Hilbert's problems

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

Hilbert's problems ranged widely, not only across many areas of pure and applied mathematics, but also in scope and precision. Some of them are propounded precisely enough to admit a clear affirmative/negative answer, like the 3rd problem (probably the easiest for a nonspecialist to understand and also the first to be solved) or the notorious 8th problem (the Riemann hypothesis). There are other problems (notably the 5th) for which experts have traditionally agreed on a single interpretation and a solution to the accepted interpretation has been given, but for which there remain open problems which are so closely related so as to be, perhaps, part of what Hilbert had in mind. 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 fit in nicely, 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 (e.g. the 11th and the 16th) call for work on what are now flourishing mathematical subdisciplines, like the theories of quadratic forms and real algebraic curves.

There are two problems which are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem calls for the axiomatization of physics, a goal that twentieth century developments in physics (including its recognition as a discipline independent from mathematics) seem to render both more remote and less important than in Hilbert's day. Also the 4th problem concerns the foundations of geometry, in a manner which is now generally judged to be too vague to admit a definitive answer.

Remarkably, 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 highest importance. Notably, Paul Cohen received the Fields Medal in 1966 for his work on the first problem, and the negative solution of the tenth problem in 1970 by Matiyasevich (completing work of Davis, Putnam and Robinson) generated similar excitement and acclaim. Aspects of these problems are still of great interest today.

Ignorabimus

Several of the Hilbert problems have been resolved (or arguably resolved) in ways that would have been profoundly surprising, and even disturbing, to Hilbert himself. Following Frege and Russell, Hilbert sought to place mathematics on a sound logical foundation using the method of formal systems, i.e., finitistic proofs from an agreed upon set of axioms. 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.

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 Gödel's theorem, but he does not seem to have written any formal response to or acknowledgement of Gödel's work. But there is no doubt that the significance of Gödel's work to mathematics as a whole (and not just the field of formal logic) was amply and dramatically illustrated by its applicability to one of Hilbert's problems.

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 the solution to this problem comes by way of showing that there can be no such algorithm would presumably have been very surprising to him.

In discussing his opinions 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. Famously, 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 no ignorabimus. It seems unclear whether he would have regarded the solution of the tenth problem as an instance of ignorabimus: what we are proving not to exist is not the integer solution, but (in a certain sense) our own ability to discern 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, a formalization which is quite reasonable but is not necessarily the only possible one.

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.

Sequels

Since 1900, other mathematicians and mathematical organizations have announced problem lists, but, with few exceptions, these later problem collections have not had nearly as much influence or generated as much work as Hilbert's problems.

One of the exceptions is furnished by 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 importance of the Weil conjectures was (and is) colossal. The first of the Weil conjectures was proved by Bernard Dwork, and a completely different proof of the first two conjectures via l-adic cohomology was given by Alexander Grothendieck. The last and deepest of the Weil conjectures (an analogue of the Riemann hypothesis) was proven by Pierre Deligne in what is widely regarded as one of the greatest mathematical achievements of all time. Both Grothendieck and Deligne were awarded the Fields medal. However, the Weil conjectures in their scope are 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 played key roles in the development of many of them.

The turn of the millennium, being also the centennial of Hilbert's announcement of his problems, was a natural occasion to propose "a new set of Hilbert problems." Several mathematicians took up the challenge, notably Fields Medalist Steve Smale, who responded to a request of Vladimir Arnold by proposing a list of 18 problems. Smale's problems have thus far not received much attention from the media, and it seems unclear how much attention they are getting from the mathematical community.

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 in 2000 by the Clay Mathematics Institute. Unlike the Hilbert problems, where the primary award was the admiration of Hilbert in particular and the worldwide mathematical community in general, each prize problem carries a million dollar bounty. As for the Hilbert problems, one of the prize problems (the Poincare conjecture) was solved relatively soon after the problems were announced.

Noteworthy for its appearance on the list of Hilbert problems, Smale's list and the list of Millennium Prize Problems -- and even, in its geometric guise, in the Weil Conjectures -- is the Riemann hypothesis. Notwithstanding some famous (and occasionally loud) recent assaults from leading mathematicians of our day, many experts believe that the Riemann hypothesis will lurk at the top of problem lists for some centuries yet to come. The maestro himself had this to say: "If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann hypothesis been proven?" — David Hilbert

Summary

Of the cleanly-formulated Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19, 20, and 21 have a resolution that is accepted by consensus. On the other hand, problems 1, 2, 5, 9, 12, 15, 18⁺, 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 conjecture 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.

