## Quick Reference

(1885–1955) German mathematician

Born at Elmshorn in Germany, Weyl studied at Göttingen, where he was one of David Hilbert's most oustanding students. He became a coworker of Hilbert, who influenced his particular interests and his general outlook on mathematics. Weyl taught at the Federal Institute of Technology in Zurich from 1913, and this too had a decisive influence in directing his mathematical interests through the presence there of Albert Einstein. In 1930 he returned to Göttingen to take up the chair vacated by Hilbert. With the Nazis' rise to power in 1933, Weyl, with many other members of the Göttingen scientific community, went into exile in America. Weyl found a post at the Institute for Advanced Study in Princeton along with other exiles, such as Einstein and Kurt Gödel.

Weyl's mathematical interests, like those of Hilbert, were exceptionally wide, ranging from mathematical physics to the foundations of mathematics. He worked on two areas of pure mathematics: group theory and the theory of Hilbert space and operators, which, although developed for purely mathematical purposes, later turned out to be precisely the mathematical framework needed for the revolutionary physical ideas of quantum mechanics. Weyl also wrote a number of books on the theory of groups and he was particularly interested in symmetry and its relation to group theory. One of his most important results in group theory was a key theorem about the application of representations to Lie algebras. Weyl's work on Hilbert space had grown out of his interest in Hilbert's work on integral equations and operators. The theory of Hilbert space (infinite-dimensional space) was recognized by Erwin Schrödinger and Walter Heisenberg in the mid-1920s as the necessary unifying systematization of their theories of quantum mechanics.

Weyl's contact with Einstein at Zurich was responsible for an interest in the mathematics of relativity, and especially Riemannian geometry, which plays a central role. Weyl initiated the whole project of trying to generalize Riemannian geometry. He himself worked chiefly on the geometry of affinely connected spaces, but this was only one of many generalizations that resulted from his work. Weyl also did similar work on generalizing and refining the basic concepts of differential geometry. All this work was to be of importance for relativity. Weyl's views on relativity were expounded in his book Raum-Zeit-Materie (1919; Space-Time-Matter).

Weyl, like his teacher Hilbert, was always interested in the philosophical aspects of mathematics. However, in contrast to Hilbert, his general attitude was similar to that of L. E. J. Brouwer with whom he shared constructivist leanings developed from work in analysis. Weyl expounded his philosophical ideas in another book, Philosophy of Mathematics and Natural Sciences (1949). Unlike Brouwer, however, Weyl was less rigorous in avoiding nonconstructive mathematics, and doubtless his interest in physics contributed to this.

*Subjects:*
Science and Mathematics — Philosophy.