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.
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:-
For 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.
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. Minkowski, after reading the thesis, wrote to Hilbert (see [8]):-
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.
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.
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]):-
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.
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]:-
... he set off in March 1888 on 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.
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]:-
... the two hit it off splendidly, as both loved nothing more than to talk about mathematics.
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.
In 1892Schwarz 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.
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:-
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.
However, Hilbert had learnt through his friend Hurwitz about Gordan's letter to Klein and Hilbert wrote himself to Klein in forceful terms:-
... 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.
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.
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:-
I do not doubt that this is the most important work on general algebra that the 'Annalen' has ever published.
In 1893 while still at Königsberg Hilbert began a work 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:-
... 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.
An extract from Hilbert's Preface to 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 20th 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:-
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...
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.
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:-
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). 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. 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.
In 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 on 20 November 1915, revised on [any date after 2 December 1915, the date of Einstein's conclusive paper]," no later priority question would have arisen.
In 1934 and 1939 two volumes of 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.
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]:-
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:-
Wir müssen wissen, wir werden wissen - We must know, we shall know.
M Ackerman and R Hermann (trs.), Hilbert's invariant theory papers(Lie Groups: History, Frontiers and Applications, VIII. Math Sci Press, Brookline, Mass., 1978).
L Corry, David Hilbert and the axiomatization of physics (1898-1918). From Grundlagen der Geometrie to Grundlagen der Physik. Archimedes: New Studies in the History and Philosophy of Science and Technology, 10(Kluwer Academic Publishers, Dordrecht, 2004).
J Fang, Hilbert. Towards a philosophy of modern mathematics.II(Paideia Press, New York 1970).
M Hallett and U Majer (eds.), David Hilbert's lectures on the foundations of geometry, 1891-1902(Springer-Verlag, Berlin, 2004).
V Peckhaus, Hilbertprogramm und Kritische Philosophie. Das Göttinger Modell interdisziplinärer Zusammenarbeit zwischen Mathematik und Philosophie (Studien zur Wissenschafts-, Sozial- und Bildungsgeschichte der Mathematik 7. Vandenhoeck & Ruprecht, Göttingen, 1990).
C Reid, Hilbert(Berlin- Heidelberg- New York, 1970).
C Reid, Hilbert-Courant( New York, 1986).
K Reidemeister (ed.), Hilbert. Gedenkband(Springer-Verlag, Berlin, 1971).
T Sauer, U Majer, A Schirrmacher and H-J Schmidt (eds.), David Hilbert's lectures on the foundations of physics 1915-1927(German), Relativity, quantum theory and epistemology (Springer-Verlag, Berlin, 2009).
H Wussing and W Arnold, Biographien bedeutender Mathematiker(Volk und Wissen Verlag, Berlin, 1983).
M V Abrusci, Hilbert and the foundations of mathematics: the axiomatic method (Italian), in The ideas of David Hilbert (Italian), Catania, 1999, Matematiche (Catania)55(suppl. 1)(2000), 127-142.
M V Abrusci, 'Proof', 'theory', and 'foundations' in Hilbert's mathematical work from 1885 to 1900, in Italian studies in the philosophy of science(Boston Stud. Philos. Sci., 47, Reidel, Dordrecht-Boston, Mass., 1981), 453-491.
P S Aleksandrov, A few words of recollections of Hilbert (Russian), Uspekhi Mat. Nauk36(1)(217)(1981), 233-236.
G L Alexanderson, About the cover: Hilbert and the Paris ICM, Bull. Amer. Math. Soc. (N.S.)51(2)(2014), 329-334.
A M Anile, Current problems in mathematical physics: the original contribution of David Hilbert, in The ideas of David Hilbert (Italian), Catania, 1999, Matematiche (Catania)55(suppl. 1)(2000), 59-74.
B Artmann, Euclid's and Hilbert's foundations of geometry, in The ideas of David Hilbert (Italian), Catania, 1999, Matematiche (Catania)55(suppl. 1)(2000), 143-160.
T A A B, Review: Anschauliche Geometrie, by D Hilbert and S Cohn-Vossen, The Mathematical Gazette 36(317)(1952), 231-232.
