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David Hilbert (; ; 23 January 1862 – 14 February 1943) was a
German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struct ...
and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including
invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descr ...
, the
calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...
,
commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
,
algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, ...
, the
foundations of geometryFoundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometry, non-Euclidean geometries. These are fundamental to the study and of histor ...
, spectral theory of operators and its application to
integral equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
,
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
, and the
foundations of mathematics Foundations of mathematics is the study of the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental r ...
(particularly
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
). Hilbert adopted and defended
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
's set theory and
transfinite number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...
.


Life


Early life and education

Hilbert, the first of two children and only son of Otto and Maria Therese (Erdtmann) Hilbert, was born in the
Province of Prussia The Province of Prussia ( ; pl, Prowincja Prusy ; csb, Prowincjô Prësë) was a province A province is almost always an administrative division within a country or state. The term derives from the ancient Roman '' provincia'', which was t ...
,
Kingdom of Prussia The Kingdom of Prussia (german: Königreich Preußen) was a German kingdom Kingdom may refer to: Monarchy * A type of monarchy * A realm ruled by: **A king, during the reign of a male monarch **A queen regnant, during the reign of a female ...
, either in
Königsberg Königsberg (, , ) was the name for the historic Prussian city that is now Kaliningrad, Russia. Königsberg was founded in 1255 on the site of the ancient Old Prussian settlement ''Twangste'' by the Teutonic Knights during the Northern Crusade ...

Königsberg
(according to Hilbert's own statement) or in Wehlau (known since 1946 as Znamensk) near Königsberg where his father worked at the time of his birth. In late 1872, Hilbert entered the Friedrichskolleg
Gymnasium Gymnasium may refer to: *Gymnasium (ancient Greece), educational and sporting institution *Gymnasium (school), type of secondary school that prepares students for higher education **Gymnasium (Denmark) **Gymnasium (Germany) **Gymnasium UNT, high ...
(''Collegium fridericianum'', the same school that
Immanuel Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about r ...

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 Wilhelm Gymnasium. Upon graduation, in autumn 1880, Hilbert enrolled at the
University of Königsberg The University of Königsberg (german: Albertus-Universität Königsberg) was the university A university ( la, universitas, 'a whole') is an educational institution, institution of higher education, higher (or Tertiary education, tertiary) educ ...
, the "Albertina". In early 1882,
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

Hermann Minkowski
(two years younger than Hilbert and also a native of Königsberg but had gone to Berlin for three semesters), returned to Königsberg and entered the university. Hilbert developed a lifelong friendship with the shy, gifted Minkowski.


Career

In 1884,
Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...

Adolf Hurwitz
arrived from Göttingen as an
Extraordinarius
Extraordinarius
(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 under
Ferdinand von Lindemann Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
, titled ''Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen'' ("On the invariant properties of special binary forms, in particular the ). 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 Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
, he obtained the position of Professor of Mathematics at the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded i ...
. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world. He remained there for the rest of his life.


Göttingen school

Among Hilbert's students were
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of ...

Hermann Weyl
,
chess Chess is a board game Board games are tabletop game Tabletop games are game with separate sliding drawer, from 1390–1353 BC, made of glazed faience, dimensions: 5.5 × 7.7 × 21 cm, in the Brooklyn Museum (New Yor ...

chess
champion
Emanuel Lasker Emanuel Lasker (December 24, 1868 – January 11, 1941) was a German chess Chess is a recreational and competitive board game played between two players. It is sometimes called Western or international chess to distinguish it from chess ...

Emanuel Lasker
,
Ernst Zermelo Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel set theory, Zer ...
, and
Carl Gustav Hempel Carl Gustav "Peter" Hempel (January 8, 1905 – November 9, 1997) was a German writer A writer is a person who uses written words in different styles and techniques to communicate ideas. Writers produce different forms of literary art a ...
.
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian Americans, Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was generally rega ...

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 Amalie Emmy Noether Emmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...

Emmy Noether
and
Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of America (US ...
. Among his 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including (with date of thesis):
Otto Blumenthal Ludwig Otto Blumenthal (20 July 1876 – 12 November 1944) was a Germany, German mathematician and professor at RWTH Aachen University. Biography He was born in Frankfurt, Hesse-Nassau. A student of David Hilbert, Blumenthal was an editor of ''Ma ...
(1898), Felix Bernstein (1901),
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of ...

