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Encyclopedia > David Hilbert


| name = David Hilbert | image = Hilbert1912.jpg | image_width = 300px | caption = David Hilbert (1912) | birth_date = January 23, 1862(1862-01-23) | birth_place = Wehlau, Province of Prussia | death_date = February 14, 1943 (aged 81) | death_place = Göttingen, Germany | residence = Germany | nationality = German | field = Mathematician and Philosopher | work_institutions = University of Königsberg
Göttingen University | alma_mater = University of Königsberg | doctoral_advisor = Ferdinand von Lindemann | doctoral_students = Wilhelm Ackermann
Otto Blumenthal
Richard Courant
Max Dehn
Erich Hecke
Hellmuth Kneser
Robert König
Emanuel Lasker
Erhard Schmidt
Hugo Steinhaus
Teiji Takagi
Hermann Weyl
Ernst Zermelo | known_for = Hilbert's basis theorem
Hilbert's axioms
Hilbert's problems
Hilbert's program
Einstein–Hilbert action
Hilbert space | prizes = | religion = }} is the 23rd day of the year in the Gregorian calendar. ... This article is about 1862 . ... Znamensk (Russian: , German: Wehlau, Polish: Welawa, Lithuanian: VÄ—luva) is an urban settlement in Kaliningrad Oblast in Russia. ... The Province of Prussia was a province of Poland from the 15th century until 1660, consisting of Royal Prussia and Ducal Prussia. ... is the 45th day of the year in the Gregorian calendar. ... Year 1943 (MCMXLIII) was a common year starting on Friday (the link will display full 1943 calendar) of the Gregorian calendar. ... Göttingen marketplace with old city hall, Gänseliesel fountain and pedestrian zone Göttingen ( ) is a city in Lower Saxony, Germany. ... Image File history File links Flag_of_Germany. ... Image File history File links Flag_of_Germany. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... A philosopher is a person who thinks deeply regarding people, society, the world, and/or the universe. ... The inscription upon Kants tomb in Kaliningrad. ... The Georg-August University of Göttingen (Georg-August-Universität Göttingen, often called the Georgia Augusta) was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and opened in 1737. ... The inscription upon Kants tomb in Kaliningrad. ... Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i. ... Wilhelm Ackermann (March 29, 1896, Herscheid municipality, Germany – December 24, 1962 Lüdenscheid, Germany ) was a German mathematician best known for the Ackermann function, an important example in the theory of computation. ... Ludwig Otto Blumenthal (July 20, 1876 - November 12, 1944) was a German mathematician. ... Richard Courant (born January 8, 1888 at Lublinitz, today Poland, died January 27, 1972 at New York/USA) was a German and American mathematician. ... Max Dehn (November 13, 1878 – June 27, 1952) was a German mathematician. ... Erich Hecke (September 20, 1887 – February 13, 1947) was a German mathematician. ... Hellmuth Kneser (April 16, 1898 - August 23, 1973) was a german mathematician. ... Emanuel Lasker (December 24, 1868 – January 11, 1941) was a German World Chess Champion, mathematician, and philosopher born at Berlinchen in Brandenburg (now Barlinek in Poland). ... Erhard Schmidt (January 13, 1876 - December 6, 1959) was a German mathematician born in Dorpat (now Tartu, Estonia). ... Hugo Dyonizy Steinhaus (January 14, 1887 - February 25, 1972) was a Polish mathematician, educator, and humanist. ... Teiji Takagi (高木 貞治 Takagi Teiji, April 21, 1875 - February 28, 1960) was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory. ... Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ... Ernst Friedrich Ferdinand Zermelo (July 27, 1871, Berlin, German Empire – May 21, 1953, Freiburg im Breisgau, West Germany) was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. ... In mathematics, Hilberts basis theorem, first proved by David Hilbert in 1888, states that, if k is a field, then every ideal in the ring of multivariate polynomials k[x1, x2, ..., xn] is finitely generated. ... Hilberts axioms are a set of 20 assumptions (originally 21), David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. ... Hilberts problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. ... Hilberts program, formulated by German mathematician David Hilbert in the 1920s, was to formalize all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. ... The Einstein-Hilbert action is a mathematical object (an action) that is used to derive Einsteins field equations of general relativity. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...


David Hilbert (January 23, 1862February 14, 1943) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He invented or developed a broad range of fundamental ideas, in invariant theory, the axiomatization of geometry, and with the notion of Hilbert space,[1] one of the foundations of functional analysis. is the 23rd day of the year in the Gregorian calendar. ... This article is about 1862 . ... is the 45th day of the year in the Gregorian calendar. ... Year 1943 (MCMXLIII) was a common year starting on Friday (the link will display full 1943 calendar) of the Gregorian calendar. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ... Hilberts axioms are a set of 20 assumptions (originally 21), David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...


Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century. Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Hilberts problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. ...


Hilbert and his students supplied significant portions of the mathematical infrastructure required for quantum mechanics and general relativity. He is also known as one of the founders of proof theory, mathematical logic and the distinction between mathematics and metamathematics. For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... In general, metamathematics or meta-mathematics is reflection about mathematics seen as an entity/object in human consciousness and culture. ...

Contents

Life

Hilbert, the first of two children and only son of Otto and Maria Therese (Erdtmann) Hilbert, was born in Wehlau (Znamensk) near Königsberg (Kaliningrad) in the Province of Prussia.[2] In the fall of 1872 he entered the Friedrichskolleg Gymnasium (the same school that Immanuel Kant had attended 140 years before), but after an unhappy duration he transferred (fall 1879) to and graduated from (spring 1880) the more science-oriented Wilhelm Gymnasium.[3] Upon graduation he enrolled (autumn 1880) at the University of Königsberg . In the spring of 1882 Hermann Minkowski (two years younger than Hilbert and also a native of Königsberg but so talented he had graduated early from his gymnasium and gone to Berlin for three semesters),[4] returned to Königsberg and entered the university. "Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the shy, gifted Minkowski."[5] In 1884 Adolf Hurwitz arrived from Göttingen as an Extraordinarius, i.e., an associate professor. An intense and fruitful scientific exchange between the three began and especially Minkowski and Hilbert 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, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties of special binary forms, in particular the spherical harmonic functions"). Znamensk (Russian: , German: Wehlau, Polish: Welawa, Lithuanian: VÄ—luva) is an urban settlement in Kaliningrad Oblast in Russia. ... Kaliningrad (Russian: ; Lithuanian: Karaliaučius; German  , Polish: Królewiec; briefly Russified as Kyonigsberg), is a seaport and the administrative center of Kaliningrad Oblast, the Russian exclave between Poland and Lithuania on the Baltic Sea. ... The Province of Prussia was a province of Poland from the 15th century until 1660, consisting of Royal Prussia and Ducal Prussia. ... A gymnasium (pronounced with or, in Swedish, as opposed to ) is a type of school providing secondary education in some parts of Europe, comparable to English Grammar Schools and U.S. High Schools. ... Kant redirects here. ... The inscription upon Kants tomb in Kaliningrad. ... Hermann Minkowski. ... Adolf Hurwitz Adolf Hurwitz (26 March 1859- 18 November 1919) was a German mathematician, and one of the most important figures in mathematics in the second half of the nineteenth century (according to Jean-Pierre Serre, always something good in Hurwitz). He was born in a Jewish family in Hildesheim... Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i. ... Binary form is a way of structuring a piece of music into two related sections, both of which are usually repeated. ...


Hilbert remained at the University of Königsberg as a professor from 1886 to 1895. In 1892, Hilbert married Käthe Jerosch (1864–1945), "the daughter of a Konigsberg merchant, an outspoken young lady with an independence of mind that matched his own".[6] While at Königsberg they had their one child Franz Hilbert (1893–1969). In 1895, as a result of intervention on his behalf by Felix Klein he obtained the position of Chairman of Mathematics at the University of Göttingen, at that time the best research center for mathematics in the world and where he remained for the rest of his life. Felix Christian Klein (April 25, 1849, Düsseldorf, Germany – June 22, 1925, Göttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... The Georg-August University of Göttingen (Georg-August-Universität Göttingen, often called the Georgia Augusta) was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and opened in 1737. ...


Son Franz would suffer his entire life from an (undiagnosed) mental illness, his inferior intellect a terrible disappointment to his father and this tragedy a matter of distress to the mathematicians and students at Göttingen.[7] Sadly, Minkowski — Hilbert's "best and truest friend"[8] — would die prematurely of a ruptured appendix in 1909.

Math department in Göttingen where Hilbert worked from 1895 until his retirement in 1930
Math department in Göttingen where Hilbert worked from 1895 until his retirement in 1930

Image File history File links Size of this preview: 800 × 354 pixelsFull resolution (3612 × 1600 pixel, file size: 1. ... Image File history File links Size of this preview: 800 × 354 pixelsFull resolution (3612 × 1600 pixel, file size: 1. ...