T A A B, Review: Methods of Mathematical Physics. I, by R Courant and D Hilbert, The Mathematical Gazette39(328)(1955), 175.
H Behnke, Die hohe Zeit Göttingens. Zu einer Biographie über Hilbert, Math.-Phys. Semesterber. 18(1971), 1-6.
P G Bergmann, Review: Methods of Mathematical Physics, Vol. II: Partial Differential Equations, by R Courant and D Hilbert, The American Mathematical Monthly71(3)(1964), 338.
P Bernays, David Hilbert, Encyclopedia of PhilosophyIII(New York, 1967), 496-504.
H A Bethe, Review: Methods of Mathematical Physics, by R Courant and D Hilbert, Science, New Series119(3080)(1954), 75-76.
K-R Biermann, David Hilbert und die Berliner Akademie, Math. Nachr.27(1963/1964), 377-384.
G Birkhoff, Review: Methoden der Mathematischen Physik, by R Courant and D Hilbert, Science, New Series99(2573)(1944), 322.
G Birkhoff and M K Bennett, Hilbert's 'Grundlagen der Geometrie', Rend. Circ. Mat. Palermo(2)36(3)(1987), 343-389.
M Black, Review: Grundlagen der Mathematik, by D Hilbert and P Bernays, Mind49(194)(1940), 239-248.
M Black, Review: Grundzüge der Theoretischen Logik, by D Hilbert and W Ackermann, The Mathematical Gazette23(255)(1939), 334.
O Blumenthal, David Hilbert (German), Mitt. Dtsch. Math.-Ver.20(3)(2012), 175-180.
W M Boothby, Review: Geometry and the Imagination, by David Hilbert and S Cohn-Vossen, The Mathematics Teacher 47(2)(1954), 126.
U Bottazzini, Hilbert's problems: a research program for 'future generations', in Mathematical lives (Springer, Berlin, 2011), 1-10.
K A Brading and T A Ryckman, Hilbert's 'Foundations of physics': gravitation and electromagnetism within the axiomatic method, Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys.39(1)(2008), 102-153.
T A A Broadbent, Review: Gesammelte Abhandlungen. II by David Hilbert; Gesammelte Abhandlungen. III by David Hilbert, The Mathematical Gazette52(379)(1968), 61-62.
L Cakalov, David Hilbert and his mathematical work (Bulgarian), Fiz. Mat. Spis. Bulgar. Akad. Nauk. 5(38)(1962), 126-135.
C Carathéodory, Obituary: David Hilbert, S-B Math.-Nat. Abt. Bayer. Akad. Wiss.1943(1943)350-354.
R Carnap, Review: Grundlagen der Mathematik. Band I, by D Hilbert and P Bernays, The Journal of Unified Science (Erkenntnis)8(1/3)(1939), 184-187.
F Catanese, Hilbert and the theory of invariants (Italian), in The ideas of David Hilbert (Italian), Catania, 1999, Matematiche (Catania)55(suppl. 1)(2000), 25-46.
F Catanese, Hilbert at the Georg August Universität Göttingen: yesterday and today (Italian), in The ideas of David Hilbert (Italian), Catania, 1999, Matematiche (Catania)55(suppl. 1)(2000), 7-24.
C Cerroni, Some models of geometries after Hilbert's Grundlagen, Rend. Mat. Appl.(7)30(1)(2010), 47-66.
C Cerroni, The contributions of Hilbert and Dehn to non-Archimedean geometries and their impact on the Italian school, Rev. Histoire Math. 13(2)(2007), 259-299.
A Church, Review: Grundzüge der Theoretischen Logik (3rd ed), by D Hilbert and W Ackermann, The Journal of Symbolic Logic15(1)(1950), 59.
W S Contro, Von Pasch zu Hilbert, Arch. History Exact Sci.15(3)(1975/76), 283-295.
E T Copson, Review Methoden der mathematischen Physik. II, by R Courant and D Hilbert, The Mathematical Gazette22(250)(1938), 302-306.
L Corry, On the origins of Hilbert's sixth problem: physics and the empiricist approach to axiomatization, in International Congress of MathematiciansIII(Eur. Math. Soc., Zürich, 2006), 1697-1718.
