Editing David Hilbert

{{short description|German mathematician (1862–1943)}}
{{Redirect|Hilbert}}
{{Use dmy dates|date=December 2023}}
{{Infobox scientist
| name              = David Hilbert
| image             = Hilbert.jpg
| caption           = Hilbert in 1912
| birth_date        = {{birth date|1862|1|23|df=y}}
| birth_place       = [[Königsberg]] or [[Wehlau]], [[Kingdom of Prussia]]
| death_date        = {{death date and age|1943|2|14|1862|1|23|df=y}}
| death_place       = [[Göttingen]], [[Nazi Germany]]
| field             = [[Mathematics]], [[physics]], [[philosophy]]
| work_institutions = [[University of Königsberg]]<br />[[Göttingen University]]
| education         = [[University of Königsberg]] ([[PhD]])
| thesis_title      = On Invariant Properties of Special Binary Forms, Especially of Spherical Functions
| thesis_year       = 1885
| doctoral_advisor  = [[Ferdinand von Lindemann]]<ref name="Lindemann">{{MathGenealogy|id=7298}}</ref>
| doctoral_students = {{collapsible list
|[[Wilhelm Ackermann]]
|[[Heinrich Behmann]]
|[[Felix Bernstein (mathematician)|Felix Bernstein]]
|[[Otto Blumenthal]]
|[[Anne Bosworth]]
|[[Werner Boy]]
|[[Ugo Broggi]]
|[[Richard Courant]]
|[[Haskell Curry]]
|[[Max Dehn]]
|[[Ludwig Föppl]]
|[[Rudolf Fueter]]
|[[Paul Funk]]
|[[Kurt Grelling]]
|[[Alfréd Haar]]
|[[Erich Hecke]]
|[[Earle Raymond Hedrick|Earle Hedrick]]
|[[Ernst Hellinger]]
|[[Wallie Abraham Hurwitz|Wallie Hurwitz]]
|[[Margarete Kahn]]
|[[Oliver Dimon Kellogg|Oliver Kellogg]]
|[[Hellmuth Kneser]]
|[[Robert König]]|[[Emanuel Lasker]]
|[[Klara Löbenstein]]|[[Max Mason|Charles Max Mason]]
|[[Alexander Myller]]
|[[Erhard Schmidt]]
|[[Kurt Schütte]]
|[[Andreas Speiser]]
|[[Hugo Steinhaus]]
|[[Gabriel Sudan]]
|[[Teiji Takagi]]
|[[Hermann Weyl]]
|[[Ernst Zermelo]]
|title={{nbsp}}
}}
| notable_students  = [[Edward Kasner]]<br />[[John von Neumann]]<br />[[Emanuel Lasker]]<br />[[Carl Gustav Hempel]]|
| known_for         = [[Hilbert's basis theorem]]<br />[[Hilbert's Nullstellensatz]]<br />[[Hilbert's axioms]]<br />[[Hilbert's problems]]<br />[[Hilbert's program]]<br />[[Einstein–Hilbert action]]<br />[[Hilbert space]]<br />[[Hilbert system]]<br />[[Epsilon calculus]]
| prizes            = [[Lobachevsky Prize]] <small>(1903)</small><br />[[Bolyai Prize]] <small>(1910)</small><br />[[Foreign Member of the Royal Society|ForMemRS]] <small>(1928)</small><ref name="frs">{{Cite journal | last1 = Weyl | first1 = H. | author-link = Hermann Weyl| title = David Hilbert. 1862–1943 | doi = 10.1098/rsbm.1944.0006 | journal = [[Obituary Notices of Fellows of the Royal Society]] | volume = 4 | issue = 13 | pages = 547–553| year = 1944 | s2cid = 161435959 }}</ref>
| spouse            = Käthe Jerosch
| children          = Franz (b. 1893)
}}

'''David Hilbert'''  ({{IPAc-en|ˈ|h|ɪ|l|b|ər|t}};<ref>[http://www.dictionary.com/browse/hilbert "Hilbert"]. ''[[Random House Webster's Unabridged Dictionary]]''.</ref> {{IPA|de|ˈdaːvɪt ˈhɪlbɐt|lang}}; 23 January 1862 – 14 February 1943) was a German [[mathematician]] and [[philosophy of mathematics|philosopher of mathematics]] and one of the most influential mathematicians of his time.

Hilbert discovered and developed a broad range of fundamental ideas including [[invariant theory]], the [[calculus of variations]], [[commutative algebra]], [[algebraic number theory]], the [[Hilbert's axioms|foundations of geometry]], [[spectral theory]] of operators and its application to [[integral equations]], [[mathematical physics]], and the [[foundations of mathematics]] (particularly [[proof theory]]). He adopted and defended [[Georg Cantor]]'s set theory and [[transfinite number]]s. In 1900, he presented a [[Hilbert's problems|collection of problems]] that set a course for mathematical research of the 20th century.<ref>{{cite web |last1=Joyce |first1=David|author1-link=David E. Joyce (mathematician) |title=The Mathematical Problems of David Hilbert |url=https://mathcs.clarku.edu/~djoyce/hilbert/ |website=Clark University |access-date=15 January 2021}}</ref><ref>{{cite web |last1=Hilbert |first1=David |title=Mathematical Problems |url=https://mathcs.clarku.edu/~djoyce/hilbert/problems.html |access-date=15 January 2021}}</ref>

Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a cofounder of proof theory and [[mathematical logic]].<ref>{{cite encyclopedia
|url=http://plato.stanford.edu/entries/hilbert-program/
|title=Hilbert's Program
|encyclopedia=Stanford Encyclopedia of Philosophy
|author=Zach, Richard
|author-link=Richard Zach
|date=31 July 2003
|access-date=23 March 2009
}}</ref>

==Life==

===Early life and education===
Hilbert, the first of two children and only son of Otto, a county judge, and Maria Therese Hilbert ([[née]] Erdtmann), the daughter of a merchant, was born in the [[Province of Prussia]], [[Kingdom of Prussia]], either in [[Königsberg]] (according to Hilbert's own statement) or in Wehlau (known since 1946 as [[Znamensk, Kaliningrad Oblast|Znamensk]]) near Königsberg where his father worked at the time of his birth. His paternal grandfather was David Hilbert, a judge and ''[[Geheimrat]]''. His mother Maria had an interest in philosophy, astronomy and [[prime number]]s, while his father Otto taught him [[Prussian virtues]]. After his father became a city judge, the family moved to Königsberg. David's sister, Elise, was born when he was six. He began his schooling aged eight, two years later than the usual starting age.<ref>{{Harvnb|Reid|1996|pp=[https://books.google.com/books?id=mR4SdJGD7tEC&pg=PA1 1–3]}}; also on [https://books.google.com/books?id=mR4SdJGD7tEC&pg=PA8 p.&nbsp;8], Reid notes that there is some ambiguity as to exactly where Hilbert was born. Hilbert himself stated that he was born in Königsberg.</ref>

