Hermann Weyl was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated 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 purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years.

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 examined a mathematical topic, he found that Weyl had preceded him.

Complete Dictionary of Scientific Biography

WEYL, Hermann (b. Elmshorn, near Hamburg, Germany, 9 November 1885; d. Zürich, Switzerland, 8 December 1955), mathematics, mathematical physics.

Weyl attended the Gymnasium at Altona and, on the recommendation of the headmaster of his Gymnasium, who was a cousin of Hilbert, decided at the age of eighteen to enter the University of Göttingen. Except for one year at Munich he remained at Göttingen, as a student and later as Privatdozent, until 1913, Weyl declined an offer to be his successor at Göttingen but accepted a second offer in 1930, after Hilbert had retired. In 1933 he decided he could no longer remain in Nazi Germany and accepted a position at the Institute for Advanced Study at Princeton, where he worked until his retirement in 1951. In the last years of his life he divided his time between Zurich and Princeton.

Weyl undoubtedly was the most gifted of Hilbert’s students. Hilbert’s thought dominated the first part of his mathematical career; and although later he sharply diverged from his master, particularly on questions related to foundations of mathematics, Weyl always shared his convictions that the value of abstract theories lies in their success in solving classical problems and that the proper way to approach a question is through a deep analysis of the concepts it involves rather than by blind computations.

Weyl arrived at Göttingen during the period when Hilbert was creating the spectral theory of self-adjoint operators, and spectral theory and harmonic analysis were central in his mathematical research throughout his life. Very soon, however, he considerably broadened the range of his interests, including areas of mathematics into which Hilbert had never penetrated, such as the theory of Lie groups and the analytic theory of numbers, thereby becoming one of the most universal mathematicians of his generation. He also had an important role in the development of mathematical physics, the field to which his most famous books, Raum, Zeit und Materie (1918), on the theory of relativity, and Gruppentheorie und Quantenmechanik (1928), are devoted.

Weyl’s first important work in spectral theory was his Habilitationsschrift (1910), on singular boundary conditions for second-order linear differential equations. The classical Sturm-Liouville problem consists in determining solutions of a selfadjoint differential equation

in a compact interval 0 ≤ x ≤ l, with p(x)> 0 and q real in that interval, the solutions being subject to boundary conditions

with real numbers w,h; it is known that nontrivial solutions exist only when λ takes one of an increasing sequence (λ_n) of real numbers ≥0 and tending to +∞ (the spectrum of the equation). Weyl investigated the case in which l = +∞; his idea was to give arbitrary complex values to λ. Then, for given real h, there is a unique solution satisfying (2) and (3), provided w is taken as a complex number w(λ,h).When h takes all real values, the points w(λ,h) are on a circle C_1(λ) in the complex plane. Weyl also showed that when l tends to +∞, the circles C_1(λ) (for fixed λ) from a nested family that has a circle or a point as a limit. The distinction between the two cases is independent of the choice of λ, for in the “limit circle” case all solutions of (1) are square-integrable on [o,+%E2%88%9E], whereas in the “limit point” case only one solution (up to a constant factor) has that property. This was actually the first example of the general theory of defects of an unbounded Hermitian operator, which was created later by von Neumann. Weyl also showed how the classical Fourier series development of a function in a series of multiples of the eigenfunctions of the Sturm-Liouville problem was replaced, when l = +∞, by an expression similar to the Fourier integral (the spectrum then being generally a nondiscrete subset of R); he thus anticipated the later developments of the Carleman integral operators and their applications to differential linear equations of arbitrary order and to elliptic linear partial differential equations.

In 1911 Weyl inaugurated another important chapter of spectral theory, the asymptotic study of the eigenvalues of a self-adjoint compact operator U in Hilbert space H, with special attention to applications to the theory of elasticity. For this purpose he introduced the “maxi-minimal” method for the direct computation of the nth eigenvalue λ_n of U (former methods gave the value of λ_n only after those of λ_1,λ_2,…,λ_n-1, had been determined). One considers an arbitrary linear subspace F of codimension n-1 in H and the largest value of the scalar product (U · x ǀ x) when x takes all values on the intersection of F and the unit sphere ǀǀxǀǀ = 1 of H; λn is the smallest of these largest values when F is allowed to run through all subspaces of codimension 1. This method has a very intuitive geometric interpretation in the theory of quadrics when H is finite-dimensional; it was used with great efficiency in many problems of functional analysis by Weyl himself and later by Richard Courant, who did much to popularize it and greatly extend its range of applications.

