## David Hilbert Research Paper

**David Hilbert**, (born January 23, 1862, Königsberg, Prussia [now Kaliningrad, Russia]—died February 14, 1943, Göttingen, Germany), German mathematician who reduced geometry to a series of axioms and contributed substantially to the establishment of the formalistic foundations of mathematics. His work in 1909 on integral equations led to 20th-century research in functional analysis.

The first steps of Hilbert’s career occurred at the University of Königsberg, at which, in 1884, he finished his *Inaugurel-dissertation* (Ph.D.); he remained at Königsberg as a *Privatdozent* (lecturer, or assistant professor) in 1886–92, as an *Extraordinarius* (associate professor) in 1892–93, and as an *Ordinarius* in 1893–95. In 1892 he married Käthe Jerosch, and they had one child, Franz. In 1895 Hilbert accepted a professorship in mathematics at the University of Göttingen, at which he remained for the rest of his life.

The University of Göttingen had a flourishing tradition in mathematics, primarily as the result of the contributions of Carl Friedrich Gauss, Peter Gustav Lejeune Dirichlet, and Bernhard Riemann in the 19th century. During the first three decades of the 20th century this mathematical tradition achieved even greater eminence, largely because of Hilbert. The Mathematical Institute at Göttingen drew students and visitors from all over the world.

Hilbert’s intense interest in mathematical physics also contributed to the university’s reputation in physics. His colleague and friend, the mathematician Hermann Minkowski, aided in the new application of mathematics to physics until his untimely death in 1909. Three winners of the Nobel Prize for Physics—Max von Laue in 1914, James Franck in 1925, and Werner Heisenberg in 1932—spent significant parts of their careers at the University of Göttingen during Hilbert’s lifetime.

In a highly original way, Hilbert extensively modified the mathematics of invariants—the entities that are not altered during such geometric changes as rotation, dilation, and reflection. Hilbert proved the theorem of invariants—that all invariants can be expressed in terms of a finite number. In his *Zahlbericht* (“Commentary on Numbers”), a report on algebraic number theory published in 1897, he consolidated what was known in this subject and pointed the way to the developments that followed. In 1899 he published the *Grundlagen der Geometrie* (*The Foundations of Geometry*, 1902), which contained his definitive set of axioms for Euclidean geometry and a keen analysis of their significance. This popular book, which appeared in 10 editions, marked a turning point in the axiomatic treatment of geometry.

A substantial part of Hilbert’s fame rests on a list of 23 research problems he enunciated in 1900 at the International Mathematical Congress in Paris. In his address, “The Problems of Mathematics,” he surveyed nearly all the mathematics of his day and endeavoured to set forth the problems he thought would be significant for mathematicians in the 20th century. Many of the problems have since been solved, and each solution was a noted event. Of those that remain, however, one, in part, requires a solution to the Riemann hypothesis, which is usually considered to be the most important unsolved problem in mathematics (*see*number theory).

In 1905 the first award of the Wolfgang Bolyai prize of the Hungarian Academy of Sciences went to Henri Poincaré, but it was accompanied by a special citation for Hilbert.

In 1905 (and again from 1918) Hilbert attempted to lay a firm foundation for mathematics by proving consistency—that is, that finite steps of reasoning in logic could not lead to a contradiction. But in 1931 the Austrian–U.S. mathematician Kurt Gödel showed this goal to be unattainable: propositions may be formulated that are undecidable; thus, it cannot be known with certainty that mathematical axioms do not lead to contradictions. Nevertheless, the development of logic after Hilbert was different, for he established the formalistic foundations of mathematics.

Hilbert’s work in integral equations in about 1909 led directly to 20th-century research in functional analysis (the branch of mathematics in which functions are studied collectively). His work also established the basis for his work on infinite-dimensional space, later called Hilbert space, a concept that is useful in mathematical analysis and quantum mechanics. Making use of his results on integral equations, Hilbert contributed to the development of mathematical physics by his important memoirs on kinetic gas theory and the theory of radiations. In 1909 he proved the conjecture in number theory that for any *n,* all positive integers are sums of a certain fixed number of *n*th powers; for example, 5 = 2^{2} + 1^{2}, in which *n* = 2. In 1910 the second Bolyai award went to Hilbert alone and, appropriately, Poincaré wrote the glowing tribute.