Hermann Weyl
(1908),
Richard Courant Richard Courant (January 8, 1888 – January 27, 1972) was a German American German Americans (german: Deutschamerikaner, ) are Americans Americans are the Citizenship of the United States, citizens and United States nationality law, nation ...

Richard Courant
(1910),
Erich Hecke Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...

Erich Hecke
(1910),
Hugo Steinhaus Władysław Hugo Dionizy Steinhaus (January 14, 1887 – February 25, 1972) was a Jewish-Polish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes t ...

Hugo Steinhaus
(1911), and
Wilhelm Ackermann Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a Germany, German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation. B ...
(1925). Between 1902 and 1939 Hilbert was editor of the ''
Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ance ...
'', the leading mathematical journal of the time.


Personal life

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 ilbert's" While at Königsberg they had their one child, (1893–1969). Franz suffered throughout his life from an undiagnosed mental illness. His inferior intellect was a terrible disappointment to his father and this misfortune was a matter of distress to the mathematicians and students at Göttingen. Hilbert considered the mathematician
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

Hermann Minkowski
to be his "best and truest friend". Hilbert was baptized and raised a
Calvinist Calvinism (also called the Reformed tradition, Reformed Christianity, Reformed Protestantism, or the Reformed faith) is a major branch of Protestantism Protestantism is a form of Christianity Christianity is an , based on the a ...
in the Prussian Evangelical Church.The Hilberts had, by this time, left the Calvinist Protestant church in which they had been baptized and married. – Reid 1996, p.91 He later left the Church and became an
agnostic Agnosticism is the view that the existence of God, of the divinity, divine or the supernatural is unknown or Uncertainty, unknowable. (page 56 in 1967 edition) Another definition provided is the view that "human reason is incapable of providing ...

agnostic
. 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 round 1902left the Reformed Protestant Church in which they had been baptized and married. It was told in Göttingen that when avid Hilbert's sonFranz 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 ( 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. —
He also argued that mathematical truth was independent of the existence of God or other ''
a priori ''A priori'' and ''a posteriori'' ('from the earlier' and 'from the later', respectively) are Latin phrases used in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaph ...
'' assumptions."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
When
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei ( , ; 15 February 1564 – 8 January 1642), commonly referred to as Galileo, was an astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific q ...

Galileo Galilei
was criticized for failing to stand up for his convictions on the
Heliocentric theory Heliocentrism is the astronomical model in which the Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. About 29% of Earth's surface is land consisting of continent A continent ...
, Hilbert objected: "But alileowas 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."


Later years

Like
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

Albert Einstein
, Hilbert had closest contacts with the Berlin Group whose leading founders had studied under Hilbert in Göttingen (
Kurt Grelling Kurt Grelling (2 March 1886 – September 1942) was a German logician and philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lov ...
,
Hans Reichenbach Hans Reichenbach (September 26, 1891 – April 9, 1953) was a leading philosopher of science, educator, and proponent of logical empiricism. He was influential in the areas of science, education, and of logical empiricism. He founded the ''Gesell ...
and Walter Dubislav). Around 1925, Hilbert developed
pernicious anemia Vitamin B12 deficiency anemia, of which pernicious anemia (PA) is a type, is a disease in which not enough red blood cell Red blood cells (RBCs), also referred to as red cells, red blood corpuscles (in humans or other animals not having nucl ...
, a then-untreatable vitamin deficiency whose primary symptom is exhaustion; his assistant
Eugene Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian theoretical physicist who also contributed to mathematical physics. He obtained United States of America, American citizenship in 1937, ...
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." Hilbert lived to see the Nazis purge many of the prominent faculty members at
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded i ...
in 1933. Those forced out included
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of ...

Hermann Weyl
(who had taken Hilbert's chair when he retired in 1930),
Emmy Noether Amalie Emmy Noether Emmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...

Emmy Noether
and
Edmund Landau Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as n ...