In late 1872, Hilbert entered the [[Friedrichskolleg]] [[Gymnasium (school)|Gymnasium]] (''Collegium fridericianum'', the same school that [[Immanuel Kant]] had attended 140 years before); but, after an unhappy period, he transferred to (late 1879) and graduated from (early 1880) the more science-oriented [[Wilhelmsgymnasium (Königsberg)|Wilhelm Gymnasium]].{{Sfn|Reid|1996|p=[https://books.google.com/books?id=mR4SdJGD7tEC&pg=PA4 4–7]}} Upon graduation, in autumn 1880, Hilbert enrolled at the [[University of Königsberg]], the "Albertina". In early 1882, [[Hermann Minkowski]] (two years younger than Hilbert and also a native of Königsberg but had gone to Berlin for three semesters),{{Sfn|Reid|1996|p=[https://books.google.com/books?id=mR4SdJGD7tEC&pg=PA11 11]}} returned to Königsberg and entered the university. Hilbert developed a lifelong friendship with the shy, gifted Minkowski.{{Sfn|Reid|1996|p=[https://books.google.com/books?id=mR4SdJGD7tEC&pg=PA12 12]}}<ref>{{citation|first=Hermann|last=Weyl|title=Levels of Infinity/Selected writings on Mathematics and Philosophy|chapter=David Hilbert and his Mathematical Work|page=94|year=2012|publisher=Dover|editor= Peter Pesic|isbn=978-0-486-48903-2}}</ref>

===Career===
{{Multiple image| image1 = David Hilbert 1886.jpg| image2 = David Hilbert, 1907.jpg| caption2 = Hilbert in 1907| caption1 = Hilbert in 1886| direction = horizontal| align = left| total_width = 390}}
In 1884, [[Adolf Hurwitz]] arrived from Göttingen as an [[Professor|Extraordinarius]] (i.e., an associate professor)<!--at the Albertina in 1884-->. An intense and fruitful scientific exchange among the three began, and Minkowski and Hilbert especially would exercise a reciprocal influence over each other at various times in their scientific careers. Hilbert obtained his doctorate in 1885, with a dissertation, written under [[Ferdinand von Lindemann]],<ref name="Lindemann"/> titled ''Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen'' ("On the invariant properties of special [[binary quantic|binary forms]], in particular the [[Spherical harmonics|spherical harmonic functions"]]).

Hilbert remained at the University of Königsberg as a ''Privatdozent'' ([[senior lecturer]]) from 1886 to 1895.  In 1895, as a result of intervention on his behalf by [[Felix Klein]], he obtained the position of Professor of Mathematics at the [[University of Göttingen]]. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world.<ref>{{citation|first=Jeff|last=Suzuki|title=Mathematics in Historical Context|year=2009|publisher=Mathematical Association of America|isbn=978-0-88385-570-6|page=342|url=https://books.google.com/books?id=lew5IC5piCwC&q=gottingen+mathematics&pg=PA342}}</ref> He remained there for the rest of his life.

[[File:Mathematik Göttingen.jpg|thumb|right|The Mathematical Institute in Göttingen. Its new building, constructed with funds from the [[Rockefeller Foundation]], was opened by Hilbert and Courant in 1930.]]

===Göttingen school===
Among Hilbert's students were [[Hermann Weyl]], [[chess]] champion [[Emanuel Lasker]], [[Ernst Zermelo]], and [[Carl Gustav Hempel]]. [[John von Neumann]] was his assistant. At the [[University of Göttingen]], Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as [[Emmy Noether]] and [[Alonzo Church]].

Among his 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including (with date of thesis): [[Otto Blumenthal]] (1898), [[Felix Bernstein (mathematician)|Felix Bernstein]] (1901), [[Hermann Weyl]] (1908), [[Richard Courant]] (1910), [[Erich Hecke]] (1910), [[Hugo Steinhaus]] (1911), and [[Wilhelm Ackermann]] (1925).<ref>{{cite web|url=http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=7298| title = The Mathematics Genealogy Project – David Hilbert | access-date=7 July 2007}}</ref> Between 1902 and 1939 Hilbert was editor of the ''[[Mathematische Annalen]]'', the leading mathematical journal of the time. He was elected an International Member of the United States [[National Academy of Sciences]] in 1907.<ref>{{Cite web |title=David Hilbert |url=http://www.nasonline.org/member-directory/deceased-members/20001326.html |access-date=30 June 2023 |website=www.nasonline.org}}</ref>

===Personal life===
[[File:ConstantinCaratheodory KatheHilbert MFO633.jpg|thumb|Käthe Hilbert with [[Constantin Carathéodory]], before 1932]]
{{Multiple image| image1 = David Hilbert and Käthe Jerosch.png| image2 = FranzHilbert MFO.jpg| caption2 = Franz Hilbert| caption1 = Hilbert and his wife Käthe Jerosch (1892)| direction = horizontal| align = left| total_width = 370}}
In 1892, Hilbert married Käthe Jerosch (1864–1945), who was the daughter of a Königsberg merchant, "an outspoken young lady with an independence of mind that matched [Hilbert's]."{{Sfn|Reid|1996|p=[https://books.google.com/books?id=mR4SdJGD7tEC&pg=PA36 36]}} While at Königsberg, they had their one child, Franz Hilbert (1893–1969).
Franz suffered throughout his life from mental illness, and after he was admitted into a psychiatric clinic, Hilbert said, "From now on, I must consider myself as not having a son." His attitude toward Franz brought Käthe considerable sorrow.{{Sfn|Reid|1996|p=[https://books.google.com/books?id=mR4SdJGD7tEC&pg=PA139 139]}}

Hilbert considered the mathematician [[Hermann Minkowski]] to be his "best and truest friend".{{Sfn|Reid|1996|p=121}}