The Göttingen school

Among the students of Hilbert, there were Hermann Weyl, the champion of chess Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. John von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church. Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ... Emanuel Lasker (December 24, 1868 – January 11, 1941) was a German World Chess Champion, mathematician, and philosopher born at Berlinchen in Brandenburg (now Barlinek in Poland). ... Ernst Friedrich Ferdinand Zermelo (July 27, 1871, Berlin, German Empire – May 21, 1953, Freiburg im Breisgau, West Germany) was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. ... Carl Gustav Hempel (* January 8th, 1905 in Oranienburg, Germany † November 9th, 1997 in Princeton, New Jersey) was a philosopher of science and a student of logical positivism. ... For other persons named John Neumann, see John Neumann (disambiguation). ... Amalie Emmy Noether [1] (March 23, 1882 – April 14, 1935) was a German-born mathematician, said by Einstein in eulogy to be [i]n the judgment of the most competent living mathematicians, [...] the most significant creative mathematical genius thus far produced since the higher education of women began. ... ‹ The template below (Expand) is being considered for deletion. ...


Among his 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including (with date of thesis): Otto Blumenthal (1898), Felix Bernstein (1901), Hermann Weyl (1908), Richard Courant (1910), Erich Hecke (1910), Hugo Steinhaus (1911), Wilhelm Ackermann (1925).[9] Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, the leading mathematical journal of the time. Ludwig Otto Blumenthal (July 20, 1876 - November 12, 1944) was a German mathematician. ... Felix Bernstein (February 24, 1878, Halle, Germany – December 3, 1956, Zurich, Switzerland) was a German mathematician known for developing a theorem of the equivalence of sets in 1897, and less well known for demonstrating the correct blood group inheritance pattern of multiple alleles at one locus in 1924 through statistical... Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ... Richard Courant (born January 8, 1888 at Lublinitz, today Poland, died January 27, 1972 at New York/USA) was a German and American mathematician. ... Erich Hecke (September 20, 1887 – February 13, 1947) was a German mathematician. ... Hugo Dyonizy Steinhaus (January 14, 1887 - February 25, 1972) was a Polish mathematician, educator, and humanist. ... Wilhelm Ackermann (March 29, 1896, Herscheid municipality, Germany – December 24, 1962 Lüdenscheid, Germany ) was a German mathematician best known for the Ackermann function, an important example in the theory of computation. ... The Mathematische Annalen is a German mathematical research journal published by Springer-Verlag. ...


Later years

Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen, in 1933.[10] Among those forced out were Hermann Weyl, who had taken Hilbert's chair when he retired in 1930, Emmy Noether and Edmund Landau. One of those who had to leave Germany was Paul Bernays, Hilbert's collaborator in mathematical logic, and co-author with him of the important book Grundlagen der Mathematik (which eventually appeared in two volumes, in 1934 and 1939). This was a sequel to the Hilbert–Ackermann book Principles of Theoretical Logic from 1928. Nazism in history Nazi ideology Nazism and race Outside Germany Related subjects Lists Politics Portal         Nazism or National Socialism (German: Nationalsozialismus), refers primarily to the ideology and practices of the Nazi Party (National Socialist German Workers Party, German: Nationalsozialistische Deutsche Arbeiterpartei or NSDAP) under Adolf Hitler. ... The Georg-August University of Göttingen (Georg-August-Universität Göttingen, often called the Georgia Augusta) was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and opened in 1737. ... Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ... Amalie Emmy Noether [1] (March 23, 1882 – April 14, 1935) was a German-born mathematician, said by Einstein in eulogy to be [i]n the judgment of the most competent living mathematicians, [...] the most significant creative mathematical genius thus far produced since the higher education of women began. ... Edmund Georg Hermann (Yehezkel) Landau (February 14, 1877 – February 19, 1938) was a German Jew mathematician and author of over 250 papers on number theory. ... Paul Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who played a crucial role in the development of mathematical logic in the 20th century. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... Wilhelm Ackermann (March 29, 1896, Herscheid municipality, Germany – December 24, 1962 Lüdenscheid, Germany ) was a German mathematician best known for the Ackermann function, an important example in the theory of computation. ... Principles of Theoretical Logic is the title of the 1950 American translation of the 1938 second edition of David Hilberts and Wilhelm Ackermanns classic text Grundzüge der theoretischen Logik, on elementary mathematical logic. ...


About a year later, he attended a banquet, and was seated next to the new Minister of Education, Bernhard Rust. Rust asked, "How is mathematics in Göttingen now that it has been freed of the Jewish influence?" Hilbert replied, "Mathematics in Göttingen? There is really none any more."[11] Bernhard Rust (1883--May 1945) was Minister of Education in Nazi Germany. ...


By the time Hilbert died in 1943, the Nazis had nearly completely restructured the university, many of the former faculty being either Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics.[12]


On his tombstone, at Göttingen, one can read his epitaph:[13]

Wir müssen wissen.
Wir werden wissen.
We must know.
We will know.

