223:"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 lectureâwhich, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, is its dueâI mean the calculus of variations."
173:), 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
1299:
page: "But these proofs cannot be mirrored inside the systems that they concern, and, since they are not finitistic, they do not achieve the proclaimed objectives of
Hilbert's original program." Hofstadter rewrote the original (1958) footnote slightly, changing the word "students" to "specialists in mathematical logic". And this point is discussed again on page 109 and was not modified there by Hofstadter (p. 108).
398:. 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.
2827:
713:(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.
324:" (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: 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.
33:
1298:
See Nagel and Newman revised by
Hofstadter (2001, p. 107), footnote 37: "Moreover, although most specialists in mathematical logic do not question the cogency of proof, it is not finitistic in the sense of Hilbert's original stipulations for an absolute proof of consistency." Also see next
446:
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
1436:
It is not difficult to show that the problem has a partial 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
327:
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,
466:
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, 9, 11, 12, 15, 21, and 22 have solutions that have partial acceptance, but there exists some controversy as to
319:
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
1318:
Reid's biography of
Hilbert, written during the 1960s from interviews and letters, reports that "Godel (who never had any correspondence with Hilbert) feels that Hilbert's scheme for the foundations of mathematics 'remains highly interesting and important in spite of my negative results'
1308:
Reid reports that upon hearing about "Gödel's work from
Bernays, he was 'somewhat angry'. ... At first he was only angry and frustrated, but then he began to try to deal constructively with the problem. ... It was not yet clear just what influence Gödel's work would ultimately have"
2830:
218:
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:
1345:"This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no
435:. 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
411:
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
1397:
According to Gray, most of the problems have been solved. Some were not defined completely, but enough progress has been made to consider them "solved"; Gray lists the fourth problem as too vague to say whether it has been
1615:
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'
362:
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.
2327:
51:
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
2656:
Thiele, RĂŒdiger (2005). "On Hilbert and his twenty-four problems". In Brummelen, Glen Van; Kinyon, Michael; Van Brummelen, Glen; Canadian Society for History and Philosophy of Mathematics (eds.).
1441:(see, for example, Abhyankar Vitushkin, Chebotarev, and others). It appears from one of Hilbert's papers that this was his original intention for the problem. The language of Hilbert there is "
1919:
447:
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?"
1362:
is not excluded by Gödel's results. ... His argument does not eliminate the possibility ... But no one today appears to have a clear idea of what a finitistic proof would be like that is
1319:(p. 217). Observe the use of present tense â she reports that Gödel and Bernays among others "answered my questions about Hilbert's work in logic and foundations" (p. vii).
709:
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.
454:
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
582:, 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
227:
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.
1328:
This issue that finds its beginnings in the "foundational crisis" of the early 20th century, in particular the controversy about under what circumstances could the
2863:
1749:
1498:
722:
87:
70:
941:
290:
1358:
Nagel, Newman and Hofstadter discuss this issue: "The possibility of constructing a finitistic absolute proof of consistency for a formal system such as
169:
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
312:
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
293:
gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible. Hilbert lived for 12 years after
211:, 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
3026:
1923:
1309:(p. 198â199). Reid notes that in two papers in 1931 Hilbert proposed a different form of induction called "unendliche Induktion" (p. 199).
57:
2378:. Interscience Tracts in Pure and Applied Mathematics. Vol. 16. New York-London-Sydney: Interscience Publishers John Wiley & Sons Inc.
2796:
106:
5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.
2756:
2733:
2690:
2671:
2646:
2612:
2428:
2267:
2039:
1836:
1648:
1608:
342:
Hilbert originally included 24 problems on his list, but decided against including one of them in the published list. The "24th problem" (in
579:
408:, many of them profound. ErdĆs often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.
192:. Still other problems, such as the 11th and the 16th, concern what are now flourishing mathematical subdisciplines, like the theories of
613:
of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second?
2081:
2856:
2283:
Serrin, James (1969-05-08). "The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables".
455:
2159:
2999:
2988:
2064:
1682:
1227:
994:
337:
203:
There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the
316:". That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.
2578:
2383:
1869:
1803:
2993:
2983:
2816:"David Hilbert's "Mathematical Problems": A lecture delivered before the International Congress of Mathematicians at Paris in 1900"
1894:
1249:
1201:
482:
Hilbert's 23 problems are (for details on the solutions and references, see the articles that are linked to in the first column):
2963:
1039:
474:), 13 and 16 unresolved, and 4 and 23 as too vague to ever be described as solved. The withdrawn 24 would also be in this class.
