212:"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."
162:), 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
1288:
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).
2816:
387:. 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.
22:
702:(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.
313:" (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.
1287:
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
435:
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
1425:
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
316:
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,
455:
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
308:
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
1307:
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'
1297:
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"
2819:
207:
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:
1334:"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
424:. 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
400:
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
1386:
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
1604:
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'
351:
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.
2316:
40:
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
2645:
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.).
1430:(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 "
1908:
436:
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?"
1351:
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
1308:(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).
698:
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.
443:
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
571:, 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
216:
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.
1317:
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
1347:
Nagel, Newman and Hofstadter discuss this issue: "The possibility of constructing a finitistic absolute proof of consistency for a formal system such as
158:
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
301:
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
282:
gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible. Hilbert lived for 12 years after
200:, 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
1912:
1298:(p. 198â199). Reid notes that in two papers in 1931 Hilbert proposed a different form of induction called "unendliche Induktion" (p. 199).
2852:
2367:. Interscience Tracts in Pure and Applied Mathematics. Vol. 16. New York-London-Sydney: Interscience Publishers John Wiley & Sons Inc.
2785:
95:
5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.
1738:
1487:
711:
331:
Hilbert originally included 24 problems on his list, but decided against including one of them in the published list. The "24th problem" (in
76:
59:
930:
397:, many of them profound. ErdĆs often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.
181:. Still other problems, such as the 11th and the 16th, concern what are now flourishing mathematical subdisciplines, like the theories of
602:
of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second?
2070:
2272:
Serrin, James (1969-05-08). "The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables".
279:
2148:
192:
There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the
3015:
305:". That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.
2805:"David Hilbert's "Mathematical Problems": A lecture delivered before the International Congress of Mathematicians at Paris in 1900"
1883:
46:
471:
Hilbert's 23 problems are (for details on the solutions and references, see the articles that are linked to in the first column):
463:), 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.
2804:
1377:
Some authors consider this problem as too vague to ever be described as solved, although there is still active research on it.
2745:
2722:
2679:
2660:
2635:
2601:
2417:
2256:
2028:
1825:
1637:
1597:
568:
367:, number theory and the links between the two, the Weil conjectures were very important. The first of these was proved by
2845:
444:
1517:
1182:
Partially resolved. A significant topic of research throughout the 20th century, resulting in solutions for some cases.
317:
since these solutions apply to a certain formalization of the problems, which is not necessarily the only possible one.
2988:
2977:
2053:
1671:
1216:
983:
326:
2982:
2972:
2567:
2372:
1858:
1792:
1238:
1190:
694:(b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"
119:
13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments.
2952:
1453:
Gray also lists the 18th problem as "open" in his 2000 book, because the sphere-packing problem (also known as the
1028:
2731:
A collection of survey essays by experts devoted to each of the 23 problems emphasizing current developments.
531:, i.e., it does not contain a contradiction). There is no consensus on whether this is a solution to the problem.
2962:
2957:
2937:
2932:
1322:
1134:
1060:
941:
901:
520:
3010:
2967:
2947:
2942:
2838:
1413:
1166:
1123:
1096:
997:
977:
714:, but subsequent developments have occurred, further challenging the axiomatic foundations of quantum physics.
661:, assuming one interpretation of the original statement. If, however, it is understood as an equivalent of the
1126:, each with density approximately 74%, such as face-centered cubic close packing and hexagonal close packing.
2922:
2777:
2652:
2500:
Bolibrukh, A.A. (1992). "Sufficient conditions for the positive solvability of the Riemann-Hilbert problem".
853:
753:
576:
2927:
2902:
1968:
1199:
877:
722:
662:
237:
2907:
2887:
2877:
2772:
883:
764:
617:
539:
54:
75:
The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in the
2917:
2912:
2897:
2892:
2882:
2872:
2274:
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
1733:
959:
829:
802:
673:
641:
593:
491:
421:
2767:
2327:(2). Nicolaus Copernicus University in ToruĆ, Juliusz Schauder Center for Nonlinear Studies: 195â228.
2754:
An account at the undergraduate level by the mathematician who completed the solution of the problem.