Or better:


We have to know.


We shall know!


The finiteness theorem

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem. Twenty years earlier, Paul Gordan had demonstrated the theorem of the finiteness of generators for binary forms using a complex computational approach. The attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. Hilbert realized that it was necessary to take a completely different path. As a result, he demonstrated Hilbert's basis theorem: showing the existence of a finite set of generators, for the invariants of quantics in any number of variables, but in an abstract form. That is, while demonstrating the existence of such a set, it was not a constructive proof — it did not display "an object" — but rather, it was an existence proof[14] and relied on use of the Law of Excluded Middle in an infinite extension. Paul Albert Gordan (April 27, 1837 – December 21, 1912) was a German mathematician. ... Look up theorem in Wiktionary, the free dictionary. ... In mathematics, Hilberts basis theorem, first proved by David Hilbert in 1888, states that, if k is a field, then every ideal in the ring of multivariate polynomials k[x1, x2, ..., xn] is finitely generated. ... In the mathematics of the nineteenth century, an important role was played by the algebraic forms that generalise quadratic forms to degrees 3 and more, also known as quantics. ... In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object. ... In mathematics, a nonconstructive proof, is a mathematical proof that purports to demonstrate the existence of something, but which does not say how to construct it. ... “Excluded middle” redirects here. ...


Hilbert sent his results to the Mathematische Annalen. Gordan, the house expert on the theory of invariants for the Mathematische Annalen, was not able to appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive. His comment was: The Mathematische Annalen is a German mathematical research journal published by Springer-Verlag. ...

Das ist nicht Mathematik. Das ist Theologie.
This is not Mathematics. This is Theology.[15]

Klein, on the other hand, recognized the importance of the work, and guaranteed that it would be published without any alterations. Encouraged by Klein and by the comments of Gordan, Hilbert in a second article extended his method, 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:

Without doubt this is the most important work on general algebra that the Annalen has ever published.[citation needed]

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.[16]

For all his successes, the nature of his proof stirred up more trouble than Hilbert could imagine at the time. Although Kronecker had conceded, Hilbert would later respond to others' similar crictisms 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".[17] Not all were convinced. While Kronecker would die soon after, his constructivist banner would be carried forward in full cry by the young Brouwer and his developing intuitionist "school", much to Hilbert's torment in his later years.[18] 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".[19] Brouwer the intuitionist in particular raged against the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert would respond: Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ... In education, constructivism is a learning theory which holds that knowledge is not transmitted unchanged from teacher to student, but instead that learning is an active process of learning. ... Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis. ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ...

" 'Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists.'
"The possible loss did not seem to bother Weyl."[20]

Axiomatization of geometry

Main article: Hilbert's axioms

The text Grundlagen der Geometrie (tr.: Foundations of Geometry) published by Hilbert in 1899 proposes a formal set, the Hilbert's axioms, substituting the traditional axioms of Euclid. They avoid weaknesses identified in those of Euclid, whose works at the time were still used textbook-fashion. Independently and contemporaneously, a 19-year-old American student named Robert Lee Moore published an equivalent set of axioms. Some of the axioms coincide, while some of the axioms in Moore's system are theorems in Hilbert's and vice-versa. Hilberts axioms are a set of 20 assumptions (originally 21), David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. ... Hilberts axioms are a set of 20 assumptions (originally 21), David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... For other uses, see Euclid (disambiguation). ... Robert Lee Moore (14 November 1882, Dallas Texas – 4 October 1974 Austin, Texas) was an American mathematician, known for his work in general topology and the Moore method of teaching university mathematics. ...


Hilbert's approach signaled the shift to the modern axiomatic method. 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, line, plane, and others, could be substituted, as Hilbert says, by tables, chairs, glasses of beer and other such objects. It is their defined relationships that are discussed. In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Look up line in Wiktionary, the free dictionary. ... In mathematics, a plane is the fundamental two-dimensional object. ...


Hilbert first enumerates the undefined concepts: point, line, plane, lying on (a relation between points and planes), betweenness, congruence of pairs of points, and congruence of angles. The axioms unify both the plane geometry and solid geometry of Euclid in a single system. An example of congruence. ... This article is about angles in geometry. ... In mathematics, plane geometry may mean: geometry of the Euclidean plane; or sometimes geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others; or geometry of the hyperbolic plane or two-dimensional spherical geometry. ... In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. ...


The 23 Problems

Main article: Hilbert's problems

He put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician. Hilberts problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. ... The International Congress of Mathematicians (ICM) is the biggest congress in mathematics. ... This article is about the capital of France. ...


