David Hilbert

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David Hilbert (1912)

David Hilbert (born January 23, 1862 in Königsberg , † February 14, 1943 in Göttingen ) was a German mathematician . He is considered one of the most important mathematicians of modern times . Much of his work in the field of mathematics and mathematical physics established independent research areas. With his suggestions, he established the formalistic view of the foundations of mathematics, which is still significant today, and initiated a critical analysis of the definitions of mathematics and mathematical proof . These analyzes led to Gödel's incompleteness theorem , which shows, among other things, that the Hilbert program , the complete axiomatization of mathematics he strived for , cannot be fully fulfilled. Hilbert's programmatic speech at the International Congress of Mathematicians in Paris in 1900, in which he presented a list of 23 mathematical problems , had a lasting influence on mathematical research in the 20th century.

Life

Koenigsberg

Childhood and youth

Royal Wilhelm-Gymnasium (postcard)

Hilbert was born as the son of the district court counselor Otto Hilbert and his wife Maria Theresia, geb. Erdtmann, born. On his father's side he came from an old East Prussian family of lawyers, the mother came from a merchant family in Königsberg. The father was described as a rather one-sided lawyer who was critical of his son's career, while the mother had varied interests, including in the field of astronomy and philosophy, and applied mathematics. He had a younger sister Elise Frenzel, who married a judge and died in 1897 at the age of 28. In his hometown, Hilbert first attended the Friedrichskollegium as a schoolboy and switched to the more scientific-mathematical-oriented Wilhelms-Gymnasium a year before graduating from high school . Nothing remarkable has come down to us about his academic achievements, anecdotal rumors that young Hilbert did not write good German essays (which his mother sometimes wrote), but was able to explain mathematical problems to his teachers. His math teacher von Morstein gave him the best possible grade in his Abitur and certified him “Thorough knowledge and the ability to solve the tasks given to him on his own”. When asked about his achievements in school, Hilbert said later: "I didn't really do math at school because I knew that I would do it later."

Studies, as well as meeting and exchanging ideas with Minkowski and Hurwitz

Albertus University around 1900 (colored postcard)

In the summer semester of 1880, 18-year-old Hilbert began studying mathematics at the Albertus University in Königsberg. At that time, the Königsberg University could look back on a brilliant tradition in mathematics and was considered a top-class training center in this subject. Carl Gustav Jacob Jacobi , Friedrich Wilhelm Bessel , Friedrich Julius Richelot and the physicist Franz Ernst Neumann taught and worked here, among others . Heinrich Weber , who came from Heidelberg , was one of Hilbert's teachers . Probably through Weber's mediation, Hilbert spent his second semester in Heidelberg , but then returned to Königsberg. Weber recognized and promoted Hilbert's mathematical talent at an early stage.

During his studies, Hilbert met his fellow student, Hermann Minkowski , who was two years his junior and came from a Jewish family from Lithuania who had immigrated to East Prussia. He had a lifelong close friendship with Minkowski. In 1883 Ferdinand Lindemann became the successor to the chair (full professorial position) of Weber and in 1884 Adolf Hurwitz was appointed to the second mathematics chair (extraordinary office). Hurwitz was only 3 years older than Hilbert and Hilbert later said of him: "We, Minkowski and I, were completely stunned by his knowledge and didn't believe that we would ever get that far." The regular scientific exchange with Hurwitz and Minkowski was formative for Hilbert. In the obituary for Hurwitz, Hilbert wrote: "On numerous walks, occasionally day after day, we probably rummaged through every corner of mathematical knowledge for eight years, and Hurwitz, with his extensive and varied, well-founded and well-ordered knowledge, was always our guide" . Lindemann, on the other hand, had little influence on Hilbert, but suggested the topic of his doctoral thesis . 1885 Hilbert was the work over invariant properties of special binary forms, especially the spherical harmonics in the Faculty of Arts PhD .