The city of Königsberg in 1930, the year of his retirement from the University of Göttingen, made Hilbert an honorary citizen. For this occasion he prepared an address entitled “Naturerkennen und Logik” (“The Understanding of Nature and Logic”). The last six words of Hilbert’s address sum up his enthusiasm for mathematics and the devoted life he spent raising it to a new level: “Wir müssen wissen, wir werden wissen” (“We must know, we shall know”). In 1939 the first Mittag-Leffler prize of the Swedish Academy went jointly to Hilbert and the French mathematician Émile Picard.

The last decade of Hilbert’s life was darkened by the tragedy brought to himself and to so many of his students and colleagues by the Nazi regime.

**David Hilbert**'s father, Otto Hilbert, was the son of a judge who was a high ranking Privy Councillor. Otto was a county judge who had married Maria Therese Erdtmann, the daughter of Karl Erdtmann, a Königsberg merchant. Maria was fascinated by philosophy, astronomy and prime numbers. Otto Hilbert had a brother who was a lawyer and another who was the director of a Gymnasium. After Otto was promoted to become a senior judge, he and Maria moved to 13 Kirchenstrasse in Königsberg and this was the home in which David spent much of his childhood. He had a strict upbringing by his father who was a man who lived his life to a standard pattern, always walking the same way every day and only leaving Königsberg once a year for the annual family holiday. David was his parents' first child and only son. He was six years old when his sister Elsie was born.

The usual age for someone to begin schooling was six but David did not enter his first school, the Royal Friedrichskolleg, until he was eight years old. It is almost certain that his mother taught him at home until he was eight. The Friedrichskolleg, also known as the Collegium Fridericianum, had a junior section which David attended for two years before entering the gymnasium of the Friedrichskolleg in 1872. Although this was reputed to be the best school in Königsberg, the emphasis was on Latin and Greek with mathematics considered as less important. Science was not taught at all in the Friedrichskolleg. The main approach to learning was having pupils memorise large amounts of material, something David was not particularly good at. Perhaps surprisingly for someone who was to make a gigantic impact on mathematics, he did not shine at school. In later life he described himself as a "dull and silly" boy at the Friedrichskolleg. Although doubtless there is modesty in these words, nevertheless they probably reflect Hilbert's own feeling about his school days. In September 1879 he transferred from the Friedrichskolleg to the Wilhelm Gymnasium where he spent his final year of schooling. Here there was more emphasis on mathematics and the teachers encouraged original thinking in a way that had not happened at the Friedrichskolleg. Hilbert was much happier and his performance in all his subjects improved. He received the top grade for mathematics and his final report stated:-

After graduating from the Wilhelm Gymnasium, he entered the University of Königsberg in the autumn of 1880. In his first semester he took courses on integral calculus, the theory of determinants and the curvature of surfaces. Then following the tradition in Germany at this time, in the second semester he went to Heidelberg where he attended lectures by Lazarus Fuchs. Returning to Königsberg for the start of session 1881-82, Hilbert attended lectures on number theory and the theory of functions by Heinrich Weber. In the spring of 1882, Hermann Minkowski returned to Königsberg after studying in Berlin. Hilbert and Minkowski, who was also a doctoral student, soon became close friends and they were to strongly influence each others mathematical progress. Ferdinand von Lindemann was appointed to Königsberg to succeed Heinrich Weber in 1883 and Adolf Hurwitz was appointed as an extraordinary professor there in the spring of 1884. Hurwitz and Hilbert became close friends, another friendship which was important factor in Hilbert's mathematical development, while Lindemann became Hilbert's thesis advisor. He received his oral examination on 11 December 1884 for his thesis entitledFor mathematics he always showed a very lively interest and a penetrating understanding: he mastered all the material taught in the school in a very pleasing manner and was able to apply it with sureness and ingenuity.

*Über invariante Eigenschaften specieller binärer Formen, insbesondere der Kugelfunctionen*Ⓣ. Lindemann had suggested that Hilbert study invariant properties of certain algebraic forms and Hilbert showed great originality in devising an approach that Lindemann had not envisaged. Minkowski, after reading the thesis, wrote to Hilbert (see [8]):-