Edmund Landau
. One who had to leave Germany,
Paul Bernays Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Switzerland, Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close coll ...
, had collaborated with Hilbert in mathematical logic, and co-authored with him the important book ''
Grundlagen der Mathematik ''Grundlagen der Mathematik'' (English: ''Foundations of Mathematics'') is a two-volume work by David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians o ...
'' (which eventually appeared in two volumes, in 1934 and 1939). This was a sequel to the Hilbert– Ackermann book '' Principles of Mathematical Logic'' from 1928. Hermann Weyl's successor was
Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) was a Germany, German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic number, ''p''-adic numbers to local c ...

Helmut Hasse
. About a year later, Hilbert attended a banquet and was seated next to the new Minister of Education,
Bernhard Rust Bernhard Rust (30 September 1883 – 8 May 1945) was Minister of Science, Education and National Culture (Reichserziehungsministerium, Reichserziehungsminister) in Nazi Germany.Claudia Koonz, ''The Nazi Conscience'', p 134 A combination of schoo ...
. 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!"


Death

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 Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For ...
, a theoretical physicist and also a native of Königsberg. News of his death only became known to the wider world six months after he died. 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 The Latin maxim ''ignoramus et ignorabimus'', meaning "we do not know and will not know," represents the idea that scientific knowledge is limited. It was popularized by Emil du Bois-Reymond, a German physiologist, in his 1872 address "Über die ...
''" or "We do not know, we shall not know": The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians,
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dict ...
—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. "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
930 Year 930 (Roman numerals, CMXXX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. Events By place Europe * The Althing, the parliament of Iceland, is established at þingvellir ("Thing ...
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 Goldbach or
Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the '' Parlement'' of Toulouse Toulouse ( , ; oc, Tolosa ; la, Tolosa ) is the capital of the French departments of France, department ...

Fermat
) that, while contentually true, are unprovable in the formal system of classical mathematics (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 o-onehas 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 59"(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 on October 1930 to the Vienna Academy of Sciences by Hans Hahn" (van Heijenoort:592); this abstract and the full paper both appear in van Heijenoort:583ff.
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorem In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference r ...
show that even
elementary In computational complexity theory, the complexity class ELEMENTARY of elementary recursive functions is the union of the classes : \begin \mathsf & = \bigcup_ k\mathsf \\ & = \mathsf\left(2^n\right)\cup\mathsf\left(2^\right)\ ...
axiomatic systems such as
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have been ...
are either self-contradicting or contain logical propositions that are impossible to prove or disprove.


Contributions to mathematics and physics


Hilbert solves Gordan's Problem

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous ''finiteness theorem''. Twenty years earlier,
Paul Gordan __NOTOC__ Paul Albert Gordan (27 April 1837 – 21 December 1912) was a Jewish-German mathematician, a student of Carl Gustav Jacob Jacobi, Carl Jacobi at the University of Königsberg before obtaining his Ph.D. at the University of Breslau ...
had demonstrated the
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
'', showing the existence of a finite set of generators, for the invariants of 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 proofIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
— it did not display "an object" — but rather, it was an
existence proofIn mathematics, a constructive proof is a method of mathematical proof, proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also kn ...
and relied on use of the
law of excluded middle In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
in an infinite extension. Hilbert sent his results to the ''
Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ance ...
''. 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: 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: Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself would say: For all his successes, the nature of his proof created more trouble than Hilbert could have imagined. Although 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". Not all were convinced. While would die soon afterwards, his constructivist philosophy would continue with the young
Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'. Brouwer * Adriaen Brouwer (1605–1638), Flemish painter * Alexander Brouwer (b. 1989), Dutch beach volleyball player * Andries Brouw ...
and his developing
intuitionist In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathe ...
"school", much to Hilbert's torment in his later years. Indeed, Hilbert would lose his "gifted pupil"
Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a Germany, German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he ...

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". Brouwer the intuitionist in particular opposed the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert responded:


Axiomatization of geometry

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 axioms of Euclid. They avoid weaknesses identified in those of
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

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. 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. Hilbert's approach signaled the shift to the modern
axiomatic method In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. In this, Hilbert was anticipated by
Moritz Pasch Moritz Pasch (8 November 1843, Breslau, Prussia Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a historically prominent Germans, German state that originated in 1525 with Duchy of Prussia, a duchy centered on the Prussia (region), reg ...