1333:
531:
2973:
2968:
2948:
2943:
2815:
1145:
1071:
952:
912:
1388:
Some authors consider this problem as too vague to ever be described as solved, although there is still active research on it.
3021:
1424:
1134:
1107:
2978:
2958:
2953:
2849:
1177:
1008:
988:
764:
2933:
2788:
2663:
864:
587:
378:, number theory and the links between the two, the Weil conjectures were very important. The first of these was proved by
673:
2938:
2913:
1979:
1528:
1210:
1193:
Partially resolved. A significant topic of research throughout the 20th century, resulting in solutions for some cases.
888:
733:
328:
since these solutions apply to a certain formalization of the problems, which is not necessarily the only possible one.
248:
894:
2783:
65:
2918:
2898:
2888:
775:
705:(b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"
628:
550:
130:
13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments.
2928:
2923:
2908:
2903:
2893:
2883:
1464:
Gray also lists the 18th problem as "open" in his 2000 book, because the sphere-packing problem (also known as the
970:
840:
813:
684:
652:
604:
502:
432:
2742:
A collection of survey essays by experts devoted to each of the 23 problems emphasizing current developments.
542:, i.e., it does not contain a contradiction). There is no consensus on whether this is a solution to the problem.
1280:
853:
428:
725:, but subsequent developments have occurred, further challenging the axiomatic foundations of quantum physics.
672:, assuming one interpretation of the original statement. If, however, it is understood as an equivalent of the
321:
1137:, each with density approximately 74%, such as face-centered cubic close packing and hexagonal close packing.
796:
2511:
Bolibrukh, A.A. (1992). "Sufficient conditions for the positive solvability of the Riemann-Hilbert problem".
1126:
212:
1329:
1255:
1239:
1183:
1151:
1130:
436:
387:
922:
819:
800:
792:
785:
742:
583:
305:
86:
The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in the
2285:
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
1744:
2778:
2338:(2). Nicolaus Copernicus University in ToruĆ, Juliusz Schauder Center for Nonlinear Studies: 195â228.
2765:
An account at the undergraduate level by the mathematician who completed the solution of the problem.
2461:
2292:
2124:
1275:
1233:
846:
508:
390:. The last and deepest of the Weil conjectures (an analogue of the Riemann hypothesis) was proved by
283:
235:
in 1966 for his work on the first problem, and the negative solution of the tenth problem in 1970 by
197:
2800:
2096:
148:
19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?
700:
658:
383:
286:
was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.
61:
2536:
2493:
2198:
2140:
1995:
1961:
1722:
1636:
1420:
1187:
1100:
788:
781:
710:
471:
458:". The DARPA list also includes a few problems from Hilbert's list, e.g. the Riemann hypothesis.
443:
375:
350:
and general methods) was rediscovered in Hilbert's original manuscript notes by German historian
275:
170:
154:
21. Proof of the existence of linear differential equations having a prescribed monodromy group.
64:. The complete list of 23 problems was published later, in English translation in 1902 by
2752:
2729:
2686:
2667:
2642:
2618:
2608:
2574:
2477:
2424:
2379:
2347:
2308:
2263:
2240:
2060:
2035:
1875:
1865:
1842:
1832:
1809:
1799:
1768:
1678:
1644:
1604:
1581:
1465:
1412:
1219:
Partially resolved. Result: Yes/no/open, depending on more exact formulations of the problem.
1207:
1155:
1088:
1045:
937:
753:
718:
512:
351:
297:
published his theorem, but does not seem to have written any formal response to Gödel's work.
236:
181:
1370:(footnote 39, page 109). The authors conclude that the prospect "is most unlikely".
127:
12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality
2708:
2630:
2566:
2520:
2469:
2416:
2408:
2339:
2300:
2232:
2190:
2132:
2025:
2005:
1953:
1758:
1714:
1571:
1507:
1416:
1166:
1053:
977:
958:
746:
714:
371:
313:
264:
204:
174:
2532:
2489:
2454:
Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk
2438:
2393:
1984:"Hilbert's 6th Problem: Exact and approximate hydrodynamic manifolds for kinetic equations"
1780:
2720:
Browder, Felix Earl (1976). "Mathematical Developments Arising from Hilbert Problems". In
2528:
2485:
2434:
2389:
1776:
1214:
1162:
1017:
966:
962:
933:
918:
898:
575:
535:
417:
251:) generated similar acclaim. Aspects of these problems are still of great interest today.
