# Hilbert, David (1862–1943)

David Hilbert was a German mathematician and one of the colossi of his subject in
the twentieth century. His most important discovery was of what is now called **Hilbert space** (see below). He was also a master of
mathematical organization. During the early phase of his career, Hilbert
reorganized number theory, crystallizing
his conclusions in the classic book *Der Zahlbericht* (The Theory of
Algebraic Number Fields, 1897). He then moved into geometry and performed
a similar service by setting forth the first rigorous set of geometrical
axioms in his *Grundlagen der Geometrie* (Foundations of Geometry,
1899). He invented a simple space-filling
curve now known as the **Hilbert curve** and also proved Waring's conjecture.

At the Paris International Congress of 1900, Hilbert proposed 23 outstanding
problems in mathematics to whose solutions he believed twentieth-century
mathematicians should devote themselves. These problems have come to be
known as **Hilbert's problems**, and a number still remain
unsolved today. Hilbert's mathematical philosophy is partly revealed by
a couple remarks, one of which he made after learning that a student in
his class had dropped the subject in order to become a poet. "Good," he
said. "He did not have enough imagination to become a mathematician." Whether
he really believed the second is open to question: "Mathematics is a game
played according to certain simple rules with meaningless marks on paper."

Hilbert's dream was that it would be possible to prove or disprove all mathematical questions starting with only one set of well-defined rules and assumptions. In 1931 Kurt Gödel showed that this could not be achieved.

## Hilbert space

Hilbert space is a space of infinite dimensions in which distance is preserved by making the sum of squares of coordinates a convergent sequence; it is of crucial importance in the mathematical formulation of quantum mechanics.