Hilbert was baptized and raised a [[Calvinist]] in the [[Prussian Union of churches|Prussian Evangelical Church]].<ref group=lower-alpha>The Hilberts had, by this time, left the Calvinist Protestant church in which they had been baptized and married. – Reid 1996, p.91</ref> He later left the Church and became an [[agnostic]].<ref name=hilbertagnostic group=lower-alpha>
David Hilbert seemed to be agnostic and had nothing to do with theology proper or even religion. Constance Reid tells a story on the subject:<blockquote>The Hilberts had by this time [around 1902] left the Reformed Protestant Church in which they had been baptized and married. It was told in Göttingen that when [David Hilbert's son] Franz had started to school he could not answer the question, "What religion are you?" (1970, p.&nbsp;91)</blockquote> In the 1927 Hamburg address, Hilbert asserted: "mathematics is pre-suppositionless science (die Mathematik ist eine voraussetzungslose Wissenschaft)" and "to found it I do not need a good God ([z]u ihrer Begründung brauche ich weder den lieben Gott)" (1928, S. 85; van Heijenoort, 1967, p.&nbsp;479). However, from Mathematische Probleme (1900) to Naturerkennen und Logik (1930) he placed his quasi-religious faith in the human spirit and in the power of pure thought with its beloved child– mathematics. He was deeply convinced that every mathematical problem could be solved by pure reason: in both mathematics and any part of natural science (through mathematics) there was "no ignorabimus" (Hilbert, 1900, S. 262; 1930, S. 963; Ewald, 1996, pp. 1102, 1165). That is why finding an inner absolute grounding for mathematics turned into Hilbert's life-work. He never gave up this position, and it is symbolic that his words "wir müssen wissen, wir werden wissen" ("we must know, we shall know") from his 1930 Königsberg address were engraved on his tombstone. Here, we meet a ghost of departed theology (to modify George Berkeley's words), for to absolutize human cognition means to identify it tacitly with a divine one. —{{cite journal
 | last = Shaposhnikov  | first = Vladislav
 | year = 2016
 | title = Theological Underpinnings of the Modern Philosophy of Mathematics. Part II: The Quest for Autonomous Foundations
 | journal = Studies in Logic, Grammar and Rhetoric
 | volume = 44  | issue = 1  | pages = 147–168
 | doi = 10.1515/slgr-2016-0009  | doi-access = free
}}
</ref> He also argued that mathematical truth was independent of the existence of God or other ''[[A priori and a posteriori|a priori]]'' assumptions.<ref group=lower-alpha>"Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs." David Hilbert, ''Die Grundlagen der Mathematik'', [http://people.cs.uchicago.edu/~odonnell/OData/Courses/22C:096/Lecture_notes/Hilbert_program.html Hilbert's program, 22C:096, University of Iowa].</ref><ref group=lower-alpha>{{cite book|title=Science, Worldviews and Education|year=2009|publisher=Springer|isbn=978-90-481-2779-5|page=129|author=Michael R. Matthews|quote=As is well known, Hilbert rejected Leopold Kronecker's God for the solution of the problem of the foundations of mathematics.}}</ref> When [[Galileo Galilei]] was criticized for failing to stand up for his convictions on the [[Heliocentric theory]], Hilbert objected: "But [Galileo] was not an idiot. Only an idiot could believe that scientific truth needs martyrdom; that may be necessary in religion, but scientific results prove themselves in due time."<ref group=lower-alpha>{{cite book |author1=Constance Reid |author2=Hermann Weyl |title=Hilbert |url=https://archive.org/details/hilbert0000reid_e2z0 |url-access=registration |date=1970 |publisher=Springer-Verlag |isbn=978-0-387-04999-1 |page=[https://archive.org/details/hilbert0000reid_e2z0/page/92 92] |quote=Perhaps the guests would be discussing Galileo's trial and someone would blame Galileo for failing to stand up for his convictions. "But he was not an idiot," Hilbert would object. "Only an idiot could believe that scientific truth needs martyrdom; that may be necessary in religion, but scientific results prove themselves in due time."}}</ref>

===Later years===
Like [[Albert Einstein]], Hilbert had closest contacts with the [[Berlin Circle|Berlin Group]], whose leading founders had studied under Hilbert in Göttingen ([[Kurt Grelling]], [[Hans Reichenbach]], and [[Walter Dubislav]]).<ref>{{cite book|first1=Nikolay|last1=Milkov|first2=Volker|last2=Peckhaus|chapter=The Berlin Group and the Vienna Circle: Affinities and Divergences |url=https://philpapers.org/archive/MILTBG-2.pdf |archive-url=https://web.archive.org/web/20140820161819/http://philpapers.org/archive/MILTBG-2.pdf |archive-date=2014-08-20 |url-status=live|page=20|date=2013-01-01 |doi=10.1007/978-94-007-5485-0_1|title=The Berlin Group and the Philosophy of Logical Empiricism|access-date=2021-05-19 |series=Boston Studies un the Philosophy and History of Science|volume=273|isbn=978-94-007-5485-0|oclc=7325392474}}</ref>

Around 1925, Hilbert developed [[pernicious anemia]], a then-untreatable vitamin deficiency of which the primary symptom is exhaustion; his assistant [[Eugene Wigner]] described him as subject to "enormous fatigue" and how he "seemed quite old", and that even after eventually being diagnosed and treated, he "was hardly a scientist after 1925, and certainly not a Hilbert".<ref>{{cite book |date=1992-10-01 |first2=Andrew |last2=Szanton |first1=Eugene P. |last1=Wigner |title=The Recollections of Eugene P. Wigner |publisher=Plenum |isbn=0-306-44326-0 }}</ref>

Hilbert was elected to the [[American Philosophical Society]] in 1932.<ref>{{Cite web |title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=David+Hilbert&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=2023-06-30 |website=search.amphilsoc.org}}</ref>

Hilbert lived to see the [[Law for the Restoration of the Professional Civil Service|Nazis purge]] many of the prominent faculty members at [[Georg August University of Göttingen|University of Göttingen]] in 1933.<ref>{{cite web |first=Steve |last=Tappan |url=http://www.atomicheritage.org/index.php/component/content/167.html?task=view |title="Shame" at Göttingen| access-date=2013-06-05 | archive-date=2013-11-05 | archive-url=https://web.archive.org/web/20131105154634/http://www.atomicheritage.org/index.php/component/content/167.html?task=view| url-status=dead}} (Hilbert's colleagues exiled)</ref> Those forced out included [[Hermann Weyl]] (who had taken Hilbert's chair when he retired in 1930), [[Emmy Noether]], and [[Edmund Landau]]. One who had to leave Germany, [[Paul Bernays]], had collaborated with Hilbert in mathematical logic, and co-authored with him the important book ''[[Grundlagen der Mathematik]]''<ref>{{cite journal
	| url = https://www.nature.com/articles/136126a0
	| title = abstract for Grundlagen der Mathematik
	| last = Milne-Thomson
	| first = L
	| date = 1935
	| journal = Nature
	| volume = 136
	| issue = 3430
	| pages = 126–127
	| doi = 10.1038/136126a0
	| s2cid = 4122792
	| access-date = 2023-12-15
	| quote = This is probably the most important book on mathematical foundations that has appeared since Whitehead and Russell's "Principia Mathematica".
}}

</ref> (which eventually appeared in two volumes, in 1934 and 1939). This was a sequel to the Hilbert–[[Wilhelm Ackermann|Ackermann]] book ''[[Principles of Mathematical Logic]]'' (1928). Hermann Weyl's successor was [[Helmut Hasse]].{{fact|date=May 2025}}

About a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, [[Bernhard Rust]]. Rust asked whether "the Mathematical Institute really suffered so much because of the departure of the [[Jews]]". Hilbert replied: "Suffered? It doesn't exist any longer, does it?"<ref>{{cite book |first=Eckart |last=Menzler-Trott |title=Gentzens Problem. Mathematische Logik im nationalsozialistischen Deutschland. |publisher=Birkhäuser |date=2001 |isbn=3-764-36574-9 |location=Auflage |page=142 }}</ref><ref>{{cite book |first=Hajo G. |last=Meyer |title=Tragisches Schicksal. Das deutsche Judentum und die Wirkung historischer Kräfte: Eine Übung in angewandter Geschichtsphilosophie |publisher=Frank & Timme |date=2008 |isbn=3-865-96174-6 |page=202 }}</ref>

===Death===
[[File:Göttingen Stadtfriedhof Grab David Hilbert.jpg|thumb|Hilbert's grave:<br />''Wir müssen wissen<br />Wir werden wissen'']]
By the time Hilbert died in 1943, the Nazis had nearly completely restaffed the university, as many of the former faculty had either been Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among them [[Arnold Sommerfeld]], a theoretical physicist and also a native of Königsberg.{{Sfn|Reid|1996|p=213}} News of his death only became known to the wider world several months after he died.{{Sfn|Reid|1996|p=214}}

