Hilbert's problems are 23 problems inmathematics published by German mathematicianDavid 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 theParis conference of theInternational Congress of Mathematicians, speaking on August 8 at theSorbonne. The complete list of 23 problems was published later, and translated into English in 1902 byMary Frances Winston Newson in theBulletin of the American Mathematical Society.[1] Earlier publications (in the original German) appeared inArchiv der Mathematik und Physik.[2]
Of the cleanly formulated Hilbert problems, numbers 3, 6,[a] 7, 10, 14, 17, 18, 19, and 21 have resolutions that are accepted by consensus of the mathematical community. Problems 1, 2, 5, 9, 11, 12, 15, 20, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether they resolve the problems. That leaves 8 (theRiemann hypothesis), 13 and 16[b] 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 Problems 4 and 23 are considered as too vague to ever be described as solved; the withdrawn 24 would also be in this class.
The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation of Hilbert's presentation, published in theBulletin of the American Mathematical Society.[1]
Hilbert originally had 24 problems on his list, but decided against including one of them in the published list. The "24th problem" (inproof theory, on a criterion forsimplicity and general methods) was rediscovered in Hilbert's original manuscript notes by German historianRüdiger Thiele in 2000.[3]
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 modernnumber theorists would probably see the 9th problem as referring to theconjecturalLanglands correspondence on representations of the absoluteGalois group of anumber field.[4] Still other problems, such as the 11th and the 16th, concern flourishing mathematical subdisciplines, like the theories ofquadratic forms andreal algebraic curves.
There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns theaxiomatization ofphysics, 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 thefoundations of geometry, in a manner that is 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 theFields Medal in 1966 for his work on the first problem, and the negative solution of the tenth problem in 1970 byYuri Matiyasevich (completing work byJulia Robinson,Hilary Putnam, andMartin Davis) generated similar acclaim. Aspects of these problems remain of great interest.
FollowingGottlob Frege andBertrand Russell, Hilbert sought to define mathematics logically using the method offormal systems, i.e.,finitisticproofs from an agreed-upon set ofaxioms.[5] One of the main goals ofHilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.[c]
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 afterKurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work.[d][e]
Hilbert's tenth problem does not ask whether there exists analgorithm for deciding the solvability ofDiophantine equations, but rather asks for theconstruction 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 inrational 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.[f] 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).[g] 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.[h]
Hilbert's 23 problems, and the unpublished 24th problem, are listed below. 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 | Thecontinuum hypothesis (that is, there is noset whosecardinality is strictly between that of theintegers and that of thereal numbers) | Proven to be impossible to prove or disprove withinZermelo–Fraenkel set theory with or without theaxiom of choice (provided Zermelo–Fraenkel set theory isconsistent, 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 theaxioms ofarithmetic areconsistent. | There is no consensus on whether results ofGödel andGentzen give a solution to the problem as stated by Hilbert. Gödel'ssecond incompleteness theorem, proven in 1931, shows that no such proof can be carried out within arithmetic itself. Gentzen proved in 1936 that the consistency of arithmetic follows from thewell-foundedness of theordinal ε0. | 1931, 1936 |
| 3rd | Given any twopolyhedra 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, proven byMax Dehn usingDehn invariants. Even differentPlatonic solids of equal volume cannot be obtained this way from each other. | 1900 |
| 4th | Construction of geometries satisfying axioms of classical geometry, where lines aregeodesics. | Too vague to be stated resolved or not.[i] | — |
| 5th | Are continuousgroups automaticallydifferential groups? | Depends on the interpretation ofcontinuous group. If the term is understood as atopological group that is also atopological manifold, then the problem was proven true byAndrew Gleason.[8] If, however,continuous group is understood as a topological groupacting on a manifold, the problem becomes theHilbert–Smith conjecture, which is still unresolved. | 1953? |