1269:
842:
417:
379:. The last and deepest of the Weil conjectures (an analogue of the Riemann hypothesis) was proved by
224:
in 1966 for his work on the first problem, and the negative solution of the tenth problem in 1970 by
2789:
2085:
137:
19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?
785:
310:
275:
was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.
1115:
447:". The DARPA list also includes a few problems from Hilbert's list, e.g. the Riemann hypothesis.
394:
339:
and general methods) was rediscovered in Hilbert's original manuscript notes by German historian
201:
143:
21. Proof of the existence of linear differential equations having a prescribed monodromy group.
1318:
1244:
1228:
1172:
1140:
1119:
376:
53:. The complete list of 23 problems was published later, in English translation in 1902 by
1208:
Partially resolved. Result: Yes/no/open, depending on more exact formulations of the problem.
425:
286:
published his theorem, but does not seem to have written any formal response to Gödel's work.
1359:(footnote 39, page 109). The authors conclude that the prospect "is most unlikely".
911:
808:
789:
781:
774:
731:
572:
294:
116:
12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality
2521:
2478:
2450:
2443:
Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk
2427:
2382:
2281:
2113:
1973:"Hilbert's 6th Problem: Exact and approximate hydrodynamic manifolds for kinetic equations"
1769:
1264:
1222:
835:
497:
272:
186:
2709:
Browder, Felix Earl (1976). "Mathematical Developments Arising from Hilbert Problems". In
240:) generated similar acclaim. Aspects of these problems are still of great interest today.
204:, in a manner that is now generally judged to be too vague to enable a definitive answer.
8:
1482:
689:
647:
50:
2454:
2285:
2117:
2104:
Vitushkin, Anatoliy G. (2004). "On Hilbert's thirteenth problem and related questions".
372:
2624:
2525:
2482:
2187:
2129:
1984:
1950:
1711:
1625:
1409:
1176:
1089:
777:
770:
699:
460:
432:
364:
264:
159:
2741:
2718:
2675:
2656:
2631:
2607:
2597:
2563:
2529:
2486:
2466:
2413:
2368:
2336:
2297:
2252:
2229:
2191:
2133:
2049:
2024:
1954:
1864:
1854:
1831:
1821:
1798:
1788:
1757:
1715:
1667:
1633:
1593:
1570:
1454:
1401:
1196:
1144:
1077:
1052:. Moreover, an upper limit was established for the number of square terms necessary.
1034:
926:
742:
707:
501:
225:
170:
2462:
2125:
1999:
1972:
1752:
1501:
340:
89:
3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.
2697:
2619:
2555:
2509:
2458:
2405:
2397:
2328:
2289:
2225:
2221:
2179:
2121:
2014:
1994:
1942:
1747:
1707:
1703:
1560:
1496:
1405:
1155:
1042:
966:
947:
735:
703:
360:
302:
253:
193:
163:
2559:
2517:
2474:
2423:
2378:
2362:
1933:
Corry, L. (1997). "David Hilbert and the axiomatization of physics (1894â1905)".
1765:
1203:
1151:
1006:
955:
951:
922:
907:
887:
564:
524:
406:
2018:
1368:
Number 6 is now considered a problem in physics rather than in mathematics.
2655:
Books in Mathematics. Vol. 21. New York, NY : Springer. pp. 243â295.
2205:
2023:. Translated by Beyer, Robert T. Princeton Oxford: Princeton University Press.
1108:
859:
658:
606:
380:
229:
182:
2830:
2409:
1660:
140:
20. The general problem of boundary values (Boundary value problems in PDE's).
3004:
2710:
2693:
2591:
2470:
2340:
2301:
2233:
1761:
1574:
1427:
688:(a) axiomatic treatment of probability with limit theorems for foundation of
368:
257:
249:
233:
37:
2611:
1835:
1729:
560:
390:
356:
283:
92:
4. Problem of the straight line as the shortest distance between two points.
2701:
2546:(1976). "An Overview of Deligne's work on Hilbert's Twenty-First Problem".
2332:
2293:
1802:
1592:((pbk.) ed.). Cambridge MA: Harvard University Press. pp. 464ff.