Weyl published the famous paper on equidistribution modulo 1, one of the highlights of his career, in 1916. A sequence (x_n) of real numbers is equidistributed modulo 1 if for any interval [%CE%B1,%CE%B2] contained in [0,1], the number v(α,β_n) of elements x_k such that k ≤ n and x_k = N_k + Y_k’ with α ≤ Y_k ≤ β and N_k a (positive or negative) integer, is such that v(α,β_n)/n tends to the length β-α of the interval when n tends to +∞. Led to such questions by his previous work on the Gibbs phenomenon for series of spherical harmonics, Weyl approached the problem by a completely new–and amazingly simple–method. For any sequence (y_n) of real numbers, write M((y_n)) the limit (when it exists) of the arithmetic mean (y_1 + …+ y_n)/n when n tends to +∞; then to say that (x_n) is equidistributed means that for any function of period 1 coinciding on [0,1] with the characteristic functions of any interval [%CE%B1,%CE%B2]. Weyl’s familiarity with harmonic analysis enabled him to conclude (1) that this property was equivalent to the existence of M((f(x_n))) for all Riemann integrable functions of period 1 and (2) that it was enough to check the existence of that limit for the particular functions exp(2πikx) for any integer k∊Z. This simple criterion immediately yields the equidistribution of the sequence (nα) for irrational α (proved independently a little earlier by Weyl, Bohl, and Waclaw Sierpiński by purely arithmetic methods), as well as a quantitative form of the Kronecker theorems on simultaneous Diophantine approximations.

Weyl’s most profound result was the proof of the equidistribution of the sequence (P(n)), where P is a polynomial of arbitrary degree, the leading coefficient of which is irrational; this amounts to showing that

For N tending to +∞. To give an idea of Weyl’s ingenious proof, consider the case when P(n) = αn^2 + βn with irrational α. One writes

Where exp(2πiαrn), Ir being the interval intersection of [0,N] and [-r, N - r] in Z, This yields . One has two majorations, ǀσrǀ ≤ N + 1 and ǀσrǀ ≤ 1/sin(2παr). For a given , the number of integers r∈ [-N,N] such that 2α r is congruent to a number in the interval [-%D1%94,%D1%94] has the form 4πєN + o(N) by equidistribution; hence it is ≤ 5єN for large N. Applying to these integers r the first majoration, and the second to the others, one obtains

ǀS_Nǀ^2 ≤ 5ε N(N + 1) + (2N + 1)/sin(πє) ≤ 6εN&2

For large N, thus proving the theorem. The extension of that idea to polynomials of higher degree d is not done by induction on d, but by a more elaborate device using the equidistribution of a multilinear function of d variable. Weyl’s results, through the improvements made later by I. M. Vinogradov and his school, have remained fundamental tools in the application of the Hardy-Littlewood method in the additive theory of numbers.

Weyl’s versatility is illustrated in a particularly striking way by the fact that immediately after these original advances in number theory (which he obtained in 1914), he spent more than ten years as a geometer–a geometer in the most modern sense of the word, uniting in his methods topology, algebra, analysis, and geometry in a display of dazzling virtuosity and uncommon depth reminiscent of Riemann. His familiarity with geometry and topology had been acquired a few years earlier when, as a young Privatdozent at Göttingen, he had given a course on Riemann’s theory of functions: but instead of following his predecessors in their constant appeal to “intuition” for the definition and properties of Riemann surfaces, he set out to give to their theory the same kind of axiomatic and rigorous treatment that Hilbert had given to Euclidean geometry. Using Hilbert’s idea of defining neighborhoods by a system of axioms, and influenced by Brouwer’s clever application of Poincaré’s simplicial methods (which had just been published), he gave the first rigorous definition of a complex manifold of dimension 1 and a thorough treatment (without any appeal to intuition) of all questions regarding orientation, homology, and fundamental groups of these manifolds. Die Idee der Riemannschen Fläiche (1913) immediately became a classic and inspired all later developments of the theory of differential and complex manifolds.