Encounter with Felix Klein, habilitation and professorship

David Hilbert (1886) as a private lecturer in Königsberg

After completing his doctorate, Hilbert went on a study trip in the winter of 1885/86, which initially led him to Felix Klein at the University of Leipzig . Klein also recognized Hilbert's great talent and an intensive scientific correspondence developed between the two. On Klein's advice, Hilbert stayed in Paris for a few more months . Klein recommended such a stay to all talented students, since he himself had been in Paris with Sophus Lie in 1870, where he had received important suggestions. Hilbert came into contact with many well-known French mathematicians ( Charles Hermite , Henri Poincaré , Camille Jordan , Pierre Ossian Bonnet ). He got the best impression of Poincaré and Hermite, but on the whole he was not very impressed with French mathematics.

In 1886 Hilbert completed his habilitation in Königsberg with a thesis on invariant theoretical investigations in the field of binary forms and became a private lecturer . After Hurwitz had accepted a call to Zurich in 1892 , Hilbert was his successor in the associate's office. In 1893 Lindemann followed a call to Munich and Hilbert became a full professor. Hilbert was able to get his friend Minkowski appointed to the vacant extraordinary position in Königsberg.

family

David Hilbert with Käthe Jerosch 1892

On October 12, 1892, David Hilbert married Käthe Jerosch, who had been friends with him for a long time (born March 31, 1864 in Braunsberg in Warmia , † January 17, 1945 in Göttingen ). Throughout her life, Käthe was an essential pillar of Hilbert's scientific work. Since the beginning of the marriage, she used her best handwriting to write fair copies of correspondence and book manuscripts for submission to the printer. She retained this obligation even after the stressful events surrounding their son Franz. Käthe died almost blind and lonely. The only child Franz Hilbert was born on August 11, 1893. All his life he suffered from an undiagnosed mental disorder. His father's poor mental faculties were a burden. Richard Courant , from 1909 private teacher at a girls' school in Göttingen and assistant to David Hilbert, was given the task of tutoring Franz in order to improve his academic performance. Attempts to gain a foothold in a profession have failed. One day Franz came home with symptoms of a delusional disorder and was then taken to a clinic for the mentally ill near the University of Göttingen. This was the trigger for the (false) assumptions that emerged in the Göttingen society that David and Käthe Hilbert were first cousins. Hilbert subsequently largely broke off contact with his son and is said to never have visited him during his stay in the clinic. When his mother brought him back home some time later, the peace in the Hilbert house was disturbed. The father could not bear his son's illness, the mother did not want to give up her son. Tensions developed between the couple. Franz was at home on Hilbert's 60th (1922) and 75th (1937) birthday.

Goettingen

The heyday of Göttingen mathematics

The Mathematical Institute in Göttingen. The new building was built in 1926-29 with funds from the Rockefeller Foundation and opened on December 2, 1929 by Hilbert and Courant.

At the instigation of Felix Klein, the University of Göttingen appointed Hilbert as professor in 1895. The Prussian Ministry of Culture had set itself the goal of establishing a focus of mathematical research in Göttingen, in the tradition of Carl Friedrich Gauß and Bernhard Riemann . The driving force was the State Secretary Friedrich Althoff , who was actively supported by Klein in this endeavor. Hilbert was 33 years old at the time and Klein was accused of making the call of such a young man easy. Thereupon he replied: “You are wrong, I am calling the most inconvenient.” However, Klein’s personal relationship with Hilbert remained friendly even after the appointment. In 1902, as a result of a call to Berlin in the stay negotiations , Hilbert was able to get Minkowski appointed to the extraordinary position in Göttingen, so that the two mathematicians friends were reunited in one place. The early death of his friend and work colleague in 1909, at the age of 44, was a severe personal blow to Hilbert. After his death, Hilbert acted as the editor of his work under the title Collected Treatises by Hermann Minkowski .

