# Hermann Weyl

**Hermann Weyl** (November 9 1885 - December 8 1955) was a German mathematician. Although much of his working life was spent in Zürich and then Princeton, he is closely identified with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as pure disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and a key member of the Institute for Advanced Study in its early years, in terms of creating an integrated and international view.

Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the 'universalism' of Henri Poincaré or Hilbert, Weyl came as close as anyone. Michael Atiyah, in particular, has commented that whenever he looked into an area, he found that Weyl had preceded him.

The similarity of the names sometimes led to his being confused with André Weil. A communal joke for mathematicians was that, each being of great stature, this was a rare example where such mistakes would not cause offence on either side.

## Contents

- 1 Early life and interests
- 2 Geometric foundations of manifolds and physics
- 3 Foundations of mathematics
- 4 Mathematics of relativity
- 5 Topological groups, Lie groups and representation theory
- 6 Harmonic analysis and analytic number theory
- 7 Later career
- 8 Personality
- 9 Quotes
- 10 See also
- 11 Published works
- 12 External links and references

## Early life and interests

Weyl was born in Elmshorn (a town near Hamburg), Germany.

From 1904 to 1908 he studied in Göttingen and Munich, mainly mathematics and physics. His doctorate was awarded at Göttingen under the direction of Hilbert and Minkowski. In 1910, he obtained a teaching post of private lecturer at Göttingen. He took a professorship at the Technische Hochschule in Zürich, Switzerland in 1913, where he remained until 1930.

## Geometric foundations of manifolds and physics

*See Weyl transformation, Weyl tensor*

In 1913, Weyl published *Die Idee der Riemannschen Fläche* (*The Concept of a Riemann Surface*), which gave a unified treatment of Riemann surfaces.

In 1918, he introduced the notion of gauge, and gave the first example of what is now known as a gauge theory. Weyl's gauge theory was an unsuccessful attempt to model electromagnetic field and the gravitational field as geometrical properties of spacetime. The Weyl tensor in Riemannian geometry is of major importance in understanding the nature of conformal geometry.

## Foundations of mathematics

He became very interested in the foundational questions raised by the intuitionists. George Pólya and Weyl, during a mathematicians' gathering in Zürich (February 9, 1918), made a bet concerning the future direction of mathematics. Weyl predicted that in the subsequent 20 years, mathematicians would come to realize the total vagueness of notions such as real numbers, sets, and countability, and moreover, that asking about the truth or falsity of the least upper bound property of the real numbers was as meaningful as asking about truth of the basic assertions of Georg Hegel on the philosophy of nature.

The existence of this bet is documented in a letter discovered by Yuri Gurevich in 1995, and it is said that when the friendly bet ended, the individuals gathered cited Pólya as the victor (with Kurt Gödel not in concurrence). After about 1928 Weyl had apparently decided that mathematical intuitionism was not to be reconciled with his enthusiasm for the thought of Husserl.

## Mathematics of relativity

Weyl tracked the development of this field in physics in his *Raum, Zeit, Materie* (*Space, Time, Matter*) from 1918, reaching a 4th edition in 1922. His approach was based on the phenomenological philosophy of Edmund Husserl, specifically his 1913 *Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Erstes Buch: Allgemeine Einführung in die reine Phänomenologie *. Apparently this was Weyl's way of dealing with Einstein's controversial dependence on the phenomenological physics of Ernst Mach. Husserl had reacted strongly to Frege's criticism of his first work on the philosophy of arithmetic and was investigating the sense of mathematical and other structures, which Frege had distinguished from empirical reference. Hence there is good reason for viewing gauge theory as it developed from Weyl's ideas as a formalism of physical measurement and not a theory of anything physical, i.e. as scientific formalism.

## Topological groups, Lie groups and representation theory

*See main articles Peter-Weyl theorem, Weyl group, Weyl spinor,Weyl algebra*

From 1923 to 1938, Weyl developed the theory of compact groups, in terms of matrix representations. In the compact Lie group case he proved a fundamental character formula.

These results are foundational in understanding the symmetry structure of quantum mechanics, which he put on a group-theoretic basis. This included spinors. Together with the mathematical formulation of quantum mechanics, in large measure due to John von Neumann, this gave the treatment familiar since about 1930. Non-compact groups and their representations, particularly the Heisenberg group, were also deeply involved. From this time, and certainly much helped by Weyl's expositions, Lie groups and Lie algebras became a mainstream part both of pure mathematics and theoretical physics.