The first geometric problem that Weyl attempted to solve (1915) was directly inspired by Hilbert’s previous work on the rigidity of convex surfaces. Hilbert had shown how the “mixed volumes” considered by Minkowski could be expressed in terms of a second-order elliptic differential linear operator L_H attached to the “Stützfunktion” H of a given convex body; Blaschke had observed that this operator was the one that intervened in the theory of infinitesimal deformation of surfaces, and this knowledge had enabled Hilbert to deduce from his results that such infinitesimal deformations for a convex body could only be Euclidean isometries. Weyl attempted to prove that not only infinitesimal deformations, but finite deformations of a convex surface as well, were necessarily Euclidean isometries. His very original idea, directly inspired by his work on two-dimensional “abstract” Riemannian manifolds, was to prove simultaneously this uniqueness property and an existence statement, namely that any two-dimensional Riemannian compact manifold with everywhere positive curvature was uniquely (up to isometries) imbeddable in Euclidean three-dimensional space. The bold method he proposed for the proof was to proceed by continuity, starting from the fact the (by another result of Hilbert’s) the problem of existence and uniqueness was already solved for the ds^2 of the sphere, and using a family of ds^2 depending continuously on a real parameter linking the given ds^2 to that of the sphere and having all positive curvature. This led him to a “functional differential equation” that he did not completely solve, but later work by L. Nirenberg showed that a complete proof of the theorem could be obtained along these lines.

Interrupted in this work by mobilization into the German army, Weyl did not resume it when he was allowed to return to civilian life in 1916. At Zurich he had worked with Einstein for one year, and he became keenly interested in the general theory of relativity, which had just been published; with his characteristic enthusiasm he devoted most of the next five years to exploring the mathematical framework of the theory. In these investigations Weyl introduced the concept of what is now called a linear connection, linked not to the Lorentz group of orthogonal transformations of a quadratic form of signature (1, 3) but to the enlarged group of similitudes (reproducing the quadratic form only up to a factor); he even thought for a time that this would give him a “unitary theory” encompassing both gravitation and electromagnetism. Although these hopes did not materialize, Weyl’s ideas undoubtedly were the source from which E. Cartan, a few years later, developed his general theory of connections (under the name of “generalized spaces”).

Weyl’s use of tensor calculus in his work on relativity led him to reexamine the basic methods of that calculus and, more generally, of classical invariant theory that had been its forerunner but had fallen into near oblivion after Hilbert’s work of 1890. On the other hand, his semiphilosophical, semimathematical ideas on the general concept of “space” in connection with Einstein’s theory had directed his investigations to generalizations of Helmholtz’s problem of characterizing Euclidean geometry by properties of “free mobility.” From these two directions Weyl was brought into contact with the theory of linear representations of Lie groups; his papers on the subject (1925-1927) certainly represent his masterpiece and must be counted among the most influential ones in twentieth-century mathematics.

In the early 1900’s Frobenius, I. Schur, and A. Young had completely determined the irreducible rational linear representations of the general linear group GL(n,C) of complex matrices of order n; it was easy to deduce from Schur’s results that all rational linear representations of the special linear group SL(n,C) (matrices of determinant 1) were completely reducible–that is, direct sums of irreducible representations. Independently, E. Cartan in 1913 had described all irreducible linear representations of the simple complex Lie algebras without paying much attention to the exact relation between these representations and the corresponding ones for the simple groups, beyond exhibiting examples of group representations for each type of Lie algebra representations. Furthermore, Cartan apparently had assumed without proof that any (finite-dimensional) linear representation of a semisimple Lie algebra is completely reducible.

Weyl inaugurated a new approach by deliberately focusing his attention on global groups, the Lie algebras being reduced to the status of technical devices. In 1897 Hurwitz had shown how one may form invariants for the orthogonal or unitary group by substituting, for the usual averaging process on finite groups, integration on the (compact) group with respect to an invariant measure. He also had observed that this yields invariants not only of the special unitary group SU(n) but also of the special linear group SL(n,C) (the first example of what Weyl later called the “unitarian trick”). Using Hurwitz’s method, I. Schur in 1924 had proved the complete reducibility of all continuous linear representations of SU(n) by showing the existence, on any representation space of that group, of a Hermitian scalar product invariant under the action of SU(n); by using the “unitarian trick” he also was able to prove the complete reducibility of the continuous linear representations of SL(n,C) and to obtain orthogonality relations for the characters of SU(n), generalizing the well-known Frobenius relations for the characters of a finite group. These relations led to the explicit determination of the characters of SL(n,C), which Schur had obtained in 1905 by purely algebraic methods.