The early years in Göttingen were not always easy for Hilbert, as the small town of Göttingen did not have such a cosmopolitan, liberal spirit as in Königsberg. The class arrogance of the local university circles was very pronounced. For example, it was seen as a scandal when Hilbert, the professor, played billiards with assistants in a pub. Years later Albert Einstein gave his friend Max Born , who had to choose between calling to Frankfurt or Göttingen, the advice: “When I think about the situation, it seems to me that I would rather stay in Frankfurt. Because it would be unbearable for me to be so completely dependent on a small circle of inflated and mostly narrow-minded (and -thinking) scholars (no other intercourse). Think about what Hilbert endured from this society. ”But then Born decided on Göttingen and soon belonged to Hilbert's circle of friends, whose assistant he had already been. After the initial difficulties, however, Hilbert settled in well in Göttingen and enjoyed great admiration from his students. His later doctoral student Otto Blumenthal reported about the impression he made on the students:

“I still remember exactly the unusual impression that made me - in the second semester - this medium-sized, agile, very unprofessor-looking, inconspicuously dressed man with the broad reddish beard, who stood out so strangely from Heinrich Weber's venerable, stooped figure and Klein's commanding appearance with the radiant look. […] Hilbert's lectures were plain. Strictly factual, with a tendency to repeat important sentences, even hesitantly, he delivered, but the rich content and the simple clarity of the presentation made the form forgotten. He brought a lot of new and original things without emphasizing them. He made an obvious effort to be understandable to everyone; he read for the students, not for himself. [...] In order to get to know his seminar people in detail, he took them to a forest economy for a while after each seminar, where mathematics was discussed. [...] A persistent pedestrian, he would take long walks with you every week in the mountains of Göttingen, everyone could ask his questions, but mostly Hilbert himself talked about his work that was occupying him. "

The 70-year-old Hilbert at a lecture in 1932

During his time in Göttingen, Hilbert supervised a total of 69 doctoral students, including a. (with year of doctorate): Otto Blumenthal (1898), Felix Bernstein (1901), Hermann Weyl (1908), Richard Courant (1910), Erich Hecke (1910), Hugo Steinhaus (1911), Wilhelm Ackermann (1925). Many of his former students later became professors.

Six women were among the 69 doctoral students, which was anything but natural at the time. Women in Prussia were not generally admitted to higher education until 1908 . Hilbert and Klein's commitment to mathematician Emmy Noether is known , who - although undoubtedly highly qualified - as a woman was only able to get a teaching position in Göttingen with great difficulty. For years she could only announce her lectures under Hilbert's name. In connection with the discussions about Noether's application for a habilitation, Hilbert's much-quoted saying "a faculty is not a bathing establishment !"

Arnold Sommerfeld repeatedly sent his colleagues to Göttingen as assistants to support Hilbert's work on basic problems in physics. For example, these were Paul Peter Ewald in 1912 , Alfred Landé in 1913 and Adolf Kratzer in 1920/21 .

In the first third of the 20th century Hilbert played a major role in the development of the University of Göttingen into a leading mathematical and scientific teaching and research center; he remained loyal to her, despite numerous offers from other universities and academies (1898 Leipzig: successor to Sophus Lie , 1902 Berlin: successor to Lazarus Immanuel Fuchs , 1912 Heidelberg: successor to Leo Koenigsberger , 1919: Bern and 1917: again Berlin) until his retirement in 1930. Until 1934 he still held lectures at the Göttingen University. Even in his Göttingen years, Hilbert remained closely connected to his East Prussian homeland and regularly spent his holidays in the seaside resort of Rauschen , "the paradise of our childhood".

In 1900 he was President of the German Mathematicians Association . In 1903 he was elected a corresponding member of the Bavarian Academy of Sciences .

1902–1939 Hilbert was co-editor of the Mathematische Annalen , the most important mathematical journal in the world at the time. In this activity he was supported by his long-time assistant Otto Blumenthal.

Although fundamentally politically liberal, Hilbert could not escape the enthusiasm for the war of August 1914 . He was not one of the signatories of the Manifesto of 93 , but about two weeks later, like his friend Max von Laue , he agreed to the no less nationalistic declaration of the university professors of the German Reich .

In 1928 he gave a plenary lecture at the International Congress of Mathematicians in Bologna (Problems of the Foundations of Mathematics). On the occasion of the congress of the Society of German Natural Scientists and Doctors in Koenigsberg, he gave his famous speech on September 8, 1930 with the title Natural Knowledge and Logic . At that time, a four-minute excerpt was broadcast on the radio and has been preserved on a record to this day.