On 7 February 1885 he defended two propositions in a public disputation. One of Hilbert's chosen propositions was on physics, the other on philosophy. This was the final stage of his doctorate, which was then duly awarded. He spent the month following the award of his doctorate taking, and passing, the Staatsexamen so that he was qualified to teach in a Gymnasium, and he also attended Lindemann's geometry course on Plücker's line geometry and Lie's sphere geometry, and he also attended Hurwitz's lectures on modular functions. Hurwitz suggested that Hilbert make a research visit to Leipzig to speak with Felix Klein. Taking this advice, he went to Leipzig and attended Klein's lectures. He also got to know Georg Pick and Eduard Study. Klein suggested that both Hilbert and Study should visit Erlangen and discuss their research with Paul Gordan who was the leading expert on invariant theory. However, the visit did not take place at that time. Klein then told both Study and Hilbert that they should visit Paris. They both went in early 1886, Hilbert at the end of March. Klein had given them instructions as to which of the Paris mathematicians they should visit and they did as he told them, alternately writing to Klein about their experiences. One of the first mathematicians they visited was Henri Poincaré who returned their visit a few days later. The two young visitors read their letters to Klein out loud to each other so that they would not both tell him the same things. He replied to each in turn, making clear that he was treating them equally. In Paris, Camille Jordan gave a dinner for Hilbert and Study to which George-Henri Halphen, Amédée Mannheim and Gaston Darboux were invited. On this occasion the French mathematicians all spoke German out of politeness to their German guests who complained to Klein afterwards that the mathematical conversation had been very superficial. They were also disappointed with their meeting with Pierre Bonnet who they felt was too old for mathematical discussions. The mathematician with whom they seemed to get on best was Charles Hermite. Although they considered him very old (he was 64), he was "extraordinarily friendly and hospitable" and discussed the big problems of invariant theory. Since they had found their visit especially useful, they returned to Hermite's home for a second visit a few days later. It is clear that Hilbert's thoughts were entirely on mathematics during his time in Paris and he wrote nothing of any sightseeing. Towards the end of his visit he suffered an illness and was probably homesick. Certainly by the spring of 1886 he was in good spirits as he returned to Germany. On his way back to Königsberg he visited Göttingen, where Klein was about to take up the chair, where he met Hermann Amandus Schwarz. Telling Schwarz that he was next going to Berlin, Hilbert was advised to expect a cold reception by Leopold Kronecker. However, Hilbert described his welcome in Berlin as very friendly.I studied your work with great interest and rejoiced over all the processes which the poor invariants had to pass through before they manage to disappear. I would not have supposed that such a good mathematical theorem could have been obtained in Königsberg.

From Berlin, Hilbert continued back to Königsberg where he prepared to submit his habilitation paper on invariant theory. He also had to give an inaugural lecture in the main auditorium of the Albertina and, from the two options offered by Hilbert, he was asked to deliver the lecture *The most general periodic functions*. Klein had told Hilbert that Königsberg may not be a good place for him to habilitate but Hilbert was happy to do so. He wrote to Klein (see for example [8]):-

He was a member of staff at Königsberg from 1886 to 1895, being a Privatdozent until 1892, then as Extraordinary Professor for one year before being appointed a full professor in 1893. The tour that he spoke about after habilitating at Königsberg happened in 1888 [126]:-I am content and full of joy to have decided myself for Königsberg. The constant association with Professor Lindemann and, above all, with Hurwitz is not less interesting than it is advantageous to myself and stimulating. The bad part about Königsberg being so far away from things I hope I will be able to overcome by making some trips again next year, and perhaps then I will get to meet Herr Gordan.

In Berlin he met Kronecker and Weierstrass who presented the young Hilbert with two rather different views of the future. Next, in Leipzig, he finally met Paul Gordan [126]:-... he set off in March1888on a tour of several leading mathematical centres in Germany, including Berlin, Leipzig, and Göttingen. During the course of a month, he spoke with some twenty mathematicians from whom he gained a stimulating overview of current research interests throughout the country.

Hilbert spent eight days in Göttingen before returning to Königsberg. He married his second cousin, Käthe Jerosch, on 12 October 1892; they had one son Franz Hilbert born on 11 August 1893.... the two hit it off splendidly, as both loved nothing more than to talk about mathematics.

In 1892 Schwarz moved from Göttingen to Berlin to occupy Weierstrass's chair and Klein wanted to offer Hilbert the vacant Göttingen chair. However Klein failed to persuade his colleagues and Heinrich Weber was appointed to the chair. Klein was probably not too unhappy when Weber moved to a chair at Strasbourg three years later since on this occasion he was successful in his aim of appointing Hilbert. So, in 1895, Hilbert was appointed to the chair of mathematics at the University of Göttingen, where he continued to teach for the rest of his career.