Starting from these results, Weyl first made the connection between the methods of Schur and those of E. Cartan for the representations of the Lie algebra of SL (n,C) by pointing out for the first time that the one-to-one correspondence between both types of representations was due to the fact that SU(n) is simply connected. He next extended the same method to the orthogonal and symplectic complex groups, observing, apparently for the first time, the existence of the two-sheeted covering group of the orthogonal group (the “spin” group, for which Cartan had only obtained the linear representations by spinors). Finally, Weyl turned to the global theory of all semisimple complex groups. First he showed that the “unitarian trick” had a validity that was not limited to the classical groups by proving that every semisimple complex Lie algebra ℑ could be considered as obtained by complexification from a well-determined real Lie algebra ℑ_u, which was the Lie algebra of a compact group G_u; E. Cartan had obtained that result through a case-by-case examination of all simple complex Lie groups, whereas Weyl obtained a general proof by using the properties of the roots of the semisimple algebra. This established a one-to-one correspondence between linear representations of ℑ and linear representations of ℑ_u; but to apply Hurwitz’s method, one had to have a compact Lie group having ℑ_u as Lie algebra and being simply connected. This is not necessarily the case for the group G_u, and to surmount that difficulty, one had to prove that the universal covering group of G_u is also compact; the a priori proof that such is the case is one of the deepest and most original parts of Weyl’s paper. It is linked to a remarkable geometric interpretation of the roots of the Lie algebra ℑ_u relative to a maximal commutative subalgebra t, which is the Lie algebra of a maximal torus T of G_u. Each root vanishes on a hyperplane of t, and the connected components of the complement of the union of these hyperplanes in the vector space t are polyhedrons that are now called Weyl chambers; each of these chambers has as boundary a number of “walls” equal to the dimension of t.

Using this description (and some intuitive considerations of topological dimension that he did not bother to make rigorous), Weyl showed simultaneously that the fundamental group of G_u was finite (hence was compact) and that for G_u the maximal torus T played a role similar to that of the group of diagonal matrices in SU (n): every element of G_u is a conjugate of an element of T. Furthermore, he proved that the Weyl chambers are permuted in a simply transitive way by the finite group generated by the reflections with respect to their walls (now called the Weyl group of ℑ or of G_u); this proof gave him not only a new method of recovering Cartan’s “dominat weights” but also the explicit determination of the character of a representation as a function of its dominant weight.

In this determination Weyl had to use the orthogonality relations of the characters of Gu (obtained through an extension of Schur’s method) and a property that would replace Frobenius’ fundamental theorem in the theory of linear representations of finite groups: that all irreducible representations are obtained by “decomposing” the regular representation. Weyl conceived the extraordinarily bold idea (for the time) of obtaining all irreducible representations of a semisimple group by “decomposing” an infinite-dimensional linear representation of Gu. To replace the group algebra introduced by Frobenius, he considered the continuous complexvalued functions on Gu and took as “product” of two such functions f,g what we now call the convolution f * g, defined by (f * g)(t) = ∫ f(st-1)g(t)dt, integration being relative to an invariant measure. To each continuous function f the operator R(f): g → f * g is then associated; the “decomposition” is obtained by considering the space of continuous functions on Gu as a pre-Hilbert space and by showing that for suitable f (those of the form h * h, where , R(f) is Hermitian and compact, so that the classical Schmidt-Riesz theory of compact operators can be applied. It should be noted that in this substitute for the group algebra formed by the continuous functions on G_u, there is no unit element if G_u is not trivial (in contrast with what happens for finite groups); again it was Weyl who saw the way out of this difficulty by using the “regularizing” property of the convolution to introduce “approximate units” —that is, sequences (u_n) of functions that are such that the convolutions u_n * f tend to f for every continuous function f.

Very few of Weyl’s 150 published books and papers—even those chiefly of an expository character—lack an original idea or a fresh viewpoint. The influence of his works and of his teaching was considerable: he proved by his example that an “abstract” approach to mathematics is perfectly compatible with “hard” analysis and, in fact, can be one of the most powerful tools when properly applied.

Weyl had a lifelong interest in philosophy and metaphysics, and his mathematical activity was seldom free from philosophical undertones or afterthoughts. At the height of the controversy over the foundations of mathematics, between the formalist school of Hilbert and the intuitionist school of Brouwer, he actively fought on Brouwer’s side; and if he never observed too scrupulously the taboos of the intuitionists, he was careful in his papers never to use the axiom of choice. Fortunately, he dealt with theories in which he could do so with impunity.