After the National Socialist seizure of power in 1933

Grave site at the Göttingen city cemetery

After the National Socialist seizure of power in 1933, Hilbert had to see that the mathematics center and the physical faculty of Göttingen University were destroyed by the National Socialists . All “non-Aryan” mathematicians such as Edmund Landau , Richard Courant , Max Born , Felix Bernstein , Emmy Noether , Otto Blumenthal and politically dissenters such as Hermann Weyl were forced to give up their work, and quite a few emigrated. When Hilbert was asked at a banquet in 1934 by the new Prussian Minister of Education, Bernhard Rust , whether it was true that his institute had suffered “from the departure of Jews and friends of Jews”, he replied: “The institute - it no longer exists . "

In 1942 he became an honorary member of the DMV .

David Hilbert died in 1943. His death was only casually registered by the German scientific public at the height of the World War . Barely a dozen people attended his funeral. Arnold Sommerfeld , who was present, also from Königsberg, wrote an obituary in The Natural Sciences . Quite different in America: There were numerous commemorative events at many universities where former graduates and emigrants of the Göttingen Mathematical Seminar worked. Hermann Weyl wrote an obituary in Princeton, among others .

Hilbert's grave is in the Göttingen city cemetery on Groner Landstrasse.

His bust is among the busts of the important Georgia-Augusta professors in the auditorium on Wilhelmsplatz.

His estate is kept by the Central Archives of German Mathematicians' bequests at the Lower Saxony State and University Library in Göttingen .

plant

Hilbert's most important contributions to individual areas of mathematics are described in more detail below.

Algebraic Geometry

Until about 1893 Hilbert made contributions to the invariant theory . Among other things, he proved Hilbert's basis theorem , which says that every ideal in a polynomial ring over a body is finite. In his zero theorem he showed the clear connection between zeros of polynomial equations and polynomial ideals. With this he combined geometry and algebra, which led to the development of algebraic geometry .

Number theory

In his important work number report from 1897 ( algebraic number theory ) he summarized the work of Ernst Eduard Kummer , Leopold Kronecker and Richard Dedekind with his own ideas. An important sentence from this work is still quoted under the numbering used there: Hilbert's sentence 90 on the structure of certain body extensions .

geometry

Hilbert's endeavor was to replace the geometry , which had hitherto been very graphic and still essentially go back to Euclid , as completely as possible from concepts from the world of perception and to justify it purely axiomatically . Such an axiomatic justification seemed to Hilbert and many mathematical contemporaries absolutely necessary, since the terms previously used from the visual world did not have the necessary mathematical accuracy and the mathematical structure of geometry built on it seemed to stand on "shaky feet".

In his fundamental work, Fundamentals of Geometry , published in 1899 to celebrate the unveiling of the Gauß-Weber monument in Göttingen, he designed a complete system of axioms for Euclidean geometry and, based on this, developed a strictly axiomatic geometry. The terms “point”, “straight line”, “plane” etc. used by Hilbert no longer relate to intuition, as Euclid tried to do (e.g. “A point is that has no parts”), but rather are defined purely axiomatically. Hilbert is credited with saying that instead of “points, straight lines and planes” one could also say “tables, chairs and beer mugs” at any time; all that matters is that the axioms are fulfilled.

From Hilbert's book it follows in particular that every geometry that satisfies Hilbert's system of axioms is uniquely determined except for isomorphism , namely isomorphic to the three-dimensional real vector space in which the vectors are the points and the subclasses of one-dimensional subspaces are the straight lines, and in which the Distance between two points is measured as in classical analytical geometry, namely with the help of the Pythagorean theorem.

