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David Hilbert

David Hilbert was born on January 23, 1862, in Wehlau (near modern-day
Kaliningrad) in what was then East Prussia.
He received his doctorate from the University of Königsberg in 1884.
He taught at Königsberg from 1886 to 1895, and then moved to
Göttingen, where he taught until his death in 1943.
Hilbert made deep and
fundamental contributions to algebra, number theory, and
geometry. In 1900 he was invited to give an address at the International
Congress of Mathematicians in Paris. His address, entitled
*Mathematical Problems*, listed 23 important problems he
felt deserved the attention of mathematicians of the coming century.

Hilbert's Tenth Problem asked if there was a "process" by which "it can
be deterined by a finite number of operations whether [a Diophantine]
equation can be solved in ... integers". (A Diophantine equation is
a multivariate polynomial equation.)
In his later career, Hilbert became more
interested in the foundations of mathematics and the possibility
of resolving, by completely mechanical means, any well-posed mathematical
problem. In 1928 he asked:

- Is mathematics
*complete*, in the sense that every statement
can either be proved or disproved?
- Is mathematics
*consistent*, in the sense that one can never
arrive at a contradiction such as 0 = 1 by a sequence of valid steps
of reasoning?
- Is mathematics
*decidable*, in the sense that there exists
a mechanical procedure that will always determine the truth or falsity
of a statement?

Hilbert believed that the answer to all these questions was "yes". But
in 1931 the Czech mathematician Kurt Gödel proved that any
sufficiently powerful mathematical system must be either inconsistent
or incomplete. Also, such a system cannot be proved consistent within
its own axiom system. But the third question remained open, with
"provability" substituted for "truth". This was finally resolved
negatively by Alan Turing in 1936.
Hilbert died in Königsberg on February 14, 1943.

## Sources

- Constance Reid,
*Hilbert*, Springer-Verlag, New York, 1970.
- David Hilbert, (transl. Mary Winston Newson), Mathematical
problems,
*Bull. Amer. Math. Soc.* **8** (1902), 437-479.
- Andrew Hodges,
*Alan Turing: The Enigma*, Simon and Schuster, 1983.

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September 10 1997