L Corry, Axiomatics, empiricism, and Anschauung in Hilbert's conception of geometry: between arithmetic and general relativity, in The architecture of modern mathematics(Oxford Univ. Press, Oxford, 2006), 133-156.
L Corry, David Hilbert and his empiricist philosophy of geometry (Spanish), History and philosophy of mathematics (Spanish), Bol. Asoc. Mat. Venez.9(1)(2002), 27-43.
L Corry, Leo Hilbert and physics (1900-1915), in The symbolic universe, Milton Keynes, 1996(Oxford Univ. Press, New York, 1999), 145-188.
L Corry, David Hilbert between mechanical and electromagnetic reductionism (1910-1915), Arch. Hist. Exact Sci.53(6)(1999), 489-527.
L Corry, The influence of David Hilbert and Hermann Minkowski on Einstein's views over the interrelation between physics and mathematics, Endeavour22(3)(1998), 95-97.
L Corry, Hilbert on kinetic theory and radiation theory (1912-1914), Math. Intelligencer20(3)(1998), 52-58.
L Corry, David Hilbert and the axiomatization of physics (1894-1905), Arch. Hist. Exact Sci.51(2)(1997), 83-198.
L Corry, Axiomatics and structural algebra in the works of David Hilbert (Spanish), Mathesis11(4)(1995), 291-329.
L Corry, J Renn and J Stachel, Belated Decision in the Hilbert-Einstein Priority Dispute, Science278(5341)(1997), 1270-1273.
R Courant, Reminiscences from Hilbert's Göttingen, Math. Intelligencer3(4)(1980/81), 154-164.
A Császár, Wie sah Hilbert die Zukunft der Mathematik?, in Grosse Augenblicke aus der Geschichte der Mathematik (Bibliographisches Inst., Mannheim, 1990), 167-192.
J B Diaz, Review: Methods of Mathematical Physics, Volume II, by R Courant and D Hilbert, SIAM Review6(4)(1964), 463-466.
J B Diaz, Review: Methods of Mathematical Physics Vol 2 by R. Courant and D. Hilbert, Mathematics of Computation18(88)(1964), 680-683.
J Dieudonné, David Hilbert (1862-1943)(Spanish), Rev. Integr. Temas Mat.2(1)(1983), 27-33.
A V Dorofeeva, Hilbert's development of the theory of infinite quadratic forms (Russian), in History and methodology of the natural sciencesXX(Russian), Moskov. Gos. Univ., Moscow, 1978), 74-80.
A Drago, Poincaré versus Peano and Hilbert about the mathematical principle of induction, in Henri Poincaré: science et philosophie, Nancy, 1994(Publ. Henri-Poincaré-Arch., Akademie Verlag, Berlin, 1996), 513-527; 586.
B Dreben and A Kanamori, Hilbert and set theory, in A Symposium on David Hilbert, Boston, MA, 1993, Synthese 110(1)(1997), 77-125.
R Dvornicich, The influence of David Hilbert in number theory (Italian), in The ideas of David Hilbert (Italian), Catania, 1999, Matematiche (Catania)55(suppl. 1)(2000), 75-91.
J Earman and C Glymour, Einstein and Hilbert: two months in the history of general relativity, Arch. Hist. Exact Sci. 19(3)(1978/79), 291-308.
U Felgner, Hilberts 'Grundlagen der Geometrie' und ihre Stellung in der Geschichte der Grundlagendiskussion, Jahresber. Dtsch. Math.-Ver.115(3-4)(2014), 185-206.
J Ferreirós, Hilbert, logicism, and mathematical existence, Synthese170(1)(2009), 33-70.
F H Fischer, Review: Grundzüge der Theoretischen Logik (4th ed), by D Hilbert and W Ackermann, The Journal of Symbolic Logic25(2)(1960), 158.
H G Forder, Review: Grundlagen der Geometrie (7th edition), by D Hilbert, The Mathematical Gazette 15(213)(1931), 397-400.
H G Forder, Review: Grundlagen der Mathematik. II, by D Hilbert and P Bernays, The Mathematical Gazette24(260)(1940), 225-227.