1412:; the non-abelian case remains unsolved, if one interprets that as meaning
1014:
1003:
815:
623:
384:
332:
221:
178:
174:
146:
22. Uniformization of analytic relations by means of automorphic functions.
2646:
2587:
2071:"Mathematicians Find Long-Sought Building Blocks for Special Polynomials"
1066:
1010:
863:
553:
528:
513:
505:
402:
33:
2404:. Aspects of Mathematics, E22. Braunschweig: Friedr. Vieweg & Sohn.
1565:
1548:
2740:. Foundations of computing (3. ed.). Cambridge, Mass.: MIT Press.
2513:
2209:
2183:
1946:
1397:
1049:
599:
549:
336:
217:
167:
2317:"Leray-Schauder degree: a half century of extensions and applications"
416:
21st century analogue of Hilbert's problems is the list of seven
2543:
2093:. Séminaires et CongrÚs. Vol. 2. Société Mathématique de France.
1776:
1688:
915:
651:
290:
256:, Hilbert sought to define mathematics logically using the method of
149:
23. Further development of the methods of the calculus of variations.
122:
14. Proof of the finiteness of certain complete systems of functions.
2648:
Mathematics and the historian's craft: the Kenneth O. May lectures;
2240:
107:
9. Proof of the most general law of reciprocity in any number field.
2825:
1590:
From Frege to Gödel: A source book in mathematical logic, 1879â1931
1549:"Reciprocity laws and Galois representations: recent breakthroughs"
1038:
627:
261:
2689:
1989:
1868:
921:
Unresolved. The continuous variant of this problem was solved by
812:
683:
509:
197:
1624:
Nagel, Ernest; Newman, James R.; Hofstadter, Douglas R. (2001).
428:) was solved relatively soon after the problems were announced.
110:
10. Determination of the solvability of a Diophantine equation.
1875:
2151:[On certain questions of the problem of resolvents].
679:
567:
give a solution to the problem as stated by Hilbert. Gödel's
545:
440:
289:
Hilbert's tenth problem does not ask whether there exists an
268:
128:
16. Problem of the topology of algebraic curves and surfaces.
113:
11. Quadratic forms with any algebraic numerical coefficients
42:
2688:
A wealth of information relevant to Hilbert's "program" and
2170:
Hilbert, David (1927). "Ăber die Gleichung neunten Grades".
309:
always can know this, that in mathematics there is not any "
83:
1. Cantor's problem of the cardinal number of the continuum.
63:. Earlier publications (in the original German) appeared in
125:
15. Rigorous foundation of Schubert's enumerative calculus.
2251:. Berlin New York: Springer Science & Business Media.
1457:) was unsolved, but a solution to it has now been claimed.
965:
Resolved. Result: No, a counterexample was constructed by
834:
Find an algorithm to determine whether a given polynomial
21:
2441:
Bolibrukh, A. A. (1990). "The Riemann-Hilbert problem".
1020:
Unresolved, even for algebraic curves of degree 8.
1002:
Describe relative positions of ovals originating from a
104:
8. Problems of prime numbers (The "Riemann Hypothesis").
2308:
2249:
Elliptic Partial Differential Equations of Second Order
1851:
Mathematical Mysteries: The beauty and magic of numbers
1809:
1785:
Mathematical developments arising from Hilbert problems
1426:
and being a question about a possible extension of the
1065:(a) Are there only finitely many essentially different
166:
would probably see the 9th problem as referring to the
153:
2735:
2626:
The honors class: Hilbert's problems and their solvers
1632:(Rev. ed.). New York: New York University Press.
1444:
functions"). As such, the problem is still unresolved.
101:
7. Irrationality and transcendence of certain numbers.
2736:
MatijaseviÄ, Jurij V.; MatijaseviÄ, Jurij V. (1993).
2265:
1623:
784:
is 1/2") and other prime-number problems, among them
371:; a completely different proof of the first two, via
1734:"Numbers of solutions of equations in finite fields"
519:
Proven to be impossible to prove or disprove within
355:
One exception consists of three conjectures made by
2717:. Providence (R.I): American Mathematical Society.