215:, in a manner that is now generally judged to be too vague to enable a definitive answer.
1493:
2465:
2296:
2128:
2115:
Vitushkin, Anatoliy G. (2004). "On Hilbert's thirteenth problem and related questions".
2635:
2216:
1119:
870:
669:
617:
391:
240:
193:
3015:
2721:
2704:
2540:
2497:
2202:
2144:
1965:
1740:
1726:
1438:
1063:. Moreover, an upper limit was established for the number of square terms necessary.
571:
405:
401:
379:
367:
294:
268:
260:
244:
48:
2473:
2136:
2010:
1983:
1763:
1512:
100:
3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.
2712:
2236:
1718:
1025:
1014:
826:
634:
395:
343:
232:
189:
185:
2570:
2373:
1944:
Corry, L. (1997). "David Hilbert and the axiomatization of physics (1894â1905)".
2657:
2598:
2029:
1379:
Number 6 is now considered a problem in physics rather than in mathematics.
1077:
1021:
874:
564:
539:
524:
516:
413:
44:
2666:
Books in Mathematics. Vol. 21. New York, NY : Springer. pp. 243â295.
2034:. Translated by Beyer, Robert T. Princeton Oxford: Princeton University Press.
17:
2841:
2420:
2220:
1671:
1408:
1060:
610:
560:
347:
228:
178:
151:
20. The general problem of boundary values (Boundary value problems in PDE's).
2602:
2481:
2351:
2312:
2244:
1772:
1585:
699:(a) axiomatic treatment of probability with limit theorems for foundation of
2622:
2554:
1846:
926:
662:
301:
103:
4. Problem of the straight line as the shortest distance between two points.
2557:(1976). "An Overview of Deligne's work on Hilbert's Twenty-First Problem".
2343:
2304:
1813:
1603:((pbk.) ed.). Cambridge MA: Harvard University Press. pp. 464ff.
1423:; the non-abelian case remains unsolved, if one interprets that as meaning
157:
22. Uniformization of analytic relations by means of automorphic functions.
2836:
2700:
2082:"Mathematicians Find Long-Sought Building Blocks for Special Polynomials"
1049:
638:
272:
2415:. Aspects of Mathematics, E22. Braunschweig: Friedr. Vieweg & Sohn.
1576:
1559:
2751:. Foundations of computing (3. ed.). Cambridge, Mass.: MIT Press.
2524:
2194:
1957:
823:
694:
520:
208:
2328:"Leray-Schauder degree: a half century of extensions and applications"
427:
21st century analogue of Hilbert's problems is the list of seven
2104:. Séminaires et CongrÚs. Vol. 2. Société Mathématique de France.
1699:
267:, Hilbert sought to define mathematics logically using the method of
160:
23. Further development of the methods of the calculus of variations.
133:
14. Proof of the finiteness of certain complete systems of functions.
2659:
Mathematics and the historian's craft: the Kenneth O. May lectures;
118:
9. Proof of the most general law of reciprocity in any number field.
1601:
From Frege to Gödel: A source book in mathematical logic, 1879â1931
1560:"Reciprocity laws and Galois representations: recent breakthroughs"
32:
2000:
690:
556:
451:
279:
53:
31:
1879:
932:
Unresolved. The continuous variant of this problem was solved by
1635:
Nagel, Ernest; Newman, James R.; Hofstadter, Douglas R. (2001).
439:) was solved relatively soon after the problems were announced.
2845:
121:
10. Determination of the solvability of a Diophantine equation.
2162:[On certain questions of the problem of resolvents].
578:
give a solution to the problem as stated by Hilbert. Gödel's
300:
Hilbert's tenth problem does not ask whether there exists an
139:
16. Problem of the topology of algebraic curves and surfaces.
124:
11. Quadratic forms with any algebraic numerical coefficients
2699:
A wealth of information relevant to Hilbert's "program" and
2181:
Hilbert, David (1927). "Ăber die Gleichung neunten Grades".
