David Hilbert

"Hilbert" redirects here. For other uses, seeHilbert (disambiguation).

David Hilbert
Hilbert in 1912
Born	(1862-01-23)23 January 1862 KönigsbergorWehlau,Kingdom of Prussia
Died	14 February 1943(1943-02-14)(aged 81) Göttingen,Nazi Germany
Education	University of Königsberg(PhD)
Known for	Hilbert's basis theorem Hilbert's Nullstellensatz Hilbert's axioms Hilbert's problems Hilbert's program Einstein–Hilbert action Hilbert space Hilbert system Epsilon calculus
Spouse	Käthe Jerosch
Children	Franz (b. 1893)
Awards	Lobachevsky Prize(1903) Bolyai Prize(1910) ForMemRS
Scientific career
Fields	Mathematics,PhysicsandPhilosophy
Institutions	University of Königsberg Göttingen University
Thesis	On Invariant Properties of Special Binary Forms, Especially of Spherical Functions(1885)
Doctoral advisor	Ferdinand von Lindemann
Doctoral students	Wilhelm Ackermann Heinrich Behmann Felix Bernstein Otto Blumenthal Anne Bosworth Werner Boy Ugo Broggi Richard Courant Haskell Curry Max Dehn Ludwig Föppl Rudolf Fueter Paul Funk Kurt Grelling Alfréd Haar Erich Hecke Earle Hedrick Ernst Hellinger Wallie Hurwitz Margarete Kahn Oliver Kellogg Hellmuth Kneser Robert König Emanuel Lasker Klara Löbenstein Charles Max Mason Alexander Myller Erhard Schmidt Kurt Schütte Andreas Speiser Hugo Steinhaus Gabriel Sudan Teiji Takagi Hermann Weyl Ernst Zermelo
Other notable students	Edward Kasner John von Neumann Carl Gustav Hempel

David Hilbert(/ˈhɪlbərt/;German:[ˈdaːvɪtˈhɪlbɐt];23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas includinginvariant theory,thecalculus of variations,commutative algebra,algebraic number theory,thefoundations of geometry,spectral theoryof operators and its application tointegral equations,mathematical physics,and thefoundations of mathematics(particularlyproof theory).

Hilbert adopted and defendedGeorg Cantor's set theory andtransfinite numbers.In 1900, he presented acollection of problemsthat set a course for mathematical research of the 20th century.

Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. Hilbert was one of the founders of proof theory andmathematical logic.

Life

Early life and education

Hilbert, the first of two children and only son of Otto, a county judge, and Maria Therese Hilbert (néeErdtmann), the daughter of a merchant, was born in theProvince of Prussia,Kingdom of Prussia,either inKönigsberg(according to Hilbert's own statement) or in Wehlau (known since 1946 asZnamensk) near Königsberg where his father worked at the time of his birth. His paternal grandfather was David Hilbert, a judge andGeheimrat.His mother Maria had an interest in philosophy, astronomy andprime numbers,while his father Otto taught himPrussian virtues.After his father became a city judge, the family moved to Königsberg. David's sister, Elise, was born when he was six. He began his schooling aged eight, two years later than the usual starting age.

In late 1872, Hilbert entered theFriedrichskollegGymnasium(Collegium fridericianum,the same school thatImmanuel Kanthad attended 140 years before); but, after an unhappy period, he transferred to (late 1879) and graduated from (early 1880) the more science-orientedWilhelm Gymnasium.Upon graduation, in autumn 1880, Hilbert enrolled at theUniversity of Königsberg,the "Albertina". In early 1882,Hermann Minkowski(two years younger than Hilbert and also a native of Königsberg but had gone to Berlin for three semesters),returned to Königsberg and entered the university. Hilbert developed a lifelong friendship with the shy, gifted Minkowski.

Career

Hilbert in 1886

Hilbert in 1907

In 1884,Adolf Hurwitzarrived from Göttingen as anExtraordinarius(i.e., an associate professor). An intense and fruitful scientific exchange among the three began, and Minkowski and Hilbert especially would exercise a reciprocal influence over each other at various times in their scientific careers. Hilbert obtained his doctorate in 1885, with a dissertation, written underFerdinand von Lindemann,titledÜber invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen( "On the invariant properties of specialbinary forms,in particular thespherical harmonic functions ").