1154:and, independently and using different methods, by
838:with integer coefficients has an integer solution.
98:
6. Mathematical treatment of the axioms of physics.
2715:Proceedings of Symposia in Pure Mathematics XXVIII
2623:
1816:Chung, Fan R. K.; Graham, Ronald L. (1999-06-01).
1659:
393:posed hundreds, if not thousands, of mathematical
134:18. Building up of space from congruent polyhedra.
2672:Logical dilemmas: the life and work of Kurt Gödel
2247:Gilbarg, David; Trudinger, Neil S. (2001-01-12).
1619:
1617:
1615:
1613:
1221:Uniformization of analytic relations by means of
383:. Both Grothendieck and Deligne were awarded the
3002:
2692:'s impact on the Second Question, the impact of
2389:
2246:
2220:(10). American Mathematical Society: 1061â1082.
1818:Erdös on Graphs: his legacy of unsolved problems
1559:(1). American Mathematical Society (AMS): 1â39.
86:2. The compatibility of the arithmetical axioms.
2860:
2395:
2361:Plemelj, Josip (1964). Radok., J. R. M. (ed.).
2068:
1901:
1610:
1587:
1476:
1474:
1472:
1088:(b) Is there a polyhedron that admits only an
2846:
2674:(Reprint ed.). Wellesley, Mass: Peters.
2596:. Oxford; New York: Oxford University Press.
2204:
2020:Mathematical foundations of quantum mechanics
1977:Bulletin of the American Mathematical Society
1787:. Providence: American Mathematical Society.
1739:Bulletin of the American Mathematical Society
1553:Bulletin of the American Mathematical Society
1488:Bulletin of the American Mathematical Society
1139:Are the solutions of regular problems in the
712:Mathematical Foundations of Quantum Mechanics
710:on a rigorous mathematical basis in his book
77:Bulletin of the American Mathematical Society
70:
60:Bulletin of the American Mathematical Society
2786:"Original text of Hilbert's talk, in German"
1967:
1431:
933:), but the algebraic variant is unresolved.
559:There is no consensus on whether results of
131:17. Expression of definite forms by squares.
2669:
2548:Proceedings of Symposia in Pure Mathematics
2013:
1815:
1653:
1651:
1649:
1509:
1469:
2853:
2839:
2493:
2434:
2364:Problems in the sense of Riemann and Klein
2149:"Đ ĐœĐ”ĐșĐŸŃĐŸŃŃŃ
ĐČĐŸĐżŃĐŸŃĐ°Ń
ĐżŃĐŸĐ±Đ»Đ”ĐŒŃ ŃĐ”Đ·ĐŸĐ»ŃĐČĐ”ĐœŃ"
2043:
2037:
1581:
752:Resolved. Result: Yes, illustrated by the
2499:
2440:
2321:Topological Methods in Nonlinear Analysis
2112:(1). Russian Academy of Sciences: 11â25.
2103:
2083:
1998:
1988:
1751:
1564:
1546:
1500:
845:implies that there is no such algorithm.
527:(provided ZermeloâFraenkel set theory is
2077:
1884:"The world's 23 toughest math questions"
1646:
1540:
1321:be employed in proofs. See much more at
1249:Too vague to be stated resolved or not.
931:KolmogorovâArnold representation theorem
633:Too vague to be stated resolved or not.
47:International Congress of Mathematicians
20:
2708:
2618:
2360:
2354:
2169:
2146:
2069:Houston-Edwards, Kelsey (25 May 2021).
1848:
1842:
1782:
1527:
1515:
1480:
1122:). Result: Highest density achieved by
409:to propose a list of 18 problems.
16:23 mathematical problems stated in 1900
3003:
2644:
2314:
2271:
1881:
1849:Clawson, Calvin C. (8 December 1999).
1820:. Natick, Mass: A K Peters/CRC Press.
1686:
1680:
910:using algebraic (variant: continuous)
412:At least in the mainstream media, the
2834:
2670:Dawson, John W.; Gödel, Kurt (1997).
2048:. Vol. 6. Elsevier. p. 69.
1932:
1882:Cooney, Michael (30 September 2008).