H G Forder, Review: Grundzüge der Theoretischen Logik, by D Hilbert and W Ackermann, The Mathematical Gazette14(197)(1928), 273-274.
R Fritsch, Hilberts Beweis der Transzendenz der Ludolphschen Zahl pi, Differentsial'naya Geom. Mnogoobraz. Figur No.34(2003), 144-148.
M Giaquinta, Hilbert and the calculus of variations (Italian), in The ideas of David Hilbert (Italian), Catania, 1999, Matematiche (Catania)55(suppl. 1)(2000), 47-58.
E N Giovannini, Completeness and continuity in Hilbert's foundations of geometry: on the Vollständigkeitsaxiom (Spanish), Theoria (San Sebastián)(2)28(76)(2013), 139-163.
R L Goodstein, Review: Principles of Mathematical Logic, by D Hilbert and W Ackermann, The Mathematical Gazette35(314)(1951), 293-294.
I Grattan-Guinness, A sideways look at Hilbert's twenty-three problems of 1900, Notices Amer. Math. Soc.47(7)(2000), 752-757.
J Gray, Poincaré replies to Hilbert: on the future of mathematics ca. 1908, Math. Intelligencer34(3)(2012), 15-29.
J Hachaj, David Hilbert in the light of his correspondence with Felix Klein (Polish), in Hilbert's Problems (Polish), Miedzyzdroje, 1993(Polsk. Akad. Nauk, Warsaw, 1997), 263-269.
G B Halsted, Review: The Foundations of Geometry, by David Hilbert, Science, New Series16(399)(1902), 307-308.
C O Hill, Husserl and Hilbert on completeness, in From Dedekind to Gödel, Boston, MA, 1992(Synthese Lib., 251, Kluwer Acad. Publ., Dordrecht, 1995), 143-163.
H N Jahnke, Hilbert, Weyl und die Philosophie der Mathematik, Math. Semesterber. 37(2)(1990), 157-179.
S C Kleene, Review: Grundlagen der Mathematik Vol 2, by D Hilbert and P Bernays, The Journal of Symbolic Logic5(1)(1940), 16-20.
G T Kneebone, Review: Grundlagen der Mathematik I (2nd edition), by D Hilbert and P Bernays, The Journal of Symbolic Logic35(2)(1970), 321-323.
G T Kneebone, Review: Principles of Mathematical Logic, by D Hilbert, W Ackermann and Robert E Luce, Philosophy 27(103)(1952), 375-376.
G T Kneebone, Review: Grundlagen der Mathematik II (2nd ed), by D Hilbert and Bernays, The Journal of Symbolic Logic39(2)(1974), 357.
E F Kondrat'ev, David Hilbert and the axiomatization of physics (Russian), in Proceedings of an International Mathematics Seminar(Russian)(Izd. Kaliningrad. Gos. Univ., Kaliningrad, 2002), 84-90.
M Laserna, Kant, Frege, Hilbert and the sources of geometrical knowledge, Rev. Acad. Colombiana Cienc. Exact. Fís. Natur.24(93)(2000), 535-546.
D W Lewis, David Hilbert and the theory of algebraic invariants, Irish Math. Soc. Bull.33(1994), 42-54.
P-L Lions, Du Courant & Hilbert Methoden der mathematischen Physik aux simulations numériques, Gaz. Math. No.108(2006), 5-16.
X H Liu, The disagreement between Gauss and Hilbert on Fermat's last theorem (Chinese), J. Northwest Univ.30(2)(2000), 176-180.
G Lolli, Hilbert and logic (Italian), in The ideas of David Hilbert (Italian), Catania, 1999, Matematiche (Catania)55(suppl. 1)(2000), 93-126.
C C MacDuffee, Review: Geometry and the Imagination, by D Hilbert and S Cohn-Vossen, Science, New Series116(3023)(1952), 643-644.
B Machado, David Hilbert (Portuguese), Gaz. Mat., Lisboa4(14)(1943), 1-2.
U Majer, Husserl and Hilbert on completeness. A neglected chapter in early twentieth century foundations of mathematics, in A Symposium on David Hilbert, Boston, MA, 1993, Synthese110(1)(1997), 37-56.