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. This article is about the group of mathematicians named Nicolas Bourbaki. ... Giuseppe Peano Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ...


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. Here is the introduction of the speech that Hilbert gave:

Who among us would not be happy to lift the veil behind which is hidden the future; to gaze at the coming developments of our science and at the secrets of its development in the centuries to come? What will be the ends toward which the spirit of future generations of mathematicians will tend? What methods, what new facts will the new century reveal in the vast and rich field of mathematical thought?

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. 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 formalist school, one of three major schools of mathematics of the 20th century. According to the formalist, mathematics is a game devoid of meaning in which one plays with symbols devoid of meaning according to formal rules which are agreed upon in advance. 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. The term formalism describes an emphasis on form over content or meaning in the arts, literature, or philosophy. ...


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. 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: In general, metamathematics or meta-mathematics is reflection about mathematics seen as an entity/object in human consciousness and culture. ... Hilberts program, formulated by German mathematician David Hilbert in the 1920s, was to formalize all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...

  1. all of mathematics follows from a correctly-chosen finite system of axioms; and
  2. 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 article is about a logical statement. ... Hilberts epsilon calculus is an extension of a formal language by the epsilon operator, where the epsilon operator substitutes for quantifiers in that language as a method leading to a proof of consistency for the extended formal language. ... The ignorabimus, short for the Latin tag ignoramus et ignorabimus meaning we do not know and will not know, stood for a pessimistic (in one sense) position on the limits on scientific knowledge, in the thought of the nineteenth century. ... 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 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. // Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. ... Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ...


Gödel's work

Hilbert and the talented 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, was however to end in failure.


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 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. Kurt Gödel (IPA: ) (April 28, 1906 Brünn, Austria-Hungary (now Brno, Czech Republic) – January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher. ... In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ... In mathematics or logic, a finitary operation is one, like those of arithmetic, that take a number of input values to produce an output. ...


Nevertheless, the subsequent achievements of proof theory at the very least clarified consistency as it relates to theories of central concern to mathematicians. Hilbert's work had started logic on this course of clarification; the need to understand Gödel's work then led to the development of recursion theory and then mathematical logic as an autonomous discipline in the 1930s. The basis for later theoretical computer science, in Alonzo Church and Alan Turing also grew directly out of this 'debate'. Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... Recursion theory, or computability theory, is a branch of mathematical logic dealing with generalizations of the notion of computable function, and with related notions such as Turing degrees and effective descriptive set theory. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... Computer science (informally, CS or compsci) is, in its most general sense, the study of computation and information processing, both in hardware and in software. ... ‹ The template below (Expand) is being considered for deletion. ... Alan Mathison Turing, OBE, FRS (23 June 1912 – 7 June 1954) was an English mathematician, logician, and cryptographer. ...


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 space is the most important single idea in the area of functional analysis that grew up around it during the 20th century. In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... Stefan Banach Stefan Banach (March 30, 1892 in Kraków, Austria-Hungary now Poland– August 31, 1945 in Lwów, Soviet Union - occupied Poland), was an eminent Polish mathematician, one of the moving spirits of the Lwów School of Mathematics in pre-war Poland. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...


Physics

Until 1912, Hilbert was almost exclusively a "pure" mathematician. When planning a visit from Bonn, where he was immersed in studying physics, his fellow mathematician and friend Hermann Minkowski joked he had to spend 10 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 in the subject in 1905. Hermann Minkowski. ...


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.[21] He started studying 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 Einstein and others were followed closely. Kinetic theory or kinetic theory of gases attempts to explain macroscopic properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion. ... For other uses, see Radiation (disambiguation). ... “Einstein” redirects here. ...


Hilbert invited Einstein to Göttingen to deliver a week of lectures in June-July 1915 on general relativity and his developing theory of gravity.[22] The exchange of ideas led to the final form of the field equations of General Relativity, namely the Einstein field equations and the Einstein-Hilbert action. In spite of the fact that Einstein and Hilbert never engaged in a public priority dispute, there has been some dispute about the discovery of the field equations. For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... The Einstein field equations (EFE) or Einsteins equations are a set of ten equations in Einsteins theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen... Albert Einstein presented the theories of Special Relativity and General Relativity in groundbreaking publications that did not include references to the work of others. ...


Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics. His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation and his namesake Hilbert space plays an important part in quantum theory. In 1926 von Neuman showed that if atomic states were understood as vectors in Hilbert space, then they would correspond with both Schrodinger's wave function theory and Heisenberg's matrices.[23] The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ... For other persons named John Neumann, see John Neumann (disambiguation). ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ... Schrödinger in 1933, when he was awarded the Nobel Prize in Physics Bust of Schrödinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...