| 6th | Mathematical treatment of theaxioms ofphysics. In later explanation given by Hilbert:[1] (a) axiomatic treatment of probability with limit theorems for foundation ofstatistical physics (b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua" | Resolved.Kolmogorov's axiomatics was accepted as standard for the foundations of probability theory. | 1933 |
Depends on the interpretation of the problem. If treated as a physical problem: since the publication of Hilbert's list, new discoveries challengedclassical mechanics and led to the formulation ofquantum field theory, which holds an "atomistic view" of physical laws; andgeneral relativity, which describes "motion of continua" at large scales. Despite many attempts to unify them into atheory of everything, it is still not obvious how to make clear link between them. Some authors tried to solve this as a mathematical problem in a classical mechanics framework, which was the dominant physical theory during the publication of the list. In March 2025, Deng, Hani, and Ma published a paper claiming to have solved this problem in by deriving continuous fluid equations and Boltzman kinetic equation from Newton's laws applied for particles.[9] The paper is currently in peer review.[10] | 2025? | ||
| 7th | Isabtranscendental, foralgebraica ≠ 0,1 andirrational algebraicb ? | Resolved. Result: Yes, illustrated by theGelfond–Schneider theorem. | 1934 |
| 8th | (a) TheRiemann hypothesis: the real part of any non-trivialzero of theRiemann zeta function is1⁄2. | Unresolved. Partial results involve much weaker estimations that at least5⁄12 of non-trivial zeros satisfy this condition and almost all of them have real part arbitrarily close to1⁄2. | — |
| (b) For pairwise coprime integers: determine solvability of diophantine equation: forx andy being prime numbers.Goldbach's conjecture and thetwin prime conjecture are special cases of this problem. | Unresolved, even the special cases of this equation are hard open problems. Partial results includeYitang Zhang's proof of bounded gaps between primes, later improved by thePolymath Project. | — | |
| (c) Generalize results using Riemann zeta function for distribution of prime numbers in integers, to apply them toDedekind zeta functions for distribution of prime ideals inring of integers for any number field. | Depends on the interpretation of expected results. In 1917,Erich Hecke constructed an analytic continuation for Dedekind zeta functions and proven functional equation, which allowed for obtaining results similar to that currently accessible using Riemann zeta function. However, if understood as proving anextended Riemann hypothesis, then the problem is still unsolved. | 1917? | |
| 9th | Find the most general law of thereciprocity theorem in anyalgebraicnumber field. | Unresolved. Partial results involve theArtin reciprocity law for abelian extensions of number fields, key result inclass field theory. Development ofnon-abelian class field theory that would work for the general case of number fields is still a largely conjectural area. | — |
| 10th | Find an algorithm to determine whether a given polynomialDiophantine equation with integer coefficients has an integer solution. | Resolved. Result: Impossible;Matiyasevich's theorem implies that there is no such algorithm. | 1970 |
| 11th | Solvingquadratic forms with any number of variables and coefficients over any number field. | Resolved.Helmut Hasse in 1924 created a general theory of classification and deciding solvability of quadratic forms over number fields using thelocal-global principle. His methodology was later simplified byErnst Witt usingWitt rings.[11] | 1924 |
| 12th | Extend theKronecker–Weber theorem on abelian extensions of therational numbers to abelian extensions of any base number field. | Unresolved. Partial results involve construction using Hilbert modular forms forCM-fields byGoro Shimura and special cases of totally real fields using Brumer-Stark units by Dasgupta and Kadke.[12][13] | — |
| 13th | Prove that the general case of7th-degree equation cannot be solved using finite composition ofcontinuous functions (variant:algebraic functions) of twoparameters. For continuous variant: at least constructanalytic function of three variables that cannot be represented as such composition. | Depends on interpretation of the problem. The stronger, continuous variant was disproven byVladimir Arnold in 1957. TheKolmogorov–Arnold representation theorem states that every multivariate continuous function can be obtained using composition of two-variable continuous functions. Some authors argue that Hilbert intended for a solution within the space of algebraic functions and possible extension of theGalois theory, thus continuing their own work on the algebraic case.[14][15][16] It appears from one of later Hilbert's papers that this was his original intention for the problem.[17]For the algebraic variant, the problem is unresolved. | 1957? |