The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians on 8 September 1930. The words were given in response to the Latin maxim: "''[[Ignoramus et ignorabimus]]''" or "We do not know and we shall not know":{{Sfn|Reid|1996|p=192}}

{{verse translation|lang=ger|
Wir müssen wissen.
Wir werden wissen.
|
We must know.
We shall know.
}}

The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, [[Kurt Gödel]]—in a round table discussion during the Conference on Epistemology held jointly with the Society meetings—tentatively announced the first expression of his incompleteness theorem.<ref group=lower-alpha>
"The Conference on Epistemology of the Exact Sciences ran for three days, from 5 to 7 September" (Dawson 1997:68). "It ... was held in conjunction with and just before the ninety-first annual meeting of the Society of German Scientists and Physicians ... and the sixth Assembly of German Physicists and Mathematicians.... Gödel's contributed talk took place on Saturday, 6 September [1930], from 3 until 3:20 in the afternoon, and on Sunday the meeting concluded with a round table discussion of the first day's addresses. During the latter event, without warning and almost offhandedly, Gödel quietly announced that "one can even give examples of propositions (and in fact of those of the type of [[Christian Goldbach|Goldbach]] or [[Pierre de Fermat|Fermat]]) that, while contentually true, are unprovable in the formal system of classical mathematics [153]" (Dawson:69) "... As it happened, Hilbert himself was present at Königsberg, though apparently not at the Conference on Epistemology. The day after the roundtable discussion he delivered the opening address before the Society of German Scientists and Physicians – his famous lecture ''Naturerkennen und Logik'' (Logic and the knowledge of nature), at the end of which he declared: 'For the mathematician there is no Ignorabimus, and, in my opinion, not at all for natural science either. ... The true reason why [no-one] has succeeded in finding an unsolvable problem is, in my opinion, that there is ''no'' unsolvable problem. In contrast to the foolish Ignorabimus, our credo avers: We must know, We shall know [159]'"(Dawson:71). Gödel's paper was received on November 17, 1930 (cf Reid p.&nbsp;197, van Heijenoort 1976:592) and published on 25 March 1931 (Dawson 1997:74). But Gödel had given a talk about it beforehand... "An abstract had been presented in October 1930 to the Vienna Academy of Sciences by [[Hans Hahn (mathematician)|Hans Hahn]]" (van Heijenoort:592); this abstract and the full paper both appear in van Heijenoort:583ff.</ref> [[Gödel's incompleteness theorems]] show that even [[elementary proof|elementary]] axiomatic systems such as [[Peano arithmetic]] are either self-contradicting or contain logical propositions that are impossible to prove or disprove within that system.

==Contributions to mathematics and physics==
===Solving Gordan's Problem<!-- predominantly capitalized in the literature; "Gordan's problem" and "Gordan's Problem" redirect here-->===
Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous ''finiteness theorem''. Twenty years earlier, [[Paul Gordan]] had demonstrated the [[theorem]] of the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. To solve what had become known in some circles as ''Gordan's Problem'', Hilbert realized that it was necessary to take a completely different path. As a result, he demonstrated ''[[Hilbert's basis theorem]]'', showing the existence of a finite set of generators, for the invariants of [[algebraic form|quantics]] in any number of variables, but in an abstract form. That is, while demonstrating the existence of such a set, it was not a [[constructive proof]]—it did not display "an object"—but rather, it was an [[existence proof]]{{Sfn|Reid|1996|p=[https://books.google.com/books?id=mR4SdJGD7tEC&pg=PA36 36–37]}} and relied on use of the [[law of excluded middle]] in an infinite extension.

Hilbert sent his results to the ''[[Mathematische Annalen]]''. Gordan, the house expert on the theory of invariants for the ''Mathematische Annalen'', could not appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive. His comment was:

{{verse translation|lang=ger|
Das ist nicht Mathematik. Das ist Theologie.
|
This is not Mathematics. This is Theology.{{Sfn|Reid|1996|p=34}}}}

[[Felix Klein|Klein]], on the other hand, recognized the importance of the work, and guaranteed that it would be published without any alterations. Encouraged by Klein, Hilbert extended his method in a second article, providing estimations on the maximum degree of the minimum set of generators, and he sent it once more to the ''Annalen''. After having read the manuscript, Klein wrote to him, saying:

{{blockquote|Without doubt this is the most important work on general algebra that the ''Annalen'' has ever published.{{Sfn|Reid|1996|p=195}}}}

Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself would say:

{{blockquote|I have convinced myself that even theology has its merits.<ref name=":0">{{harvnb|Reid|1996|p=[https://books.google.com/books?id=mR4SdJGD7tEC&pg=PA37 37].}}</ref>}}

For all his successes, the nature of his proof created more trouble than Hilbert could have imagined. Although [[Leopold Kronecker|Kronecker]] had conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea"—in other words (to quote Reid): "Through a proof of existence, Hilbert had been able to obtain a construction"; "the proof" (i.e. the symbols on the page) ''was'' "the object".<ref name=":0" /> Not all were convinced. While [[Leopold Kronecker|Kronecker]] would die soon afterwards, his [[Constructivism (mathematics)|constructivist]] philosophy would continue with the young [[Luitzen Egbertus Jan Brouwer|Brouwer]] and his developing [[intuitionist]] "school", much to Hilbert's torment in his later years.<ref>cf. {{harvnb|Reid|1996|pp=148–149.}}</ref> Indeed, Hilbert would lose his "gifted pupil" [[Hermann Weyl|Weyl]] to intuitionism—"Hilbert was disturbed by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker".{{Sfn|Reid|1996|p=148}} Brouwer the intuitionist in particular opposed the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert responded:

{{blockquote|Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists.{{Sfn|Reid|1996|p=150}}}}

=== Nullstellensatz ===
{{Main|Hilbert's Nullstellensatz}}
In the subject of [[algebra]], a [[field (algebra)|field]] is called ''[[Algebraically closed field|algebraically closed]]'' if and only if every polynomial over it has a root in it. Under this condition, Hilbert gave a criterion for when a collection of polynomials <math>(p_\lambda)_{\lambda \in \Lambda}</math> of <math>n</math> variables has a ''common'' root: This is the case if and only if there do not exist polynomials <math>q_1, \ldots, q_k</math> and indices <math>\lambda_1, \ldots, \lambda_k</math> such that
:<math>1 = \sum_{j=1}^k p_{\lambda_j}(\vec x) q_j(\vec x)</math>.
This result is known as the '''Hilbert root theorem''', or "Hilberts Nullstellensatz" in German. He also proved that the correspondence between vanishing ideals and their vanishing sets is bijective between [[Affine variety|affine varieties]] and [[radical ideal]]s in <math>\C[x_1, \ldots, x_n]</math>.