280:Gödel's second incompleteness theorem
2586:
2542:
2536:
2017:(2018). Wheeler, Nicholas A. (ed.).
1728:
1657:
466:
154:Nature and influence of the problems
1588:van Heijenoort, Jean, ed. (1976) .
1114:Widely believed to be resolved, by
605:Resolved. Result: No, proved using
456:whether they resolve the problems.
320:
173:on representations of the absolute
49:, speaking on August 8 at the
13:
2579:
2198:
36:published by German mathematician
14:
3027:
2760:
1689:"Hilbert's twenty-fourth problem"
1666:. New York, NY: Springer-Verlag.
1355:capable of being mirrored inside
1150:Resolved. Result: Yes, proven by
807:Find the most general law of the
3016:Unsolved problems in mathematics
2814:
1687:Thiele, RĂŒdiger (January 2003).
1530:Archiv der Mathematik und Physik
508:is strictly between that of the
405:, who responded to a request by
293:for deciding the solvability of
65:Archiv der Mathematik und Physik
2463:10.1070/RM1990v045n02ABEH002350
2315:Mawhin, Jean (1 January 1999).
2163:
2159:(2). Kazan University: 173â187.
2153:Proceedings of Kazan University
2140:
2126:10.1070/RM2004v059n01ABEH000698
2097:
2084:Abhyankar, Shreeram S. (1997).
2062:
2007:
2000:10.1090/S0273-0979-2013-01439-3
1961:
1926:
1909:"DARPA Mathematical Challenges"
1753:10.1090/S0002-9904-1949-09219-4
1722:
1547:Weinstein, Jared (2015-08-25).
1502:10.1090/S0002-9904-1902-00923-3
1447:
1419:
1390:
1380:
1371:
1362:
1341:
1338:." (Hilbert, 1902, p. 445)
1328:
1311:
984:Schubert's enumerative calculus
706:made an early attempt to place
327:Hilbert's twenty-fourth problem
2226:10.1080/00029890.1972.11993188
1708:10.1080/00029890.2003.11919933
1414:non-abelian class field theory
1301:
1291:
1281:
1073:-dimensional Euclidean space?
1048:Resolved. Result: Yes, due to
841:Resolved. Result: Impossible;
678:Mathematical treatment of the
243:
1:
2630:. Natick, Mass: A.K. Peters.
2214:American Mathematical Monthly
2212:(1972). "Schubert Calculus".
2147:Morozov, Vladimir V. (1954).
1696:American Mathematical Monthly
1463:
1396:Problem 9 has been solved by
1200:linear differential equations
886:on Abelian extensions of the
569:second incompleteness theorem
346:
2106:Russian Mathematical Surveys
2087:Hilbert's Thirteenth Problem
2044:Hazewinkel, Michiel (2009).
1911:. 2008-09-26. Archived from
1853:. Basic Books. p. 258.
7:
2824:public domain audiobook at
2773:Encyclopedia of Mathematics
2402:The Riemann-Hilbert problem
1528:Hilbert, David (1901). "".
1323:BrouwerâHilbert controversy
1258:
1243:Further development of the
773:("the real part of any non-
521:ZermeloâFraenkel set theory
271:. One of the main goals of
267:from an agreed-upon set of
55:Mary Frances Winston Newson
10:
3032:
1783:Browder, Felix E. (1976).
1408:during the development of
1195:Proof of the existence of
1095:Resolved. Result: Yes (by
1076:Resolved. Result: Yes (by
890:to any base number field.
450:
422:Clay Mathematics Institute
420:chosen during 2000 by the
324:
297:, but rather asks for the
71:List of Hilbert's Problems
2868:
2410:10.1007/978-3-322-92909-9
1432:
1270:Millennium Prize Problems
1059:
925:in 1957 based on work by
862:with algebraic numerical
754:GelfondâSchneider theorem
418:Millennium Prize Problems
2704:on Hilbert's philosophy.
2560:10.1090/pspum/028.2/9904
1658:Reid, Constance (1996).
1518:"Mathematische Probleme"
1275:
1107:(c) What is the densest
665:, it is still unsolved.
