**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 made contributions in many areas of mathematics and physics.

- 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.
- 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.
- The main approach to learning was having pupils memorise large amounts of material, something David was not particularly good at.
- Although doubtless there is modesty in these words, nevertheless they probably reflect Hilbert's own feeling about his school days.
- Hilbert was much happier and his performance in all his subjects improved.
- Returning to Königsberg for the start of session 1881-82, Hilbert attended lectures on number theory and the theory of functions by Heinrich Weber.
- Hilbert and Minkowski, who was also a doctoral student, soon became close friends and they were to strongly influence each others mathematical progress.
- Hurwitz and Hilbert became close friends, another friendship which was important factor in Hilbert's mathematical development, while Lindemann became Hilbert's thesis advisor.
- 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.
- One of Hilbert's chosen propositions was on physics, the other on philosophy.
- Hurwitz suggested that Hilbert make a research visit to Leipzig to speak with Felix Klein.
- 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.
- Klein then told both Study and Hilbert that they should visit Paris.
- They both went in early 1886, Hilbert at the end of March.
- In Paris, Camille Jordan gave a dinner for Hilbert and Study to which George-Henri Halphen, Amédée Mannheim and Gaston Darboux were invited.
- It is clear that Hilbert's thoughts were entirely on mathematics during his time in Paris and he wrote nothing of any sightseeing.
- 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.
- 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.
- In Berlin he met Kronecker and Weierstrass who presented the young Hilbert with two rather different views of the future.
- Hilbert spent eight days in Göttingen before returning to Königsberg.
- In 1892 Schwarz moved from Göttingen to Berlin to occupy Weierstrass's chair and Klein wanted to offer Hilbert the vacant Göttingen 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's 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.
- Hilbert himself tried at first to follow Gordan's approach but soon realised that a new line of attack was necessary.
- 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.
- Hilbert has scorned to present his thoughts following formal rules, he thinks it suffices that no one contradict his proof ...
- 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 Annalen without any changes whatsoever, as indeed it did.
- In 1893 while still at Königsberg Hilbert began a work Zahlbericht Ⓣ(A number theory report) on algebraic number theory.
- 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.
- 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.
- 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.
- 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.
- 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.
- Today Hilbert's name is often best remembered through the concept of Hilbert space.
- 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.
- Many have claimed that in 1915 Hilbert discovered the correct field equations for general relativity before Einstein but never claimed priority.
- 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 contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations.
- In Hilbert's case, his greatness lies in an immensely powerful insight that penetrates into the depths of a question.
- But when it comes to penetrating insight, only a few of the very greatest were the equal of Hilbert.
- Among Hilbert's students were Hermann Weyl, the famous world chess champion Emanuel Lasker, and Ernst Zermelo.
- 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.
- Hilbert, although retired, had still been giving a few lectures.
- Hilbert received many honours.
- In 1905 the Hungarian Academy of Sciences gave a special citation for Hilbert.
- In 1930 Hilbert retired and the city of Königsberg made him an honorary citizen of the city.
- Hilbert was elected an honorary member of the London Mathematical Society in 1901 and of the German Mathematical Society in 1942.

Born 23 January 1862, Wehlau, near Königsberg, Prussia (now Kaliningrad, Russia). Died 14 February 1943, Göttingen, Germany.

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Algebra, Analysis, Group Theory, Origin Russia, Physics, Set Theory, Topology

Branches: 1

Chapters: 2

Definitions: 3 4

Parts: 5 6

Sections: 7 8

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive