Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a co-founder of proof theory andmathematical logic.[6]
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 theProvince of Prussia,Kingdom of Prussia, either inKönigsberg, now Kaliningrad, (according to Hilbert's own statement) or in Wehlau (known since 1946 asZnamensk) near Königsberg where his father worked at the time of his birth. His paternal grandfather was David Hilbert, a judge andGeheimrat. His mother Maria had an interest in philosophy, astronomy andprime numbers, while his father Otto taught himPrussian 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.[7]
In late 1872, Hilbert entered theFriedrichskollegGymnasium (Collegium fridericianum, the same school thatImmanuel Kant had attended 140 years before); but, after an unhappy period, he transferred to (late 1879) and graduated from (early 1880) the more science-orientedWilhelm Gymnasium.[8] Upon graduation, in autumn 1880, Hilbert enrolled at theUniversity 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),[9] returned to Königsberg and entered the university. Hilbert developed a lifelong friendship with the shy, gifted Minkowski.[10][11]
In 1884,Adolf Hurwitz arrived from Göttingen as anExtraordinarius (i.e., an associate professor). 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 underFerdinand von Lindemann,[2] titledÜber invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties of specialbinary forms, in particular thespherical harmonic functions").
Hilbert remained at the University of Königsberg as aPrivatdozent (senior lecturer) from 1886 to 1895. In 1895, as a result of intervention on his behalf byFelix Klein, he obtained the position of Professor of Mathematics at theUniversity of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world.[12] He remained there for the rest of his life.
The Mathematical Institute in Göttingen. Its new building, constructed with funds from theRockefeller Foundation, was opened by Hilbert and Courant in 1930.
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]."[15] 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.[16]
Hilbert considered the mathematicianHermann Minkowski to be his "best and truest friend".[17]
Hilbert was baptized and raised aCalvinist in thePrussian Evangelical Church.[a] He later left the Church and became anagnostic.[b] He also argued that mathematical truth was independent of the existence of God or othera priori assumptions.[c][d] WhenGalileo Galilei was criticized for failing to stand up for his convictions on theHeliocentric 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."[e]
Around 1925, Hilbert developedpernicious anemia, a then-untreatable vitamin deficiency of which the primary symptom is exhaustion; his assistantEugene 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".[19]
About a year after the purge, 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 theJews". Hilbert replied: "Suffered? It doesn't exist any longer, does it?"[24][25]
Hilbert's grave: Wir müssen wissen 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 themArnold Sommerfeld, a theoretical physicist and also a native of Königsberg.[26] News of his death only became known to the wider world several months after he died.[27]
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":[28]
Wir müssen wissen. Wir werden wissen.
Translation:
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.[f]Gödel's incompleteness theorems show that evenelementary axiomatic systems such asPeano arithmetic are either self-contradicting or contain logical propositions that are impossible to prove or disprove within that system.
Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famousfiniteness theorem. Twenty years earlier,Paul Gordan had demonstrated thetheorem 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 asGordan's Problem, Hilbert realized that it was necessary to take a completely different path. As a result, he demonstratedHilbert's basis theorem, showing the existence of a finite set of generators, for the invariants ofquantics in any number of variables, but in an abstract form. That is, while demonstrating the existence of such a set, it was not aconstructive proof—it did not display "an object"—but rather, it was anexistence proof[29] and relied on use of thelaw of excluded middle in an infinite extension.
Hilbert sent his results to theMathematische Annalen. Gordan, the house expert on the theory of invariants for theMathematische 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:
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 theAnnalen. After having read the manuscript, Klein wrote to him, saying:
Without doubt this is the most important work on general algebra that theAnnalen has ever published.[31]
Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself would say:
I have convinced myself that even theology has its merits.[32]
For all his successes, the nature of his proof created more trouble than Hilbert could have imagined. AlthoughKronecker 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".[32] Not all were convinced. WhileKronecker would die soon afterwards, hisconstructivist philosophy would continue with the youngBrouwer and his developingintuitionist "school", much to Hilbert's torment in his later years.[33] Indeed, Hilbert would lose his "gifted pupil"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".[34] Brouwer the intuitionist in particular opposed the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert responded:
Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists.[35]
In the subject ofalgebra, afield is calledalgebraically 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 of variables has acommon root: This is the case if and only if there do not exist polynomials and indices such that
.