Hilbert's 23 problems

In 1900, from 6 to 12 August, the second international mathematicians' congress took place parallel to the world exhibition in Paris . The congress met in 6 sections: arithmetic and algebra, analysis, geometry, mechanics and mathematical physics, history and bibliography of mathematics and teaching and methodology of mathematics. 226 scholars from all over the world took part in the congress. The then 39-year-old Hilbert was considered one of the leading German mathematicians and was asked to give a keynote speech in a joint meeting of the 5th and 6th sections. Many expected him to give a kind of “celebratory speech” at the turn of the century to review the great successes in the development of mathematics over the past century. However, Hilbert made a completely different decision. Instead of looking back at the past century, he dared to look boldly into the future. The opening words in his lecture on August 8, 1900 express this:

“Who of us would not like to lift the veil under which the future lies, to glimpse the impending advances in our science and the mysteries of its development over the centuries to come! What special goals will the leading mathematical minds of the coming generations pursue? What new methods and new facts will the new centuries discover - in the wide and rich field of mathematical thought? "

For his lecture he had compiled a list of 23 unsolved mathematical problems from very different sub-areas of mathematics (geometry, number theory, logic, topology, arithmetic, algebra, etc.), of which he presented 10. In this selection of problems Hilbert showed his impressive comprehensive overview of all mathematics. He had chosen these problems because they seemed to be of central importance to him and because he hoped that the solution of these problems would result in substantial progress in the relevant areas. These later so-called Hilbert problems became the guiding lines for entire generations of mathematicians, and the solution to each problem was seen as a great achievement. Of the problems, 15 are currently (2012) considered to be solved, 3 to be unsolved and 5 to be fundamentally unsolvable, the latter partly also because the formulation is too imprecise. The most famous case of such an unsolvable (but precisely formulated) problem is the demand for a proof of the consistency of the axioms of arithmetic (Hilbert's 2nd problem), a demand whose unsolvability was proven by Kurt Gödel in 1930. The most famous unsolved problem is the question of the zeros of the Riemann zeta function , Hilbert's 8th problem.

Logic and basics of mathematics

Hilbert is considered to be the founder and most exposed representative of the direction of formalism in mathematics. In the list of unsolved problems, Hilbert pointed out that the consistency of arithmetic had not been clarified. At the beginning of the 1920s, as a reaction to the fundamental crisis in mathematics, he made the demand that mathematics be based entirely on a system of axioms that should be demonstrably free of contradictions. In Hilbert's words:

"But that is what I demand: in mathematical matters there should be no doubts in principle, there should be no half-truths and also no truths of principally different kinds [...]."

and further:

“The goal of establishing mathematics with certainty is also mine; I want to restore mathematics to the old reputation of incontestable truth, which it seems to have lost through the paradoxes of set theory; but I believe that this is possible with full preservation of their acquis. "

The intuitionistic approach Brouwer had called the Hilbert Weyl students as "revolutionary," Hilbert declined sharply, mainly because of that, also because he was her previous "acquis" would have deprived the math of a large part:

“What Weyl and Brouwer are doing basically means that they are following the former paths of Kronecker : They try to justify mathematics by throwing everything that seems uncomfortable to them overboard and establishing a ban dictatorship à la Kronecker. But this means dismembering and mutilating our science, and we run the risk of losing a large part of our most precious treasures if we follow such reformers. [...] no, Brouwer is not, as Weyl thinks, the revolution, but the repetition of an attempted coup with old means, which [...] is doomed to failure from the outset. "

Hilbert's stated aim was to base arithmetic and ultimately all of the mathematics based on it on a system of consistent axioms. This endeavor became known as the " Hilbert program ". As part of this program, Hilbert formulated the Hilbert calculus, which was later named after him . The Hilbert program ultimately turned out to be impracticable in the form Hilbert intended, as Kurt Gödel was able to show with his incompleteness sentence published in 1930 . Nevertheless, the Hilbert program was very fruitful for mathematics, as it contributed to a deeper understanding of the structure of formal systems with their limits and to the clarification of terms in wide areas of mathematics and logic.

Analysis

In the calculus of variations , Hilbert placed the Dirichlet principle used by Riemann in his illustration theorem on firm foundations. In the integral equations he closed some of Fredholm's gaps in the proof of the Fredholm alternative . These topics flowed significantly into the development of functional analysis . The important Hilbert area in particular is inextricably linked with his name.