BIBLIOGRAPHY

• Weyl’s writings were brought together in his Gesammelte Abhandlungen, K. Chandrasekharan, ed., 4 vols. (Berlin-Heidelberg-New York, 1968). See also Selecta Hermann Weyl (Basel-Stuttgart, 1956).

J. Dieudonné

The New Dictionary of Scientific Biography

The original DSB article focuses on Weyl’s contributions to analysis, analytical number theory, and the representation theory of Lie groups. Other fields of study are mentioned only in passing. This appendix presents some aspects of Weyl’s ideas in the foundations of mathematics, differential geometry, and mathematical physics.

It was already mentioned in the main article that Weyl changed his research orientation under the impact of the experience of World War I and the crisis in its aftermath. As Weyl later said, he looked for safeness in his main fields of research. He turned toward the foundations of analysis and to the newly founded theory of general relativity with its strong appeal to philosophically intriguing interrelations of mathematics and physics. In 1918 Weyl published two books on these two rather distinct topics: The Continuum and Space, Time, Matter.

Constructivist Sympathies and Transitional Turn to Intuitionism. In The Continuum Weyl proposed a constructivist arithmetical foundation of analysis or, more properly, for a part of analysis. This began his public intervention into the foundations of mathematics, in clear opposition to David Hilbert’s program of a purely formalistic axiomatic foundation. He sketched how a reduced part of real analysis could be secured by constructions in a semiformalized arithmetical language, respecting the restrictions of predicativity, although the constructivist (reduced) continuum stood in stark contrast to the intuitive continuum and physical ideas of space-time. That seems to have contributed to his change of mind already a year after the publication of the book. He started to support Luitzen Egbertus Jan Brouwer’s more radical intuitionistic program and attacked Hilbert’s foundational views even more strongly in a programmatic article on the recent foundational crisis of mathematics (1921). But his initial hopes of a unified intuitionist foundation of mathematics, which would also serve the purpose of a deeper understanding of the physical space-time were not fulfilled.

Weyl soon realized that the intuitionist foundation of mathematics led to undesirable technical complications and relied too much on a kind of evidence that was difficult to reconcile with his interests in mathematical physics. Hilbert’s foundational program seemed to be better adapted to providing the symbolical tools needed in most advanced contemporary mathematical physics, in particular the rising quantum mechanics. In the late 1920s Weyl showed more sympathy with Hilbert’s foundational view, but could not avoid being irritated by Kurt Gödel’s negative result of 1930 for Hilbert’s strategy of founding modern analysis on a finitistic consistency proof. He even returned after World War II to a weak preference of his arithmetical constructive approach of 1918.

Linking Geometry and Physics. Parallel to the work on the foundations of analysis and the concept of the continuum, Weyl started to analyze and to deepen the links between differential geometry and the newly established general theory of relativity. Space, Time, Matter was one of the first text books on the subject and among the most influential ones over decades to come. It was revised in five successive editions until 1923. In the first part of this book, Weyl gave an up-to-date introduction to Riemannian and Lorentzian geometry, which made it an important introductory monograph to modern differential geometry. At the time when the book first went to print, Weyl developed a generalization of Riemannian geometry, which was more firmly built on what he called a “purely infinitesimal’’ point of view. He avoided a direct comparison of lengths and other quantities at different points of the manifold and introduced the seminal concept of point-dependent gauges and a gauge field(a scale connection) to allow the transfer of metrical concepts from one point to another. The scale connection had formal properties that made it appear as an appropriate mathematical expression for the potential of the electromagnetic field. It became the embryonic core of a tradition of physical field theories which—in a more general form—became important in high energy physics of the last third of the twentieth century.

The gauge geometry of 1918 served Weyl as a starting point for an attempt at unifying electromagnetism and gravity. In this respect Weyl took up and tried to improve Hilbert’s program to derive the basic matter structures from purely field theoretical considerations (extending an approach of Gustav Mie; see Weyl, “Gravitation und Elektrizität”). For about two years he believed he had found a clue to the problem of understanding matter by such a classical field theoretical approach. In the third edition of Space, Time, Matter he added a passage on a gauge geometric generalization of semi-Riemannian geometry and his unified field theory. But already a year later he realized that the derivation of matter structures from pure field theory had no chance for success.