U Majer, Hilbert's criticism of Poincaré's conventionalism, in Henri Poincaré: science et philosophie, Nancy, 1994(Publ. Henri-Poincaré-Arch., Akademie Verlag, Berlin, 1996),
V S Malakhovskii, The prominent role of David Hilbert in the development of mathematics in the 20th century (Russian), in Proceedings of an International Mathematics Seminar(Russian)(Izd. Kaliningrad. Gos. Univ., Kaliningrad, 2002), 60-64.
E A Maxwell, Review: Geometry and the Imagination, by D Hilbert, S Cohn-Vossen and P Nemenyi, The Mathematical Gazette 37(322)(1953), 295.
D C McCarty, David Hilbert and Paul du Bois-Reymond: limits and ideals, in One hundred years of Russell's paradox(de Gruyter Ser. Log. Appl., 6, de Gruyter, Berlin, 2004), 517-532.
W H McCrea, Review: Principles of Mathematical Logic, by D Hilbert, W Ackermann, The British Journal for the Philosophy of Science2(8)(1952), 332-333.
C McLarty, Hilbert on theology and its discontents: the origin myth of modern mathematics, in Circles disturbed(Princeton Univ. Press, Princeton, NJ, 2012), 105-129.
H A Medicus, A comment on the relations between Einstein and Hilbert, Amer. J. Phys.52(3)(1984), 206-208.
P Modesti, Hilbert and the 'Grundlagen der Geometrie', in Seminars in Complex Analysis and Geometry, 1989/90, Arcavacata (Sem. Conf., 10, Mediterranean, Rende, 1993), 65-92.
G H Moore, Hilbert on the infinite: the role of set theory in the evolution of Hilbert's thought, Historia Math.29(1)(2002), 40-64.
L J Mordell, Review: Gesammelte Abhandlungen. III. Analysis. Grundlagen der Mathematik. Physik. Verschiedenesnebst einer Lebensgeschichte, by David Hilbert, The Mathematical Gazette20(239)(1936), 213-215.
E N, Review: Grundzüge der Theoretischen Logik, by D Hilbert and W Ackermann, The Journal of Philosophy35(14)(1938), 390-391.
B H Neumann, David Hilbert, Mathematical Spectrum25(1992/93), 70-73.
J D North, Review: Gesammelte Abhandlungen, by David Hilbert; Hilbert Gedenkband, by K Reidemeister, The British Journal for the History of Science6(2)(1972), 208-209.
A J F de Oliveira, Hilbert's formalism and intuitive thought (Portuguese), Bol. Soc. Port. Mat. No.52(2005), 1-25.
V Pambuccian, Review: David Hilbert's lectures on the foundations of geometry, 1891-1902, Philos. Math.(3)21(2)(2013), 255-277.
A N Parsin, David Hilbert and the theory of invariants (Russian), Istor.-Mat. Issled. Vyp.20(1975), 171-197; 380.
C Parsons, Review: Beiträge zur Philosophie der Logik und Mathematik, by Leonard Nelson, Wilhelm Ackermann, Paul Bernays and David Hilbert, The Journal of Philosophy59(9)(1962), 242-246.
L Patton, Hilbert's objectivity, Historia Math.41(2)(2014), 188-203.
A L Pavlova, D Hilbert: friends and teachers (Russian), in Proceedings of an International Mathematics Seminar(Russian)(Izd. Kaliningrad. Gos. Univ., Kaliningrad, 2002), 95-99.
V Peckhaus, Hilbert's axiomatic programme and philosophy, in The history of modern mathematicsIII(Academic Press, Boston, MA, 1994), 90-112.
V Peckhaus, Hilbert, Zermelo und die Institutionalisierung der mathematischen Logik in Deutschland, Ber. Wiss.-Gesch.15(1)(1992), 27-38.
V Peckhaus and R Kahle, Hilbert's paradox, Historia Math.29(2)(2002), 157-175.
A Pietsch, D Hilbert and E Schmidt or A landmark in the history of mathematics (Czech), Pokroky Mat. Fyz. Astronom.39(2)(1994), 65-94.