Hilbert remained at the University of Königsberg as aPrivatdozent(senior lecturer) from 1886 to 1895. In 1895, as a result of intervention on his behalf byFelix Klein,he obtained the position of Professor of Mathematics at theUniversity of Göttingen.During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world.He remained there for the rest of his life.

The Mathematical Institute in Göttingen. Its new building, constructed with funds from theRockefeller Foundation,was opened by Hilbert and Courant in 1930.

Göttingen school

Among Hilbert's students wereHermann Weyl,chesschampionEmanuel Lasker,Ernst Zermelo,andCarl Gustav Hempel.John von Neumannwas his assistant. At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such asEmmy NoetherandAlonzo Church.

Among his 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including (with date of thesis):Otto Blumenthal(1898),Felix Bernstein(1901),Hermann Weyl(1908),Richard Courant(1910),Erich Hecke(1910),Hugo Steinhaus(1911), andWilhelm Ackermann(1925).Between 1902 and 1939 Hilbert was editor of theMathematische Annalen,the leading mathematical journal of the time. He was elected an International Member of the United StatesNational Academy of Sciencesin 1907.

Personal life

Käthe Hilbert withConstantin Carathéodory,before 1932

Hilbert and his wife Käthe Jerosch (1892)

Franz Hilbert

In 1892, Hilbert married Käthe Jerosch (1864–1945), who was the daughter of a Königsberg merchant, "an outspoken young lady with an independence of mind that matched [Hilbert's]."While at Königsberg they had their one child, Franz Hilbert (1893–1969). Franz suffered throughout his life from mental illness, and after he was admitted into a psychiatric clinic, Hilbert said, "From now on, I must consider myself as not having a son." His attitude toward Franz brought Käthe considerable sorrow.

Hilbert considered the mathematicianHermann Minkowskito be his "best and truest friend".

Hilbert was baptized and raised aCalvinistin thePrussian Evangelical Church.He later left the Church and became anagnostic.He also argued that mathematical truth was independent of the existence of God or othera prioriassumptions.WhenGalileo Galileiwas criticized for failing to stand up for his convictions on theHeliocentric theory,Hilbert objected: "But [Galileo] was not an idiot. Only an idiot could believe that scientific truth needs martyrdom; that may be necessary in religion, but scientific results prove themselves in due time."

Later years

LikeAlbert Einstein,Hilbert had closest contacts with theBerlin Groupwhose leading founders had studied under Hilbert in Göttingen (Kurt Grelling,Hans ReichenbachandWalter Dubislav).

Around 1925, Hilbert developedpernicious anemia,a then-untreatable vitamin deficiency whose primary symptom is exhaustion; his assistantEugene Wignerdescribed him as subject to "enormous fatigue" and how he "seemed quite old," and that even after eventually being diagnosed and treated, he "was hardly a scientist after 1925, and certainly not a Hilbert."

Hilbert was elected to theAmerican Philosophical Societyin 1932.

Hilbert lived to see theNazis purgemany of the prominent faculty members atUniversity of Göttingenin 1933.Those forced out includedHermann Weyl(who had taken Hilbert's chair when he retired in 1930),Emmy NoetherandEdmund Landau.One who had to leave Germany,Paul Bernays,had collaborated with Hilbert in mathematical logic, and co-authored with him the important bookGrundlagen der Mathematik(which eventually appeared in two volumes, in 1934 and 1939). This was a sequel to the Hilbert–AckermannbookPrinciples of Mathematical Logicfrom 1928. Hermann Weyl's successor wasHelmut Hasse.

About a year later, Hilbert attended a banquet and was seated next to the new Minister of Education,Bernhard Rust.Rust asked whether "the Mathematical Institute really suffered so much because of the departure of the Jews." Hilbert replied, "Suffered? It doesn't exist any longer, does it?"

Death

Hilbert's tomb:
Wir müssen wissen
Wir werden wissen

By the time Hilbert died in 1943, the Nazis had nearly completely restaffed the university, as many of the former faculty had either been Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among themArnold Sommerfeld,a theoretical physicist and also a native of Königsberg.News of his death only became known to the wider world several months after he died.