320:
always can know this, that in mathematics there is not any "
94:
1. Cantor's problem of the cardinal number of the continuum.
74:. Earlier publications (in the original German) appeared in
136:
15. Rigorous foundation of Schubert's enumerative calculus.
2262:. Berlin New York: Springer Science & Business Media.
1468:) was unsolved, but a solution to it has now been claimed.
976:
Resolved. Result: No, a counterexample was constructed by
845:
Find an algorithm to determine whether a given polynomial
2452:
Bolibrukh, A. A. (1990). "The Riemann-Hilbert problem".
1031:
Unresolved, even for algebraic curves of degree 8.
1013:
Describe relative positions of ovals originating from a
115:
8. Problems of prime numbers (The "Riemann Hypothesis").
2260:
Elliptic Partial Differential Equations of Second Order
1862:
Mathematical Mysteries: The beauty and magic of numbers
1796:
Mathematical developments arising from Hilbert problems
1437:
and being a question about a possible extension of the
1076:(a) Are there only finitely many essentially different
177:
would probably see the 9th problem as referring to the
2637:
The honors class: Hilbert's problems and their solvers
1643:(Rev. ed.). New York: New York University Press.
1455:
functions"). As such, the problem is still unresolved.
112:
7. Irrationality and transcendence of certain numbers.
2747:
MatijaseviÄ, Jurij V.; MatijaseviÄ, Jurij V. (1993).
795:
is 1/2") and other prime-number problems, among them
382:; a completely different proof of the first two, via
1745:"Numbers of solutions of equations in finite fields"
530:
Proven to be impossible to prove or disprove within
366:
One exception consists of three conjectures made by
2728:. Providence (R.I): American Mathematical Society.
1165:and, independently and using different methods, by
849:with integer coefficients has an integer solution.
109:
6. Mathematical treatment of the axioms of physics.
2726:Proceedings of Symposia in Pure Mathematics XXVIII
2634:
1827:Chung, Fan R. K.; Graham, Ronald L. (1999-06-01).
1670:
404:posed hundreds, if not thousands, of mathematical
145:18. Building up of space from congruent polyhedra.
2683:Logical dilemmas: the life and work of Kurt Gödel
2258:Gilbarg, David; Trudinger, Neil S. (2001-01-12).
1630:
1628:
1626:
1624:
1232:Uniformization of analytic relations by means of
394:. Both Grothendieck and Deligne were awarded the
2703:'s impact on the Second Question, the impact of
2231:(10). American Mathematical Society: 1061â1082.
1829:Erdös on Graphs: his legacy of unsolved problems
1570:(1). American Mathematical Society (AMS): 1â39.
97:2. The compatibility of the arithmetical axioms.
2372:Plemelj, Josip (1964). Radok., J. R. M. (ed.).
221:
1487:
1485:
1483:
1099:(b) Is there a polyhedron that admits only an
2857:
2685:(Reprint ed.). Wellesley, Mass: Peters.
2607:. Oxford; New York: Oxford University Press.
2031:Mathematical foundations of quantum mechanics
1988:Bulletin of the American Mathematical Society
1798:. Providence: American Mathematical Society.
1750:Bulletin of the American Mathematical Society
1564:Bulletin of the American Mathematical Society
1499:Bulletin of the American Mathematical Society
1150:Are the solutions of regular problems in the
723:Mathematical Foundations of Quantum Mechanics
721:on a rigorous mathematical basis in his book
88:Bulletin of the American Mathematical Society
71:Bulletin of the American Mathematical Society
8:
2797:"Original text of Hilbert's talk, in German"
1442:
944:), but the algebraic variant is unresolved.
570:There is no consensus on whether results of
142:17. Expression of definite forms by squares.
2559:Proceedings of Symposia in Pure Mathematics
1664:
1662:
1660:
2864:
2850:
2842:
2375:Problems in the sense of Riemann and Klein
2160:"Đ ĐœĐ”ĐșĐŸŃĐŸŃŃŃ
ĐČĐŸĐżŃĐŸŃĐ°Ń
ĐżŃĐŸĐ±Đ»Đ”ĐŒŃ ŃĐ”Đ·ĐŸĐ»ŃĐČĐ”ĐœŃ"
763:Resolved. Result: Yes, illustrated by the
484:
2332:Topological Methods in Nonlinear Analysis
2123:(1). Russian Academy of Sciences: 11â25.
2009:
1999:
1762:
1575:
1511:
856:implies that there is no such algorithm.
