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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.

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:

1. all of mathematics follows from a correctly-chosen finite system of axioms; and
2. that some such axiom system is probably 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.

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

• O'Connor, John J.; Robertson, Edmund F., "David Hilbert", MacTutor History of Mathematics Archive, University of St Andrews
• Hilbert's 23 Problems Address
• Hilbert's Program Template:Link FA

• 1862 births
• 1943 deaths
• German mathematicians
• Number theorists
• Contributors to general relativity