This result is known as theHilbert root theorem, or "Hilberts Nullstellensatz" in German. He also proved that the correspondence between vanishing ideals and their vanishing sets is bijective betweenaffine varieties andradical ideals in.
In 1890,Giuseppe Peano had published an article in theMathematische Annalen describing the historically firstspace-filling curve. In response, Hilbert designed his own construction of such a curve, which is now called theHilbert 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.
The textGrundlagen der Geometrie (tr.:Foundations of Geometry) published by Hilbert in 1899 proposes a formal set, called Hilbert's axioms, substituting for the traditionalaxioms of Euclid. They avoid weaknesses identified in those ofEuclid, 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 theGrundlagen 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.[36][37] 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.[g]
Hilbert's approach signaled the shift to the modernaxiomatic method. In this, Hilbert was anticipated byMoritz Pasch's work from 1882. Axioms are not taken as self-evident truths. Geometry may treatthings, about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such aspoint,line,plane, and others, could be substituted, as Hilbert is reported to have said toSchoenflies andKötter, by tables, chairs, glasses of beer and other such objects.[38] 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 segments), andcongruence ofangles. The axioms unify both theplane geometry andsolid geometry of Euclid in a single system.
Hilbert put forth a highly influential list consisting of 23 unsolved problems at theInternational 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?]
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 contemporaryGiuseppe 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:
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?[39]
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 alsoHilbert'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.
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 theformalist 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.
In 1920, Hilbert proposed a research project inmetamathematics 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 ofaxioms; and
that some such axiom system is provably consistent through some means such as theepsilon 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 theignorabimus, still an active issue in his time in German thought, and traced back in that formulation toEmil du Bois-Reymond.[40]
This program is still recognizable in the most popularphilosophy of mathematics, where it is usually calledformalism. For example, theBourbaki 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 theaxiomatic 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:
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.[41]
Hilbert published his views on the foundations of mathematics in the 2-volume work,Grundlagen der Mathematik.
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.
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, hisincompleteness 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 genuinelyfinitary.
Nevertheless, the subsequent achievements of proof theory at the very leastclarified 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 ofrecursion theory and thenmathematical logic as an autonomous discipline in the 1930s. The basis for latertheoretical computer science, in the work ofAlonzo Church andAlan Turing, also grew directly out of this "debate".[42]
Around 1909, Hilbert dedicated himself to the study of differential andintegral equations; 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 dimensionalEuclidean space, later calledHilbert 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, definingBanach spaces. Hilbert spaces are an important class of objects in the area offunctional analysis, particularly of thespectral theory of self-adjoint linear operators, that grew up around it during the 20th century.
Until 1912, Hilbert was almost exclusively apure mathematician. When planning a visit from Bonn, where he was immersed in studying physics, his fellow mathematician and friendHermann Minkowski joked he had to spend 10 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.[43] He started studyingkinetic gas theory and moved on to elementaryradiation theory and the molecular theory of matter. Even after the war started in 1914, he continued seminars and classes where the works ofAlbert Einstein and others were followed closely.
By 1907, Einstein had framed the fundamentals of the theory ofgravity, but then struggled for nearly 8 years to put the theory intoits final form.[44] MeetingEmmy Noether at Göttinger was instrumental in his breakthrough.[45] By early summer 1915, Hilbert's interest in physics had focused ongeneral relativity, and he invited Einstein to Göttingen to deliver a week of lectures on the subject.[46] Einstein received an enthusiastic reception at Göttingen.[47] Over the summer, Einstein learned that Hilbert was also working on the field equations and redoubled his own efforts. During November 1915, Einstein published several papers culminating inThe Field Equations of Gravitation (seeEinstein field equations).[h] Nearly simultaneously, Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations (seeEinstein–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.[i] See more atpriority.
Additionally, Hilbert's work anticipated and assisted several advances in themathematical formulation of quantum mechanics. His work was a key aspect ofHermann Weyl andJohn von Neumann's work on the mathematical equivalence ofWerner Heisenberg'smatrix mechanics andErwin Schrödinger'swave equation, and his namesake Hilbert space plays an important part in quantum theory. In 1926, von 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.[j]
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 ofintegral equations. When his colleague Richard Courant wrote the now classicMethoden 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.
Hilbert unified the field ofalgebraic number theory with his 1897 treatiseZahlbericht (literally "report on numbers"). He also resolved a significant number-theoryproblem formulated by Waring in 1770. As withthe 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.[48] He then had little more to publish on the subject; but the emergence ofHilbert modular forms in the dissertation of a student means his name is further attached to a major area.
Hilbert did not work in the central areas ofanalytic number theory, but his name has become known for theHilbert–Pólya conjecture, for reasons that are anecdotal.Ernst Hellinger, a student of Hilbert, once toldAndré Weil that Hilbert had announced in his seminar in the early 1900s that he expected the proof of theRiemann Hypothesis would be a consequence of Fredholm's work on integral equations with a symmetric kernel.[49]
His collected works (Gesammelte Abhandlungen) have been published several times. The original versions of his papers contained "many technical errors of varying degree";[50] 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 thecontinuum hypothesis.[51][52] The errors were nonetheless so numerous and significant that it tookOlga Taussky-Todd three years to make the corrections.[52]
^The Hilberts had, by this time, left the Calvinist Protestant church in which they had been baptized and married. – Reid 1996, p.91
^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:
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. 91)
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. 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. —Shaposhnikov, Vladislav (2016)."Theological Underpinnings of the Modern Philosophy of Mathematics. Part II: The Quest for Autonomous Foundations".Studies in Logic, Grammar and Rhetoric.44 (1):147–168.doi:10.1515/slgr-2016-0009.
^"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,Hilbert's program, 22C:096, University of Iowa.
^Michael R. Matthews (2009).Science, Worldviews and Education. Springer. p. 129.ISBN978-90-481-2779-5.As is well known, Hilbert rejected Leopold Kronecker's God for the solution of the problem of the foundations of mathematics.
^Constance Reid; Hermann Weyl (1970).Hilbert. Springer-Verlag. p. 92.ISBN978-0-387-04999-1.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."
^"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 ofGoldbach orFermat) 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 lectureNaturerkennen 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 isno 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. 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 byHans Hahn" (van Heijenoort:592); this abstract and the full paper both appear in van Heijenoort:583ff.
^Independently and contemporaneously, a 19 year-old American student namedRobert 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]
^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. 437)
^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. 141–142, also Isaacson 2007:222 quoting Thorne p. 119).
^In 1926, the year after the matrix mechanics formulation of quantum theory byMax Born andWerner Heisenberg, the mathematicianJohn von Neumann became an assistant to Hilbert at Göttingen. When von Neumann left in 1932, von Neumann's book on the mathematical foundations of quantum mechanics, based on Hilbert's mathematics, was published under the titleMathematische Grundlagen der Quantenmechanik. See: Norman Macrae (1999)John von 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).
^This work established Takagi as Japan's first mathematician of international stature.
^Reid 1996, pp. 1–3; also onp. 8, Reid notes that there is some ambiguity as to exactly where Hilbert was born. Hilbert himself stated that he was born in Königsberg.