538:(provided ZermeloâFraenkel set theory is
1895:"The world's 23 toughest math questions"
1332:be employed in proofs. See much more at
1260:Too vague to be stated resolved or not.
942:KolmogorovâArnold representation theorem
644:Too vague to be stated resolved or not.
58:International Congress of Mathematicians
2080:Houston-Edwards, Kelsey (25 May 2021).
1479:
1291:
1133:). Result: Highest density achieved by
490:
420:to propose a list of 18 problems.
27:23 mathematical problems stated in 1900
1860:Clawson, Calvin C. (8 December 1999).
1831:. Natick, Mass: A K Peters/CRC Press.
921:using algebraic (variant: continuous)
423:At least in the mainstream media, the
2681:Dawson, John W.; Gödel, Kurt (1997).
2059:. Vol. 6. Elsevier. p. 69.
1893:Cooney, Michael (30 September 2008).
291:Gödel's second incompleteness theorem
7:
2028:(2018). Wheeler, Nicholas A. (ed.).
165:Nature and influence of the problems
1599:van Heijenoort, Jean, ed. (1976) .
1125:Widely believed to be resolved, by
616:Resolved. Result: No, proved using
467:whether they resolve the problems.
184:on representations of the absolute
60:, speaking on August 8 at the
47:published by German mathematician
25:
1700:"Hilbert's twenty-fourth problem"
1677:. New York, NY: Springer-Verlag.
1366:capable of being mirrored inside
1161:Resolved. Result: Yes, proven by
818:Find the most general law of the
3027:Unsolved problems in mathematics
2825:
1698:Thiele, RĂŒdiger (January 2003).
1541:Archiv der Mathematik und Physik
519:is strictly between that of the
416:, who responded to a request by
304:for deciding the solvability of
76:Archiv der Mathematik und Physik
2474:10.1070/RM1990v045n02ABEH002350
2326:Mawhin, Jean (1 January 1999).
2170:(2). Kazan University: 173â187.
2164:Proceedings of Kazan University
2137:10.1070/RM2004v059n01ABEH000698
2095:Abhyankar, Shreeram S. (1997).
2011:10.1090/S0273-0979-2013-01439-3
1920:"DARPA Mathematical Challenges"
1764:10.1090/S0002-9904-1949-09219-4
1558:Weinstein, Jared (2015-08-25).
1513:10.1090/S0002-9904-1902-00923-3
1349:." (Hilbert, 1902, p. 445)
995:Schubert's enumerative calculus
717:made an early attempt to place
338:Hilbert's twenty-fourth problem
2237:10.1080/00029890.1972.11993188
1719:10.1080/00029890.2003.11919933
1425:non-abelian class field theory
1084:-dimensional Euclidean space?
1059:Resolved. Result: Yes, due to
852:Resolved. Result: Impossible;
689:Mathematical treatment of the
1:
2641:. Natick, Mass: A.K. Peters.
2225:American Mathematical Monthly
2223:(1972). "Schubert Calculus".
2158:Morozov, Vladimir V. (1954).
1707:American Mathematical Monthly
1407:Problem 9 has been solved by
1211:linear differential equations
897:on Abelian extensions of the
580:second incompleteness theorem
2117:Russian Mathematical Surveys
2098:Hilbert's Thirteenth Problem
2055:Hazewinkel, Michiel (2009).
1922:. 2008-09-26. Archived from
1864:. Basic Books. p. 258.
2835:public domain audiobook at
2784:Encyclopedia of Mathematics
2413:The Riemann-Hilbert problem
1539:Hilbert, David (1901). "".
1334:BrouwerâHilbert controversy
1254:Further development of the
784:("the real part of any non-
532:ZermeloâFraenkel set theory
282:. One of the main goals of
278:from an agreed-upon set of
66:Mary Frances Winston Newson
3043:
1794:Browder, Felix E. (1976).
1419:during the development of
1206:Proof of the existence of
1106:Resolved. Result: Yes (by
1087:Resolved. Result: Yes (by
901:to any base number field.
433:Clay Mathematics Institute
431:chosen during 2000 by the
335:
308:, but rather asks for the
82:List of Hilbert's Problems
2879:
2421:10.1007/978-3-322-92909-9
1443:
1281:Millennium Prize Problems
1070:
936:in 1957 based on work by
873:with algebraic numerical
765:GelfondâSchneider theorem
429:Millennium Prize Problems
2715:on Hilbert's philosophy.