His book *The Classical Groups*, a seminal if difficult text, reconsidered invariant theory. It covered symmetric groups, full linear groups, orthogonal groups, and symplectic groups and results on their invariants and representations.

## Harmonic analysis and analytic number theory

Weyl also showed how to use exponential sums in diophantine approximation, with his criterion for uniform distribution mode 1, which was fundamental step in analytic number theory. This work applied to the Riemann zeta function, as well as additive number theory. It was developed by many others.

## Later career

In 1928 and 1929, he was a visiting professor at Princeton University.

Weyl left the professorship at the Technische Hochschule in Zürich, Switzerland, in the year of 1930 and he became Hilbert's successor at Göttingen where he held the chair of mathematics. The rise of the National Socialism in Germany in 1933, resulted in Weyl accepting an offer of going to the Institute for Advanced Study; Weyl's wife Hella (nee Joseph) was Jewish and subject to the Nazi racial legislation.

There at Princeton he worked with Einstein. Weyl researched a unification of gravitation and electromagnetism. Weyl tried to incorporate electromagnetism in the geometrical formalism of general relativity.

Weyl's research was the framework for later explanations of the violation of nonconservation of parity, a characteristic of weak interactions between leptons, in particle physics.

Weyl worked at the IAS until retirement in 1952. He died in Zürich, Switzerland.

## Personality

Weyl's own comment, although half a joke, sums up his personality.

*My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.*

## Quotes

- "
*The question for the ultimate foundations and the ultimate meaning of mathematics remains open; we do not know in which direction it will find its final solution nor even whether a final objective answer can be expected at all. "Mathematizing" may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization.*" -- Hermann Weyl (Gesammelte Abhandlungen)

- "
*The problems of mathematics are not problems in a vacuum ...*" -- Hermann Weyl

- "
*[Impredicative definition's] vicious circle, which has crept into analysis through the foggy nature of the usual set and function concepts, is not a minor, easily avoided form of error in analysis*". -- Hermann Weyl

- "
*In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.*"

## See also

**Main**:
Weyl algebra,
Weyl group,
Weyl's postulate,
Weyl tensor,
Weyl spinor,
Peter-Weyl theorem

## Published works

- Weyl, Hermann, "
*The Continuum : A Critical Examination of the Foundation of Analysis*". 1918. ISBN 0486679829 - Weyl, Hermann, "
*Mathematische Analyse des Raumproblems*". 1923. - Weyl, Hermann, "
*Was ist Materie?*". 1924. - Weyl, Hermann, "
*Gruppentheorie und Quantenmechanik*". 1928. - Weyl, Hermann, "
*Space Time Matter*". June 1952. ISBN 0486602672- original title : "
*Raum, Zeit, Materie*"

- original title : "
- Weyl, Hermann, "
*On generalized Riemann matrices*". Ann. of Math. 35, Vol. III, pp.~400--415, 1934. - Weyl, Hermann, "
*Elementary Theory of Invariants*". 1935 - Weyl, Hermann, "
*Symmetry*". Princeton University Press, 1952. ISBN 0691023743 - Weyl, Hermann, "
*Philosophy of Mathematics and Natural Science*". 1949. - Weyl, Hermann, "
*The Concept of a Riemann Surface*" Addison-Wesley, 1955. - Weyl, Hermann (and Herausgegeben von K. Chandrasekharan
*ed*), "*Gesammelte Abhandlungen*". Vol IV. Springer 1968.

- Weyl, Hermann, "
*Classical Groups: Their Invariants And Representations*". ISBN 0691057567

## External links and references

- National Academy of Sciences biography
- Biography by Atiyah
- The MacTutor History of Mathematics archive, "
*Hermann Klaus Hugo Weyl*". School of Mathematics and Statistics, University of St Andrews, Scotland - Weisstein, Eric W., "
*Weyl, Hermann (1885-1955)*". Eric Weisstein's World of Science. - Bell, John L., "
*Hermann Weyl on intuition and the continuum*" (PDF) - Gurevich, Yuri, "
*Platonism, Constructivism and Computer Proofs vs Proofs by Hand*" (PDF), Bulletin of the European Association of Theoretical Computer Science, 1995. - Kilmister, C. W. Zeno, "
*Aristotle, Weyl and Shuard: two-and-a-half millennia of worries over number.*" 1980.

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