# David Hilbert

**HilbertHilbert, DavidD. HilbertHilbert, D.Hilbertian**

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential and universal mathematicians of the 19th and early 20th centuries.wikipedia

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### Hilbert's problems

**Hilbert problems23 problems23 unsolved problems**

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's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.

### Mathematics

**mathematicalmathmathematician**

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.

Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.

### Mathematical logic

**formal logicsymbolic logiclogic**

Hilbert is known as one of the founders of proof theory and mathematical logic. 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).

In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories.

### Georg Cantor

**CantorCantor, Georgavoids the paradoxes**

Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers.

David Hilbert defended it from its critics by declaring, "No one shall expel us from the paradise that Cantor has created."

### Königsberg

**Königsberg in PrussiaKönigsberg, PrussiaKöningsberg**

Hilbert, the first of two children and only son of Otto and Maria Therese (Erdtmann) Hilbert, was born in the Province of Prussia, Kingdom 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.

A university city, home of the Albertina University (founded in 1544), Königsberg developed into an important German intellectual and cultural centre, being the residence of Simon Dach, Immanuel Kant, Käthe Kollwitz, E. T. A. Hoffmann, David Hilbert, Agnes Miegel, Hannah Arendt, Michael Wieck and others.

### Emmy Noether

**NoetherAmalie "Emmy" NoetherE. Noether**

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.

In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research.

### Hugo Steinhaus

**SteinhausHugo Dyonizy Steinhaussection below**

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).

Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz University in Lwów (now Lviv, Ukraine), where he helped establish what later became known as the Lwów School of Mathematics.

### Proof theory

**proof-theoreticProof-theoreticallyderive**

Hilbert is known as one of the founders of proof theory and mathematical logic.

Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Giuseppe Peano, Bertrand Russell, and Richard Dedekind, the story of modern proof theory is often seen as being established by David Hilbert, who initiated what is called Hilbert's program in the foundations of mathematics.

### Eugene Wigner

**Eugene Paul WignerWignerEugene P. Wigner**

Around 1925, Hilbert developed pernicious anemia, a then-untreatable vitamin deficiency whose primary symptom is exhaustion; his assistant Eugene Wigner described 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."

A graduate of the Technical University of Berlin, Wigner worked as an assistant to Karl Weissenberg and Richard Becker at the Kaiser Wilhelm Institute in Berlin, and David Hilbert at the University of Göttingen.

### Erich Hecke

**HeckeE. T. HeckeHecke, Erich**

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).

He obtained his doctorate in Göttingen under the supervision of David Hilbert.

### Principles of Mathematical Logic

**Grundzüge der theoretischen Logik**

This was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928.

Principles of Mathematical Logic is the 1950 American translation of the 1938 second edition of David Hilbert's and Wilhelm Ackermann's classic text Grundzüge der theoretischen Logik, on elementary mathematical logic.

### Paul Bernays

**BernaysP. BernaysBernays, Paul**

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).

He was an assistant and close collaborator of David Hilbert.

### Calculus of variations

**variationalvariational calculusvariational methods**

Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and foundations of mathematics (particularly proof theory).

In the 20th century David Hilbert, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques Hadamard among others made significant contributions.

### Kurt Gödel

**GödelGödel, KurtGodel, Kurt**

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.

Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, and David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor.

### John von Neumann

**von NeumannJ. von NeumannNeumann, John von**

John von Neumann was his assistant.

He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.

### Hermann Minkowski

**MinkowskiMinkowski, HermannHerman Minkowski**

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.

David Hilbert's obituary of Minkowski illustrates the deep friendship between the two mathematicians (translated):

### Carl Gustav Hempel

**Carl HempelCarl G. HempelHempel**

Among Hilbert's students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel.

In Göttingen, he encountered David Hilbert and was impressed by his program attempting to base all mathematics on solid logical foundations derived from a limited number of axioms.

### Felix Klein

**KleinFelix Christian KleinC. Felix Klein**

In 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen.

During 1895, Klein hired David Hilbert away from the University of Königsberg; this appointment proved fateful, because Hilbert continued Göttingen's good reputation until his own retirement during 1932.

### Arnold Sommerfeld

**SommerfeldArnold Johannes Wilhelm SommerfeldA. Sommerfeld**

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.

and he also benefited from classes with mathematicians Adolf Hurwitz and David Hilbert and physicist Emil Wiechert.

### Otto Blumenthal

**Blumenthal, Otto**

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).

A student of David Hilbert, Blumenthal was an editor of Mathematische Annalen.

### Richard Courant

**R. CourantCourantCourant, Richard**

He became David Hilbert's assistant in Göttingen and obtained his doctorate there in 1910.

### Grundlagen der Mathematik

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).

Grundlagen der Mathematik (English: Foundations of Mathematics) is a two-volume work by David Hilbert and Paul Bernays.

### Invariant theory

**algebraic invariantinvariantstheory of invariants**

Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and foundations of mathematics (particularly proof theory).

David Hilbert's work on the question of the finite generation of the algebra of invariants (1890) resulted in the creation of a new mathematical discipline, abstract algebra.

### Adolf Hurwitz

**HurwitzAdolph HurwitzHurwitz, Adolf**

In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius (i.e., an associate professor).

Following two years at the University of Göttingen, in 1884 he was invited to become an Extraordinary Professor at the Albertus Universität in Königsberg; there he encountered the young David Hilbert and Hermann Minkowski, on whom he had a major influence.

### Hilbert's axioms

**Grundlagen der GeometrieHilbertaxiomatization of geometry**

Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and foundations of mathematics (particularly proof theory).

Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr.