In 1900 David Hilbert went to the second International Congress of Mathematicians in Paris to give an invited paper. He spoke on The Problems of Mathematics, to such effect that Hermann Weyl later referred to anyone who solved one of the 23 problems that Hilbert presented as entering the honours class of mathematicians. Throughout the 20th century the solution of a problem was the occasion for praise and celebration. David Hilbert around 1900

Hilbert in 1900
By 1900 Hilbert had emerged as the leading mathematician in Germany. He was famous for his solution of the major problems of invariant theory, and for his great Zahlbericht, or Report on the theory of numbers, published in 1896. In 1899, at Klein's request, Hilbert published The foundations of geometry as part of the commemorations of Gauss and Weber in Göttingen. Hurwitz saw clearly that the implications of that little book reached far beyond its immediate field. As he put in a letter to Hilbert: You have opened up an immeasurable field of mathematical investigation which can be called the "mathematics of axioms" and which goes far beyond the domain of geometry.

Hermann Minkowski

Hilbert was therefore poised to lead the international community of mathematicians. He consulted with his friends Minkowski and Hurwitz, and Minkowski advised him to seize the moment, writing: Most alluring would be the attempt to look into the future, in other words, a characterisation of the problems to which the mathematicians should turn in the future. With this, you might conceivably have people talking about your speech even decades from now. And if the Congress itself was something of a shambles, Hilbert's written text nonetheless made its impact, just as Minkowski had predicted.

One reason is undoubtedly that, for Hilbert, problem solving and theory formation went hand in hand. Indeed, some of his problems are not problems at all, but whole programmes of research: Hilbert's 6th problem, for example, calls for the axiomatisation of physics. But he gave good reasons for caring about his problems, and bound it all up with his inspiring optimism. In opposition to Emil du Bois-Reymond's fashionable academic pessimism, Hilbert insisted that in mathematics: We can know, and we shall know.

Hilbert presents his problems
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.
The Hilbert problems very quickly succeeded in getting young mathematicians to create the future that Hilbert had conjectured - not always accurately, however, as we shall see. The Russian mathematician Serge Bernstein travelled from Paris to Göttingen in 1904 to present his proof that, under the conditions Hilbert had stated, elliptic partial differential equations do have analytic solutions. It would take more than a single book to describe the results produced in this mushrooming field in the 20th century. The opportunistic and power-seeking Bieberbach was another man drawn to the Hilbert problems for the fame that they could confer. In 1908 he showed that there are only a finite number of crystallographic groups in a Euclidean space of any dimension, thus solving part of Hilbert's 18th problem. His results confirmed that there are 17 patterns of this kind for the Euclidean plane, and 219 patterns (or crystal structures) for Euclidean 3-dimensional space.

Hilbert and axiomatisation
Hilbert himself did not work exclusively on the problems, and nor did most of his many students, who were instead often drawn in the early 1900s to the study of `Hilbert space'. After 1909, when his friend Minkowski died, Hilbert became more and more interested in questions of axiomatisation and the foundations of mathematics. His interest in axiomatising physics was to lead to several lecture courses but few publications, and has accordingly been much misunderstood (see Leo Corry's work for the new picture). Hilbert lectured regularly and well, and he promoted the view that an axiom was a fundamental idea from which many others followed. So axioms played a crucial role in organising theories, as they had done in his Foundations of geometry.
As the years went by, the consistency of arithmetic seemed more of a challenge. In 1917 Hilbert wrote: Since the examination of the consistency is a task that cannot be avoided, it appears necessary to axiomatise logic itself and to prove that number theory and set-theory are only parts of logic. He imposed high standards on the task. What was needed, he said, were proofs that:

By 1922 Hilbert became locked into a dispute with Brouwer, who believed that the human mind was strikingly limited in its ability to deal with infinite sets. Hilbert's hopes for the future of mathematics were further darkened by the fact that his best student, Hermann Weyl, also found Brouwer's ideas attractive, and for a time seemed willing to forgo certain mathematical arguments on philosophical grounds. Hilbert attempted to get round their arguments in 1922 by defining mathematics as a system of signs and distinguishing carefully between mathematics and meta-mathematics. Mathematics was to be identified with the stock of provable formulae, while inference about the content was only admissible at the level of a new meta-mathematics. On the basis of this distinction between valid formulae and their interpretations he called for a proof theory - truly a remarkable idea.
In the event Brouwer withdrew from the contest, Weyl found that he needed classical analysis for his work on Lie groups, and the crisis passed, although Hilbert's ideas about proof theory did not convince the experts. But Hilbert continued, with his assistants Ackermann and Bernays, and in 1931 he published a forceful re-statement of his views on the occasion of his becoming an honorary citizen of his native Königsberg. The lecture ends with a moving affirmation of his deepest belief about mathematics: There are absolutely no unsolvable problems. Instead of the foolish ignorabimus, our answer is on the contrary: We must know, We shall know.
Ironically, the day before Hilbert lectured, the young Austrian logician Kurt Gödel also lectured in Königsberg on his incompleteness theorem, the work that is popularly said to have killed Hilbert's programme, even though, as Gödel said in the famous paper: I wish to note expressly that [this theorem does] not contradict Hilbert's formalistic viewpoint.
The problem is that finding such proofs has proved elusive. No agreed place to stand has been found which compels universal assent (such as the elementary rules of logic) and which delivers all of set theory. On the other hand, powerful negative results continued to accumulate. When Alan Turing showed in 1936 that the decision problem is also unsolvable, the original hopes for Hilbert's programme were all in tatters.