H Poincaré, Review of Hilbert's Foundations of geometry, in Real numbers, generalizations of the reals, and theories of continua(Synthese Lib., 242, Kluwer Acad. Publ., Dordrecht, 1994, 147-168.
C J Posy, Brouwer versus Hilbert: 1907-1928, Sci. Context11(2)(1998), 291-325.
M F Ranada, David Hilbert, Hermann Minkowski, the axiomatization of physics and the Sixth Problem (Spanish), Gac. R. Soc. Mat. Esp.6(3)(2003), 641-664.
C Reid, The Hilbert mentality (Spanish), Bol. Mat. 9(3)(1975), 187-188.
Review: The Foundations of Geometry, by D Hilbert and E J Townsend, The Mathematical Gazette2(38)(1903), 268-269.
P Roket, David Hilbert in Königsberg (Russian), in Proceedings of an International Mathematics Seminar (Russian)(Izd. Kaliningrad. Gos. Univ., Kaliningrad, 2002), 3-30.
W V Quine, Review: Grundzüge der Theoretischen Logik, by D Hilbert and W Ackermann, The Journal of Symbolic Logic3(2)(1938), 83-84.
D E Rowe, Klein, Hilbert, and the Göttingen mathematical tradition, Osiris(2)5(1989), 186-213.
D E Rowe, David Hilbert on Poincaré, Klein, and the world of mathematics, The Mathematical Intelligencer8(1)(1986), 75-77.
D E Rowe, Hilbert's early career: encounters with allies and rivals, Math. Intelligencer27(1)(2005), 72-82.
D E Rowe, From Königsberg to Göttingen: a sketch of Hilbert's early career, Math. Intelligencer25(2)(2003), 44-50.
D E Rowe, Mathematics made in Germany: on the background to Hilbert's Paris lecture, Math. Intelligencer 35(3)(2013), 9-20.
D E Rowe, Making mathematics in an oral culture: Göttingen in the era of Klein and Hilbert, Sci. Context17(1-2)(2004), 85-129.
D E Rowe, Einstein meets Hilbert: at the crossroads of physics and mathematics, Phys. Perspect.3(4)(2001), 379-424.
D E Rowe, Perspective on Hilbert, Perspect. Sci.5(4)(1997), 533-570.
D E Rowe, The philosophical views of Klein and Hilbert, in The intersection of history and mathematics(Sci. Networks Hist. Stud., 15, Birkhäuser, Basel, 1994), 187-202.
T Sauer, The relativity of discovery: Hilbert's first note on the foundations of physics, Arch. Hist. Exact Sci.53(6)(1999), 529-575.
T Sauer and D E Rowe, Otto Blumenthal über David Hilbert, Mitt. Dtsch. Math.-Ver.20(3)(2012), 173-174.
M M Schiffer, Review: Methods of Mathematical Physics. vol. 2, Partial Differential Equations, by R Courant and D Hilbert, Science, New Series137(3527)(1962), 334.
A Schirrmacher, Planting in his neighbor's garden: David Hilbert and early Göttingen quantum physics, Phys. Perspect.5(1)(2003), 4-20.
W Sieg, Hilbert's programs: 1917-1922, Bull. Symbolic Logic5(1)(1999), 1-44.
S Sigurdsson, Review: Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919-1920 in Gottingen, by David Hilbert and David E Rowe, Isis84(3)(1993), 600-602.
H Sinaceur, Le rôle de Poincaré dans la genèse de la métamathématique de Hilbert, in Henri Poincaré: science et philosophie, Nancy, 1994(Publ. Henri-Poincaré-Arch., Akademie Verlag, Berlin, 1996), 493-511.
I Smadja, Local axioms in disguise: Hilbert on Minkowski diagrams, Synthese186(1)(2012), 315-370.
I Smadja, Erratum to: Local axioms in disguise: Hilbert on Minkowski diagrams, Synthese186(1)(2012), 441-442.
Snz, Review: Grundzüge der theoretischen Logik, by D Hilbert and W Ackermann, Annalen der Philosophie und philosophischen Kritik7(1928), 157.
