# Weyl, (Claus Hugo) Hermann (1885–1955)

# WEYL, (CLAUS HUGO) HERMANN

*(1885–1955)*

(Claus Hugo) Hermann Weyl, the German-American mathematician, physicist, and philosopher of science, was born in Elmshorn, Germany, and died in Zürich. He studied at Munich and received his Ph.D. in 1908 from Göttingen, where he was Privatdozent from 1910 to 1913. He taught at the Eidgenossische Technische Hochschule in Zürich from 1913 to 1930, lecturing at Princeton in 1928-1929. He taught at Göttingen again from 1930 to 1933 and then returned to Princeton, remaining at the Institute for Advanced Study until 1953, when he became emeritus. He became a naturalized citizen in 1939. In 1925 he received the Lobachevski Prize for his research in geometrical theory. Weyl received many honorary degrees and was a member of numerous scientific societies and a civilian member of the Office of Scientific Research and Development in 1944.

Weyl's *Raum, Zeit, Materie* (Berlin, 1918; translated by H. L. Brose from the 4th German edition as *Space-Time-Matter,* London, 1922) is a classic in relativity theory. Weyl also made significant contributions to the formalization of quantum theory (*Gruppentheorie und Quantenmechanik,* Leipzig, 1928; translated by H. P. Robertson as *Theory of Groups and Quantum Mechanics,* London, 1931). Perhaps his most important contribution of philosophical interest in this book was his attempted solution to the problem of a unified field theory in relativity. Such a theory would ultimately express in one general invariant mathematical tensor equation or law the characteristics of gravitational, electric, and magnetic fields, and show the so-called elementary particles (such as electrons or protons) as derivative from that equation. That is, the discontinuous "particles" would be generated and controlled by the continuous unified field. In 1950, in a new preface to *Space-Time-Matter,* Weyl wrote that after his own first attempt at formulating such a theory, "Quite a number of unified field theories have sprung up in the meantime. They are all based on mathematical speculation and, as far as I can see, none has had a conspicuous success." He explained that "a unitary field theory … should encompass at least three fields: electromagnetic, gravitational, and electronic. Ultimately the wave fields of other elementary particles will have to be included too, unless quantum physics succeeds in interpreting them all as different quantum states of one particle." (In quantum theory all particles have associated wave fields.) No such theory has as yet been successfully formulated, despite even Albert Einstein's final heroic and desperate attempts along this line.

Weyl also showed the validity in general relativity of a variational principle of least action. He dealt in some detail with the problem of action at a distance by examining and defining more precisely the notion of gravitational waves propagated at a finite speed (the speed of light), as is held in general relativity, in contrast to the older Newtonian theory of an infinite or indefinitely high speed for all gravitational influences. Weyl also espoused a cosmological model in which all observers located on different galaxies anywhere would have equivalent overall views of the universe.

Weyl's *Das Kontinuum* (Leipzig, 1918) consists, first, of a logical and mathematical analysis of groups and functions and deals with such questions as the axiomatic method (in the manner of David Hilbert), the natural numbers (including Richard's antinomy), and the iteration and substitution principles of formal mathematical systems. Second, Weyl analyzed the concept of number in general, in conjunction with the notion of the continuum: the logical foundations of the infinitesimal calculus, with applications to spatial and temporal continua, magnitudes and measures, curves and surfaces. In all of this he explicitly used the ideas of Georg Cantor, Bertrand Russell, A. N. Whitehead, Jules Henri Poincaré, Augustin-Louis Cauchy, Richard Dedekind, Gottlob Frege, Ernst Zermelo, and Henri Bergson. Throughout, he attempted to distinguish the abstract, idealized, schematized ("objective") mathematical continua of space and time from the intuitive, phenomenal ("subjective") space and time personally and immediately experienced by each individual. Weyl acknowledged a debt to the ideas of Bergson concerning "duration" as given in phenomenal or intuitive time.

Weyl's definitive work in the philosophy of science, *Philosophie der Mathematik und Naturwissenschaft* (Munich, 1927; translated by O. Helmer, revised and augmented, as *Philosophy of Mathematics and Natural Science,* Princeton, NJ, 1949), dealt with pure and applied mathematics. In pure mathematics, he discussed mathematical logic and axiomatics, number theory and the continuum, the infinite, and geometry. In the natural sciences, he explained basic questions concerning space, time, and the transcendental world, with special concern for the epistemological problem of subject and object. The transcendental world is, of course, the Kantian idea with Weyl's added notion that this world might be knowable by the physicist. But the question of knowing was precisely the epistemological problem that troubled Weyl, as will be seen below.

In this work Weyl also discussed methodological problems in the theory of measurement and in the formation of scientific concepts and theories. Finally, he attempted to offer a general "physical picture of the world" in the course of analyzing the ideas of matter and causality.

The first German edition of *Philosophy of Mathematics and Natural Science* was written just before the broader philosophical implications of quantum theory had been recognized; hence Weyl added several appendices to the English edition in which he coped with the newer problems. In Appendix C he declared that "whatever the future may bring, the road will not lead back to the old classical scheme." Thus, Weyl had no real hope that a classical mechanical model would ever again be established as the basis of objective reality, and he explicitly emphasized that in quantum theory the relations between subject and object "are more closely tied together than classical physics had recognized." Weyl's notion of the vagueness of the distinction between subject and object in quantum theory has deeper metaphysical implications, of which fact he was clearly aware. How could we know the real world apart from our interactions with it and apart from the consequent indeterminacy in such "knowledge"? What, then, is the physical "object" apart from our subjective knowledge of it?

Weyl's final work was *Symmetry* (Princeton, NJ, 1952), published on the eve of his retirement from the institute. In it Weyl related the precise geometrical concept of symmetry to the vaguer artistic ideas of proportion, harmony, and beauty. In this account he was sensitive to the ideas of Plato and other great Greek classical aestheticians. His illustrated survey ranged from Sumerian art forms through the ancient Greeks and the medievals, and down to contemporary physicists, crystallographers, and biologists, briefly mentioning modern women's fashions.

** See also ** Bergson, Henri; Cantor, Georg; Confirmation Theory; Frege, Gottlob; Hilbert, David; Mathematics, Foundations of; Philosophy of Science, History of; Philosophy of Science, Problems of; Plato; Poincaré, Jules Henri; Relativity Theory; Russell, Bertrand Arthur William; Whitehead, Alfred North.

## Bibliography

Other works by Weyl of interest to philosophers of science are: *Die Idee der Riemannschen Fläche* (Leipzig, 1913); *The Classical Groups* (Princeton, NJ: Princeton University Press, 1939); *Algebraic Theory of Numbers* (Princeton, NJ: Princeton University Press, 1940); *Metamorphic Functions and Analytic Curves* (Princeton: Princeton University Press, 1943); and *The Structure and Representation of Continuous Groups* (Princeton: Princeton University Press, 1955).

For further works by Weyl and for works on him, see *Biographical Memoirs of the Fellows of the Royal Society* 3 (1957): 305–328.

*Carlton W. Berenda (1967)*

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# Weyl, (Claus Hugo) Hermann (1885–1955)

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**Weyl, (Claus Hugo) Hermann (1885–1955)**