# 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. Earlier publications (in the original German) appeared in Archiv der Mathematik und Physik.
Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, 21, and 20 have resolutions that are accepted by consensus of the mathematical community. Problems 1, 2, 5, 6, 9, 11, 12, 15, 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 unresolved. Problems 4 and 23 are considered as too vague to ever be described as solved; the withdrawn 24 would also be in this class.

## List of Hilbert's Problems

The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in the Bulletin of the American Mathematical Society.
    1. Cantor's problem of the cardinal number of the continuum.
    2. The compatibility of the arithmetical axioms.
    3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.
    4. Problem of the straight line as the shortest distance between two points.
    5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.
    6. Mathematical treatment of the axioms of physics.
    7. Irrationality and transcendence of certain numbers.
    8. Problems of prime numbers (The "Riemann Hypothesis").
    9. Proof of the most general law of reciprocity in any number field.
    10. Determination of the solvability of a Diophantine equation.
    11. Quadratic forms with any algebraic numerical coefficients
    12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality
    13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments.
    14. Proof of the finiteness of certain complete systems of functions.
    15. Rigorous foundation of Schubert's enumerative calculus.
    16. Problem of the topology of algebraic curves and surfaces.
    17. Expression of definite forms by squares.
    18. Building up of space from congruent polyhedra.
    19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?
    20. The general problem of boundary values (Boundary value problems in PD)
    21. Proof of the existence of linear differential equations having a prescribed monodromy group.
    22. Uniformization of analytic relations by means of automorphic functions.
    23. Further development of the methods of the calculus of variations.

## The 24th problem

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.

## Nature and influence of the problems

Hilbert's problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer. For other problems, such as the 5th, experts have traditionally agreed on a single interpretation, and a solution to the accepted interpretation has been given, but closely related unsolved problems exist. Some of Hilbert's statements were not precise enough to specify a particular problem, but were suggestive enough that certain problems of contemporary nature seem to apply; for example, 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, such as the 11th and the 16th, concern what are now flourishing mathematical subdisciplines, like the theories of quadratic forms and real algebraic curves.
There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the axiomatization of physics, a goal that 20th-century developments seem to render both more remote and less important than in Hilbert's time. Also, the 4th problem concerns the foundations of geometry, in a manner that is now generally judged to be too vague to enable a definitive answer.
The 23rd problem was purposefully set as a general indication by Hilbert to highlight the calculus of variations as an underappreciated and understudied field. In the lecture introducing these problems, Hilbert made the following introductory remark to the 23rd problem:
   "So far, I have generally mentioned problems as definite and special as possible, in the opinion that it is just such definite and special problems that attract us the most and from which the most lasting influence is often exerted upon science. Nevertheless, I should like to close with a general problem, namely with the indication of a branch of mathematics repeatedly mentioned in this lecture—which, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, is its due—I mean the calculus of variations."
The other 21 problems have all received significant attention, and late into the 20th century work on these problems was still considered to be of the greatest importance. 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 Yuri Matiyasevich (completing work by Julia Robinson, Hilary Putnam, and Martin Davis) generated similar acclaim. Aspects of these problems are still of great interest today.

## Knowability

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. 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 Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work.
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. 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). It seems unclear whether he would have regarded the solution of the tenth problem as an instance of ignorabimus.
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.

## Follow-ups

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. 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.
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 (Smale's 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?"
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". The DARPA list also includes a few problems from Hilbert's list, e.g. the Riemann hypothesis.