=== Curve ===
[[File:Hilbert_curve_production_rules!.svg|thumb|The replacement rules]]
{{Main|Hilbert curve}}

In 1890, [[Giuseppe Peano]] had published an article in the [[Mathematische Annalen]] describing the historically first [[space-filling curve]]. In response, Hilbert designed his own construction of such a curve, which is now called the ''Hilbert curve''. Approximations to this curve are constructed iteratively according to the replacement rules in the first picture of this section. The curve itself is then the pointwise limit.

[[File:Hilbert_curve.svg|thumb|center|The first six approximations to the Hilbert curve]]

===Axiomatization of geometry===
{{Main|Hilbert's axioms}}

The text ''[[Grundlagen der Geometrie]]'' (tr.: ''Foundations of Geometry'') published by Hilbert in 1899 proposes a formal set, called Hilbert's axioms, substituting for the traditional [[Euclid's elements|axioms of Euclid]]. They avoid weaknesses identified in those of [[Euclid]], whose works at the time were still used textbook-fashion. It is difficult to specify the axioms used by Hilbert without referring to the publication history of the ''Grundlagen'' since Hilbert changed and modified them several times. The original monograph was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902.<ref>{{harvnb|Hilbert|1950}}</ref><ref>[[G. B. Mathews]](1909) [http://www.nature.com/nature/journal/v80/n2066/pdf/080394a0.pdf The Foundations of Geometry] from [[Nature (journal)|Nature]] 80:394,5 (#2066)</ref> This translation incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. New editions followed the 7th, but the main text was essentially not revised.{{efn|Independently and contemporaneously, a 19&nbsp;year-old American student named [[Robert Lee Moore]] published an equivalent set of axioms. Some of the axioms coincide, while some of the axioms in Moore's system are theorems in Hilbert's and vice versa. {{citation needed|date=December 2020}}}}

Hilbert's approach signaled the shift to the modern [[axiomatic method]]. In this, Hilbert was anticipated by [[Moritz Pasch]]'s work from 1882.  Axioms are not taken as self-evident truths. Geometry may treat ''things'', about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as [[point (geometry)|point]], [[Line (geometry)|line]], [[plane (geometry)|plane]], and others, could be substituted, as Hilbert is reported to have said to [[Schoenflies]] and [[Ernst Kötter|Kötter]], by tables, chairs, glasses of beer and other such objects.<ref>{{cite book |author=Otto Blumenthal |title=Lebensgeschichte |year=1935 |volume=3 |pages=388–429 |publisher=Julius Springer |editor=David Hilbert | series=Gesammelte Abhandlungen |url=http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?action=pdf&metsFile=PPN237834022&divID=LOG_0001&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN237834022%7C&targetFileName=PPN237834022_LOG_0001.pdf& |access-date=6 September 2018 |archive-url=https://web.archive.org/web/20160304122623/http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?action=pdf&metsFile=PPN237834022&divID=LOG_0001&pagesize=original&pdfTitlePage=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fpdftitle%2F%3FmetsFile%3DPPN237834022%7C&targetFileName=PPN237834022_LOG_0001.pdf& |archive-date=4 March 2016 |url-status=dead |author-link=Otto Blumenthal}} Here: p.402-403</ref> It is their defined relationships that are discussed.

Hilbert first enumerates the undefined concepts: point, line, plane, lying on (a relation between points and lines, points and planes, and lines and planes), betweenness, congruence of pairs of points ([[line segment]]s), and [[criteria of congruence of angles|congruence]] of [[angle]]s. The axioms unify both the [[Euclidean geometry|plane geometry]] and [[solid geometry]] of Euclid in a single system.

===23 problems===
{{Main|Hilbert's problems}}
Hilbert put forth a highly influential list consisting of 23 unsolved problems at the [[International Congress of Mathematicians]] in [[Paris]] in 1900. This is generally reckoned as the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician.{{By whom|date=February 2021}}

After reworking the foundations of classical geometry, Hilbert could have extrapolated to the rest of mathematics. His approach differed from the later "foundationalist" Russell–Whitehead or "encyclopedist" [[Nicolas Bourbaki]], and from his contemporary [[Giuseppe Peano]]. The mathematical community as a whole could engage in problems of which he had identified as crucial aspects of important areas of mathematics.

The problem set was launched as a talk, "The Problems of Mathematics", presented during the course of the Second International Congress of Mathematicians, held in Paris. The introduction of the speech that Hilbert gave said:

{{blockquote|Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries ? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive ? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?<ref name="BAMSProblems">{{cite journal | last=Hilbert | first=David |translator-last1=Winston Newson |translator-first1=Mary  |translator-link1=Mary Frances Winston Newson| title=Mathematical problems | journal=Bulletin of the American Mathematical Society | volume=8 | issue=10 | date=1902 | issn=0273-0979 | doi=10.1090/S0002-9904-1902-00923-3 | doi-access=free | pages=437–479}}</ref>}}

He presented fewer than half the problems at the Congress, which were published in the acts of the Congress. In a subsequent publication, he extended the panorama, and arrived at the formulation of the now-canonical 23 Problems of Hilbert (see also [[Hilbert's twenty-fourth problem]]). The full text is important, since the exegesis of the questions still can be a matter of debate when it is asked how many have been solved.

Some of these were solved within a short time. Others have been discussed throughout the 20th century, with a few now taken to be unsuitably open-ended to come to closure. Some continue to remain challenges.

The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in the [[Bulletin of the American Mathematical Society]].

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

===Formalism===
In an account that had become standard by the mid-century, Hilbert's problem set was also a kind of manifesto that opened the way for the development of the [[formalism (mathematics)|formalist]] school, one of three major schools of mathematics of the 20th century. According to the formalist, mathematics is manipulation of symbols according to agreed upon formal rules. It is therefore an autonomous activity of thought.

====Program====
{{Main|Hilbert's program}}
In 1920, Hilbert proposed a research project in [[metamathematics]] that became known as Hilbert's program. He wanted mathematics to be formulated on a solid and complete logical foundation. He believed that in principle this could be done by showing that:

# all of mathematics follows from a correctly chosen finite system of [[axiom]]s; and
# that some such axiom system is provably consistent through some means such as the [[epsilon calculus]].

He seems to have had both technical and philosophical reasons for formulating this proposal. It affirmed his dislike of what had become known as the [[ignorabimus]], still an active issue in his time in German thought, and traced back in that formulation to [[Emil du Bois-Reymond]].<ref>{{Cite book |last=Finkelstein |first=Gabriel |title=Emil du Bois-Reymond: Neuroscience, Self, and Society in Nineteenth-Century Germany |date=2013 |publisher=The MIT Press |isbn=978-0262019507 |location=Cambridge; London |pages=265–289 |language=English}}</ref>

This program is still recognizable in the most popular [[philosophy of mathematics]], where it is usually called ''formalism''. For example, the [[Bourbaki group]] adopted a watered-down and selective version of it as adequate to the requirements of their twin projects of (a) writing encyclopedic foundational works, and (b) supporting the [[axiomatic method]] as a research tool. This approach has been successful and influential in relation with Hilbert's work in algebra and functional analysis, but has failed to engage in the same way with his interests in physics and logic.