Throughout this immersion in physics, Hilbert worked on putting rigor into the mathematics of physics. While highly dependent on higher math, the physicist 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 the physics and how the physicists were using mathematics, he developed a coherent mathematical theory for what he found, most importantly in the area of integral equations. When his colleague Richard Courant wrote the now classic 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. In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... Richard Courant (born January 8, 1888 at Lublinitz, today Poland, died January 27, 1972 at New York/USA) was a German and American mathematician. ... Methoden der mathematischen Physik was a 1924 book, in two volumes totalling around 1000 pages, published under the names of David Hilbert and Richard Courant. ...


Number theory

Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He disposed of Waring's problem in the wide sense. 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. This article or section does not cite its references or sources. ... In number theory, Warings problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers. ... In mathematics, a Hilbert modular form is a generalization of the elliptic modular forms, to functions of two or more variables. ...


He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution is seen in the names of the Hilbert class field and the Hilbert symbol of local class field theory. Results on them were mostly proved by 1930, after breakthrough work by Teiji Takagi that established him as Japan's first mathematician of international stature. In mathematics, class field theory is a major branch of algebraic number theory. ... In algebraic number theory, the Hilbert class field E of a number field is the maximal abelian unramified extension of Note that in this context, unramified is meant not only for the finite places (the classical ideal theoretic interpretation) but also for the infinite places. ... This article lacks information on the importance of the subject matter. ... In mathematics, local classfield theory is the study in number theory of the abelian extensions of local fields. ... Teiji Takagi (高木 貞治 Takagi Teiji, April 21, 1875 - February 28, 1960) was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory. ...


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. Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ... In mathematics, the Hilbert-Pólya conjecture is a possible approach to the Riemann hypothesis, by means of spectral theory. ...


Miscellaneous talks, essays, and contributions

Hilberts paradox of the Grand Hotel was a mathematical paradox about infinity presented by German mathematician David Hilbert (1862 – 1943): In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... The Erdős number, honouring the late Hungarian mathematician Paul Erdős, one of the most prolific writers of mathematical papers, is a way of describing the collaborative distance, in regard to mathematical papers, between an author and Erdős. ... Foreign Member of the Royal Society is an honourary postition within the Royal Society. ...

See also

The Einstein-Hilbert action is a mathematical object (an action) that is used to derive Einsteins field equations of general relativity. ... In algebraic number theory, the Hilbert class field E of a number field is the maximal abelian unramified extension of Note that in this context, unramified is meant not only for the finite places (the classical ideal theoretic interpretation) but also for the infinite places. ... In mathematics, the Hilbert cube is a topological space that provides an instructive example of some ideas in topology. ... Hilbert curve, first order Hilbert curves, first and second order Hilbert curves, first to third order A Hilbert curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). ... In linear algebra, a Hilbert matrix is a matrix with elements Hij: = 1 /(i + j − 1) For example, this is the 5 × 5 Hilbert matrix: The Hilbert matrix can be regarded as derived from the integral i. ... In mathematics, the Hilbert polynomial of a graded commutative algebra A = ⊕An over a field k that is generated by the finite dimensional space A1 is the unique polynomial f(x) with rational coefficients such that f(n) = dimk An for all but finitely many positive integers n. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... This article is considered orphaned, since there are very few or no other articles that link to this one. ... This article lacks information on the importance of the subject matter. ... The Hilbert transform, in red, of a square wave, in blue In mathematics and in signal processing, the Hilbert transform, here denoted , of a real-valued function, , is obtained by convolving signal with to obtain . ... An algebraic approach introduced by German mathematician David Hilbert for Poincaré disk model of hyperbolic geometry [1]. One can also set up a hyperbolic analytic geometry and hyperbolic trigonometry, whereby any geometric problem can be translated into an algebraic problem in the field. ... Hilberts axioms are a set of 20 assumptions (originally 21), David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. ... In mathematics, Hilberts basis theorem, first proved by David Hilbert in 1888, states that, if k is a field, then every ideal in the ring of multivariate polynomials k[x1, x2, ..., xn] is finitely generated. ... In mathematics, Hilberts irreducibility theorem is a result of David Hilbert, stating that an irreducible polynomial in two variables and having rational number coefficients will remain irreducible as a polynomial in one variable, when a rational number is substituted for the other variable, in infinitely many ways. ... Hilberts Nullstellensatz (German: theorem of zeros) is a theorem in algebraic geometry that relates varieties and ideals in polynomial rings over algebraically closed fields. ... Hilberts paradox of the Grand Hotel was a mathematical paradox about infinity presented by German mathematician David Hilbert (1862 – 1943): In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. ... In differential geometry, Hilberts theorem (1901) states that there exists no complete regular surface of constant negative Gaussian curvature immersed in . ... In number theory, Hilberts Theorem 90 tells us that if L/K is a cyclic extension of number fields generated by an element s and if α is an element of L of relative norm 1, then then there exists β in L such that α = β/βs. ... In mathematics, Hilberts syzygy theorem is a result of commutative algebra, first proved by David Hilbert (1890) in connection with the syzygy (relation) problem of invariant theory. ... The phrase Hilbert-style deduction system denotes a specific formalization of notion deduction in first-order logic [1], attributed to Gottlob Frege and David Hilbert. ... In mathematics, the Hilbert-Pólya conjecture is a possible approach to the Riemann hypothesis, by means of spectral theory. ... In mathematics, a Hilbert-Schmidt operator is a bounded operator A on a Hilbert space H with finite Hilbert-Schmidt norm, meaning that there exists an orthonormal basis of H with the property If this is true for one orthonormal basis, it is true for any other orthonormal basis. ... In mathematics, the Hilbert-Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithfully) on a (topological) manifold M. Restricting to G which are locally compact and have a continuous, faithful group action on M... In mathematics, the Hilbert-Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. ... Principles of Theoretical Logic is the title of the 1950 American translation of the 1938 second edition of David Hilberts and Wilhelm Ackermanns classic text Grundzüge der theoretischen Logik, on elementary mathematical logic. ... Albert Einstein presented the theories of Special Relativity and General Relativity in groundbreaking publications that did not include references to the work of others. ...

