User:Charles Matthews/Hilbert problems

Nature and influence of the problems

The list, which was put forth tentatively, proved a resounding success. This 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. In 1900 Hilbert could not appeal to axiomatic set theory, the Lebesgue integral, topological spaces or Church's thesis, each of which would permanently change 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 had it, about spectral theory.

In that sense the list was not predictive. 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 could be given, or the question could 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. 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 the reader 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.

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.

Sequels

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. Landau's problems are only in prime number theory. John von Neumann produced a list, but not to universal acclaim. Smale's problems from 2000 are too recent for any verdicts.

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.

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, the list poses no problems directly about spectral theory. Hilbert 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.