Hilbert discovered and developed a range of fundamental ideas in many areas. He worked on invariant theory, the axiomization of geometry, and the notion of Hilbert space. This is one of the foundations of functional analysis. Hilbert and his students supplied much of the mathematics needed for quantum mechanics and general relativity. He was one of the founders of proof theory and mathematical logic. He was also one of the first people to make the distinction between mathematics and metamathematics, and warmly defended Cantor's set theory and transfinite numbers.
The Göttingen school[change]
In 1895 Hilbert became Chairman of Mathematics at the University of Göttingen, at that time the best research center for mathematics in the world. He remained for the rest of his life. Among his students were: Hermann Weyl, the champion of chess 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.
Axioms and problems[change]
The text Grundlagen der Geometrie (Foundations of Geometry) was published by Hilbert in 1899. It proposed a formal set, Hilbert's axioms, instead of the traditional axioms of Euclid. They avoid weaknesses in those of Euclid, whose works at the time were still used textbmathematics is his 1900 presentation of a set of problems that set the course for much of the mathematical research of the 20th century.
He put forward a number of unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is reckoned the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician. Later he expanded his lit to 23 problems.
In 1920 he proposed explicitly a research project in metamathematics, which 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.
He seems to have had both technical and philosophical reasons for formulating this proposal.
- Hilbert's 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 23 Problems Address
- Hilbert's Program
- Works by David Hilbert at Project Gutenberg
- Hilberts radio speech recorded in Königsberg 1930 (in German), with English translation
- Today in Kalingrad Oblast, Russia.
- Entities that are not altered during such geometric changes as rotation
- Reid, Constance 1996. Hilbert. Springer-Verlag, New York. p74; see footnote 1. ISBN 0387946748.
- A reliable source of Hilbert's axiomatic system, his comments on them and on the foundational "crisis" that was on-going at the time (translated into English), appears as Hilbert's 1927 "The foundations of mathematics". This can be found on p. 464ff in Jean van Heijenoort (editor) 1976/1966, From Frege to Gödel: A Source Book in Mathematical Logic, 1979–1931, Harvard University Press, Cambridge MA, ISBN 0-674-32449-8(pbk.).
- Ewald, William B. (ed) 1996. From Kant to Hilbert: a source book in the Foundations of Mathematics. 2 vols, Oxford.
- Jean van Heijenoort, 1967. From Frege to Godel: a source book in Mathematical Logic, 1879–1931. Harvard Univ. Press.
- David Hilbert; Cohn-Vossen S. 1999. Geometry and Imagination. American Mathematical Society. ISBN 0-8218-1998-4. An accessible set of lectures originally for the citizens of Göttingen.
- [David Hilbert] Michael Hallett and Ulrich Majer. eds. 2004. David Hilbert's Lectures on the foundations of Mathematics and Physics, 1891–1933. Springer-Verlag Berlin Heidelberg. ISBN 3-540-64373-7.
- Rowe, David; Gray, Jeremy J 2000. The Hilbert challenge. Oxford University Press. ISBN 0-19-850651-1.