He also resolved a significant number-theory problem formulated by Waring in 1770.

David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries.

The Search for Mathematical Roots 1870–1940.

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), returned to Königsberg and entered the university.

In this, Hilbert was anticipated by Moritz Pasch's work from 1882.

Hilbert developed a lifelong friendship with the shy, gifted Minkowski. ===Career=== In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius (i.e., an associate professor).

Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties of special binary forms, in particular the spherical harmonic functions"). Hilbert remained at the University of Königsberg as a Privatdozent (senior lecturer) from 1886 to 1895.

Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties of special binary forms, in particular the spherical harmonic functions"). Hilbert remained at the University of Königsberg as a Privatdozent (senior lecturer) from 1886 to 1895.

Gödel's incompleteness theorems show that even elementary axiomatic systems such as Peano arithmetic 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.

Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, 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 [Hilbert's]." While at Königsberg they had their one child, Franz Hilbert (1893–1969). Franz suffered throughout his life from an undiagnosed mental illness.

Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties of special binary forms, in particular the 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.

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").

prohibiting the boxer the use of his fists. ===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.

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.

The axioms unify both the 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.

Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, 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 [Hilbert's]." While at Königsberg they had their one child, Franz Hilbert (1893–1969). Franz suffered throughout his life from an undiagnosed mental illness.

Townsend and copyrighted in 1902.

"The grounding of elementary number theory," 1148–1156. * 1904.

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.

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 years with a confounding problem of putting the theory into final form.

The basis for later theoretical computer science, in the work of Alonzo Church 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.

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.

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.

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 years with a confounding problem of putting the theory into final form.

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.

There is, however, room to doubt whether Hilbert's own views were simplistically formalist in this sense. ====Hilbert's program==== In 1920 he proposed explicitly a research project (in metamathematics, as it was then termed) that became known as Hilbert's program.

"Axiomatic thought," 1114–1115. * 1922.

"The new grounding of mathematics: First report," 1115–1133. * 1923.

"On the foundations of logic and arithmetic," 129–138. * 1925.

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. Throughout this immersion in physics, Hilbert worked on putting rigor into the mathematics of physics.

"On the infinite," 367–392. * 1927.

This was a sequel to the Hilbert–Ackermann book Principles of Mathematical Logic from 1928.

Those forced out included Hermann Weyl (who had taken Hilbert's chair when he retired in 1930), Emmy Noether and Edmund Landau.

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.

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.

Results were mostly proved by 1930, after work by Teiji Takagi. 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 logical foundations of mathematics," 1134–1147. * 1930.

In 1931 his incompleteness theorem showed that Hilbert's grand plan was impossible as stated.

"Logic and the knowledge of nature," 1157–1165. * 1931.

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 (which eventually appeared in two volumes, in 1934 and 1939).

Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, 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 [Hilbert's]." While at Königsberg they had their one child, Franz Hilbert (1893–1969). Franz suffered throughout his life from an undiagnosed mental illness.

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 (which eventually appeared in two volumes, in 1934 and 1939).

David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries.

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.

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.

Mehra, Jagdish, 1974.

(in English) Thorne, Kip, 1995.

A clear exposition of the "errors" of Euclid and of the solutions presented in the Grundlagen der Geometrie, with reference to non-Euclidean geometry. Reid, Constance, 1996.

UTET, Corry, L., Renn, J., and Stachel, J., 1997, "Belated Decision in the Hilbert-Einstein Priority Dispute," Science 278: nn-nn. Dawson, John W.

. Folsing, Albrecht, 1998.

Penguin. Grattan-Guinness, Ivor, 2000.

Press. Gray, Jeremy, 2000.

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.

Reidel. Piergiorgio Odifreddi, 2003.

The definitive English-language biography of Hilbert. Sieg, Wilfried, and Ravaglia, Mark, 2005, "Grundlagen der Mathematik" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics.

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