## Table of problems

Hilbert's 23 problems are (for details on the solutions and references, see the articles that are linked to in the first column):
| Problem | Brief explanation | Status | Year solved |
| 1st | The continuum hypothesis (that is, there is no set whose cardinality is strictly between that of the integers and that of the real numbers) | Proven to be impossible to prove or disprove within Zermelo–Fraenkel set theory with or without the axiom of choice (provided Zermelo–Fraenkel set theory is consistent, i.e., it does not contain a contradiction). There is no consensus on whether this is a solution to the problem. | 1940, 1963 |
| 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 ε$_0$. | 1931, 1936 |
| 3rd | Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second? | Resolved. Result: No, proved using Dehn invariants. | 1900 |
| 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, assuming one interpretation of the original statement. If, however, it is understood as an equivalent of the Hilbert–Smith conjecture, it is still unsolved. | 1953? |
| 6th | Mathematical treatment of the axioms of physics: (a) axiomatic treatment of probability with limit theorems for foundation of statistical physics (b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua" | Unresolved, or partially resolved, depending on how the original statement is interpreted. Items (a) and (b) were two specific problems given by Hilbert in a later explanation. Kolmogorov's axiomatics (1933) is now accepted as standard for the foundations of probability theory. There is some success on the way from the "atomistic view to the laws of motion of continua", but the transition from classical to quantum physics means that there would have to be two axiomatic formulations, with a clear link between them. John von Neumann made an early attempt to place Quantum Mechanics on a rigorous mathematical basis in his book Mathematical Foundations of Quantum Mechanics, but subsequent developments have occurred, further challenging the axiomatic foundations of quantum physics. | 1933–2002? |
| 7th | Is a$^b$ transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? | Resolved. Result: Yes, illustrated by the Gelfond–Schneider theorem. | 1934 |
| 8th | The Riemann hypothesis ("the real part of any non-trivial zero of the Riemann zeta function is 1/2") and other prime-number problems, among them Goldbach's conjecture and the twin prime conjecture | Unresolved. | — |
| 9th | Find the most general law of the reciprocity theorem in any algebraic number field. | Partially resolved. Solved by Emil Artin in 1927 for Abelian extensions of the rational numbers during the development of class field theory. The non-abelian case remains unsolved, if one interprets that as meaning non-abelian class field theory. | — |
| 10th | Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. | Resolved. Result: Impossible; Matiyasevich's theorem implies that there is no such algorithm. | 1970 |
| 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. | Partially resolved. | — |
| 13th | Solve 7th-degree equation using algebraic (variant: continuous) functions of two parameters. | Unresolved. The continuous variant of this problem was solved by Vladimir Arnold in 1957 based on work by Andrei Kolmogorov (see Kolmogorov–Arnold representation theorem), but the algebraic variant is unresolved. | — |
| 14th | Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? | Resolved. Result: No, a counterexample was constructed by Masayoshi Nagata. | 1959 |
| 15th | Rigorous foundation of Schubert's enumerative calculus. | Partially resolved. Haibao Duan and Xuezhi Zhao claimed that this problem is actually resolved. | — |
| 16th | Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane. | Unresolved, even for algebraic curves of degree 8. | — |
| 17th | Express a nonnegative rational function as quotient of sums of squares. | Resolved. Result: Yes, due to Emil Artin. Moreover, an upper limit was established for the number of square terms necessary. | 1927 |
| 18th | (a) Are there only finitely many essentially different space groups in n-dimensional Euclidean space? | Resolved. Result: Yes (by Ludwig Bieberbach) | 1910 |
| 18th | (b) Is there a polyhedron that admits only an anisohedral tiling in three dimensions? | Resolved. Result: Yes (by Karl Reinhardt). | 1928 |
| 18th | (c) What is the densest sphere packing? | Widely believed to be resolved, by computer-assisted proof (by Thomas Callister Hales). Result: Highest density achieved by close packings, each with density approximately 74%, such as face-centered cubic close packing and hexagonal close packing. | 1998 |
| 19th | Are the solutions of regular problems in the calculus of variations always necessarily analytic? | Resolved. Result: Yes, proven by Ennio De Giorgi and, independently and using different methods, by John Forbes Nash. | 1957 |
| 20th | Do all variational problems with certain boundary conditions have solutions? | Partially resolved. A significant topic of research throughout the 20th century, resulting in solutions for some cases. | ? |
| 21st | Proof of the existence of Fuchsian linear differential equations having a prescribed monodromy group | Resolved. Result: No, shown by A. Bolibrukh. | 1989 |
| 22nd | Uniformization of analytic relations by means of automorphic functions | Partially resolved. Uniformization theorem | ? |
| 23rd | Further development of the calculus of variations | Too vague to be stated resolved or not. | — |