Mathematical physics

Hilbert's work on function spaces ( Hilbert space ) and partial differential equations are among the foundations of mathematical physics today. From 1912 Hilbert began to turn intensively to physics (initially in the application of integral equations to the kinetic gas theory), with the mathematical treatment of which he was dissatisfied. A well-known quote from Hilbert reads: Physics is actually much too difficult for physicists . His pupil and assistant Richard Courant suggested in 1918 that he start a book project on this subject, which was largely realized by Courant himself, but - as Courant wrote in the foreword - was based on Hilbert's treatises and lectures and was permeated by the spirit of the Hilbert School which is why he (Courant) insisted on listing Hilbert as a co-author. According to Hilbert's biographer Constance Reid , Hilbert showed an interest in his former student's book, but otherwise did not participate in any way . The first volume appeared in 1924, the second in 1937. The book became a fundamental work on mathematical physics in the first half of the 20th century (and was completely revised by Courant in the 1950s and 1960s), as the successor to the Theory of Sound by Lord Rayleigh . It was and is generally known as the Courant / Hilbert and, in the stormy development of quantum mechanics that began soon after, proved to be an important source from which theoretical physicists learned the necessary new mathematics.

Hilbert also pursued a program on the axiomatic foundations of physics, one of Hilbert's problems. One of the fruits of this was his work on general relativity. With the development of quantum mechanics in Göttingen around 1925, he also began to be interested in it, partly in collaboration with John von Neumann and his physical assistant (who Arnold Sommerfeld regularly selected for Hilbert) Lothar Nordheim . In 1928 this resulted in the essay The Basics of Quantum Mechanics by Nordheim, Hilbert and von Neumann.

general theory of relativity

On November 20, 1915, five days before Einstein , Hilbert submitted a work on general relativity that was equivalent to Einstein's theory, but without Einstein's field equations , which are included in Hilbert's principle of variation . However, his work appeared only after Einstein's work. Hilbert never claimed authorship for the general theory of relativity and there was no public "priority dispute" between Einstein and Hilbert. However, there was briefly an upset on the part of Einstein, which was soon cleared up by Hilbert, who gave Einstein complete priority in the physical field. Various historians of science, however, have speculated about the priority. Fölsing considers a significant influence of Hilbert on Einstein when setting up the field equations to be unlikely. Conversely, Leo Corry / Renn / Stachel questioned the independent perfecting of the equations by Hilbert on the basis of a discovery of proofs in 1997, which, however, is in turn denied by others.

Against the ignorabimus

Hilbert always resisted a view of the limits of science in the sense of an ignoramus et ignorabimus . His belief that man can understand the world is shown in his saying: We must know and we will know. What Hilbert was trying to say is clear from the following quote:

“The philosopher Comte once said - with the intention of calling a certain insoluble problem - that science would never succeed in unraveling the mystery of the chemical composition of the heavenly bodies. A few years later this problem was solved by spectral analysis by Kirchhoff and Bunsen , and today we can say that we use the most distant stars as the most important physical and chemical laboratories, as we do not find any on earth. The real reason Comte failed to find an unsolvable problem, in my opinion, is that it doesn't even exist. "

Or in other words:

“This conviction that every mathematical problem can be solved is a powerful incentive for us during our work; we have the constant call within us: there is the problem, look for the solution. You can find it through pure thinking; because in mathematics there is no ignorance. "

Hilbert advocates an optimism in research that rejects self-imposed restrictions on thinking. The motto can also be found as an epitaph on his tombstone:

"We have to know.
We will know. "

Appreciation

The following mathematical terms, objects or sentences are named after David Hilbert:

In addition, the lunar crater Hilbert and the asteroid Hilbert are named after the mathematician. The foyer of the old mathematical faculty in Göttingen is named Hilbert-Raum, and a street in the city is named after him.

In 1906 Hilbert received the Cothenius Medal of the German Academy of Natural Scientists Leopoldina ; In 1907 he became an honorary foreign member of the National Academy of Sciences and in 1911 a corresponding member of the Académie des Sciences ; in 1932 Hilbert was elected a member and honorary member of the Leopoldina and a member of the American Philosophical Society .