Space, Time, Matter was only a small part of the contributions to the interchange between differential geometry and general relativity. Weyl pursued it in his own peculiar way based on a broad conceptual and philosophical view. He studied the interrelation of conformal and projective differential geometric structure and realized that both together specify a (Weylian) metric uniquely. An even deeper conceptual approach was contained in his Mathematische Analyse des Raumproblems (1923; Analysis of the Problem of Space), in which the older space problem of the nineteenth century was transplanted into the context of purely infinitesimal geometry. Here Weyl (sketchily) introduced concepts of infinitesimal group operations and connections. Independently developed by Élie Cartan in 1922, they were later turned into the language of fiber bundles and led to the study of geometries characterized by gauge structures.

Group Theory and Physics. The study of differential geometry and the problem of space led Weyl to studying the representation theory of Lie groups. In the mid-1920s he delved into what became his most influential contributions to pure mathematics, the study of the representations of semisimple Lie groups (1925–1926). Extended and refined, this work formed the core of his later book The Classical Groups (1939), written as a harvest of his work and his lecturing activities on this topic during his Princeton years.

During his work on the representation theory of Lie groups, Weyl actively followed the turn toward the new quantum mechanics of Werner Heisenberg, Max Born, and Pascual Jordan and started to explore the possibilities opened up by the interplay of infinitesimal and finite group operations in quantum mechanics. In 1927–1928 he gave a lecture at Zürich Eidgenössische Technische Hochschule (ETH) on the topic, which gave rise to his second, again very influential, book on mathematical physics, The Theory of Groups and Quantum Mechanics (1928; Engl. trans. 1931). Here he emphasized the conceptual role of group methods in the symbolic representation of quantum structures, in particular the intriguing interplay of representations of the special linear group and permutations groups. In this interplay he also saw the mathematical clue to understanding the phenomenon of spin coupling, studied at the time by some of the young physicists turning toward quantum chemistry (Fritz London, Walter Heitler, and others). Moreover, Weyl explored the central role of the discrete symmetries (parity, charge conjugation, and time inversion) in the first steps toward a quantized version of electrodynamics, thus anticipating structural elements that turned into important questions for physics three decades later.

Not covered in the book, but published separately, was a second step for his gauge theory of the electromagnetic field. Quantum theorists had proposed reconsidering the gauge idea for the phase of wave functions rather than for scale gauge as in Weyl’s original idea of 1918. Weyl (1929) took up the proposal and explored it after the rise of Paul Dirac’s spinor fields for the electron (as was done similarly and independently by Vladimir Fock). He came to the conclusion that the new context demanded considering symmetry extension of the Lorentz group by unitary transformations in U(1). That gave rise to a modified gauge theory of electromagnetism, which was endorsed by leading theoretical physicists (among them Wolfgang Pauli, Erwin Schrödinger, and Fock) and served as a starting point for the next generation of physicists, who molded the symbolic frame of gauge field theories in the 1950s and 1960s.

Weyl’s Philosophical Writings. Hermann Weyl was not only a philosophically interested researcher in mathematics and physics. He had close, but changing relations to philosophers such as Edmund Husserl, Fritz Medicus, and later to the existential philosophies of Karl Jaspers and Martin Heidegger. He gave active literary expression to his philosophical reflections of scientific activity in many of his publications. Most influential was his 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. A central topic of his thought was the mode and the role of symbolic construction in scientific knowledge of the world. It reappeared in his later reflections on twentieth-century science (1953). Like others of his generation, Weyl was shocked not only by the atrocities of the Nazi regime but also by the destructiveness of the nuclear weapons developed during the war. He considered it as a kind of hubris of modern techno-scientific culture, which had started to step beyond the circle of activities that can be accounted for. He hesitatingly hoped that it might perhaps be contained, if at all, by a slow rise of moral awareness of users and producers of scientific knowledge.