The Hilbert problems between the Wars
The Hilbert problems themselves have perhaps proved a more enduring part of his legacy than Hilbert's own work on mathematical logic. Those on number theory have turned out, given Hilbert's own interests, to have been particularly well put. Ironically, Hilbert's own formulation of Kronecker's Jugendtraum was misleading, but the Japanese mathematician Takagi, who had studied under Hilbert in Göttingen in 1900 while writing his thesis for the University of Tokyo, succeeded with the crucial generalisation to the abelian case in 1920. In 1923 Emil Artin bumped into Takagi's work almost by accident and the work he did as a result enabled Hasse to solve Hilbert's 9th Problem (calling for a general reciprocity law) in 1927. Hasse had meanwhile also solved Hilbert's 11th problem, on quadratic forms, in 1923. In 1926 Artin solved Hilbert's 17th problem in a restricted, but nonetheless very general, case. Hilbert's 7th Problem (to show that a^b is an irrational transcendental number when a is algebraic and not equal to 0 or 1, and b is irrational and algebraic) was solved by the Soviet mathematician A.O.Gelfond in 1934, and by Th.Schneider independently later the same year.

The Bourbaki connection
After the Second World War the struggle for the heart of mathematics was won by the pure mathematicians. In this context a vigorous contest developed for the mantle of Hilbert. Should it go to mathematical logicians, or to applied mathematicians working in the tradition of Courant and Hilbert, or to the number theorists and algebraists? The most powerful advocate of the last of these views, both by word and deed, was Bourbaki. André Weil and Jean Dieudonné shared a view of mathematics that put Hilbert centre stage, although it was a Hilbert created in their own image. They espoused the axiomatic method, which Dieudonné claimed has revealed unsuspected analogies and permitted extended generalizations; the origin of the modern developments of algebra, topology and group theory is to he found only in the employment of axiomatic methods. André Weil made the connection to Hilbert even more forcefully. In 1947 he quoted Hilbert: A branch of science is full of life, as long as it offers an abundance of problems; a lack of problems is a sign of death. Great problems, said Weil, furnish the daily bread on which the mathematician thrives. And turning to Hilbert's famous list of problems he singled out the 5th problem (then still unsolved) on Lie groups, the Riemann hypothesis, and the problem of generalising the theorems of Kronecker's Jugendtraum which `still escapes us, in spite of the conjectures of Hilbert himself and the efforts of his pupils'.

The Hilbert problems after the War
Hilbert's Paris address had brilliantly united problems with their theoretical context and so, for a generation, did Bourbaki. The stock of the Hilbert problems rose with theirs. The remaining ones acquired an extra cachet for having held out, and they too began to fall. The 5th problem, on characterising Lie groups, was solved through the work of Gleason and of Montgomery and Zippin in 1952. The 14th problem on rings of invariants was interpreted geometrically by Zariski and then solved in the negative by Nagata in 1959. In the 1960s and 1970s mathematical logicians turned back to the Hilbert problems. A high point was reached with the award of a Fields Medal in 1966 to Paul Cohen for showing that the axiom of choice and the continuum hypothesis are independent of the other axioms of set theory. The axiom of choice is not one of Hilbert's problems, but in calling for a consistent set of axioms for arithmetic, Hilbert had opened the way to similar analyses of all of mathematics, and indeed by establishing the independence of the continuum hypothesis, Cohen did indeed settle Hilbert's 1st problem.
In the early 1970s the Russian mathematician Yuri Matijasevich solved Hilbert's 10th problem (Is there a finite process which determines if a polynomial equation is solvable in integers?) in the negative, using earlier papers by the American mathematician Martin Davis, Hilary Putnam, and Julia Robinson. Julia Robinson had been close to solving this problem herself, and a powerful collaboration developed between her and Matijasevich, despite the Cold War. It follows from their work that had Hilbert's 10th problem been answered positively, Goldbach's conjecture (mentioned by Hilbert in Paris) would have been answered in the negative - a connection that Hilbert surely had not suspected.
Few of Hilbert's problems have dwindled with the years. Perhaps the complete solution of the 5th problem is a case in point. In 1986 Jean-Pierre Serre said: Still, it is true that sometimes a theory can be killed. A well-known example is Hilbert's 5th problem. ... When I was a young topologist, that was a problem I really wanted to solve - but I could get nowhere. It was Gleason, and Montgomery-Zippin, who solved it, and their solution all but killed the problem. What else is there to find in this direction? I can only think of one question: [but it] seems quite hard - but a solution would have no application whatsoever, as far as I can see.
Yet there are numerous examples where the Hilbert touch has proved beneficial. The great development of algebraic number theory was surely animated by Hilbert's problems. The topic of partial differential equations was re-opened by Hilbert with his 19th problem, and much of that rich theory can be traced back to the work it inspired. Not only are the implications of the solutions and reformulations of the problems still to be worked out, some of the problems are still alive. Russian mathematicians have recently shown that in two cases the original `solutions' were flawed, and have given different and rigorous accounts. Aspects of both halves of the 16th problem (on real algebraic curves and on vector fields) are still open questions. Hilbert's deepest vision, of the intricate dance of theory and problems in mathematics, is one that all mathematicians share, but few have articulated as well as he.