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Individual evidence

  1. ↑ Constance Reid Hilbert , Springer Verlag 1972, gives Wehlau bei Königsberg
  2. ↑ 150th birthday of the mathematician David Hilbert - Faster than Einstein . In: Süddeutsche Zeitung, January 22, 2012 at: sueddeutsche.de
  3. ↑ The Einstein of Mathematics . In: Die Zeit, January 12, 2012 at: zeit.de
  4. ↑ a b c d e Otto Blumenthal: Life story. In: David Hilbert. Collected Treatises. Volume III, Springer-Verlag, 1970, 2nd edition, p. 388ff digitized full text
  5. ↑ Felix Klein: Lectures on the development of mathematics in the 19th century. P. 112 ff: The Königsberg School. In: Basic Teachings of the Mathematical Sciences. 24/25. Berlin [ua], Springer-Verlag (Reprint 1979) digitized full text
  6. ^ David Hilbert: Adolf Hurwitz. In: Mathem. Annalen Vol. 83, pp. 161-168 (1921) digitized full text
  7. ↑ David Hilbert in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used
  8. ^ Letters of April 2 and April 21, 1886 to Felix Klein. In: The exchange of letters David Hilbert - Felix Klein (1886-1918) . Göttingen, Vandenhoeck & Ruprecht 1985, ISBN 3-525-85457-9
  9. ↑ Constance Reid 1972, p. 40
  10. ^ Gravestone, http://www.w-volk.de/museum/grave34.htm
  11. ↑ Constance Reid 1972, p. 46
  12. ↑ Constance Reid 1972, p. 52
  13. ↑ Constance Reid 1972, pp. 139/140
  14. ↑ Constance Reid 1972, p. 215
  15. ↑ in Cranz , Constance Reid 1972, p. 43
  16. ↑ Constance Reid 1972, p. 139
  17. ↑ Apostolos Doxiadis, Christos H. Papadimitriou 2012, pp. 282 and 330
  18. ↑ Constance Reid 1972, pp. 151/152
  19. ↑ Fig. 60th birthday, Constance Reid 1972, p. 238
  20. ↑ Constance Reid 1972, p. 210
  21. ↑ see Hilbert's commemorative speech, held in the public meeting of the Kgl. Gesellschaft zu Göttingen on May 1, 1909 (published: D. Hilbert: Hermann Minkowski . Göttinger Nachrichten, Geschäftsliche Mitteilungen 1909, pp. 72-101, and Math. Ann. Vol. 68, pp. 445-471 (1910), also contained in the Collected Treatises , Vol. 3. digitized full text )
  22. ↑ David Hilbert (ed.) With the assistance of Andreas Speiser and Hermann Weyl: Collected treatises by Hermann Minkowski , Leipzig and Berlin, Teubner, 1911
  23. ^ Letter of March 3, 1920 In: Albert Einstein - Max Born Briefwechsel 1916–1955 . Langen / Müller publishing house; March 2005; ISBN 3-7844-2997-1
  24. ^ Günter M. Ziegler and Andreas Loos: The Einstein of Mathematics . For the 150th birthday of David Hilbert. In: Die Zeit , January 12, 2012, page 32.
  25. ^ Declaration by the university teachers of the German Reich of October 23, 1914 Declaration by the university teachers of the German Reich
  26. ↑ An original record is kept in the Mathematisches Institut Göttingen and a copy of the record is contained in the book by Kurt Reidemeister (1971). The audio document in MP3 format can be found at: https://www.swr.de/swr2/wissen/archivradio/08/-/id=2847740/did=22966576/nid=2847740/1295vl0/index.html and http: //math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.mp3 (1.7 MB)
  27. ^ Norbert Schappacher: The Mathematical Institute of the University of Göttingen 1929–1950 , in: Becker, Dahms, Wegeler (ed.): The University of Göttingen under National Socialism , Munich (KG Saur) 1987, 345–373 - second expanded edition: Munich ( KG Saur) 1998, pp. 523-551. Full text