The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians on 8 September 1930. The words were given in response to the Latin maxim: "Ignoramus et ignorabimus"or" We do not know and we shall not know ":

Wir müssen wissen.
Wir werden wissen.

We must know.
We shall know.

The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians,Kurt Gödel—in a round table discussion during the Conference on Epistemology held jointly with the Society meetings—tentatively announced the first expression of his incompleteness theorem.Gödel's incompleteness theoremsshow that evenelementaryaxiomatic systems such asPeano arithmeticare either self-contradicting or contain logical propositions that are impossible to prove or disprove within that system.

Contributions to mathematics and physics

Solving Gordan's Problem

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famousfiniteness theorem.Twenty years earlier,Paul Gordanhad demonstrated thetheoremof the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. To solve what had become known in some circles asGordan's Problem,Hilbert realized that it was necessary to take a completely different path. As a result, he demonstratedHilbert's basis theorem,showing the existence of a finite set of generators, for the invariants ofquanticsin any number of variables, but in an abstract form. That is, while demonstrating the existence of such a set, it was not aconstructive proof—it did not display "an object" —but rather, it was anexistence proofand relied on use of thelaw of excluded middlein an infinite extension.

Hilbert sent his results to theMathematische Annalen.Gordan, the house expert on the theory of invariants for theMathematische Annalen,could not appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive. His comment was:

Das ist nicht Mathematik. Das ist Theologie.

This is not Mathematics. This is Theology.

Klein,on the other hand, recognized the importance of the work, and guaranteed that it would be published without any alterations. Encouraged by Klein, Hilbert extended his method in a second article, providing estimations on the maximum degree of the minimum set of generators, and he sent it once more to theAnnalen.After having read the manuscript, Klein wrote to him, saying:

Without doubt this is the most important work on general algebra that theAnnalenhas ever published.

Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself would say:

I have convinced myself that even theology has its merits.

For all his successes, the nature of his proof created more trouble than Hilbert could have imagined. AlthoughKroneckerhad conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea" —in other words (to quote Reid): "Through a proof of existence, Hilbert had been able to obtain a construction"; "the proof" (i.e. the symbols on the page)was"the object".Not all were convinced. WhileKroneckerwould die soon afterwards, hisconstructivistphilosophy would continue with the youngBrouwerand his developingintuitionist"school", much to Hilbert's torment in his later years.Indeed, Hilbert would lose his "gifted pupil"Weylto intuitionism— "Hilbert was disturbed by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker".Brouwer the intuitionist in particular opposed the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert responded:

Taking the Principle of the Excluded Middle from the mathematician... is the same as... prohibiting the boxer the use of his fists.

Axiomatization of geometry

Main article:Hilbert's axioms

The textGrundlagen der Geometrie(tr.:Foundations of Geometry) published by Hilbert in 1899 proposes a formal set, called Hilbert's axioms, substituting for the traditionalaxioms of Euclid.They avoid weaknesses identified in those ofEuclid,whose works at the time were still used textbook-fashion. It is difficult to specify the axioms used by Hilbert without referring to the publication history of theGrundlagensince Hilbert changed and modified them several times. The original monograph was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902.This translation incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. New editions followed the 7th, but the main text was essentially not revised.

Hilbert's approach signaled the shift to the modernaxiomatic method.In this, Hilbert was anticipated byMoritz Pasch's work from 1882. Axioms are not taken as self-evident truths. Geometry may treatthings,about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such aspoint,line,plane,and others, could be substituted, as Hilbert is reported to have said toSchoenfliesandKötter,by tables, chairs, glasses of beer and other such objects.It is their defined relationships that are discussed.

Hilbert first enumerates the undefined concepts: point, line, plane, lying on (a relation between points and lines, points and planes, and lines and planes), betweenness, congruence of pairs of points (line segments), andcongruenceofangles.The axioms unify both theplane geometryandsolid geometryof Euclid in a single system.

The 23 problems

Main article:Hilbert's problems

Hilbert put forth a highly influential list consisting of 23 unsolved problems at theInternational Congress of MathematiciansinParisin 1900. This is generally reckoned as the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician.

After re-working the foundations of classical geometry, Hilbert could have extrapolated to the rest of mathematics. His approach differed, however, from the later "foundationalist" Russell–Whitehead or "encyclopedist"Nicolas Bourbaki,and from his contemporaryGiuseppe Peano.The mathematical community as a whole could engage in problems of which he had identified as crucial aspects of important areas of mathematics.

The problem set was launched as a talk, "The Problems of Mathematics", presented during the course of the Second International Congress of Mathematicians held in Paris. The introduction of the speech that Hilbert gave said:

Who among us would not be happy to lift the veil behind which is hidden the future; to gaze at the coming developments of our science and at the secrets of its development in the centuries to come? What will be the ends toward which the spirit of future generations of mathematicians will tend? What methods, what new facts will the new century reveal in the vast and rich field of mathematical thought?

He presented fewer than half the problems at the Congress, which were published in the acts of the Congress. In a subsequent publication, he extended the panorama, and arrived at the formulation of the now-canonical 23 Problems of Hilbert. See alsoHilbert's twenty-fourth problem.The full text is important, since the exegesis of the questions still can be a matter of inevitable debate, whenever it is asked how many have been solved.

Some of these were solved within a short time. Others have been discussed throughout the 20th century, with a few now taken to be unsuitably open-ended to come to closure. Some continue to remain challenges.

The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in theBulletin of the American Mathematical Society.

1. Cantor's problem of the cardinal number of the continuum.

2. The compatibility of the arithmetical axioms.

3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.

4. Problem of the straight line as the shortest distance between two points.

5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.

6. Mathematical treatment of the axioms of physics.

7. Irrationality and transcendence of certain numbers.

8. Problems of prime numbers (The "Riemann Hypothesis" ).

9. Proof of the most general law of reciprocity in any number field.

10. Determination of the solvability of a Diophantine equation.

11. Quadratic forms with any algebraic numerical coefficients

12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality

13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments.

14. Proof of the finiteness of certain complete systems of functions.

15. Rigorous foundation of Schubert's enumerative calculus.

16. Problem of the topology of algebraic curves and surfaces.

17. Expression of definite forms by squares.

18. Building up of space from congruent polyhedra.

19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?

20. The general problem of boundary values (Boundary value problems in PDE's).

21. Proof of the existence of linear differential equations having a prescribed monodromy group.

22. Uniformization of analytic relations by means of automorphic functions.

23. Further development of the methods of the calculus of variations.

Formalism

In an account that had become standard by the mid-century, Hilbert's problem set was also a kind of manifesto that opened the way for the development of theformalistschool, one of three major schools of mathematics of the 20th century. According to the formalist, mathematics is manipulation of symbols according to agreed upon formal rules. It is therefore an autonomous activity of thought. There is, however, room to doubt whether Hilbert's own views were simplistically formalist in this sense.

Hilbert's program

Main article:Hilbert's program

In 1920, Hilbert proposed a research project inmetamathematicsthat became known as Hilbert's program. He wanted mathematics to be formulated on a solid and complete logical foundation. He believed that in principle this could be done by showing that:

1. all of mathematics follows from a correctly chosen finite system ofaxioms;and
2. that some such axiom system is provably consistent through some means such as theepsilon calculus.

He seems to have had both technical and philosophical reasons for formulating this proposal. It affirmed his dislike of what had become known as theignorabimus,still an active issue in his time in German thought, and traced back in that formulation toEmil du Bois-Reymond.

This program is still recognizable in the most popularphilosophy of mathematics,where it is usually calledformalism.For example, theBourbaki groupadopted a watered-down and selective version of it as adequate to the requirements of their twin projects of (a) writing encyclopedic foundational works, and (b) supporting theaxiomatic methodas a research tool. This approach has been successful and influential in relation with Hilbert's work in algebra and functional analysis, but has failed to engage in the same way with his interests in physics and logic.

Hilbert wrote in 1919:

We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.

Hilbert published his views on the foundations of mathematics in the 2-volume work,Grundlagen der Mathematik.

Gödel's work

Hilbert and the mathematicians who worked with him in his enterprise were committed to the project. His attempt to support axiomatized mathematics with definitive principles, which could banish theoretical uncertainties, ended in failure.

Gödeldemonstrated that any non-contradictory formal system, which was comprehensive enough to include at least arithmetic, cannot demonstrate its completeness by way of its own axioms. In 1931 hisincompleteness theoremshowed that Hilbert's grand plan was impossible as stated. The second point cannot in any reasonable way be combined with the first point, as long as the axiom system is genuinelyfinitary.

Nevertheless, the subsequent achievements of proof theory at the very leastclarifiedconsistency as it relates to theories of central concern to mathematicians. Hilbert's work had started logic on this course of clarification; the need to understand Gödel's work then led to the development ofrecursion theoryand thenmathematical logicas an autonomous discipline in the 1930s. The basis for latertheoretical computer science,in the work ofAlonzo ChurchandAlan Turing,also grew directly out of this "debate".

Functional analysis

Around 1909, Hilbert dedicated himself to the study of differential andintegral equations;his work had direct consequences for important parts of modern functional analysis. In order to carry out these studies, Hilbert introduced the concept of an infinite dimensionalEuclidean space,later calledHilbert space.His work in this part of analysis provided the basis for important contributions to the mathematics of physics in the next two decades, though from an unanticipated direction. Later on,Stefan Banachamplified the concept, definingBanach spaces.Hilbert spaces are an important class of objects in the area offunctional analysis,particularly of thespectral theoryof self-adjoint linear operators, that grew up around it during the 20th century.

Physics

Until 1912, Hilbert was almost exclusively apure mathematician.When planning a visit from Bonn, where he was immersed in studying physics, his fellow mathematician and friendHermann Minkowskijoked he had to spend 10 days in quarantine before being able to visit Hilbert. In fact, Minkowski seems responsible for most of Hilbert's physics investigations prior to 1912, including their joint seminar on the subject in 1905.

In 1912, three years after his friend's death, Hilbert turned his focus to the subject almost exclusively. He arranged to have a "physics tutor" for himself.He started studyingkinetic gas theoryand moved on to elementaryradiationtheory and the molecular theory of matter. Even after the war started in 1914, he continued seminars and classes where the works ofAlbert Einsteinand others were followed closely.

By 1907, Einstein had framed the fundamentals of the theory ofgravity,but then struggled for nearly 8 years to put the theory intoits final form.By early summer 1915, Hilbert's interest in physics had focused ongeneral relativity,and he invited Einstein to Göttingen to deliver a week of lectures on the subject.Einstein received an enthusiastic reception at Göttingen.Over the summer, Einstein learned that Hilbert was also working on the field equations and redoubled his own efforts. During November 1915, Einstein published several papers culminating inThe Field Equations of Gravitation(seeEinstein field equations).Nearly simultaneously, Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations (seeEinstein–Hilbert action). Hilbert fully credited Einstein as the originator of the theory and no public priority dispute concerning the field equations ever arose between the two men during their lives.See more atpriority.

Additionally, Hilbert's work anticipated and assisted several advances in themathematical formulation of quantum mechanics.His work was a key aspect ofHermann WeylandJohn von Neumann's work on the mathematical equivalence ofWerner Heisenberg'smatrix mechanicsandErwin Schrödinger'swave equation,and his namesake Hilbert space plays an important part in quantum theory. In 1926, von Neumann showed that, if quantum states were understood as vectors in Hilbert space, they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.

Throughout this immersion in physics, Hilbert worked on putting rigor into the mathematics of physics. While highly dependent on higher mathematics, physicists tended to be "sloppy" with it. To a pure mathematician like Hilbert, this was both ugly, and difficult to understand. As he began to understand physics and how physicists were using mathematics, he developed a coherent mathematical theory for what he found – most importantly in the area ofintegral equations.When his colleague Richard Courant wrote the now classicMethoden der mathematischen Physik(Methods of Mathematical Physics) including some of Hilbert's ideas, he added Hilbert's name as author even though Hilbert had not directly contributed to the writing. Hilbert said "Physics is too hard for physicists", implying that the necessary mathematics was generally beyond them; the Courant-Hilbert book made it easier for them.

