**David Hilbert**(ˈdaːvɪt ˈhɪlbɐt; January 23, 1862 –

February 14, 1943) was a German

mathematician

. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory

and the axiomatization of geometry

. He also formulated the theory of Hilbert space

s, one of the foundations of functional analysis

.

Hilbert adopted and warmly defended Georg Cantor

's set theory and transfinite number

s.

Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können.

One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.

Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.

If one were to bring ten of the wisest men in the world together and ask them what was the most stupid thing in existence, they would not be able to discover anything so stupid as astrology.

If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis|Riemann hypothesis been proven?

A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.

**David Hilbert**(ˈdaːvɪt ˈhɪlbɐt; January 23, 1862 –

February 14, 1943) was a German

mathematician

. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory

and the axiomatization of geometry

. He also formulated the theory of Hilbert space

s, one of the foundations of functional analysis

.

Hilbert adopted and warmly defended Georg Cantor

's set theory and transfinite number

s. A famous example of his leadership in mathematics

is his 1900 presentation of a collection of problems

that set the course for much of the mathematical research of the 20th century.

Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory

and mathematical logic

, as well as for being among the first to distinguish between mathematics and metamathematics

.

## Life

Hilbert, the first of two children of Otto and Maria Therese (Erdtmann) Hilbert, was born in the Province of Prussia- either in Königsberg

(according to Hilbert's own statement) or in Wehlau (known since 1946 as Znamensk

) near Königsberg where his father worked at the time of his birth. In the fall of 1872, he entered the Friedrichskolleg Gymnasium

(

*Collegium fridericianum*, the same school that Immanuel Kant

had attended 140 years before), but after an unhappy duration he transferred (fall 1879) to and graduated from (spring 1880) the more science-oriented Wilhelm Gymnasium. Upon graduation he enrolled (autumn 1880) at the University of Königsberg

, the "Albertina". In the spring of 1882, Hermann Minkowski

(two years younger than Hilbert and also a native of Königsberg but so talented he had graduated early from his gymnasium and gone to Berlin for three semesters), returned to Königsberg and entered the university. "Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the shy, gifted Minkowski." In 1884, Adolf Hurwitz

arrived from Göttingen as an Extraordinarius

, i.e., an associate professor. An intense and fruitful scientific exchange between the three began and especially Minkowski and Hilbert 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 under Ferdinand von Lindemann

, titled

*Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen*("On the invariant properties of special binary forms, in particular the spherical harmonic functions").

Hilbert remained at the University of Königsberg as a professor from 1886 to 1895. In 1892, Hilbert married Käthe Jerosch (1864–1945), "the daughter of a Konigsberg merchant, an outspoken young lady with an independence of mind that matched his own". While at Königsberg they had their one child Franz Hilbert (1893–1969). In 1895, as a result of intervention on his behalf by Felix Klein

he obtained the position of Chairman of Mathematics at the University of Göttingen, at that time the best research center for mathematics in the world and where he remained for the rest of his life.

His son Franz would suffer his entire life from an (undiagnosed) mental illness, his inferior intellect a terrible disappointment to his father and this tragedy a matter of distress to the mathematicians and students at Göttingen. Sadly, Minkowski — Hilbert's "best and truest friend" — would die prematurely of a ruptured appendix in 1909.

### The Göttingen school

Among the students of Hilbert were: Hermann Weyl, chess champion Emanuel Lasker

, Ernst Zermelo

, and Carl Gustav Hempel

. John von Neumann

was 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 as Emmy Noether

and Alonzo 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), and Wilhelm Ackermann

(1925). Between 1902 and 1939 Hilbert was editor of the

*Mathematische AnnalenMathematische AnnalenMathematische Annalen is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann...*

, the leading mathematical journal of the time.

### Later years

Hilbert lived to see the Nazispurge many of the prominent faculty members at University of Göttingen in 1933. Those forced out included Hermann Weyl

(who had taken Hilbert's chair when he retired in 1930), Emmy Noether

and Edmund Landau

. One who had to leave Germany, Paul Bernays

, had collaborated with Hilbert in mathematical logic

, and co-authored with him the important book Grundlagen der Mathematik

(which eventually appeared in two volumes, in 1934 and 1939). This was a sequel to the Hilbert-Ackermann

book

*Principles of Mathematical Logic*from 1928.