663:HilbertâSmith conjecture
171:Langlands correspondence
2738:Hilbert's tenth problem
2502:Matematicheskie Zametki
1516:Hilbert, David (1900).
1483:"Mathematical Problems"
1481:Hilbert, David (1902).
1116:computer-assisted proof
982:Rigorous foundation of
884:KroneckerâWeber theorem
700:Kolmogorov's axiomatics
359:in the late 1940s (the
202:foundations of geometry
2333:10.12775/TMNA.1999.029
2294:10.1098/rsta.1969.0033
1626:Hofstadter, Douglas R.
1319:Law of Excluded Middle
1245:calculus of variations
1229:Uniformization theorem
1141:calculus of variations
1120:Thomas Callister Hales
1033:Express a nonnegative
843:Matiyasevich's theorem
500:(that is, there is no
377:Alexander Grothendieck
214:
26:
2821:Mathematical Problems
2593:The Hilbert challenge
1971:; Karlin, I. (2014).
1935:Arch. Hist. Exact Sci
1522:Göttinger Nachrichten
1357:Principia Mathematica
1349:Principia Mathematica
1223:automorphic functions
1092:in three dimensions?
790:twin prime conjecture
786:Goldbach's conjecture
782:Riemann zeta function
335:, on a criterion for
295:Diophantine equations
210:
187:real algebraic curves
24:
1227:Partially resolved.
1202:having a prescribed
1173:variational problems
989:Partially resolved.
893:Partially resolved.
869:Partially resolved.
836:Diophantine equation
821:Partially resolved.
523:with or without the
498:continuum hypothesis
363:). In the fields of
228:(completing work by
2455:1990RuMaS..45Q...1B
2286:1969RSPTA.264..413S
2118:2004RuMaS..59...11V
2046:Handbook of Algebra
1177:boundary conditions
1143:always necessarily
908:7th-degree equation
809:reciprocity theorem
690:statistical physics
652:differential groups
459:That leaves 8 (the
426:Poincaré conjecture
32:are 23 problems in
3011:Hilbert's problems
2862:Hilbert's problems
2768:"Hilbert problems"
2514:10.1007/BF02102113
2184:10.1007/BF01447867
1947:10.1007/BF00375141
1410:class field theory
1402:Abelian extensions
1090:anisohedral tiling
960:finitely generated
948:ring of invariants
771:Riemann hypothesis
480:Brief explanation
461:Riemann hypothesis
433:Riemann hypothesis
365:algebraic geometry
160:Riemann hypothesis
45:conference of the
30:Hilbert's problems
27:
2998:
2997:
2747:978-0-262-13295-4
2724:978-0-8218-1428-4
2711:Browder, Felix E.
2681:978-1-56881-256-4
2662:978-0-387-25284-1
2637:978-1-56881-141-3
2603:978-0-19-850651-5
2419:978-3-528-06496-9
2280:(1153): 413â496.
2258:978-3-540-41160-4
2030:978-0-691-17856-1
2015:Von Neumann, John
1827:978-1-56881-111-6
1639:978-0-8147-5816-8
1599:978-0-674-32449-7
1566:10.1090/bull/1515
1536:: 44â63, 213â237.
1455:Kepler conjecture
1440:" ("existence of
1265:Landau's problems
1256:
1255:
1078:Ludwig Bieberbach
1035:rational function
927:Andrei Kolmogorov
708:Quantum Mechanics
467:Table of problems
373:â-adic cohomology
303:rational integers
273:Hilbert's program
226:Yuri Matiyasevich
3023:
2855:
2848:
2841:
2832:
2831:
2818:
2817:
2811:
2809:
2800:
2798:
2797:
2788:. Archived from
2781:
2751:
2728:
2685:
2666:
2641:
2629:
2615:
2574:
2573:
2540:
2534:
2533:
2497:
2491:
2490:
2438:
2432:
2431:
2398:Bolibruch, A. A.