2571:10.1090/pspum/028.2/9904
1669:Reid, Constance (1996).
1529:"Mathematische Probleme"
1118:(c) What is the densest
676:, it is still unsolved.
674:HilbertâSmith conjecture
182:Langlands correspondence
81:
2749:Hilbert's tenth problem
2513:Matematicheskie Zametki
1527:Hilbert, David (1900).
1494:"Mathematical Problems"
1492:Hilbert, David (1902).
1127:computer-assisted proof
993:Rigorous foundation of
895:KroneckerâWeber theorem
711:Kolmogorov's axiomatics
370:in the late 1940s (the
213:foundations of geometry
2344:10.12775/TMNA.1999.029
2305:10.1098/rsta.1969.0033
1637:Hofstadter, Douglas R.
1330:Law of Excluded Middle
1256:calculus of variations
1240:Uniformization theorem
1152:calculus of variations
1131:Thomas Callister Hales
1044:Express a nonnegative
854:Matiyasevich's theorem
511:(that is, there is no
388:Alexander Grothendieck
225:
37:
2832:Mathematical Problems
2604:The Hilbert challenge
1982:; Karlin, I. (2014).
1946:Arch. Hist. Exact Sci
1533:Göttinger Nachrichten
1368:Principia Mathematica
1360:Principia Mathematica
1234:automorphic functions
1103:in three dimensions?
801:twin prime conjecture
797:Goldbach's conjecture
793:Riemann zeta function
346:, on a criterion for
306:Diophantine equations
198:real algebraic curves
35:
1238:Partially resolved.
1213:having a prescribed
1184:variational problems
1000:Partially resolved.
904:Partially resolved.
880:Partially resolved.
847:Diophantine equation
832:Partially resolved.
534:with or without the
509:continuum hypothesis
374:). In the fields of
239:(completing work by
2466:1990RuMaS..45Q...1B
2297:1969RSPTA.264..413S
2129:2004RuMaS..59...11V
2057:Handbook of Algebra
1188:boundary conditions
1154:always necessarily
919:7th-degree equation
820:reciprocity theorem
701:statistical physics
663:differential groups
470:That leaves 8 (the
437:Poincaré conjecture
43:are 23 problems in
3022:Hilbert's problems
2873:Hilbert's problems
2779:"Hilbert problems"
2525:10.1007/BF02102113
2195:10.1007/BF01447867
1958:10.1007/BF00375141
1421:class field theory
1413:Abelian extensions
1101:anisohedral tiling
971:finitely generated
959:ring of invariants
782:Riemann hypothesis
491:Brief explanation
472:Riemann hypothesis
444:Riemann hypothesis
376:algebraic geometry
171:Riemann hypothesis
56:conference of the
41:Hilbert's problems
38:
3009:
3008:
2758:978-0-262-13295-4
2735:978-0-8218-1428-4
2722:Browder, Felix E.
2692:978-1-56881-256-4
2673:978-0-387-25284-1
2648:978-1-56881-141-3
2614:978-0-19-850651-5
2430:978-3-528-06496-9
2291:(1153): 413â496.
2269:978-3-540-41160-4
2041:978-0-691-17856-1
2026:Von Neumann, John
1838:978-1-56881-111-6
1650:978-0-8147-5816-8
1610:978-0-674-32449-7
1577:10.1090/bull/1515
1547:: 44â63, 213â237.
1466:Kepler conjecture
1451:" ("existence of
1276:Landau's problems
1267:
1266:
1089:Ludwig Bieberbach
1046:rational function
938:Andrei Kolmogorov
719:Quantum Mechanics
478:Table of problems
384:â-adic cohomology
314:rational integers
284:Hilbert's program
237:Yuri Matiyasevich
16:(Redirected from
3034:
2866:
2859:
2852:
2843:
2829:
2828:
2822:
2820:
2811:
2809:
2808:
2799:. Archived from
2792:
2762:
2739:
2696:
2677:
2652:
2640:
2626:
2585:
2584:
2551:
2545:
2544:
2508:
2502:
2501:
2449:
2443:
2442:
2409:Bolibruch, A. A.