Notes

  1. ^ David Hilbert. Encyclopædia Britannica (2007). Retrieved on 2007-09-08.
  2. ^ Reid 1996, pp. 1–2; also on p. 8, Reid notes that there is some ambiguity of exactly where Hilbert was born. Hilbert himself stated that he was born in Königsberg.
  3. ^ Reid 1996, pp. 4–7.
  4. ^ Reid 1996, p. 11.
  5. ^ Reid 1996, p. 12.
  6. ^ Reid 1996, p. 36.
  7. ^ Reid 1996, p. 139.
  8. ^ Reid 1996, p. 121.
  9. ^ The Mathematics Genealogy Project - David Hilbert. Retrieved on 2007-07-07.
  10. ^ "Shame" at Göttingen. (Hilbert's colleagues exiled)
  11. ^ Reid 1996, p. 205.
  12. ^ Reid 1996, p. 213.
  13. ^ Reid 1996, p. 220.
  14. ^ Reid 1996, pp. 36–37.
  15. ^ Reid 1996, p. 34.
  16. ^ Reid 1996, p. 37.
  17. ^ Reid 1996, p. 37.
  18. ^ cf. Reid 1996, pp. 148–149.
  19. ^ Reid 1996, p. 148.
  20. ^ Reid 1996, p. 150.
  21. ^ Reid 1996, p. 129.
  22. ^ Sauer 1999, Folsing 1998.
  23. ^ It is of interest to note that in 1926, the year after the matrix mechanics formulation of quantum theory by Max Born and Werner Heisenberg, the mathematician John von Neumann became an assistant to David 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, John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More (Reprinted by the American Mathematical Society, 1999) and Reid 1996.
  24. ^ Some Famous People with Finite Erdős Numbers.
  25. ^ Wolfram MathWorld – Cayley-Klein-Herbert metric
  26. ^ Wolfram MathWorld – Hilbert class field
  27. ^ Wolfram MathWorld – Hilbert inequality
  28. ^ Wolfram MathWorld – Hilbert Series
  29. ^ Wolfram MathWorld – Hilbert’s constants

Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 251st day of the year (252nd in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 188th day of the year (189th in leap years) in the Gregorian calendar. ... Max Born (December 11, 1882 in Breslau – January 5, 1970 in Göttingen) was a mathematician and physicist. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... For other persons named John Neumann, see John Neumann (disambiguation). ...

References

Primary literature in English translation

  • Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press.
    • 1918. "Axiomatic thought," 1115–14.
    • 1922. "The new grounding of mathematics: First report," 1115–33.
    • 1923. "The logical foundations of mathematics," 1134–47.
    • 1930. "Logic and the knowledge of nature," 1157–65.
    • 1931. "The grounding of elementary number theory," 1148–56.
    • 1904. "On the foundations of logic and arithmetic," 129–38.
    • 1925. "On the infinite," 367–92.
    • 1927. "The foundations of mathematics," with comment by Weyl and Appendix by Bernays, 464–89.
  • Jean van Heijenoort, 1967. From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press.
  • David Hilbert; Cohn-Vossen, S. (1999). Geometry and Imagination. American Mathematical Society. ISBN 0-8218-1998-4.  - an accessible set of lectures originally for the citizens of Göttingen.
  • David Hilbert (2004). in Michael Hallett and Ulrich Majer: David Hilbert's Lectures on the foundations of Mathematics and Physics, 1891–1933. Springer-Verlag Berlin Heidelberg. ISBN 3-540-64373-7. 

Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ... Edward Bernays, the most famous Bernays. ... Jean van Heijenoort (prounounced highenort) (July 23, 1912, Creil France - March 29, 1986, Mexico City) was a pioneer historian of mathematical logic. ... Stefan or Stephan Cohn-Vossen (28 May 1902-25 June 1936) was a German-Jewish mathematician, now best known for his collaboration with David Hilbert on the 1932 book Anschauliche Geometrie. ...

Secondary literature

  • B, Umberto, 2003. Il flauto di Hilbert. Storia della matematica. UTET, ISBN 88-7750-852-3
  • Corry, L., Renn, J., and Stachel, J., 1997, "Belated Decision in the Hilbert-Einstein Priority Dispute," Science 278: nn-nn.
  • Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.
  • Gray, Jeremy, 2000. The Hilbert Challenge, ISBN 0-19-850651-1
  • Piergiorgio Odifreddi, 2003. Divertimento Geometrico - Da Euclide ad Hilbert. Bollati Boringhieri, ISBN 88-339-5714-4. A clear exposition of the "errors" of Euclid and of the solutions presented in the Grundlagen der Geometrie, with reference to non-Euclidean geometry.
  • Reid, Constance, 1996. Hilbert, Springer, ISBN 0-387-94674-8. The biography in English.
  • Sauer, Tilman, 1999. "The relativity of discovery: Hilbert's first note on the foundations of physics", Arch. Hist. Exact Sci., v53, pp 529-575. (Available from Cornell University Library, as a downloadable Pdf [1])
  • Thorne, Kip, 1995. Black Holes and Time Warps: Einstein's Outrageous Legacy, W. W. Norton & Company; Reprint edition. ISBN 0-393-31276-3.
  • Folsing, Albrecht, 1998. Albert Einstein, Penguin.
  • Mehra, Jagdish, 1974. Einstein, Hilbert, and the Theory of Gravitation, Reidel.
  • Paolo Mancosu (1998). From Brouwer to Hilbert, The Debate on the Foundations of Mathematics in the 1920's. Oxford University Press. ISBN 0-19-509631-2. 

Ivor Grattan-Guinness (Born 23 June 1941, in Bakewell, England) is a prolific historian of mathematics and logic, at Middlesex University. ... Piergiorgio Odifreddi (Born in Cuneo, July 13, 1950), is an Italian mathematician, logician and afficianado of the history of science, who is also extremely active as a popular science writer and essayist. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... Springer Science+Business Media or Springer is a worldwide publishing company based in Germany. ... Kip S. Thorne Professor Kip Stephen Thorne, Ph. ... This page is a candidate for speedy deletion. ...

External links

Wikiquote has a collection of quotations related to:
David Hilbert
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David Hilbert
Logic Portal
Persondata
NAME Hilbert., David
ALTERNATIVE NAMES
SHORT DESCRIPTION Mathematician
DATE OF BIRTH January 23, 1862(1862-01-23)
PLACE OF BIRTH Königsberg, East Prussia
DATE OF DEATH February 14, 1943
PLACE OF DEATH Göttingen, Germany

Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... is the 23rd day of the year in the Gregorian calendar. ... This article is about 1862 . ... Former German name of the city of Kaliningrad. ... East Prussia (German: Ostpreu en; Polish: Prusy Wschodnie; Russian: Восточная Пруссия — Vostochnaya Prussiya) was a province of Kingdom of Prussia, situated on the territory of former Ducal Prussia. ... is the 45th day of the year in the Gregorian calendar. ... Year 1943 (MCMXLIII) was a common year starting on Friday (the link will display full 1943 calendar) of the Gregorian calendar. ... Göttingen marketplace with old city hall, Gänseliesel fountain and pedestrian zone Göttingen ( ) is a city in Lower Saxony, Germany. ...


  Results from FactBites:
 
David Hilbert (1070 words)
David Hilbert was born in Koenigsberg, East Prussia in 1862 and received his doctorate from his home town university in 1885.
Hilbert's axioms could be proved as theorems from Moore's and conversely, Moore's axioms could be proved as theorems from Hilbert's.
David Hilbert, Mathematical Problems, Paris, 1900, Translation by Mary Winston published in the Bulletin of the American Mathematical Society volume 8 (1902), pp 437-479.
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