Hilbert's eminent position in the world of mathematics after 1900 meant that other institutions would have liked to tempt him to leave Göttingen and, in 1902, the University of Berlin offered Hilbert Fuchs' chair. Hilbert turned down the Berlin chair, but only after he had used the offer to bargain with Göttingen and persuade them to set up a new chair to bring his friend Minkowski to Göttingen.

As we saw above, Hilbert's first work was on invariant theory and, in 1888, he proved his famous Basis Theorem. Twenty years earlier Gordan had proved the finite basis theorem for binary forms using a highly computational approach. Attempts to generalise Gordan's work to systems with more than two variables failed since the computational difficulties were too great. Hilbert himself tried at first to follow Gordan's approach but soon realised that a new line of attack was necessary. He discovered a completely new approach which proved the finite basis theorem for any number of variables but in an entirely abstract way. Although he proved that a finite basis existed his methods did not construct such a basis.

Hilbert submitted a paper proving the finite basis theorem to *Mathematische Annalen*. However Gordan was the expert on invariant theory for *Mathematische Annalen* and he found Hilbert's revolutionary approach difficult to appreciate. He refereed the paper and sent his comments to Klein:-

However, Hilbert had learnt through his friend Hurwitz about Gordan's letter to Klein and Hilbert wrote himself to Klein in forceful terms:-The problem lies not with the form ... but rather much deeper. Hilbert has scorned to present his thoughts following formal rules, he thinks it suffices that no one contradict his proof ... he is content to think that the importance and correctness of his propositions suffice. ... for a comprehensive work for the 'Annalen' this is insufficient.

At the time Klein received these two letters from Hilbert and Gordan, Hilbert was an assistant lecturer while Gordan was the recognised leading world expert on invariant theory and also a close friend of Klein's. However Klein recognised the importance of Hilbert's work and assured him that it would appear in the... I am not prepared to alter or delete anything, and regarding this paper, I say with all modesty, that this is my last word so long as no definite and irrefutable objection against my reasoning is raised.

*Annalen*without any changes whatsoever, as indeed it did.

Hilbert expanded on his methods in a later paper, again submitted to the *Mathematische Annalen* and Klein, after reading the manuscript, wrote to Hilbert saying:-

In 1893 while still at Königsberg Hilbert began a workI do not doubt that this is the most important work on general algebra that the 'Annalen' has ever published.

*Zahlbericht*Ⓣ on algebraic number theory. The German Mathematical Society requested this major report three years after the Society was created in 1890. The

*Zahlbericht*(1897) is a brilliant synthesis of the work of Kummer, Kronecker and Dedekind but also contains a wealth of Hilbert's own ideas. The ideas of the present day subject of 'Class field theory' are all contained in this work. Rowe, in [124], describes this work as:-

An extract from Hilbert's Preface to... not really a Bericht in the conventional sense of the word, but rather a piece of original research revealing that Hilbert was no mere specialist, however gifted. ... he not only synthesized the results of prior investigations ... but also fashioned new concepts that shaped the course of research on algebraic number theory for many years to come.

*Zahlbericht*is quote 7 in our collection

*Quotes by and about Hilbert*at THIS LINK.

Hilbert's work in geometry had the greatest influence in that area after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance. He published *Grundlagen der Geometrie* in 1899 putting geometry in a formal axiomatic setting. The book continued to appear in new editions and was a major influence in promoting the axiomatic approach to mathematics which has been one of the major characteristics of the subject throughout the 20^{th} century.

Reviews of *Grundlagen der Geometrie* and other of Hilbert's books are at THIS LINK.

More about *Grundlagen der Mathematik* is at THIS LINK.

Hilbert's famous 23 Paris problems challenged (and still today challenge) mathematicians to solve fundamental questions. Hilbert's famous speech The Problems of Mathematics was delivered to the Second International Congress of Mathematicians in Paris. It was a speech full of optimism for mathematics in the coming century and he felt that open problems were the sign of vitality in the subject:-

Hilbert's problems included the continuum hypothesis, the well ordering of the reals, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet's principle and many more. Many of the problems were solved during this century, and each time one of the problems was solved it was a major event for mathematics.The great importance of definite problems for the progress of mathematical science in general ... is undeniable. ...[for]as long as a branch of knowledge supplies a surplus of such problems, it maintains its vitality. ... every mathematician certainly shares ..the conviction that every mathematical problem is necessarily capable of strict resolution ... we hear within ourselves the constant cry: There is the problem, seek the solution. You can find it through pure thought...