Hilbert wrote in 1919:

{{blockquote|We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules.  Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.<ref>Hilbert, D. (1919–20), Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in G\"ottingen. Nach der Ausarbeitung von Paul Bernays (Edited and with an English introduction by David E. Rowe), Basel, Birkh\"auser (1992).</ref>}}

Hilbert published his views on the foundations of mathematics in the 2-volume work, [[Grundlagen der Mathematik]].

====Gödel's work====
Hilbert and the mathematicians who worked with him in his enterprise were committed to the project. His attempt to support axiomatized mathematics with definitive principles, which could banish theoretical uncertainties, ended in failure.

[[Kurt Gödel|Gödel]] demonstrated that any consistent formal system that is sufficiently powerful to express basic arithmetic cannot prove its own completeness using only its own axioms and rules of inference. In 1931, his [[Gödel's incompleteness theorem|incompleteness theorem]] showed that Hilbert's grand plan was impossible as stated. The second point cannot in any reasonable way be combined with the first point, as long as the axiom system is genuinely [[finitary]].

Nevertheless, the subsequent achievements of proof theory at the very least ''clarified'' consistency as it relates to theories of central concern to mathematicians. Hilbert's work had started logic on this course of clarification; the need to understand Gödel's work then led to the development of [[recursion theory]] and then [[mathematical logic]] as an autonomous discipline in the 1930s. The basis for later [[theoretical computer science]], in the work of [[Alonzo Church]] and [[Alan Turing]], also grew directly out of this "debate".<ref>{{Cite journal |last=Reichenberger |first=Andrea |date=31 January 2019 |title=From Solvability to Formal Decidability: Revisiting Hilbert's "Non-Ignorabimus" |url=https://scholarship.claremont.edu/jhm/vol9/iss1/5 |journal=Journal of Humanistic Mathematics |volume=9 |issue=1 |pages=49–80 |doi=10.5642/jhummath.201901.05 |s2cid=127398451 |issn=2159-8118|doi-access=free }}</ref>

===Functional analysis===
Around 1909, Hilbert dedicated himself to the study of differential and [[integral equation]]s; his work had direct consequences for important parts of modern functional analysis. In order to carry out these studies, Hilbert introduced the concept of an infinite dimensional [[Euclidean space]], later called [[Hilbert space]]. His work in this part of analysis provided the basis for important contributions to the mathematics of physics in the next two decades, though from an unanticipated direction. Later on, [[Stefan Banach]] amplified the concept, defining [[Banach spaces]]. Hilbert spaces are an important class of objects in the area of [[functional analysis]], particularly of the [[spectral theory]] of self-adjoint linear operators, that grew up around it during the 20th century.

===Physics===
Until 1912, Hilbert was almost exclusively a [[pure mathematician]]. When planning a visit from Bonn, where he was immersed in studying physics, his fellow mathematician and friend [[Hermann Minkowski]] joked he had to spend 10&nbsp;days in quarantine before being able to visit Hilbert. In fact, Minkowski seems responsible for most of Hilbert's physics investigations prior to 1912, including their joint seminar on the subject in 1905.

In 1912, three years after his friend's death, Hilbert turned his focus to the subject almost exclusively. He arranged to have a "physics tutor" for himself.{{sfn|Reid|1996|p=129}} He started studying [[Kinetic theory of gases|kinetic gas theory]] and moved on to elementary [[radiation]] theory and the molecular theory of matter. Even after the war started in 1914, he continued seminars and classes where the works of [[Albert Einstein]] and others were followed closely.

By 1907, Einstein had framed the fundamentals of the theory of [[gravity]], but then struggled for nearly 8&nbsp;years to put the theory into [[General Relativity|its final form]].<ref>Isaacson 2007:218</ref> By early summer&nbsp;1915, Hilbert's interest in physics had focused on [[general relativity]], and he invited Einstein to Göttingen to deliver a week of lectures on the subject.<ref>{{harvnb|Sauer|1999}}; {{harvnb|Fölsing|1998|p=}}{{Page needed|date=November 2021}}; Isaacson 2007:212</ref> Einstein received an enthusiastic reception at Göttingen.<ref>Isaacson 2007:213</ref> Over the summer, Einstein learned that Hilbert was also working on the field equations and redoubled his own efforts. During November&nbsp;1915, Einstein published several papers culminating in ''The Field Equations of Gravitation'' (see [[Einstein field equations]]).<ref group=lower-alpha>In time, associating the gravitational field equations with Hilbert's name became less and less common. A noticeable exception is P. Jordan (Schwerkraft und Weltall, Braunschweig, Vieweg, 1952), who called the equations of gravitation in the vacuum the Einstein–Hilbert equations. (''Leo Corry, David Hilbert and the Axiomatization of Physics'', p.&nbsp;437)</ref> Nearly simultaneously, Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations (see [[Einstein–Hilbert action]]). Hilbert fully credited Einstein as the originator of the theory and no public priority dispute concerning the field equations ever arose between the two men during their lives.<ref group=lower-alpha>Since 1971 there have been some spirited and scholarly discussions about which of the two men first presented the now accepted form of the field equations. "Hilbert freely admitted, and frequently stated in lectures, that the great idea was Einstein's: "Every boy in the streets of Gottingen understands more about four dimensional geometry than Einstein," he once remarked. "Yet, in spite of that, Einstein did the work and not the mathematicians." (Reid 1996, pp.&nbsp;141–142, also Isaacson 2007:222 quoting Thorne p.&nbsp;119).</ref> See more at [[General relativity priority dispute|priority]].

Additionally, Hilbert's work anticipated and assisted several advances in the [[mathematical formulation of quantum mechanics]]. His work was a key aspect of [[Hermann Weyl]] and [[John von Neumann]]'s work on the mathematical equivalence of [[Werner Heisenberg]]'s [[matrix mechanics]] and [[Erwin Schrödinger]]'s [[Schrödinger equation|wave equation]], and his namesake Hilbert space plays an important part in quantum theory. In 1926, von&nbsp;Neumann showed that, if quantum states were understood as vectors in Hilbert space, they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.<ref group=lower-alpha>In 1926, the year after the matrix mechanics formulation of quantum theory by [[Max Born]] and [[Werner Heisenberg]], the mathematician [[John von Neumann]] became an assistant to Hilbert at Göttingen. When von&nbsp;Neumann left in 1932, von&nbsp;Neumann's book on the mathematical foundations of quantum mechanics, based on Hilbert's mathematics, was published under the title ''Mathematische Grundlagen der Quantenmechanik''. See: Norman Macrae (1999) ''John von&nbsp;Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More'' (reprinted by the American Mathematical Society) and Reid (1996).</ref>

Throughout this immersion in physics, Hilbert worked on putting rigor into the mathematics of physics. While highly dependent on higher mathematics, physicists tended to be "sloppy" with it. To a pure mathematician like Hilbert, this was both ugly and difficult to understand. As he began to understand physics and how physicists were using mathematics, he developed a coherent mathematical theory for what he found – most importantly in the area of [[integral equations]]. When his colleague Richard Courant wrote the now classic ''[[Methoden der mathematischen Physik]]'' (''Methods of Mathematical Physics'') including some of Hilbert's ideas, he added Hilbert's name as author even though Hilbert had not directly contributed to the writing. Hilbert said "Physics is too hard for physicists", implying that the necessary mathematics was generally beyond them; the Courant–Hilbert book made it easier for them.