^Weyl, Hermann (2012), "David Hilbert and his Mathematical Work", in Peter Pesic (ed.),Levels of Infinity/Selected writings on Mathematics and Philosophy, Dover, p. 94,ISBN978-0-486-48903-2
^Milne-Thomson, L (1935)."abstract for Grundlagen der Mathematik".Nature.136 (3430):126–127.doi:10.1038/136126a0.S2CID4122792. Retrieved15 December 2023.This is probably the most important book on mathematical foundations that has appeared since Whitehead and Russell's "Principia Mathematica".
^Menzler-Trott, Eckart (2001).Gentzens Problem. Mathematische Logik im nationalsozialistischen Deutschland. Auflage: Birkhäuser. p. 142.ISBN3-764-36574-9.
^Meyer, Hajo G. (2008).Tragisches Schicksal. Das deutsche Judentum und die Wirkung historischer Kräfte: Eine Übung in angewandter Geschichtsphilosophie. Frank & Timme. p. 202.ISBN978-3-865-96174-7.
^Otto Blumenthal (1935). David Hilbert (ed.).Lebensgeschichte. Gesammelte Abhandlungen. Vol. 3. Julius Springer. pp. 388–429. Archived fromthe original on 4 March 2016. Retrieved6 September 2018. Here: p.402-403
^Finkelstein, Gabriel (2013).Emil du Bois-Reymond: Neuroscience, Self, and Society in Nineteenth-Century Germany. Cambridge; London: The MIT Press. pp. 265–289.ISBN978-0262019507.
^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).
^Endres, S.; Steiner, F. (2009), "The Berry–Keating operator on and on compact quantum graphs with general self-adjoint realizations",Journal of Physics A: Mathematical and Theoretical,43 (9): 37,arXiv:0912.3183v5,doi:10.1088/1751-8113/43/9/095204,S2CID115162684
Hilbert, David (1990) [1971].Foundations of Geometry [Grundlagen der Geometrie]. Translated by Unger, Leo (2nd English ed.). La Salle, IL: Open Court Publishing.ISBN978-0-87548-164-7.translated from the 10th German edition
Hilbert, David (2004). Hallett, Michael; Majer, Ulrich (eds.).David Hilbert's Lectures on the Foundations of Mathematics and Physics, 1891–1933. Berlin & Heidelberg: Springer-Verlag.ISBN978-3-540-64373-9.
Bottazzini Umberto, 2003.Il flauto di Hilbert. Storia della matematica.UTET,ISBN88-7750-852-3
Corry, L., Renn, J., and Stachel, J., 1997, "Belated Decision in the Hilbert-Einstein Priority Dispute,"Science 278: nn-nn.
Corry, Leo (2004).David Hilbert and the Axiomatization of Physics (1898–1918): From Grundlagen der Geometrie to Grundlagen der Physik. Springer.ISBN90-481-6719-1.
Dawson, John W. Jr 1997.Logical Dilemmas: The Life and Work of Kurt Gödel. Wellesley MA: A. K. Peters.ISBN1-56881-256-6.
Mancosu, Paolo (1998).From Brouwer to Hilbert, The Debate on the Foundations of Mathematics in 1920s. Oxford Univ. Press.ISBN978-0-19-509631-6.
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,ISBN88-339-5714-4. A clear exposition of the "errors" of Euclid and of the solutions presented in theGrundlagen der Geometrie, with reference tonon-Euclidean geometry.
Rowe, D. E. (1989). "Klein, Hilbert, and the Gottingen Mathematical Tradition".Osiris.5:186–213.doi:10.1086/368687.S2CID121068952.
Sauer, Tilman (1999). "The relativity of discovery: Hilbert's first note on the foundations of physics".Arch. Hist. Exact Sci.53:529–75.arXiv:physics/9811050.Bibcode:1998physics..11050S.
Sieg, Wilfried, and Ravaglia, Mark, 2005, "Grundlagen der Mathematik" inGrattan-Guinness, I., ed.,Landmark Writings in Western Mathematics.Elsevier: 981–99. (in English)
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.