For more information about Hilbert's problems see THIS LINK.

Today Hilbert's name is often best remembered through the concept of Hilbert space. Irving Kaplansky, writing in [2], explains Hilbert's work which led to this concept:-

Many have claimed that in 1915 Hilbert discovered the correct field equations for general relativity before Einstein but never claimed priority. The article [54] however, shows that this view is in error. In this paper the authors show convincingly that Hilbert submitted his article on 20 November 1915, five days before Einstein submitted his article containing the correct field equations. Einstein's article appeared on 2 December 1915 but the proofs of Hilbert's paper (dated 6 December 1915) do not contain the field equations.Hilbert's work in integral equations in about1909led directly to20(^{th}-century research in functional analysisthe branch of mathematics in which functions are studied collectively). This work also established the basis for his work on infinite-dimensional space, later called Hilbert space, a concept that is useful in mathematical analysis and quantum mechanics. Making use of his results on integral equations, Hilbert contributed to the development of mathematical physics by his important memoirs on kinetic gas theory and the theory of radiations.

As the authors of [54] write:-

In 1934 and 1939 two volumes ofIn the printed version of his paper, Hilbert added a reference to Einstein's conclusive paper and a concession to the latter's priority: "The differential equations of gravitation that result are, as it seems to me, in agreement with the magnificent theory of general relativity established by Einstein in his later papers". If Hilbert had only altered the dateline to read "submitted on20November1915, revised on[any date after2December1915, the date of Einstein's conclusive paper]," no later priority question would have arisen.

*Grundlagen der Mathematik*Ⓣ were published which were intended to lead to a 'proof theory', a direct check for the consistency of mathematics. Gödel's paper of 1931 showed that this aim is impossible.

See THIS LINK.

Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations. His mathematical abilities were nicely summed up by Otto Blumenthal, his first student [30]:-

Among Hilbert's students were Hermann Weyl, the famous world chess champion Emanuel Lasker, and Ernst Zermelo. But the list includes many other famour names including Wilhelm Ackermann, Felix Bernstein, Otto Blumenthal, Richard Courant, Haskell Curry, Max Dehn, Rudolf Fueter, Alfred Haar, Georg Hamel, Erich Hecke, Earle Hedrick, Ernst Hellinger, Edward Kasner, Oliver Kellogg, Hellmuth Kneser, Otto Neugebauer, Erhard Schmidt, Hugo Steinhaus, and Teiji Takagi.In the analysis of mathematical talent one has to differentiate between the ability to create new concepts that generate new types of thought structures and the gift for sensing deeper connections and underlying unity. In Hilbert's case, his greatness lies in an immensely powerful insight that penetrates into the depths of a question. All of his works contain examples from far-flung fields in which only he was able to discern an interrelatedness and connection with the problem at hand. From these, the synthesis, his work of art, was ultimately created. Insofar as the creation of new ideas is concerned, I would place Minkowski higher, and of the classical great ones, Gauss, Galois, and Riemann. But when it comes to penetrating insight, only a few of the very greatest were the equal of Hilbert.

In 1930 Hilbert retired but only a few years later, in 1933, life in Göttingen changed completely when the Nazis came to power and Jewish lecturers were dismissed. By the autumn of 1933 most had left or were dismissed. Hilbert, although retired, had still been giving a few lectures. In the winter semester of 1933-34 he gave one lecture a week on the foundations of geometry. After he finished giving this course he never set foot in the Institute again. In early 1942 he fell and broke his arm while walking in Göttingen. This made him totally inactive and this seems to have been a major factor in his death a year after the accident.

Hilbert received many honours. In 1905 the Hungarian Academy of Sciences gave a special citation for Hilbert. He was awarded the Bolyai Prize in 1910 and elected a fellow of the Royal Society of London in 1928. In 1930 Hilbert retired and the city of Königsberg made him an honorary citizen of the city. He gave an address which ended with six famous words showing his enthusiasm for mathematics and his life devoted to solving mathematical problems:-

See quote 3 at THIS LINK.Wir müssen wissen, wir werden wissen - We must know, we shall know.

In 1939 he was awarded the Mittag-Leffler prize by the Swedish Academy of Sciences. He shared this Prize with Émile Picard. Hilbert was elected an honorary member of the London Mathematical Society in 1901 and of the German Mathematical Society in 1942.

For quotes which describe Hilbert's personality and hobbies see 5 and 10 at THIS LINK.

**Article by:***J J O'Connor* and *E F Robertson*

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