===Number theory===
Hilbert unified the field of [[algebraic number theory]] with his 1897 treatise ''[[Zahlbericht]]'' (literally "report on numbers"). He also resolved a significant number-theory [[Waring's problem|problem formulated by Waring]] in 1770.  As with [[#The finiteness theorem|the finiteness theorem]], he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.{{sfn|Reid|1996|p=114}} He then had little more to publish on the subject; but the emergence of [[Hilbert modular form]]s in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on [[class field theory]]. The concepts were highly influential, and his own contribution lives on in the names of the [[Hilbert class field]] and of the [[Hilbert symbol]] of [[local class field theory]]. Results were mostly proved by 1930, after work by [[Teiji Takagi]].<ref group=lower-alpha>This work established Takagi as Japan's first mathematician of international stature.</ref>

Hilbert did not work in the central areas of [[analytic number theory]], but his name has become known for the [[Hilbert–Pólya conjecture]], for reasons that are anecdotal. [[Ernst Hellinger]], a student of Hilbert, once told [[André Weil]] that Hilbert had announced in his seminar in the early 1900s that he expected the proof of the [[Riemann hypothesis|Riemann Hypothesis]] would be a consequence of Fredholm's work on integral equations with a symmetric kernel.<ref name="Weil anecdote">{{citation|first1=S. |last1=Endres|first2=F.|last2=Steiner|year=2009|title=The Berry–Keating operator on <math>L^2({\mathbb R}_>,{\rm d}x)</math> and on compact quantum graphs with general self-adjoint realizations|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=9| page = 37|doi=10.1088/1751-8113/43/9/095204|arxiv=0912.3183v5|s2cid=115162684}}</ref>

==Works==
His collected works (''Gesammelte Abhandlungen'') have been published several times. The original versions of his papers contained "many technical errors of varying degree";<ref>{{harvnb|Reid|1996}}, chap. 13.</ref> when the collection was first published, the errors were corrected and it was found that this could be done without major changes in the statements of the theorems, with one exception—a claimed proof of the [[continuum hypothesis]].{{sfn|Sieg|2013|p=284-285}}<ref name="Rota97">[[Gian-Carlo Rota|Rota G.-C.]] (1997), "[https://www.ams.org/notices/199701/comm-rota.pdf Ten lessons I wish I had been taught]", ''[[Notices of the AMS]]'', 44: 22–25.</ref> The errors were nonetheless so numerous and significant that it took [[Olga Taussky-Todd]] three years to make the corrections.<ref name="Rota97"/>

==See also==
{{Portal|Biography|Philosophy}}

===Concepts===
{{Div col|colwidth=15em}}
* [[List of things named after David Hilbert]]
* [[Foundations of geometry]]
* [[Hilbert C*-module]]
* [[Hilbert cube]]
* [[Hilbert curve]]
* [[Hilbert matrix]]
* [[Hilbert metric]]
* [[Hilbert–Mumford criterion]]
* [[Hilbert number]]
* [[Hilbert ring]]
* [[Hilbert–Poincaré series]]
* [[Hilbert series and Hilbert polynomial]]
* [[Hilbert space]]
* [[Hilbert spectrum]]
* [[Hilbert system]]
* [[Hilbert transform]]
* [[Hilbert's arithmetic of ends]]
* [[Hilbert's paradox of the Grand Hotel]]
* [[Hilbert–Schmidt operator]]
* [[Hilbert–Smith conjecture]]

===Theorems===
* [[Hilbert–Burch theorem]]
* [[Hilbert's irreducibility theorem]]
* [[Hilbert's Nullstellensatz]]
* [[Hilbert's theorem (differential geometry)]]
* [[Hilbert's Theorem 90]]
* [[Hilbert's syzygy theorem]]
* [[Hilbert–Speiser theorem]]

===Other===
* [[Brouwer–Hilbert controversy]]
* [[Direct method in the calculus of variations]]
* [[Entscheidungsproblem]]
* ''[[Geometry and the Imagination]]''
* [[General relativity priority dispute]]
{{Div col end}}

==Footnotes==
{{notelist|1}}

==Citations==
{{reflist|25em}}

==Sources==

===Primary literature in English translation===
* {{cite book |editor=Ewald, William B. |year=1996 |title=From Kant to Hilbert: A Source Book in the Foundations of Mathematics |publisher=Oxford University Press |location=Oxford, UK}}
** 1918. "Axiomatic thought," 1114–1115.
** 1922. "The new grounding of mathematics: First report," 1115–1133.
** 1923. "The logical foundations of mathematics," 1134–1147.
** 1930. "Logic and the knowledge of nature," 1157–1165.
** 1931. "The grounding of elementary number theory," 1148–1156.
** 1904. "On the foundations of logic and arithmetic," 129–138.
** 1925. "On the infinite," 367–392.
** 1927. "The foundations of mathematics," with comment by [[Weyl]] and Appendix by [[Bernays]], 464–489.
* {{cite book |author-link=Jean van Heijenoort |first=Jean |last=van Heijenoort |year=1967 |title=From Frege to Gödel: A source book in mathematical logic, 1879–1931 |publisher=Harvard University Press}}
* {{cite book |last=Hilbert |first=David |author-link=David Hilbert |title=The Foundations of Geometry [Grundlagen der Geometrie] |edition=2nd |publisher=Open Court Publishing |place=La Salle, IL |year=1950 |orig-year=1902 |translator=Townsend, E.J. |url=http://www.gutenberg.org/files/17384/17384-pdf.pdf |archive-url=https://web.archive.org/web/20051228050734/http://www.gutenberg.org/files/17384/17384-pdf.pdf |archive-date=28 December 2005 |url-status=live}}
* {{cite book |last=Hilbert |first=David |author-link=David Hilbert |title=Foundations of Geometry [Grundlagen der Geometrie] |edition=2nd English |publisher=Open Court Publishing |place=La Salle, IL |year=1990 |orig-year=1971 |translator=Unger, Leo |quote=translated from the 10th German edition |isbn=978-0-87548-164-7}}
* {{cite book |title=Geometry and Imagination |author1=Hilbert, David |author-link1=David Hilbert |author2=Cohn-Vossen, Stephan |author-link2=Stephan Cohn-Vossen |year=1999 |publisher=American Mathematical Society |isbn=978-0-8218-1998-2 |quote=An accessible set of lectures originally for the citizens of Göttingen.}}
* {{cite book |title=David Hilbert's Lectures on the Foundations of Mathematics and Physics, 1891–1933 |author=Hilbert, David |editor1=Hallett, Michael |editor2=Majer, Ulrich |year=2004 |publisher=Springer-Verlag |location=Berlin & Heidelberg |isbn=978-3-540-64373-9}}