| 14th | Is thering of invariants of analgebraic group acting on apolynomial ring alwaysfinitely generated? | Resolved. Result: No, a counterexample was constructed byMasayoshi Nagata. | 1959 |
| 15th | Rigorous foundation ofSchubert's enumerative calculus. | Major developments for resolving this problem have been made since the publication of the list:
Duan and Zhao claimed that their result actually resolved this problem. Currently there is no consensus whether the problem is resolved completely or partially. | 1987–2020? |
| 16th | Describe relative positions of ovals originating from arealalgebraic curve and aslimit cycles of a polynomialvector field on the plane. | Unresolved, even for algebraic curves of degree 8. | — |
| 17th | Express a nonnegativerational function asquotient of sums ofsquares. | Resolved. Result: Yes, due toEmil Artin. Moreover, an upper limit was established for the number of square terms necessary. | 1927 |
| 18th | (a) Are there only finitely many essentially differentspace groups inn-dimensional Euclidean space? | Resolved. Result: Yes (byLudwig Bieberbach) | 1910 |
| (b) Is there a polyhedron that admits only ananisohedral tiling in three dimensions? | Resolved. Result: Yes (byKarl Reinhardt). | 1928 | |
| (c) What is the densestsphere packing? | Widely believed to be resolved, bycomputer-assisted proof (byThomas Callister Hales). Result: Highest density achieved byclose packings, each with density approximately 74%, such asface-centered cubic close packing andhexagonal close packing.[j] | 1998 | |
| 19th | Are the solutions of regular problems in thecalculus of variations always necessarilyanalytic? | Resolved. Result: Yes, proven byEnnio De Giorgi and, independently and using different methods, byJohn Forbes Nash. | 1957 |
| 20th | Do allvariational problems with certainboundary conditions have solutions? | Unresolved. A significant topic of research throughout the 20th century, resulting in solutions for some cases.[22][23][24] | — |
| 21st | Proof of the existence ofFuchsianlinear differential equations having a prescribedmonodromy group | Resolved. Result: No, a counterexample was shown byAndrei Bolibrukh.[25][26][27] Despite a negative answer in the most general case, Fuchsian equations may exist in special cases under some additional assumptions.[28] | 1989 |
| 22nd | Uniformization of analytic relations by means ofautomorphic functions | Unresolved. Partial results involveuniformization theorem forRiemann surfaces. | — |
| 23rd | Further development of thecalculus of variations | Too vague to be stated resolved or not. Since the list was proposed, Hilbert and many other mathematicians have made numerous contributions to the calculus of variations.[29] Thedynamic programming ofRichard Bellman is considered an alternative to the calculus of variations.[30][31][32][k] | — |
| Unpublished 24th problem | |||
| 24th | Development of a theory of proof simplicity | Recovered from Hilbert's unpublished notes. Too vague to be stated resolved or not. | — |
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 four conjectures made byAndré Weil in the late 1940s (theWeil conjectures). In the fields ofalgebraic geometry, number theory and the links between the two, the Weil conjectures were very important.[33][34] The first of these was proven byBernard Dwork; a completely different proof of the first two, viaℓ-adic cohomology, was given byAlexander Grothendieck. The last and deepest of the Weil conjectures (an analogue of the Riemann hypothesis) was proven byPierre Deligne. Both Grothendieck and Deligne were awarded theFields 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 mathematicalproblems, many of them profound. Erdős often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.[35]
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 MedalistSteve Smale, who responded to a request byVladimir Arnold to propose a list of 18 problems (Smale's problems).
At least in the mainstream media, thede facto 21st century analogue of Hilbert's problems is the list of sevenMillennium Prize Problems chosen during 2000 by theClay 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 (thePoincaré 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?"[36]
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 theDoD".[37][38] The DARPA list also includes a few problems from Hilbert's list, e.g. the Riemann hypothesis.
A reliable source of Hilbert's axiomatic system, his comments on them and on the foundational 'crisis' that was on-going at the time (translated into English), appears as Hilbert's 'The Foundations of Mathematics' (1927).
Mathematical Problems public domain audiobook atLibriVox| International | |
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