A Sommerfeld, Zum Andenken an David Hilbert, Naturwissenschaften 31(1943), 213-214.
E Stepinska, The debate between Hilbert and Frege (Polish), Wiadom. Mat. 34(1998), 105-122.
R Thiele, Hilbert and his twenty-four problems, in Mathematics and the historian's craft (CMS Books Math./Ouvrages Math. SMC, 21, Springer, New York, 2005), 243-295.
R Thiele, Hilbert und Hamburg, Mitt. Math. Ges. Hamburg 22(2003), 99-126.
M Toepell, On the origins of David Hilbert's 'Grundlagen der Geometrie', Archive for History of Exact Science35(4)(1986), 329-344.
M Toepell, 100 Jahre Grundlagen der Geometrie - David Hilberts entscheidender Beitrag zur Formalisierung der Mathematik., Mitt. Dtsch. Math.-Ver.1999(1)(1999), 10-15.
H W Turnbull, Review: Anschauliche Geometrie, by D Hilbert and S Cohn-Vossen, The Mathematical Gazette17(225)(1933), 277-280.
D van Dalen, Der Grundlagenstreit zwischen Brouwer und Hilbert, in Vorlesungen zum Gedenken an Felix Hausdorff(Berliner Studienreihe Math., 5, Heldermann, Berlin, 1994), 207-212.
B L van der Waerden, The school of Hilbert and Emmy Noether, Bull. London Math. Soc.15(1)(1983), 1-7.
S C van Veen, In memoriam David Hilbert (1862-1943)(Dutch), Mathematica, Zutphen. B 11(1943), 159-169.
V Vinnikov, We shall know: Hilbert's apology, Math. Intelligencer21(1)(1999), 42-46.
V P Vizgin, Einstein, Hilbert, and Weyl: the genesis of the geometrical unified field theory program, in Einstein and the history of general relativity, North Andover, MA, 1986(Einstein Stud., 1, Birkhäuser Boston, Boston, MA, 1989), 300-314.
V P Vizgin, Hilbert and the problem of general covariant equations of gravitation (Russian), Istor.-Mat. Issled. No. 24(1979), 226-246; 388.
Jan von Plato, Review: David Hilbert's Lectures on the Foundations of Geometry 1891-1902, edited by Michael Hallett and Ulrich Majer, The Bulletin of Symbolic Logic12(3)(2006), 492-494.
C Wang, Comparative study of theories of abstract spaces introduced by Fréchet and Hilbert (Chinese), J. Northwest Univ. Nat. Sci.43(1)(2013), 163-167.
J C Webb, Hilbert's formalism and arithmetization of mathematics, in A Symposium on David Hilbert, Boston, MA, 1993, Synthese110(1)(1997), 1-14.
K F Wehmeier, Aspekte der Frege-Hilbert-Korrespondenz, Hist. Philos. Logic18(4)(1997), 201-209.
H Weyl, David Hilbert and his mathematical work, Bull. Amer. Math. Soc.50(1944), 612-654.
H Weyl, Hermann Obituary: David Hilbert. 1862-1943, Obituary Notices of Fellows of the Royal Society of London4(1944), 547-553.
H Weyl, David Hilbert and his mathematical work (Portuguese), Bol. Soc. Mat. Sao Paulo1(1946), 76-104.
H Weyl, David Hilbert and his mathematical work (Portuguese), Bol. Soc. Mat. Sao Paulo2(1947), 37-60.
O Zich, On the thirtieth anniversary of the death of David Hilbert (Czech), Pokroky Mat. Fyz. Astronom.18(1973), 301-305.
H J Zassenhaus, On the Minkowski-Hilbert dialogue on mathematization, Canad. Math. Bull.18(3)(1975), 443-461.
O Zich, On the thirtieth anniversary of the death of David Hilbert (Czech), Pokroky Mat. Fyz. Astronom.18(1973), 301-305.
D Zitarelli, Hilbert in Missouri, Math. Mag.84(5)(2011), 351-364.
G Zubieta R, Review: Principles of Mathematical Logic, by D Hilbert, W Ackermann, The Journal of Symbolic Logic16(1)(1951), 52-53.
MacTutor
