Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski.

[4][5] Weyl contributed to an exceptionally[6] wide range of fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics.

Freeman Dyson wrote that Weyl alone bore comparison with the "last great universal mathematicians of the nineteenth century", Henri Poincaré and David Hilbert.

[7] Hermann Weyl was born in Elmshorn, a small town near Hamburg, in Germany, and attended the Gymnasium Christianeum in Altona.

Helene was a daughter of Dr. Bruno Joseph (13 December 1861 – 10 June 1934), a physician who held the position of Sanitätsrat in Ribnitz-Damgarten, Germany.

Speakers at her memorial service included her son Fritz Joachim Weyl and mathematicians Oswald Veblen and Richard Courant.

[24] For the academic year 1928–1929, he was a visiting professor at Princeton University,[25] where he wrote a paper, "On a problem in the theory of groups arising in the foundations of infinitesimal geometry," with Howard P.

[26] Weyl left Zürich in 1930 to become Hilbert's successor at Göttingen, leaving when the Nazis assumed power in 1933, particularly as his wife was Jewish.

He had been offered one of the first faculty positions at the new Institute for Advanced Study in Princeton, New Jersey, but had declined because he did not desire to leave his homeland.

In it Weyl utilized point set topology, in order to make Riemann surface theory more rigorous, a model followed in later work on manifolds.

Together with the mathematical formulation of quantum mechanics, in large measure due to John von Neumann, this gave the treatment familiar since about 1930.

Weyl also showed how to use exponential sums in diophantine approximation, with his criterion for uniform distribution mod 1, which was a fundamental step in analytic number theory.

In The Continuum Weyl developed the logic of predicative analysis using the lower levels of Bertrand Russell's ramified theory of types.

Weyl appealed[clarification needed] in this period to the radical constructivism of the German romantic, subjective idealist Fichte.

George Pólya and Weyl, during a mathematicians' gathering in Zürich (9 February 1918), made a bet concerning the future direction of mathematics.

In the last decades of his life Weyl emphasized mathematics as "symbolic construction" and moved to a position closer not only to Hilbert but to that of Ernst Cassirer.

It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness.

And the mathematician watches with pain the greater part of his towering edifice which he believed to be built of concrete blocks dissolve into mist before his eyes."

As John L Bell puts it: "It seems to me a great pity that Weyl did not live to see the emergence in the 1970s of smooth infinitesimal analysis, a mathematical framework within which his vision of a true continuum, not “synthesized” from discrete elements, is realized.

Although the underlying logic of smooth infinitesimal analysis is intuitionistic — the law of excluded middle not being generally affirmable — mathematics developed within avoids the “unbearable awkwardness” to which Weyl refers above."

and German idealism (Fichte's dialectic) in Weyl's philosophical interpretation of the physical concept of matter in the context of quantum theory and general relativity and with regard to interaction of a symbol with its surroundings in a mathematical theory structure also in Weyl's philosophy of mathematics (debate between formalism and intuitionism under the influence of Brouwer).

According to Sieroka, Fichte and Ernst Cassirer were also an important influence in Weyl's late philosophy (science as a “symbolic construction”).