VI.80 Hermann Weyl

b. Elmshorn, Germany, 1885; d. Zürich, 1955 Analysis; geometry; topology; foundations; mathematical physics

Weyl studied mathematics at Göttingen under HILBERT [VI.63], KLEIN [VI.57], and MINKOWSKI [VI.64] between 1904 and 1908. His first teaching positions were at Göttingen (1910–13) and ETH Zürich (1913–30). In 1930 he accepted the call to Göttingen as Hilbert’s successor. After the rise to power of the Nazis, he emigrated to the United States and became a member of the newly founded Institute of Advanced Studies at Princeton (1933–51).

Weyl made contributions to real and complex analysis, geometry and topology, LIE GROUPS [III.48 §1], number theory, the foundations of mathematics, mathematical physics, and philosophy. He contributed at least one book to each of these fields, publishing thirteen in total. Together with his other technical and conceptual innovations, these books were all of lasting influence: many had a pronounced and immediate effect.

His early research dealt with integral operators and differential equations with singular boundary conditions. His fame came later, with his book The Concept of a Riemann Surface (1913). This grew out of a lecture course in the winter of 1910-11 and built upon Klein’s intuitive treatment of RIEMANN’s [VI.49] geometric function theory and Hilbert’s justification of the DIRICHLET PRINCIPLE [IV.12 §3.5]. Here Weyl gave a new presentation of the properties of RIEMANN SURFACES [III.79], which became highly influential for the geometric function theory of the twentieth century.

His second book, The Continuum (1918), marked the beginning of Weyl’s interest in the foundations of mathematics. He was critical of Hilbert’s “formalist” program for an axiomatic foundation of mathematics, and explored the possibility of a semi-formalized arithmetical approach to a strictly constructivist foundation of real analysis. Shortly thereafter he shifted toward BROUWER’s [VI.75] intuitionistic program and attacked Hilbert’s foundational views even more strongly in a famous article of 1921. In the late 1920s he developed a more balanced view of the foundational questions. After World War II he returned to a weak preference for his arithmetical constructive approach of 1918.

At the same time as he was working on foundational questions Weyl took up Einstein’s theory of general relativity and wrote his third book, Space-Time-Matter. It was first published in 1918, and appeared in five successive editions until 1923. This was one of the first monographs on relativity theory and was among the most influential. The book represented only the tip of the iceberg of his contributions to differential geometry and general relativity. Weyl undertook this research within a broad conceptual and philosophical framework. One of the outcomes of this approach was his Analysis of the Problem of Space (1923), in which he sketched ideas that would later be analyzed in terms of the geometry of FIBER BUNDLES [IV.6 §5] and the study of gauge fields. He had already introduced gauge fields (and the key idea of a point-dependent rescaling of the metric) in 1918 for a generalization of RIEMANNIAN GEOMETRY [I.3 §6.10] and a geometrically unified field theory of gravity and electromagnetism.

Weyl made his most influential contributions to pure mathematics around the middle of the 1920s with his work on the REPRESENTATION THEORY [IV.9] of semisimple Lie groups. Combining CARTAN’s [VI.69] insights into the representation theory of LIE ALGEBRAS [III.48 §2] with methods developed by Hurwitz and Schur, Weyl used his knowledge of the topology of manifolds and developed the core of the general theory of representations of Lie groups in a blend of geometric, algebraic, and analytic methods. He extended and refined this work and it formed the core of his later book The Classical Groups (1939)—a harvest of his work and lectures on this topic during his Princeton years.

Along with all this work, Weyl actively followed the rise of the new quantum mechanics. In 1927–28 he gave a lecture course at ETH on the topic, which gave rise to his next book on mathematical physics, Group Theory and Quantum Mechanics (1928). Weyl emphasized the conceptual role of group methods in the symbolic representation of quantum structures, in particular the intriguing interplay between representations of the special linear group and PERMUTATION GROUPS [III.68]. A second step in his gauge theory of the electromagnetic field was published separately, which gave rise to a modified gauge theory of electromagnetism. This was endorsed by leading theoretical physicists, including Pauli, Schrödinger, and Fock. It served as a starting point for the next generation of physicists who developed gauge field theories in the 1950s and 1960s.

Weyl’s research in mathematics and physics was shaped by his philosophical outlook and he included his philosophical reflections on scientific activity in many of his publications. Most influential was his contribution to a philosophical handbook, Philosophy of Mathematics and Natural Science, originally published in German in 1927 and translated into English in 1949. It became a classic in the philosophy of science.

Further Reading

Chandrasekharan, K., ed. 1986. Hermann Weyl: 1885–1985. Centenary Lectures delivered by C. N. Yang, R. Penrose, and A. Borel at the Eidgenössische Technische Hochschule Zürich. Berlin: Springer.

Deppert, W., K. Hübner, A. Oberschelp, and V. Weidemann, eds. 1988. Exact Sciences and Their Philosophical Foundations. Frankfurt: Peter Lang.

Hawkins, T. 2000. Emergence of the Theory of Lie Groups. An Essay in the History of Mathematics 1869–1926. Berlin: Springer.

Scholz, E., ed. 2001. Hermann Weyl’s Raum-Zeit-Materie and a General Introduction to His Scientific Work. Basel: Birkhäuser.

Weyl, H. 1968. Gesammelte Abhandlungen, edited by K. Chandrasekharan, four volumes. Berlin: Springer.

Erhard Scholz