===Secondary literature===
* {{Citation
| last = Bertrand
| first = Gabriel
| author-link = Gabriel Bertrand
| title = Allocution
| journal = [[Comptes rendus de l'Académie des sciences|Comptes rendus hebdomadaires des séances de l'Académie des sciences]]
| place = Paris
| volume = 217
| pages =625–640
| date = 20 December 1943b
| language = fr
| url =http://gallica.bnf.fr/ark:/12148/bpt6k31698/f629.image
}}, available at [[Gallica]]. The "Address" of Gabriel Bertrand of 20 December 1943 at the French Academy: he gives biographical sketches of the lives of recently deceased members, including [[Pieter Zeeman]], David Hilbert and [[Georges Giraud]].
* Bottazzini Umberto, 2003. ''Il flauto di Hilbert. Storia della matematica''. [[UTET]], {{isbn|88-7750-852-3}}
* Corry, L., Renn, J., and Stachel, J., 1997, "Belated Decision in the Hilbert-Einstein Priority Dispute," ''Science 278'': nn-nn.
* {{cite book | title = David Hilbert and the Axiomatization of Physics (1898–1918): From Grundlagen der Geometrie to Grundlagen der Physik| first = Leo |last= Corry| year = 2004 | publisher = Springer | isbn = 90-481-6719-1}}
* Dawson, John W. Jr 1997. ''Logical Dilemmas: The Life and Work of Kurt Gödel''. Wellesley MA: A. K. Peters. {{isbn|1-56881-256-6}}.
* {{Cite book|last=Fölsing|first=Albrecht|title=Albert Einstein|publisher=Penguin|year=1998|author-link=Albrecht Fölsing}}
* [[Ivor Grattan-Guinness|Grattan-Guinness, Ivor]], 2000. ''The Search for Mathematical Roots 1870–1940''. Princeton Univ. Press.
* [[Jeremy Gray|Gray, Jeremy]], 2000. ''The Hilbert Challenge''. {{isbn|0-19-850651-1}}
* {{cite book | title = From Brouwer to Hilbert, The Debate on the Foundations of Mathematics in 1920s| first = Paolo |last= Mancosu| year = 1998 | publisher = Oxford Univ. Press| isbn = 978-0-19-509631-6}}
* [[Jagdish Mehra|Mehra, Jagdish]], 1974. ''Einstein, Hilbert, and the Theory of Gravitation''. Reidel.
* [[Piergiorgio Odifreddi]], 2003. ''Divertimento Geometrico. Le origini geometriche della logica da Euclide a Hilbert''. Bollati Boringhieri, {{isbn|88-339-5714-4}}. A clear exposition of the "errors" of Euclid and of the solutions presented in the ''Grundlagen der Geometrie'', with reference to [[non-Euclidean geometry]].
* {{Cite book|last=Reid|first=Constance.|url=https://books.google.com/books?id=mR4SdJGD7tEC|title=Hilbert|publisher=[[Springer Publishing|Springer]]|year=1996|isbn=0-387-94674-8|location=New York|author-link=Constance Reid}} The definitive English-language biography of Hilbert.
* {{Cite journal | last1 = Rowe | first1 = D. E. | doi = 10.1086/368687 | title = Klein, Hilbert, and the Gottingen Mathematical Tradition | journal = Osiris | volume = 5 | pages = 186–213 | year = 1989 | s2cid = 121068952 }}
* {{cite journal | last1 = Sauer | first1 = Tilman | year = 1999 | title = The relativity of discovery: Hilbert's first note on the foundations of physics | journal = Arch. Hist. Exact Sci. | volume = 53 | pages = 529–75 | arxiv = physics/9811050 | bibcode = 1998physics..11050S }}
* {{Cite book|last=Sieg|first=Wilfried|url=https://books.google.com/books?id=4lDrwqo-8TkC|title=Hilbert's Programs and Beyond|publisher=Oxford University Press|year=2013|isbn=978-0-19-537222-9}}
*Sieg, Wilfried, and Ravaglia, Mark, 2005, "Grundlagen der Mathematik" in [[Ivor Grattan-Guinness|Grattan-Guinness, I.]], ed., ''Landmark Writings in Western Mathematics''. [[Elsevier]]: 981–99. (in English)
* [[Kip Thorne|Thorne, Kip]], 1995. ''[[Black Holes and Time Warps|Black Holes and Time Warps: Einstein's Outrageous Legacy]]'',  W. W. Norton & Company; Reprint edition. {{isbn|0-393-31276-3}}.
* Georg von Wallwitz: ''Meine Herren, dies ist keine Badeanstalt. Wie ein Mathematiker das 20. Jahrhundert veränderte.'' Berenberg Verlag, Berlin 2017, ISBN 978-3-946334-24-8. The definitive German-language biography of Hilbert.

==External links==
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{{Wikiquote}}
* [https://web.archive.org/web/20110517092213/http://www.ags.uni-sb.de/~cp/p/hilbertbernays/goal.htm Hilbert Bernays Project]
* [http://aleph0.clarku.edu/~djoyce/hilbert/problems.html Hilbert's 23 Problems Address]
* [http://mathematics.conference-site.com// ICMM 2014 dedicated to the memory of D.Hilbert]
* {{Gutenberg author |id=7340| name=David Hilbert}}
* {{Internet Archive author |sname=David Hilbert}}
* {{Librivox author |id=3033}}
* [http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.mp3 Hilbert's radio speech recorded in Königsberg 1930 (in German)] {{Webarchive|url=https://web.archive.org/web/20060214172824/http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.mp3 |date=14 February 2006 }}, with English [http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.pdf translation] {{Webarchive|url=https://web.archive.org/web/20201112012103/http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.pdf |date=12 November 2020 }}
* [http://mathworld.wolfram.com/HilbertsConstants.html Wolfram MathWorld – Hilbert'Constant]
* {{MathGenealogy |id=7298}}
* {{MacTutor Biography|id=Hilbert}}
* [https://web.archive.org/web/20080514013255/http://www.gresham.ac.uk/event.asp?PageId=45&EventId=628 'From Hilbert's Problems to the Future'], lecture by Professor Robin Wilson, [[Gresham College]], 27 February 2008 (available in text, audio and video formats).
* {{PM20|FID=pe/007811}}

{{Authority control}}

{{DEFAULTSORT:Hilbert, David}}
[[Category:David Hilbert| ]]
[[Category:1862 births]]
[[Category:1943 deaths]]
[[Category:Scientists from Königsberg]]
[[Category:People from the Province of Prussia]]
[[Category:19th-century German mathematicians]]
[[Category:20th-century German mathematicians]]
[[Category:Foreign members of the Royal Society]]
[[Category:Foreign associates of the National Academy of Sciences]]
[[Category:German agnostics]]
[[Category:Formalism (deductive)]]
[[Category:Former Protestants]]
[[Category:German geometers]]
[[Category:German mathematical analysts]]
[[Category:German number theorists]]
[[Category:Operator theorists]]
[[Category:Recipients of the Pour le Mérite (civil class)]]
[[Category:German relativity theorists]]
[[Category:Academic staff of the University of Göttingen]]
[[Category:University of Königsberg alumni]]
[[Category:Academic staff of the University of Königsberg]]
[[Category:Philosophers of mathematics]]
[[Category:Members of the American Philosophical Society]]
[[Category:Recipients of the Cothenius Medal]]
[[Category:Presidents of the German Mathematical Society]]