# David Hilbert

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**David Hilbert** (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. His own discoveries alone would have given him that honor, yet it was his leadership in the field of mathematics throughout his later life that distinguishes him. He held a professorship in mathematics at the University of Göttingen for most of his life.

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## Major contributions

Hilbert solved several important problems in the theory of invariants. Hilbert's basis theorem solved the principal problem in nineteenth century invariant theory by showing that any form of a given number of variables and of a given degree has a finite, yet complete system of independent rational integral invariants and covariants.

He also unified the field of algebraic number theory with his 1897 treatise *Zahlbericht* (literally "report on numbers").

Famous for his ability to make discoveries in various mathematical fields, Hilbert went on to provide the first correct and complete axiomatization of Euclidean geometry to replace Euclid's axiomatization of geometry, in his 1899 book *Grundlagen der Geometrie* ("Foundations of Geometry"). See Hilbert's axioms.

He also laid the foundations of functional analysis by studying integral equations and formulating a first version, in terms of quadratic forms in infinitely many variables, of what would be called Hilbert space. This work turned out in the 1920s to be foundational for quantum mechanics.

His interest in physics, in the decade 1900-1910, was not as important as later contacts with Albert Einstein and formulations of general relativity that helped its mathematical respectability (see also Einstein-Hilbert action).

Hilbert helped provide the basis for the theory of automata which was later built upon by computer scientist Alan Turing.

## Miscellaneous talks, essays, and contributions

He put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in 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.

Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics. These include his introduction of Hilbert space, and Hermann Weyl's proof of the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation.

His paradox of the Grand Hotel, a meditation on strange properties of the infinite, is often used in popular accounts of infinite cardinal numbers.

### Hilbert's program

In 1920 he proposed explicitly a research project (in *metamathematics*, 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 axioms; and
- that some such axiom system is provably consistent.

There seem to have been both technical and psychological reasons why he formulated this proposal. It affirmed his dislike of what had become known as the *ignorabimus*, 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 recognisable in the most popular philosophy of mathematics, amongst working mathematicians that is, usually called formalism. For example, the Bourbaki group adopted a milk-and-water 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.

Gödel's incompleteness theorem showed, however, in 1931 that Hilbert's grand plan was impossible, as stated. The point 2 cannot in any reasonable way be combined with the point 1, as long as the axiom system is genuinely finitary.

## Later years

Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen, in 1933. [1]. Among those forced out were Hermann Weyl, who had taken Hilbert's chair when he retired in 1930, Emmy Noether and Edmund Landau. One of those who had to leave Germany was Paul Bernays, Hilbert's collaborator in mathematical logic, and co-author with him of 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 Theoretical Logic* from 1928.

About a year later, he 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" (Reid, 205).

By the time Hilbert died in 1943, the Nazis had nearly completely restructured the university, many of the former faculty being either Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics (Reid, 213).

## See also

- Einstein-Hilbert action
- Hilbert's Nullstellensatz
- Hilbert's syzygy theorem
- Hilbert's Theorem 90
- Hilbert space
- Hilbert-Speiser theorem
- Hilbert's irreducibility theorem
- Principles of Theoretical Logic

## References

*Hilbert*, Constance Reid, Springer, April 1996, ISBN 0387946748. (biography)

## Further reading

- Jeremy Gray, 2000,
*The Hilbert Challenge*, ISBN 0198506511

## External links