About a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust

. Rust asked, "How is mathematics in Göttingen now that it has been freed of the Jewish influence?" Hilbert replied, "Mathematics in Göttingen? There is really none any more."

By the time Hilbert died in 1943, the Nazis had nearly completely restaffed the university, inasmuch 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 them Arnold Sommerfeld

, a theoretical physicist and also a native of Königsberg. News of his death only became known to the wider world six months after he had died.

The epitaph on his tombstone in Göttingen is the famous lines he had spoken at the conclusion of his retirement address to the Society of German Scientists and Physicians in the fall of 1930:

*Wir müssen wissen.**Wir werden wissen.*

In English:

- We must know.
- We will 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 roundtable discussion during the Conference on Epistemology held jointly with the Society meetings—tentatively announced the first expression of his incompleteness theorem.

## Hilbert Solves Gordan's Problem

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous*finiteness theorem*. Twenty years earlier, Paul Gordan had demonstrated the theorem

of 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. In order to solve what had become known in some circles as

*Gordan's Problem*, Hilbert realized that it was necessary to take a completely different path. As a result, he demonstrated

*Hilbert's basis theoremHilbert's basis theoremIn mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the...*

: showing the existence of a finite set of generators, for the invariants of quantics in any number of variables, but in an abstract form. That is, while demonstrating the existence of such a set, it was not a constructive proof

— it did not display "an object" — but rather, it was an existence proof and relied on use of the Law of Excluded Middle

in an infinite extension.

Hilbert sent his results to the

*Mathematische AnnalenMathematische AnnalenMathematische Annalen is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann...*

. Gordan, the house expert on the theory of invariants for the

*Mathematische Annalen*, was not able to 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 and by the comments of Gordan, Hilbert in a second article extended his method, providing estimations on the maximum degree of the minimum set of generators, and he sent it once more to the

*Annalen*. After having read the manuscript, Klein wrote to him, saying:

*Without doubt this is the most important work on general algebra that the*Annalen*has 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 stirred up more trouble than Hilbert could have imagined at the time. Although Kronecker had 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. While Kronecker would die soon after, his constructivist

philosophy would continue with the young Brouwer

and his developing intuitionist "school", much to Hilbert's torment in his later years. Indeed Hilbert would lose his "gifted pupil" Weyl to 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 would respond:

*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

The text*Grundlagen der Geometrie*(tr.:

*Foundations of Geometry*) published by Hilbert in 1899 proposes a formal set, the Hilbert's axioms

, substituting the traditional axioms of Euclid

. They avoid weaknesses identified in those of Euclid

, whose works at the time were still used textbook-fashion. Independently and contemporaneously, a 19-year-old American student named Robert Lee Moore

published an equivalent set of axioms. Some of the axioms coincide, while some of the axioms in Moore's system are theorems in Hilbert's and vice-versa.

Hilbert's approach signaled the shift to the modern axiomatic method. In this, Hilbert was anticipated by Peano's work from 1889. Axioms are not taken as self-evident truths. Geometry may treat

*things*, about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as point

, line

, plane, and others, could be substituted, as Hilbert says, 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 planes), betweenness, congruence of pairs of points, and congruence of angle

s. The axioms unify both the plane geometry

and solid geometry

of Euclid in a single system.

## The 23 Problems

Hilbert put forth a most influential list of 23 unsolved problems at the International Congress of Mathematiciansin Paris

in 1900. This is generally reckoned 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 contemporary Giuseppe Peano

. The mathematical community as a whole could enlist in problems, which he had identified as crucial aspects of the areas of mathematics he took to be key.

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. Here is the introduction of the speech that Hilbert gave:

- 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. 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 even continue to this day to remain a challenge for mathematicians.

## 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 the formalistschool, 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

In 1920 he proposed explicitly a research project (in*metamathematicsMetamathematicsMetamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...*

, as it was then termed) that 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:

- all of mathematics follows from a correctly chosen finite system of axiomAxiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s; and - that some such axiom system is provably consistent through some means such as the epsilon calculusEpsilon calculusHilbert's epsilon calculus is an extension of a formal language by the epsilon operator, where the epsilon operator substitutes for quantifiers in that language as a method leading to a proof of consistency for the extended formal language...

