that the axioms of arithmetic are consistent (that is, that arithmetic is a formal system that does not prove a contradiction).||Partially resolved: Some hold it has been shown impossible to establish in a consistent, finitistic axiomatic system  - However, Gentzen proved in 1936 that consistency of arithmetic followed from the well-foundedness of the ordinal , a fact amenable to combinatorial intuition.|
|3rd||Can two tetrahedra be proved to have equal volume (under certain assumptions)?||Resolved - no, using Dehn invariants.|
|4th||Construct all metrics where lines are geodesics.||Too vague|
|5th||Are continuous groups automatically differential groups?||Resolved|
|6th||Axiomatize all of physics||Non-mathematical|
|7th||Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ?||Resolved - Yes - Gelfond's theorem or the Gelfond–Schneider theorem).|
|8th||The Riemann hypothesis (the real part of any non-trivial zero of the Riemann zeta function is ½) and Goldbach's conjecture (every even number greater than 2 can be written as the sum of two prime numbers).||Open|
|9th||Find most general law of the reciprocity theorem in any algebraic number field||Partially resolved|
|10th||Determination of the solvability of a Diophantine equation||Resolved - Matiyasevich's theorem implies that this is impossible.|
|11th||Solving quadratic forms with algebraic numerical coefficients.||Resolved|
|12th||Extend Kronecker's theorem on abelian extensions of the rational numbers to any base number field.||Open|
|13th||Solve all 7-th degree equations using functions of two parameters.||Resolved|
|14th||Proof of the finiteness of certain complete systems of functions.||Resolved|
|15th||Rigorous foundation of Schubert's enumerative calculus.||Resolved|
|16th||Topology of algebraic curves and surfaces.||Open|
|17th||Expression of definite rational function as quotient of sums of squares||Resolved|
|18th||Is there a non-regular, space-filling polyhedron? What is the densest sphere packing?||Resolved|
|19th||Are the solutions of Lagrangians always analytic?||Resolved|
|20th||Do all variational problems with certain boundary conditions have solutions?||Resolved|
|21st||Proof of the existence of linear differential equations having a prescribed monodromic group||Resolved|
|22nd||Uniformization of analytic relations by means of automorphic functions||Resolved|
|23rd||Further development of the calculus of variations||Resolved|
- ^ Cohen's independence result, showing the continuum hypothesis to be independent of ZFC (Zermelo-Frankel set theory, extended to include the axiom of choice) is often cited to justify the assertion that the first problem has been solved. One contemporary view is that it may be the case that set theory should have additional axioms, capable of settling the problem.
- ^ A matter of opinion, not shared by all. Gentzen's result shows rather precisely how much needs to be assumed to prove that Peano arithmetic is consistent. It is widely held that Gödel's second incompleteness theorem shows that there is no finitistic proof that PA is consistent (though Gödel himself disclaimed this inference [this needs a better reference-- but cf Dawson p.71ff "...Gödel too [like Hilbert] believed that no mathematical problems lay beyond the reach of human reason. Yet his results showed that the program that Hilbert had proposed to validate that belief -- his proof theory -- could not be carried through as Hilbert had envisioned" (p.71) See also p. 98ff for more discussion of 'finite procedure').
- ^ According to Rowe & Gray (see reference below), most of the problems have been solved. Some were not completely defined, but enough progress has been made to consider them "solved"; Rowe & Gray lists the fourth problem as too vague to say whether it has been solved.
- ^ Problem 8 contains two famous problems, both of which remain unsolved. The first of them, the Riemann hypothesis, is one of the seven Millennium Prize Problems, which were intended to be the "Hilbert Problems" of the 21st century.
- ^ Problem 9 has been solved in the abelian case, by the development of class field theory; the non-abelian case remains unsolved, if one interprets that as meaning non-abelian class field theory.
- ^ Rowe & Gray also list the 18th problem as "open" in their 2000 book, because the sphere-packing problem (also known as the Kepler conjecture) was unsolved, but a solution to it has now been claimed (see reference below).
- Listing of the 23 problems, with descriptions of which have been solved
- Original text of Hilbert's talk, in German
- English translation of Hilbert's 1900 address
- Details on the solution of the 18th problem
- The Mathematical Gazette, March 2000 (page 2-8) "100 Years On"
- "On Hilbert's 24th Problem: Report on a New Source and Some Remarks."
- Rowe, David; Gray, Jeremy J. (2000). The Hilbert Challenge. Oxford University Press. ISBN 0198506511
- Yandell, Benjamin H. (2002). The Honors Class. Hilbert's Problems and Their Solvers. A K Peters. ISBN 1568811411
- On Hilbert and his 24 Problems. In: Proceedings of the Joint Meeting of the CSHPM 13(2002)1-22 (26th Meeting; ed. M. Kinyon)
- Nagel, Ernest and Newman, James R., Godel's Proof, New York University Press, 1958. A wonderful (readable, thorough) presentation of Gödel's Proof, with commentary.
- John W. Dawson, Jr, Logical Dilemmas, The Life and Work of Kurt Gödel, AK Peters, Wellesley, Mass., 1997. Difficult, confusing biography. But a wealth of information relevant to Hilbert's "program" and Gödel's impact on the Second Question, the impact of Arend Heyting's and Brouwer's Intuitionism on Hilbert's philosophy. Dawson is Professor of Mathematics at Penn State U, cataloguer of Godel's papers for the Institute for Advanced Study in Princeton, and a co-editor of Godel's Collected Works.