2404:
2398:
2397:
2369:
2363:
2362:
2360:
2358:
2323:
2317:
2316:
2280:
2274:
2273:
2255:
2249:
2248:
2213:
2207:
2206:
2178:
2172:
2171:
2155:
2149:
2148:
2112:
2106:
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2092:
2086:
2085:
2077:
2071:
2070:
2052:
2046:
2045:
2022:
2016:
2015:
2013:
2003:
1976:
1970:
1969:
1941:
1935:
1934:
1932:
1931:
1916:
1910:
1909:
1907:
1905:
1890:
1884:
1883:
1857:
1851:
1850:
1824:
1818:
1817:
1791:
1785:
1784:
1766:
1737:
1731:
1730:
1704:
1695:
1689:
1688:
1676:
1666:
1655:
1654:
1632:
1619:
1618:
1596:
1590:
1589:
1579:
1555:
1549:
1548:
1536:
1524:
1518:
1517:
1515:
1489:
1469:
1462:
1456:
1450:
1449:
1434:
1428:
1417:rational numbers
1405:
1399:
1395:
1389:
1386:
1380:
1377:
1371:
1356:
1350:
1343:
1337:
1326:
1320:
1316:
1310:
1306:
1300:
1296:
1190:have solutions?
1167:John Forbes Nash
1024:of a polynomial
978:Masayoshi Nagata
899:rational numbers
715:John von Neumann
637:where lines are
584:well-foundedness
523:and that of the
485:
372:Weil conjectures
332:The 24th problem
265:Bertrand Russell
175:number theorists
21:
3042:
3041:
3037:
3036:
3035:
3033:
3032:
3031:
3012:
3011:
3010:
3005:
2875:
2870:
2826:
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2814:
2806:
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2795:
2777:
2774:
2769:
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2746:
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2736:
2719:
2697:
2693:
2680:
2674:
2655:
2649:
2629:
2615:
2597:
2593:
2591:Further reading
2588:
2581:
2553:
2552:
2548:
2510:
2509:
2505:
2451:
2450:
2446:
2431:
2407:Anosov, D. V.;
2406:
2405:
2401:
2386:
2371:
2370:
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2356:
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2019:
1978:
1977:
1973:
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1611:
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1593:
1557:
1556:
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1538:
1526:
1525:
1521:
1506:(10): 437â479.
1491:
1490:
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1477:
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1435:
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1396:
1392:
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1340:
1327:
1323:
1317:
1313:
1307:
1303:
1297:
1293:
1289:
1272:
1215:monodromy group
1163:Ennio De Giorgi
1018:algebraic curve
967:polynomial ring
963:algebraic group
934:Vladimir Arnold
871:quadratic forms
657:Are continuous
618:Dehn invariants
594:
555:Prove that the
536:axiom of choice
480:
464:
418:Vladimir Arnold
386:, was given by
360:
340:
334:
257:
194:quadratic forms
167:
84:
28:
23:
22:
18:Hilbert problem
15:
12:
11:
5:
3040:
3038:
3030:
3029:
3024:
3014:
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3007:
3006:
3004:
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2996:
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2793:
2773:
2772:External links
2770:
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2647:
2627:
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2594:
2592:
2589:
2587:
2586:
2579:
2546:
2519:(2): 110â117.
2515:(in Russian).
2503:
2456:(in Russian).
2444:
2429:
2399:
2384:
2364:
2318:
2275:
2268:
2250:
2208:
2173:
2166:(in Russian).
2150:
2107:
2087:
2072:
2066:978-0080932811
2065:
2047:
2040:
2017:
1994:(2): 186â246.
1971:
1936:
1911:
1885:
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1852:
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1757:(5): 497â508.
1732:
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1684:978-0387946740
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1135:close packings
1123:
1120:sphere packing
1115:
1114:
1111:
1108:Karl Reinhardt
1104:
1096:
1095:
1092:
1085:
1074:
1068:
1067:
1064:
1057:
1042:
1036:
1035:
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1029:
1028:on the plane.
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743:transcendental
736:
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726:
707:
687:
681:
680:
677:
670:Andrew Gleason
666:
661:automatically
655:
649:
648:
645:
642:
633:Construct all
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614:
609:Given any two
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392:Pierre Deligne
359:
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352:RĂŒdiger Thiele
336:Main article:
333:
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269:formal systems
256:
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241:Julia Robinson
205:axiomatization
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2838:
2834:
2833:
2824:
2817:
2813:
2803:on 2012-02-05
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2706:
2705:Arend Heyting
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2217:Kleiman, S.L.