Table of problems

Hilbert's twenty-three problems are:

Problem	Brief explanation	Status
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)	Proven to be impossible to prove or disprove within the Zermelo-Frankel set theory with or without the Axiom of Choice. There is no consensus on whether this is a solution to the problem.
2nd	Prove that the axioms of arithmetic are consistent.	There is no consensus on whether results of Gödel and Gentzen give a solution to the problem as stated by Hilbert. Gödel's second incompleteness theorem, proved in 1931, shows that no proof of its consistency can be carried out within arithmetic itself. Gentzen proved in 1936 that the consistency of arithmetic follows from the well-foundedness of the ordinal ε0.
3rd	Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?	Resolved. Result: no, proved using Dehn invariants.
4th	Construct all metrics where lines are geodesics.	Too vague to be stated resolved or not.
5th	Are continuous groups automatically differential groups?	Resolved by Andrew Gleason, depending on how the original statement is interpreted. If, however, it is understood as an equivalent of the Hilbert-Smith conjecture, it is still unsolved.
6th	Axiomatize all of physics	Unresolved.
7th	Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ?	Resolved. Result: yes, illustrated by 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).	Unresolved.
9th	Find most general law of the reciprocity theorem in any algebraic number field	Partially resolved.
10th	Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.	Resolved. Result: no, Matiyasevich's theorem implies that there is no such algorithm.
11th	Solving quadratic forms with algebraic numerical coefficients.	Partially resolved.
12th	Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field.	Unresolved.
13th	Solve all 7-th degree equations using functions of two parameters.	A variant of this problem, looking for a solution within the universe of continuous functions, was solved (negatively) by Andrei Kolmogorov and Vladimir Arnold. It is not difficult to show that the problem has a positive solution within the space of single-valued analytic functions (Raudenbush). Some authors argue that Hilbert intended for a solution within the space of (multi-valued) algebraic functions, thus continuing his own work on algebraic functions and being a question about a possible extension of the Galois theory (see, for example, Abyankar , Vitushkin , Chebotarev and others). It appears from one of the Hilbert's papers that this was the his original intention for the problem. As such, the problem is still unresolved.
14th	Proof of the finiteness of certain complete systems of functions.	Resolved. Result: no, generally, due to counterexample made by Masayoshi Nagata.
15th	Rigorous foundation of Schubert's enumerative calculus.	Partially resolved.
16th	Topology of algebraic curves and surfaces.	Unresolved.
17th	Expression of definite rational function as quotient of sums of squares	Resolved. Result: An upper limit was established for the number of square terms necessary.
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. Result: yes, proven by Ennio de Giorgi and, independently and using different methods, by John Forbes Nash.
20th	Do all variational problems with certain boundary conditions have solutions?	Resolved. A significant area of research throughout the 20th century, culminating in solutions for the non-linear case.
21st	Proof of the existence of linear differential equations having a prescribed monodromic group	Resolved. Result: Yes or no, depending on more exact formulations of the problem.
22nd	Uniformization of analytic relations by means of automorphic functions	Resolved.
23rd	Further development of the calculus of variations	Resolved.

External links

• 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
• "On Hilbert's 24th Problem: Report on a New Source and Some Remarks."
• The Paris Problems
• Hilbert's Tenth Problem page!

References

• Rowe, David; Gray, Jeremy J. (2000). The Hilbert Challenge. Oxford University Press. ISBN 0-19-850651-1
• Yandell, Benjamin H. (2002). The Honors Class. Hilbert's Problems and Their Solvers. A K Peters. ISBN 1-56881-141-1
• On Hilbert and his 24 Problems. In: Proceedings of the Joint Meeting of the CSHPM 13(2002)1-22 (26th Meeting; ed. M. Kinyon)
• John Dawson, Jr Logical Dilemmas, The Life and Work of Kurt Gödel, AK Peters, Wellesley, Mass., 1997. 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 Gödel's papers for the Institute for Advanced Study in Princeton, and a co-editor of Gödel's Collected Works.
• Felix E. Browder (editor), Mathematical Developments Arising from Hilbert Problems, Proceedings of Symposia in Pure Mathematics XXVIII (1976), American Mathematical Society. A collection of survey essays by experts devoted to each of the 23 problems emphasizing current developments.
• Yuri Matiyasevich, Hilbert's Tenth Problem, MIT Press, Cambridge, Massachusetts, 1993. An account at the undergraduate level by the mathematican who completed the solution of the problem.