2393:
2387:
2386:
2358:
2352:
2351:
2349:
2347:
2312:
2306:
2305:
2269:
2263:
2262:
2244:
2238:
2237:
2202:
2196:
2195:
2167:
2161:
2160:
2144:
2138:
2137:
2101:
2095:
2094:
2092:
2081:
2075:
2074:
2066:
2060:
2059:
2041:
2035:
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2011:
2005:
2004:
2002:
1992:
1965:
1959:
1958:
1930:
1924:
1923:
1921:
1920:
1905:
1899:
1898:
1896:
1894:
1879:
1873:
1872:
1846:
1840:
1839:
1813:
1807:
1806:
1780:
1774:
1773:
1755:
1726:
1720:
1719:
1693:
1684:
1678:
1677:
1665:
1655:
1644:
1643:
1621:
1608:
1607:
1585:
1579:
1578:
1568:
1544:
1538:
1537:
1525:
1513:
1507:
1506:
1504:
1478:
1458:
1451:
1445:
1439:
1438:
1423:
1417:
1406:rational numbers
1394:
1388:
1384:
1378:
1375:
1369:
1366:
1360:
1345:
1339:
1332:
1326:
1315:
1309:
1305:
1299:
1295:
1289:
1285:
1179:have solutions?
1156:John Forbes Nash
1013:of a polynomial
967:Masayoshi Nagata
888:rational numbers
704:John von Neumann
626:where lines are
573:well-foundedness
512:and that of the
474:
473:
361:Weil conjectures
321:The 24th problem
254:Bertrand Russell
164:number theorists
3031:
3030:
3026:
3025:
3024:
3022:
3021:
3020:
3001:
3000:
2999:
2994:
2864:
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2815:
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2795:
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2758:
2752:
2748:
2729:
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2686:
2682:
2663:
2638:
2604:
2582:
2580:Further reading
2577:
2570:
2541:
2537:
2498:
2494:
2439:
2435:
2420:
2396:Anosov, D. V.;
2394:
2390:
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2359:
2355:
2345:
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2309:
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2266:
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2090:
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2042:
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2012:
2008:
1966:
1962:
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1647:
1640:
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1611:
1600:
1586:
1582:
1545:
1541:
1514:
1510:
1495:(10): 437â479.
1479:
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1461:
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1448:
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1420:
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1381:
1376:
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1306:
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1296:
1292:
1286:
1282:
1278:
1261:
1204:monodromy group
1152:Ennio De Giorgi
1007:algebraic curve
956:polynomial ring
952:algebraic group
923:Vladimir Arnold
860:quadratic forms
646:Are continuous
607:Dehn invariants
583:
544:Prove that the
525:axiom of choice
469:
453:
407:Vladimir Arnold
375:, was given by
349:
329:
323:
246:
183:quadratic forms
156:
73:
17:
12:
11:
5:
3029:
3019:
3018:
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2782:
2762:
2761:External links
2759:
2757:
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2636:
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2583:
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2578:
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2535:
2508:(2): 110â117.
2504:(in Russian).
2492:
2445:(in Russian).
2433:
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2388:
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2307:
2264:
2257:
2239:
2197:
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2155:(in Russian).
2139:
2096:
2076:
2061:
2055:978-0080932811
2054:
2036:
2029:
2006:
1983:(2): 186â246.
1960:
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1746:(5): 497â508.
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1673:978-0387946740
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1109:sphere packing
1104:
1103:
1100:
1097:Karl Reinhardt
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1081:
1074:
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1021:
1018:
1017:on the plane.
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732:transcendental
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669:
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659:Andrew Gleason
655:
650:automatically
644:
638:
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622:Construct all
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598:Given any two
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381:Pierre Deligne
348:
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341:RĂŒdiger Thiele
325:Main article:
322:
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258:formal systems
245:
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230:Julia Robinson
194:axiomatization
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2792:on 2012-02-05
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2206:Kleiman, S.L.
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1991:
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1978:
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1970:
1969:Gorban, A. N.
1964:
1956:
1952:
1948:
1944:
1941:(2): 83â198.