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1997:
1993:
1989:
1985:
1981:
1980:Gorban, A. N.
1975:
1972:
1967:
1963:
1959:
1955:
1952:(2): 83â198.
1951:
1947:
1940:
1937:
1926:on 2019-01-12
1925:
1921:
1915:
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1900:
1899:Network World
1896:
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1871:9780738202594
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1805:0-8218-1428-1
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1641:Gödel's proof
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1446:algebraischen
1444:Existenz von
1440:
1439:Galois theory
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2801:the original
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2713:Intuitionism
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1924:the original
1914:
1902:. Retrieved
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875:coefficients
827:number field
805:Unresolved.
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589:
525:real numbers
497:Year solved
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365:
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233:Fields Medal
226:
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190:number field
186:Galois group
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2460:(2): 3â47.
2221:Laksov, Dan
2189:: 243â250.
1741:Weil, André
1347:ignorabimus
1052:of sums of
893:Extend the
728:1933â2002?
599:1931, 1936
545:1940, 1963
517:cardinality
414:Steve Smale
322:ignorabimus
255:Knowability
179:conjectural
45:mathematics
3016:Categories
2807:2005-02-05
2555:Katz, N.M.
1930:2021-03-31
1535:: 253â297.
1475:References
1448:Funktionen
1409:Emil Artin
1061:Emil Artin
927:parameters
756:algebraic
754:irrational
752:â 0,1 and
565:consistent
561:arithmetic
540:consistent
402:Paul ErdĆs
368:André Weil
358:Follow-ups
348:simplicity
295:Kurt Gödel
273:finitistic
259:Following
229:Paul Cohen
2789:EMS Press
2541:121743184
2498:250853546
2482:0042-1316
2352:1230-3429
2313:0080-4614
2245:0377-9017
2203:179178089
2183:Math. Ann
2145:250837749
2001:1310.0406
1966:122709777
1880:99-066854
1773:0002-9904
1727:123061382
1586:0273-0979
1453:algebraic
923:functions
824:algebraic
747:algebraic
639:geodesics
611:polyhedra
450:In 2008,
354:in 2000.
302:algorithm
289:However,
2837:LibriVox
2791:. 2001 .
2633:(2002).
2623:44153228
2601:(2000).
2411:(1994).
1847:42809520
1743:(1949).
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1270:See also
1208:Fuchsian
1156:analytic
1050:quotient
869:Solving
799:and the
760: ?
521:integers
488:Problem
425:de facto
406:problems
271:, i.e.,
62:Sorbonne
2724:(ed.).
2709:Brouwer
2707:'s and
2533:1165460
2490:1069347
2462:Bibcode
2439:1276272
2394:0174815
2357:8 April
2293:Bibcode
2125:Bibcode
1904:7 April
1814:2331329
1781:0029393
1673:Hilbert
1639:(ed.).
1616:(1927).
1415:of the
1398:solved.
1182:Do all
1054:squares
1020:and as
969:always
957:Is the
925:of two
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791:of the
786:trivial
695:physics
635:metrics
586:of the
576:Gentzen
494:Status
462:Summary
209:physics
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679:1953?
659:groups
557:axioms
515:whose
280:axioms
276:proofs
247:, and
2819:(PDF)
2701:Gödel
2537:S2CID
2494:S2CID
2199:S2CID
2141:S2CID
2102:(PDF)
1996:arXiv
1962:S2CID
1723:S2CID
1703:(PDF)
1543:. 3.
1287:Notes
1172:1957
1140:1998
1113:1928
1094:1910
1066:1927
983:1959
940:(see
859:1970
770:1934
623:1900
572:Gödel
452:DARPA
188:of a
54:Paris
2753:ISBN
2730:ISBN
2687:ISBN
2668:ISBN
2643:ISBN
2619:OCLC
2609:ISBN
2575:ISBN
2478:ISSN
2425:ISBN
2380:ISBN
2359:2024
2348:ISSN
2309:ISSN
2264:ISBN
2241:ISSN
2061:ISBN
2036:ISBN
1906:2024
1876:LCCN
1866:ISBN
1843:OCLC
1833:ISBN
1810:OCLC
1800:ISBN
1769:ISSN
1679:ISBN
1645:ISBN
1605:ISBN
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1537:and
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1228:22nd
1202:21st
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841:10th
789:zero
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507:The
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