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 Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
,
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

line
,
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
, 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. 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), and criteria of congruence of angles, congruence of angles. The axioms unify both the Euclidean geometry, plane geometry and solid geometry of Euclid in a single system.


The 23 problems

Hilbert put forth a most influential list 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. After re-working the foundations of classical geometry, Hilbert could have extrapolated to the rest of mathematics. His approach differed, however, from the later 'foundationalist' Russell–Whitehead or 'encyclopedist' Nicolas Bourbaki, and from his contemporary Giuseppe Peano. The mathematical community as a whole could enlist in problems, which he had identified as crucial aspects of the areas of mathematics he took to be key. 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: 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 inevitable debate, whenever 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 even continue to this day to remain a challenge for mathematicians.


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. There is, however, room to doubt whether Hilbert's own views were simplistically formalist in this sense.


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 axioms; 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. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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: Hilbert published his views on the foundations of mathematics in the 2-volume work,
Grundlagen der Mathematik ''Grundlagen der Mathematik'' (English: ''Foundations of Mathematics'') is a two-volume work by David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians o ...
.


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 non-contradictory formal system, which was comprehensive enough to include at least arithmetic, cannot demonstrate its completeness by way of its own axioms. 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 Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...
as an autonomous discipline in the 1930s. The basis for later theoretical computer science, in the work of
Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of America (US ...
and Alan Turing, also grew directly out of this 'debate'.


Functional analysis

Around 1909, Hilbert dedicated himself to the study of differential and integral 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 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 Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

Hermann 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. 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 Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

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 years to put the theory into General Relativity, its final form. By early summer 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. Einstein received an enthusiastic reception at Göttingen. 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 in ''The Field Equations of Gravitation'' (see Einstein field equations).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) Nearly simultaneously, David 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.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). See more at relativity priority dispute#General Relativity 3, 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 Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of ...

Hermann Weyl
and
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian Americans, Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was generally rega ...

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 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.In 1926, the year after the matrix mechanics formulation of quantum theory by Max Born and Werner Heisenberg, the mathematician
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian Americans, Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was generally rega ...

John 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 title ''Mathematische 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).
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, ...
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. He then had little more to publish on the subject; but the emergence of Hilbert modular forms 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.This work established Takagi as Japan's first mathematician of international stature. 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.


Works

His collected works (''Gesammelte Abhandlungen'') have been published several times. The original versions of his papers contained "many technical errors of varying degree";, chap. 13. 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.Gian-Carlo Rota, Rota G.-C. (1997),
Ten lessons I wish I had been taught
, ''Notices of the AMS'', 44: 22-25.
The errors were nonetheless so numerous and significant that it took Olga Taussky-Todd three years to make the corrections.


See also


Concepts

* 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'' * Relativity priority dispute


Footnotes


Citations


Sources


Primary literature in English translation

* ** 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. * * * * *


Secondary literature

* , 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, * Corry, L., Renn, J., and Stachel, J., 1997, "Belated Decision in the Hilbert-Einstein Priority Dispute," ''Science 278'': nn-nn. * * Dawson, John W. Jr 1997. ''Logical Dilemmas: The Life and Work of Kurt Gödel''. Wellesley MA: A. K. Peters. . * * 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''. * * 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, . A clear exposition of the "errors" of Euclid and of the solutions presented in the ''Grundlagen der Geometrie'', with reference to non-Euclidean geometry. * The definitive English-language biography of Hilbert. * * * *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. .


External links


Hilbert Bernays Project



ICMM 2014 dedicated to the memory of D.Hilbert
* * *
Hilbert's radio speech recorded in Königsberg 1930 (in German)
with Englis
translation


* *
'From Hilbert's Problems to the Future'
lecture by Professor Robin Wilson, Gresham College, 27 February 2008 (available in text, audio and video formats). * {{DEFAULTSORT:Hilbert, David David Hilbert, 1862 births 1943 deaths 19th-century German mathematicians 20th-century German mathematicians Foreign Members of the Royal Society Foreign associates of the National Academy of Sciences German agnostics Formalism (deductive) Former Protestants Geometers Mathematical analysts Number theorists Operator theorists Scientists from Königsberg People from the Province of Prussia Recipients of the Pour le Mérite (civil class) German relativity theorists University of Göttingen faculty University of Königsberg alumni University of Königsberg faculty