Hilbert's problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved 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 Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.[1] Earlier publications (in the original German) appeared in Archiv der Mathematik und Physik.[2]
Knowability[edit]
Following Gottlob Frege and Bertrand Russell, Hilbert sought to define mathematics logically using the method of formal systems, i.e., finitistic proofs from an agreed-upon set of axioms.[4] One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.[a]
However, Gödel's second incompleteness theorem gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible. Hilbert lived for 12 years after Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work.[b][c]
Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers". That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.
In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible.[d] He stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is not any "ignorabimus" (statement whose truth can never be known).[e] It seems unclear whether he would have regarded the solution of the tenth problem as an instance of ignorabimus: what is proved not to exist is not the integer solution, but (in a certain sense) the ability to discern in a specific way whether a solution exists.
On the other hand, the status of the first and second problems is even more complicated: there is no clear mathematical consensus as to whether the results of Gödel (in the case of the second problem), or Gödel and Cohen (in the case of the first problem) give definitive negative solutions or not, since these solutions apply to a certain formalization of the problems, which is not necessarily the only possible one.[f]
Follow-ups[edit]
Since 1900, mathematicians and mathematical organizations have announced problem lists but, with few exceptions, these have not had nearly as much influence nor generated as much work as Hilbert's problems.
One exception consists of three conjectures made by André Weil in the late 1940s (the Weil conjectures). In the fields of algebraic geometry, number theory and the links between the two, the Weil conjectures were very important.[8][9] The first of these was proved by Bernard Dwork; a completely different proof of the first two, via ℓ-adic cohomology, was given by Alexander Grothendieck. The last and deepest of the Weil conjectures (an analogue of the Riemann hypothesis) was proved by Pierre Deligne. Both Grothendieck and Deligne were awarded the Fields medal. However, the Weil conjectures were, in their scope, more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics. This is somewhat ironic, since arguably Weil was the mathematician of the 1940s and 1950s who best played the Hilbert role, being conversant with nearly all areas of (theoretical) mathematics and having figured importantly in the development of many of them.
Paul Erdős posed hundreds, if not thousands, of mathematical problems, many of them profound. Erdős often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.[10]
The end of the millennium, which was also the centennial of Hilbert's announcement of his problems, provided a natural occasion to propose "a new set of Hilbert problems". Several mathematicians accepted the challenge, notably Fields Medalist Steve Smale, who responded to a request by Vladimir Arnold to propose a list of 18 problems.
At least in the mainstream media, the de facto 21st century analogue of Hilbert's problems is the list of seven Millennium Prize Problems chosen during 2000 by the Clay Mathematics Institute. Unlike the Hilbert problems, where the primary award was the admiration of Hilbert in particular and mathematicians in general, each prize problem includes a million-dollar bounty. As with the Hilbert problems, one of the prize problems (the Poincaré conjecture) was solved relatively soon after the problems were announced.
The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise. Although it has been attacked by major mathematicians of our day, many experts believe that it will still be part of unsolved problems lists for many centuries. Hilbert himself declared: "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proved?"[11]
In 2008, DARPA announced its own list of 23 problems that it hoped could lead to major mathematical breakthroughs, "thereby strengthening the scientific and technological capabilities of the DoD".[12][13][14] The DARPA list also includes a few problems from Hilbert's list, e.g. the Riemann hypothesis.
Summary[edit]
Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, and 20 have resolutions that are accepted by consensus of the mathematical community. Problems 1, 2, 5, 6,[g] 9, 11, 12, 15, 21, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether they resolve the problems.
That leaves 8 (the Riemann hypothesis), 13 and 16[h] unresolved, and 4 and 23 as too vague to ever be described as solved. The withdrawn 24 would also be in this class.
