## Quick Reference

(1863–1943)

German mathematician, a major contributor to most branches of modern mathematics.

Born in Königsberg (East Prussia; now Kaliningrad, Russia), the son of a judge, Hilbert was educated at the universities of Königsberg and Heidelberg. He taught briefly at Königsberg before moving to the University of Göttingen, where he was professor of mathematics from 1895 until his retirement in 1930.

Hilbert established his reputation as a mathematician by his work in the 1880s on the theory of invariants. This was followed by his Grundlagen der Geometrie (1899; ‘The Foundations of Geometry’) in which, some two millennia after its initial appearance, Hilbert offered the first rigorous treatment of Euclidean geometry. In the following year Hilbert became known to a much larger audience by his presentation at the International Congress of Mathematicians in Paris of twenty-three outstanding unsolved problems. In maths, he declared, ‘there is no Ignorabimus’, all problems are solvable. The first problem (on the continuum hypothesis) was solved only in 1963, while the second (on the consistency of arithmetic) and the eighth (on the Riemann hypothesis) remain, with several others, essentially unsolved. On the principle that ‘physics is too difficult to be left to physicists,’ Hilbert spent much of his time on his sixth problem, the axiomatization of physics. Although this may now look somewhat presumptuous, it yielded concepts and results that have profoundly influenced the development of modern physics. The notion of Hilbert space, for example, was an important constituent in the evolution of quantum mechanics.

Hilbert also tackled the problem of the foundations of mathematics. In opposition to the intuitionist philosophy of Luitzen Brouwer and the logic of Bertrand Russell he developed a formalist approach in which the axiomatization and consistency of all mathematics was sought. The programme was most fully developed, in collaboration with Paul Bernays (1888–1977), in their Grundlagen der Mathematik (2 vols, 1934–39). The work of Gödel, however, has thrown doubt on the viability of the formalist approach.

*Subjects:*
Mathematics.