Number theory

Hilbert unified the field ofalgebraic number theorywith his 1897 treatiseZahlbericht(literally "report on numbers" ). He also resolved a significant number-theoryproblem formulated by Waringin 1770. As withthe finiteness theorem,he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.He then had little more to publish on the subject; but the emergence ofHilbert modular formsin the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures onclass field theory.The concepts were highly influential, and his own contribution lives on in the names of theHilbert class fieldand of theHilbert symboloflocal class field theory.Results were mostly proved by 1930, after work byTeiji Takagi.

Hilbert did not work in the central areas ofanalytic number theory,but his name has become known for theHilbert–Pólya conjecture,for reasons that are anecdotal.Ernst Hellinger,a student of Hilbert, once toldAndré Weilthat Hilbert had announced in his seminar in the early 1900s that he expected the proof of the Riemann Hypothesis would be a consequence of Fredholm's work on integral equations with a symmetric kernel.

Works

His collected works (Gesammelte Abhandlungen) have been published several times. The original versions of his papers contained "many technical errors of varying degree";when the collection was first published, the errors were corrected and it was found that this could be done without major changes in the statements of the theorems, with one exception—a claimed proof of thecontinuum hypothesis.The errors were nonetheless so numerous and significant that it tookOlga Taussky-Toddthree years to make the corrections.

See also

Concepts

• List of things named after David Hilbert
• Foundations of geometry
• Hilbert C*-module
• Hilbert cube
• Hilbert curve
• Hilbert matrix
• Hilbert metric
• Hilbert–Mumford criterion
• Hilbert number
• Hilbert ring
• Hilbert–Poincaré series
• Hilbert series and Hilbert polynomial
• Hilbert space
• Hilbert spectrum
• Hilbert system
• Hilbert transform
• Hilbert's arithmetic of ends
• Hilbert's paradox of the Grand Hotel
• Hilbert–Schmidt operator
• Hilbert–Smith conjecture

Theorems

• Hilbert–Burch theorem
• Hilbert's irreducibility theorem
• Hilbert's Nullstellensatz
• Hilbert's theorem (differential geometry)
• Hilbert's Theorem 90
• Hilbert's syzygy theorem
• Hilbert–Speiser theorem

Other

• Brouwer–Hilbert controversy
• Direct method in the calculus of variations
• Entscheidungsproblem
• Geometry and the Imagination
• General relativity priority dispute

Sources

Primary literature in English translation

• Ewald, William B., ed. (1996).From Kant to Hilbert: A Source Book in the Foundations of Mathematics.Oxford, UK: Oxford University Press.
  • 1918. "Axiomatic thought," 1114–1115.
  • 1922. "The new grounding of mathematics: First report," 1115–1133.
  • 1923. "The logical foundations of mathematics," 1134–1147.
  • 1930. "Logic and the knowledge of nature," 1157–1165.
  • 1931. "The grounding of elementary number theory," 1148–1156.
  • 1904. "On the foundations of logic and arithmetic," 129–138.
  • 1925. "On the infinite," 367–392.
  • 1927. "The foundations of mathematics," with comment byWeyland Appendix byBernays,464–489.
• van Heijenoort, Jean(1967).From Frege to Gödel: A source book in mathematical logic, 1879–1931.Harvard University Press.
• Hilbert, David(1950) [1902].The Foundations of Geometry [Grundlagen der Geometrie](PDF).Translated by Townsend, E.J. (2nd ed.). La Salle, IL: Open Court Publishing.Archived(PDF)from the original on 28 December 2005.
• Hilbert, David(1990) [1971].Foundations of Geometry [Grundlagen der Geometrie].Translated by Unger, Leo (2nd English ed.). La Salle, IL: Open Court Publishing.ISBN978-0-87548-164-7.translated from the 10th German edition
• Hilbert, David;Cohn-Vossen, Stephan(1999).Geometry and Imagination.American Mathematical Society.ISBN978-0-8218-1998-2.An accessible set of lectures originally for the citizens of Göttingen.
• Hilbert, David (2004). Hallett, Michael; Majer, Ulrich (eds.).David Hilbert's Lectures on the Foundations of Mathematics and Physics, 1891–1933.Berlin & Heidelberg: Springer-Verlag.ISBN978-3-540-64373-9.