1940:
1936:
1929:
1915:on 2019-01-12
1914:
1910:
1904:
1889:
1888:Network World
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1860:9780738202594
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1435:algebraischen
1433:Existenz von
1429:
1428:Galois theory
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241:
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25:David Hilbert
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2790:the original
2771:
2753:
2737:
2730:
2714:
2702:Intuitionism
2687:
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2620:Yandell, Ben
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2588:Gray, Jeremy
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1976:
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1928:
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1913:the original
1903:
1891:. Retrieved
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1015:vector field
1011:limit cycles
954:acting on a
864:coefficients
816:number field
794:Unresolved.
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578:
514:real numbers
486:Year solved
470:
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385:Fields medal
354:
350:
333:proof theory
330:
315:
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299:construction
298:
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277:
247:
238:Martin Davis
222:Fields Medal
215:
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179:number field
175:Galois group
157:
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2554:: 537â557.
2449:(2): 3â47.
2210:Laksov, Dan
2178:: 243â250.
1730:Weil, André
1336:ignorabimus
1041:of sums of
882:Extend the
717:1933â2002?
588:1931, 1936
534:1940, 1963
506:cardinality
403:Steve Smale
311:ignorabimus
244:Knowability
168:conjectural
34:mathematics
3005:Categories
2796:2005-02-05
2544:Katz, N.M.
1919:2021-03-31
1524:: 253â297.
1464:References
1437:Funktionen
1398:Emil Artin
1050:Emil Artin
916:parameters
745:algebraic
743:irrational
741:â 0,1 and
554:consistent
550:arithmetic
529:consistent
391:Paul ErdĆs
357:André Weil
347:Follow-ups
337:simplicity
284:Kurt Gödel
262:finitistic
248:Following
218:Paul Cohen
2778:EMS Press
2530:121743184
2487:250853546
2471:0042-1316
2341:1230-3429
2302:0080-4614
2234:0377-9017
2192:179178089
2172:Math. Ann
2134:250837749
1990:1310.0406
1955:122709777
1869:99-066854
1762:0002-9904
1716:123061382
1575:0273-0979
1442:algebraic
912:functions
813:algebraic
736:algebraic
628:geodesics
600:polyhedra
439:In 2008,
343:in 2000.
291:algorithm
278:However,
2826:LibriVox
2780:. 2001 .
2622:(2002).
2612:44153228
2590:(2000).
2400:(1994).
1836:42809520
1732:(1949).
1702:: 1â24.
1259:See also
1197:Fuchsian
1145:analytic
1039:quotient
858:Solving
788:and the
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510:integers
477:Problem
414:de facto
395:problems
260:, i.e.,
51:Sorbonne
2713:(ed.).
2698:Brouwer
2696:'s and
2522:1165460
2479:1069347
2451:Bibcode
2428:1276272
2383:0174815
2346:8 April
2282:Bibcode
2114:Bibcode
1893:7 April
1803:2331329
1770:0029393
1662:Hilbert
1628:(ed.).
1605:(1927).
1404:of the
1387:solved.
1171:Do all
1043:squares
1009:and as
958:always
946:Is the
914:of two
811:in any
780:of the
775:trivial
684:physics
624:metrics
575:of the
565:Gentzen
483:Status
451:Summary
198:physics
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668:1953?
648:groups
546:axioms
504:whose
269:axioms
265:proofs
236:, and
2808:(PDF)
2690:Gödel
2526:S2CID
2483:S2CID
2188:S2CID
2130:S2CID
2091:(PDF)
1985:arXiv
1951:S2CID
1712:S2CID
1692:(PDF)
1532:. 3.
1276:Notes
1161:1957
1129:1998
1102:1928
1083:1910
1055:1927
972:1959
929:(see
848:1970
759:1934
612:1900
561:Gödel
441:DARPA
177:of a
43:Paris
2742:ISBN
2719:ISBN
2676:ISBN
2657:ISBN
2632:ISBN
2608:OCLC
2598:ISBN
2564:ISBN
2467:ISSN
2414:ISBN
2369:ISBN
2348:2024
2337:ISSN
2298:ISSN
2253:ISBN
2230:ISSN
2050:ISBN
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1865:LCCN
1855:ISBN
1832:OCLC
1822:ISBN
1799:OCLC
1789:ISBN
1758:ISSN
1668:ISBN
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1526:and
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1217:22nd
1191:21st
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978:15th
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496:The
431:The
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