Influential Mathematicians: David Hilbert
Whilst we’re working to finalise the structure of the quantum formalism lectures, I’ve been tasked with introducing you to the key mathematicians who directly & indirectly played a role in the mathematical formalism of quantum mechanics. So we're starting with David Hilbert, the man for whom Hilbert spaces were named after!
Who was David Hilbert?
The School of Mathematics and Statistics University of St Andrew has a pretty comprehensive bio on David Hilbert that I've summarised for you below.
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. 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.
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 entitled 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.
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.
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'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.
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 contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations.
Hilbert’s Quantum Formalism Influence
Although Hilbert did not directly contribute to the quantum formalism, his work on integral equations laid the foundations for the likes of John von Neumann (his student!) to create the abstract mathematical formalism of quantum mechanics centered around the concept of Hilbert spaces.
Hilbert himself was very interested in applying axiomatic methods to physics. This indeed motivated his famous Sixth problem:
Mathematical Treatment of the Axioms of Physics
‘‘The investigations on the foundations of geometry suggest the problem: to treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.’’ David Hilbert
This it for today, I hope you enjoyed this brief biography of one of the greatest mathematicians of 20th century. I’ll end with a recommendation of some influential books written by Hilbert:
‘The Foundations of Geometry’ (free pdf English translation from Berkeley)
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Author: Bambordé, co-founder at Zaiku Group. Please take the survey if you haven’t already done so - we’ll keep it up until the end of this month!