.

He seems to have had both technical and philosophical reasons for formulating this proposal. It affirmed his dislike of what had become known as the

*ignorabimusIgnorabimusThe Latin maxim ignoramus et ignorabimus, meaning "we do not know and will not know", stood for a position on the limits of scientific knowledge, in the thought of the nineteenth century...*

, still an active issue in his time in German thought, and traced back in that formulation to Emil du Bois-Reymond

.

This program is still recognizable in the most popular philosophy of mathematics

, where it is usually called

*formalism*. For example, the Bourbaki group adopted 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 the axiomatic method as 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, was however to end in failure.Gödel

demonstrated 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 his incompleteness theorem showed 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 genuinely finitary

.

Nevertheless, the subsequent achievements of proof theory

at the very least

*clarified*consistency 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 of recursion theory

and then mathematical logic

as an autonomous discipline in the 1930s. The basis for later theoretical computer science

, in Alonzo Church

and Alan Turing

also grew directly out of this 'debate'.

## Functional analysis

Around 1909, Hilbert dedicated himself to the study of differential and integral 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 dimensional Euclidean space

, later called Hilbert 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 Banach

amplified the concept, defining Banach spaces. Hilbert spaces are an important class of objects in the area of functional analysis

, particularly of the spectral theory

of self-adjoint linear operators, that grew up around it during the 20th century.

## Physics

Until 1912, Hilbert was almost exclusively a "pure" mathematician. When planning a visit from Bonn, where he was immersed in studying physics, his fellow mathematician and friend Hermann 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 in 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 studying kinetic gas theory

and moved on to elementary radiation

theory and the molecular theory of matter. Even after the war started in 1914, he continued seminars and classes where the works of Albert Einstein

and others were followed closely.

By 1907 Einstein had framed the fundamentals of the theory of gravity, but then struggled for nearly 8 years with a confounding problem of putting the theory into final form. By early summer 1915, Hilbert's interest in physics had focused on general 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 in "The Field Equations of Gravitation" (see Einstein field equations

). Nearly simultaneously David Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations (see Einstein–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 at priority).

Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics

. His work was a key aspect of Hermann Weyl

and John von Neumann

's work on the mathematical equivalence of Werner Heisenberg

's matrix mechanics

and Erwin Schrödinger

's wave equation

and his namesake Hilbert space

plays an important part in quantum theory. In 1926 von Neuman showed that if atomic states were understood as vectors in Hilbert space, then 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 math, 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 of integral equations. When his colleague Richard Courant

wrote the now classic 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 of algebraic number theorywith his 1897 treatise

*ZahlberichtZahlberichtIn mathematics, the Zahlbericht was a report on algebraic number theory by .-History: and and the English introduction to give detailed discussions of the history and influence of Hilbert's Zahlbericht....*

(literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring

in 1770. As with the 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 of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on class field theory

. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field

and of the Hilbert symbol

of local class field theory. Results on them were mostly proved by 1930, after work by Teiji Takagi

.

Hilbert did not work in the central areas of analytic number theory

, but his name has become known for the Hilbert–Pólya conjecture, for reasons that are anecdotal.

## Miscellaneous talks, essays, and contributions

- Hilbert's paradox of the Grand HotelHilbert's paradox of the Grand HotelHilbert's paradox of the Grand Hotel is a mathematical veridical paradox about infinite sets presented by German mathematician David Hilbert .-The paradox:...

, a meditation on strange properties of the infinite, is often used in popular accounts of infinite cardinal numberCardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

s. - He was a Foreign member of the Royal Society.
- He received the second Bolyai PrizeBolyai PrizeThe International Bolyai János Prize of Mathematics is an international prize for mathematicians founded by the Hungarian Academy of Sciences. The prize is awarded in every five years to mathematicians having published their monograph describing their own highly important new results in the past 10...

in 1910.

## Quotes

*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.*

## See also

- Brouwer–Hilbert controversyBrouwer–Hilbert controversyIn a foundational controversy in twentieth-century mathematics, L. E. J. Brouwer, a supporter of intuitionism, opposed David Hilbert, the founder of formalism.- Background :...
- Einstein–Hilbert action
- Einstein–Hilbert equationsEinstein field equationsThe Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
- Hilbert's axiomsHilbert's axiomsHilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie , as the foundation for a modern treatment of Euclidean geometry...
- Hilbert–Burch theoremHilbert–Burch theoremIn mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a projective dimension 2 quotient of a local or graded ring. proved a version of this theorem for polynomial rings, and proved a more general version. Several other authors later rediscovered and...
- Hilbert class fieldHilbert class fieldIn algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime...
- Hilbert C*-module
- Hilbert cubeHilbert cubeIn mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology...
- Hilbert curveHilbert curveA Hilbert curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling curves discovered by Giuseppe Peano in 1890....
- Hilbert function
- Hilbert inequality
- Hilbert matrix
- Hilbert metricHilbert metricIn mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the n-dimensional Euclidean space Rn...
- Hilbert modular form
- Hilbert number
- Hilbert polynomialHilbert polynomialIn commutative algebra, the Hilbert polynomial of a graded commutative algebra or graded module is a polynomial in one variable that measures the rate of growth of the dimensions of its homogeneous components...
- Hilbert's problemsHilbert's problemsHilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...
- Hilbert's programHilbert's programIn mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies...
- Hilbert ring
- Hilbert–Poincaré seriesHilbert–Poincaré seriesIn mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series , named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures...
- Hilbert spaceHilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
- Hilbert spectrumHilbert spectrumThe Hilbert spectrum , named after David Hilbert, is a statistical tool that can help in distinguishing among a mixture of moving signals. The spectrum itself is decomposed into its component sources using independent component analysis...
- Hilbert symbolHilbert symbolIn mathematics, given a local field K, such as the fields of reals or p-adic numbers, whose multiplicative group of non-zero elements is K×, the Hilbert symbol is an algebraic construction, extracted from reciprocity laws, and important in the formulation of local class field theory...
- Hilbert transformHilbert transformIn mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
- Hilbert's Arithmetic of EndsHilbert's arithmetic of endsIn mathematics, specifically in the area of hyperbolic geometry, Hilbert's arithmetic of ends is an algebraic construction introduced by German mathematician David Hilbert....
- Hilbert's basis theoremHilbert's basis theoremIn mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the...
- Hilbert's constants
- Hilbert's irreducibility theoremHilbert's irreducibility theoremIn number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers...
- Hilbert's NullstellensatzHilbert's NullstellensatzHilbert's Nullstellensatz is a theorem which establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields...
- Hilbert's paradox of the Grand HotelHilbert's paradox of the Grand HotelHilbert's paradox of the Grand Hotel is a mathematical veridical paradox about infinite sets presented by German mathematician David Hilbert .-The paradox:...
- Hilbert's theorem (differential geometry)
- Hilbert's Theorem 90Hilbert's Theorem 90In abstract algebra, Hilbert's Theorem 90 refers to an important result on cyclic extensions of fields that leads to Kummer theory...
- Hilbert's syzygy theoremHilbert's syzygy theoremIn mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert in connection with the syzygy problem of invariant theory. Roughly speaking, starting with relations between polynomial invariants, then relations between the relations, and so on, it...
- Hilbert-style deduction systemHilbert-style deduction systemIn logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege and David Hilbert...
- Hilbert–Pólya conjecture
- Hilbert–Schmidt operator
- Hilbert–Smith conjecture
- Hilbert–Speiser theorem
- Principles of Mathematical Logic
- Relativity priority disputeRelativity priority disputeAlbert Einstein presented the theories of Special Relativity and General Relativity in groundbreaking publications that either contained no formal references to previous literature, or referred only to a small number of his predecessors for fundamental results on which he based his theories, most...

### Primary literature in English translation

- Ewald, William B., ed., 1996.
*From Kant to Hilbert: A Source Book in the Foundations of Mathematics*, 2 vols. Oxford Uni. Press.- 1918. "Axiomatic thought," 1115–14.
- 1922. "The new grounding of mathematics: First report," 1115–33.
- 1923. "The logical foundations of mathematics," 1134–47.
- 1930. "Logic and the knowledge of nature," 1157–65.
- 1931. "The grounding of elementary number theory," 1148–56.
- 1904. "On the foundations of logic and arithmetic," 129–38.
- 1925. "On the infinite," 367–92.
- 1927. "The foundations of mathematics," with comment by Weyl and Appendix by BernaysBernaysBernays is a surname and may refer to:* Isaac Bernays , a German rabbi, and father of:** Jakob Bernays , a German classical linguist** Michael Bernays , a German literature historian...