SUPPLEMENTARY BIBLIOGRAPHY

WORKS BY WEYL

• The Concept of a Riemann Surface. Translated by Gerald R MacLane. Reading: MA: Addison-Wesley, 1964. Originally published 1913.
• The Continuum: A Critical Examination of the Foundation of Analysis. Translated by Stephen Pollard and Thomas Bole. Kirksville, MO: Thomas Jefferson University Press, 1987. Corrected republication, New York: Dover, 1994. First published 1918.
• “Gravitation und Elektrizität.” In The Dawning of Gauge Theory, by Lochlainn O’Raifeartaigh. Princeton, NJ: Princeton University Press, 1997. First published 1918.
• Space, Time, Matter. Translated by Henry L. Brose (from the 1921 German 4th edition). London: Methuen, 1922. First published 1918.
• “Theorie der Darstellung kontinuierlicher halbeinfacher Gruppen durch lineare Transformationen. I, II, III und Nachtrag.” Mathematische Zeitschrift 23 (1925): 271–309; 24 (1926): 328–395, 789–791.
• Philosophie der Mathematik und Naturwissenschaft. Handbuch der Philosophie, Abt. 2, Beitr. A. Munich: Oldenbourg, 1927.
• “Elektron und Gravitation.” Zeitschrift für Physik 56 (1929): 330–352. In English in O’Raifeartaigh, 1997, pp. 121–144.
• Mathematische Analyse des Raumproblems: Vorlesungen gehalten in Barcelona und Madrid; Was ist Materie?; zwei Aufsätze zur Naturphilosophie.. Darmstadt: Wissenschaftliche Buchgesellschaft, 1963. First published 1923.
• The Theory of Groups and Quantum Mechanics. Translated by H. P. Robertson (from the German 2nd edition). New York: Dutton, 1931. First German edition 1928.
• The Classical Groups: Their Invariants and Representations. Princeton, NJ: Princeton University Press, 1939.
• Philosophy of Mathematics and Natural Science. Revised and augmented English edition, based on a translation of Olaf Helmer. Princeton, NJ: Princeton University Press, 1949.
• Wissenschaft als symbolische Konstruktion des Menschen. EranosJahrbuch Zürich: Rhein-Verlag, 1949.
• “Über den Symbolismus der Mathematik und mathematischen Physik.” Studium generale 6 (1953): 219–228.

OTHER SOURCES

• Borel, Armand. Essays in the History of Lie Groups and Algebraic Groups. Providence, RI: AMS; London: Mathematical Society, 2001.
• Chandrasekharan, Komaravolu, ed. Hermann Weyl: 1885–1985: Centenary Lectures Delivered by C. N. Yang, R. Penrose, and A. Borel at the ETH. Zürich; Berlin: Springer, 1986.
• Chevalley, Claude, and André Weil. “Hermann Weyl (1885–1955).” L’Enseignement Mathématique 2, no. 3 (1957): 157–187.
• Coleman, Robert, and Herbert Korté. “Hermann Weyl: Mathematician, Physicist, Philosopher.” In Hermann Weyl’s Raum-Zeit-Materie and a General Introduction to His Scientific Work, edited by Erhard Scholz. Basel: Birkhauser Verlag, 2001.
• Frei, Günther, and Urs Stammbach. Hermann Weyl und die Mathematik an der ETH Zürich, 1913–1930. Basel: Birkhäuser, 1992.
• Hawkins, Thomas. Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics, 1869–1926. Berlin: Springer, 2000.
• Hendricks, Vincent F., Stig Andur Pedersen; and Klaus F. Jørgensen, eds. Proof Theory: History and Philosophical Significance. Dordrecht: Kluwer, 2000. Includes chapters on Weyl by Solomon Feferman, Erhard Scholz, and Dirk van Dalen.
• Mackey, George. “Hermann Weyl and the Application of Group Theory to Quantum Mechanics.” In Exact Sciences and Their Philosophical Foundations, edited by Werner Deppert et al. Frankfurt am Main and New York: Lang, 1988.
• Mancosu, Paolo. From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford: Oxford University Press, 1998.
• ———, and Thomas Ryckman. “The Correspondence between O. Becker and H. Weyl.” Philosophia Mathematica 10 (2002): 130–202.
• Newman, M. H. A. “Hermann Weyl, 1885–1955.” Biographical Memoirs of Fellows of the Royal Society London 3 (1957): 305–328.
• O’Raifeartaigh, Lochlainn. The Dawning of Gauge Theory. Princeton, NJ: Princeton University Press, 1997.
• ———, and Norbert Straumann. “Gauge Theory: Historical Origins and some Modern Developments.” Reviews of Modern Physics 72 (2000): 1–23.
• Scholz, Erhard. “Hermann Weyl’s Analysis of the ‘Problem of Space’ and the Origin of Gauge Structures.” Science in Context 17 (2004): 165–197.
• ———. “Local Spinor Structures in V. Fock’s and H. Weyl’s Work on the Dirac Equation (1929).” In Géométrie au XXième siècle, edited by D. Flament, C. Houzel, et al. Paris: Hermann, 2005.
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Erhard Scholz