Secondary literature

• Bertrand, Gabriel(20 December 1943b),"Allocution",Comptes rendus hebdomadaires des séances de l'Académie des sciences(in French),217,Paris: 625–640,available atGallica.The "Address" of Gabriel Bertrand of 20 December 1943 at the French Academy: he gives biographical sketches of the lives of recently deceased members, includingPieter Zeeman,David Hilbert andGeorges Giraud.
• Bottazzini Umberto, 2003.Il flauto di Hilbert. Storia della matematica.UTET,ISBN88-7750-852-3
• Corry, L., Renn, J., and Stachel, J., 1997, "Belated Decision in the Hilbert-Einstein Priority Dispute,"Science 278:nn-nn.
• Corry, Leo (2004).David Hilbert and the Axiomatization of Physics (1898–1918): From Grundlagen der Geometrie to Grundlagen der Physik.Springer.ISBN90-481-6719-1.
• Dawson, John W. Jr 1997.Logical Dilemmas: The Life and Work of Kurt Gödel.Wellesley MA: A. K. Peters.ISBN1-56881-256-6.
• Fölsing, Albrecht(1998).Albert Einstein.Penguin.
• Grattan-Guinness, Ivor,2000.The Search for Mathematical Roots 1870–1940.Princeton Univ. Press.
• Gray, Jeremy,2000.The Hilbert Challenge.ISBN0-19-850651-1
• Mancosu, Paolo (1998).From Brouwer to Hilbert, The Debate on the Foundations of Mathematics in 1920s.Oxford Univ. Press.ISBN978-0-19-509631-6.
• Mehra, Jagdish,1974.Einstein, Hilbert, and the Theory of Gravitation.Reidel.
• Piergiorgio Odifreddi,2003.Divertimento Geometrico. Le origini geometriche della logica da Euclide a Hilbert.Bollati Boringhieri,ISBN88-339-5714-4.A clear exposition of the "errors" of Euclid and of the solutions presented in theGrundlagen der Geometrie,with reference tonon-Euclidean geometry.
• Reid, Constance.(1996).Hilbert.New York:Springer.ISBN0-387-94674-8.The definitive English-language biography of Hilbert.
• Rowe, D. E. (1989). "Klein, Hilbert, and the Gottingen Mathematical Tradition".Osiris.5:186–213.doi:10.1086/368687.S2CID121068952.
• Sauer, Tilman (1999). "The relativity of discovery: Hilbert's first note on the foundations of physics".Arch. Hist. Exact Sci.53:529–75.arXiv:physics/9811050.Bibcode:1998physics..11050S.
• Sieg, Wilfried (2013).Hilbert's Programs and Beyond.Oxford University Press.ISBN978-0-19-537222-9.
• Sieg, Wilfried, and Ravaglia, Mark, 2005, "Grundlagen der Mathematik" inGrattan-Guinness, I.,ed.,Landmark Writings in Western Mathematics.Elsevier:981-99. (in English)
• Thorne, Kip,1995.Black Holes and Time Warps: Einstein's Outrageous Legacy,W. W. Norton & Company; Reprint edition.ISBN0-393-31276-3.
• Georg von Wallwitz:Meine Herren, dies ist keine Badeanstalt. Wie ein Mathematiker das 20. Jahrhundert veränderte.Berenberg Verlag, Berlin 2017, ISBN 978-3-946334-24-8. The definitive German-language biography of Hilbert.

External links

• Hilbert Bernays Project
• Hilbert's 23 Problems Address
• ICMM 2014 dedicated to the memory of D.Hilbert
• Works by David HilbertatProject Gutenberg
• Works by or about David Hilbertat theInternet Archive
• Works by David HilbertatLibriVox(public domain audiobooks)
• Hilbert's radio speech recorded in Königsberg 1930 (in German)Archived14 February 2006 at theWayback Machine,with EnglishtranslationArchived12 November 2020 at theWayback Machine
• Wolfram MathWorld – Hilbert'Constant
• David Hilbertat theMathematics Genealogy Project
• O'Connor, John J.;Robertson, Edmund F.,"David Hilbert",MacTutor History of Mathematics Archive,University of St Andrews
• 'From Hilbert's Problems to the Future',lecture by Professor Robin Wilson,Gresham College,27 February 2008 (available in text, audio and video formats).
• Newspaper clippings about David Hilbertin the20th Century Press Archivesof theZBW