, 464–89.

- Jean van HeijenoortJean Van HeijenoortJean Louis Maxime van Heijenoort was a pioneer historian of mathematical logic. He was also a personal secretary to Leon Trotsky from 1932 to 1939, and from then until 1947, an American Trotskyist activist.-Life:Van Heijenoort was born in Creil, France...

, 1967.*From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931*. Harvard Univ. Press. - an accessible set of lectures originally for the citizens of Göttingen.

### Secondary literature

- Bottazzini Umberto, 2003.
*Il flauto di Hilbert. Storia della matematica*. UTET, ISBN 88-7750-852-3 - Corry, L., Renn, J., and Stachel, J., 1997, "Belated Decision in the Hilbert-Einstein Priority Dispute,"
*Science 278*: nn-nn. - Dawson, John W. Jr 1997.
*Logical Dilemmas: The Life and Work of Kurt Gödel*. Wellesley MA: A. K. Peters. ISBN 1-56881-256-6. - Folsing, Albrecht, 1998.
*Albert Einstein*. Penguin. - Grattan-Guinness, IvorIvor Grattan-GuinnessIvor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...

, 2000.*The Search for Mathematical Roots 1870-1940*. Princeton Univ. Press. - Gray, Jeremy, 2000.
*The Hilbert Challenge*. ISBN 0-19-850651-1 - Mehra, Jagdish, 1974.
*Einstein, Hilbert, and the Theory of Gravitation*. Reidel. - Piergiorgio OdifreddiPiergiorgio OdifreddiPiergiorgio Odifreddi , is an Italian mathematician, logician and aficionado of the history of science, who is also extremely active as a popular science writer and essayist, especially in a perspective of philosophical atheism as a member of the Italian Union of Rationalist Atheists and...

, 2003.*Divertimento Geometrico - Da Euclide ad Hilbert*. Bollati Boringhieri, ISBN 88-339-5714-4. A clear exposition of the "errors" of Euclid and of the solutions presented in the*Grundlagen der Geometrie*, with reference to non-Euclidean geometryNon-Euclidean geometryNon-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

. - Reid, Constance, 1996.
*Hilbert*, Springer, ISBN 0-387-94674-8.*The*biography in English. - Sauer, Tilman, 1999, "The relativity of discovery: Hilbert's first note on the foundations of physics,"
*Arch. Hist. Exact Sci.*53: 529-75. - Sieg, Wilfried, and Ravaglia, Mark, 2005, "Grundlagen der Mathematik" in Grattan-Guinness, I.Ivor Grattan-GuinnessIvor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...

, ed.,*Landmark Writings in Western Mathematics*. ElsevierElsevierElsevier is a publishing company which publishes medical and scientific literature. It is a part of the Reed Elsevier group. Based in Amsterdam, the company has operations in the United Kingdom, USA and elsewhere....

: 981-99. (in English) - Thorne, KipKip ThorneKip Stephen Thorne is an American theoretical physicist, known for his prolific contributions in gravitation physics and astrophysics and for having trained a generation of scientists...

, 1995.*Black Holes and Time Warps: Einstein's Outrageous Legacy*, W. W. Norton & Company; Reprint edition. ISBN 0-393-31276-3.Black Holes and Time WarpsBlack Holes and Time Warps: Einstein's Outrageous Legacy is a popular science book by Kip Thorne. It provides an illustrated overview of the history and development of black hole theory up until the early 1990s....

## External links

- Hilbert Bernays Project
- Hilbert's 23 Problems Address
- Hilbert's Program
- Hilbert's radio speech recorded in Königsberg 1930 (in German), with English translation
- 'From Hilbert's Problems to the Future', lecture by Professor Robin Wilson, Gresham CollegeGresham CollegeGresham College is an institution of higher learning located at Barnard's Inn Hall off Holborn in central London, England. It was founded in 1597 under the will of Sir Thomas Gresham and today it hosts over 140 free public lectures every year within the City of London.-History:Sir Thomas Gresham,...

, 27 February 2008 (available in text, audio and video formats).