  28. ^ D. Nachmansohn, R. Schmidt: The great era of science in Germany 1900–1933 , 1988, p. 55.
  29. ↑ Sommerfeld, A. / Carathéodory C .: In memory of David Hilbert: died February 14, 1943. Speeches in the mourning house on the morning of the burial day in front of the coffin. Berlin 1943. In: The natural sciences. 31. pp. 213-214.
  30. ^ Hermann Weyl: David Hilbert and his mathematical work. Bulletin of the American Mathematical Society 50, 612-654 (1944) pdf
  31. ↑ Göttinger Tageblatt online from June 18, 2009 for damage on June 18, 2009
  32. ^ David Hilbert: The theory of algebraic number fields, annual report of the German Mathematicians Association, V.4 p.175-546 1897
  33. ^ [1] Franz Lemmermeyer (2018): "120 Years Hilbert's Number Report", Annual Reports of the German Mathematicians Association, 120 (1), 41–79
  34. ↑ A short overview in the preface to: D. Hilbert: Die Hilbertschen problems. Verlag Harri Deutsch, Ostwald's Classics of Exact Sciences, Volume 252. ISBN 978-3-8171-3401-4
  35. ^ A b D. Hilbert: Mathematical Problems - Lecture, given at the international mathematicians' congress in Paris 1900. In: Nachrichten von der Königl. Society of Sciences at Göttingen. Mathematical-physical class. Pp. 253–297 (1900) - full text (Uni Bielefeld) ( Memento of the original from April 8, 2012 on WebCite ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. - digitized full text @1@ 2Template: Webachiv / IABot / www.mathematik.uni-bielefeld.de
  36. ↑ a b D. Hilbert: New foundation of mathematics. First treatise. In: Abhandl. from the math seminar d. Hamb. Univ., Vol. 1, pp. 157-177 (1922), published in Gesammelte Werke , Vol. 3, Chapter 10 pdf
  37. ^ Reid Hilbert , Springer Verlag 1996, p. 127
  38. ↑ He showed an interest in the book his former student was writing bud did not participate in any other way , Constance Reid Courant , Springer / Copernicus 1996, p. 97
  39. ^ Fölsing, Albert Einstein, 1993, p. 422
  40. ^ Albrecht Fölsing: Albert Einstein, Suhrkamp 1993, p. 421f.
  41. ^ Leo Corry, Jürgen Renn, John Stachel: Belated Decision in the Hilbert-Einstein Priority Dispute , SCIENCE, Vol. 278, November 14, 1997.
  42. ↑ Daniela Wuensch , Zwei Real Kerle, News about the discovery of the gravitational equations of the general relativity theory by Einstein and Hilbert . Termessos, 2005, ISBN 3-938016-04-3
  43. ↑ Klaus P. Sommer: Who discovered the general theory of relativity? Priority dispute between Hilbert and Einstein. In: Physics in Our Time. 36, No. 5, 2005, pp. 230-235, ISSN  0031-9252
  44. ↑ David Hilbert: Knowledge of nature and logic. Naturwissenschaften 1930, pp. 959–963 (also published in: Gesammelte Abhandlungen Vol. 3, p. 378) digitized full text
  45. ↑ ssd.jpl.nasa.gov: 12022 Hilbert (1996 XH26) , accessed July 3, 2010
  46. ^ Winner of the Cothenius Medal from 1864 to 1953. Leopoldina, accessed on May 2, 2013 .
  47. ^ List of members since 1666: Letter H. Académie des sciences, accessed on November 27, 2019 (French).
  48. ^ Member History: David Hilbert. American Philosophical Society, accessed September 29, 2018 .
This version was added to the list of articles worth reading on June 11, 2008 .


personal data
SURNAME Hilbert, David
BRIEF DESCRIPTION German mathematician
DATE OF BIRTH January 23, 1862
PLACE OF BIRTH Königsberg (Prussia)
DATE OF DEATH February 14, 1943
Place of death Goettingen