# Hermann Weyl

Hermann Klaus Hugo Weyl, ForMemRS (/vaɪl/; German:; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. 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.
His research has had major significance for theoretical physics as well as purely mathematical disciplines such as 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 contributed to an exceptionally wide range of fields, including 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. Freeman Dyson wrote that Weyl alone bore comparison with the "last great universal mathematicians of the nineteenth century", Henri Poincaré and David Hilbert. Michael Atiyah, in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him.

## Biography

Hermann Weyl was born in Elmshorn, a small town near Hamburg, in Germany, and attended the Gymnasium Christianeum in Altona. His father, Ludwig Weyl, was a banker; whereas his mother, Anna Weyl (née Dieck), came from a wealthy family.
From 1904 to 1908, he studied mathematics and physics in both Göttingen and Munich. His doctorate was awarded at the University of Göttingen under the supervision of David Hilbert, whom he greatly admired.
In September 1913, in Göttingen, Weyl married Friederike Bertha Helene Joseph (30 March 1893 – 5 September 1948) who went by the name Helene (nickname "Hella"). 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. Helene was a philosopher (she was a disciple of phenomenologist Edmund Husserl) and a translator of Spanish literature into German and English (especially the works of Spanish philosopher José Ortega y Gasset). It was through Helene's close connection with Husserl that Hermann became familiar with (and greatly influenced by) Husserl's thought. Hermann and Helene had two sons, Fritz Joachim Weyl (19 February 1915 – 20 July 1977) and Michael Weyl (15 September 1917 – 19 March 2011), both of whom were born in Zürich, Switzerland. Helene died in Princeton, New Jersey, on 5 September 1948. A memorial service in her honor was held in Princeton on 9 September 1948. Speakers at her memorial service included her son Fritz Joachim Weyl and mathematicians Oswald Veblen and Richard Courant. In 1950. Hermann married sculptor Ellen Bär (née Lohnstein) (17 April 1902 – 14 July 1988), who was the widow of professor Richard Josef Bär (11 September 1892 – 15 December 1940) of Zürich.
After taking a teaching post for a few years, Weyl left Göttingen in 1913 for Zürich to take the chair of mathematics at the ETH Zürich, where he was a colleague of Albert Einstein, who was working out the details of the theory of general relativity. Einstein had a lasting influence on Weyl, who became fascinated by mathematical physics. In 1921, Weyl met Erwin Schrödinger, a theoretical physicist who at the time was a professor at the University of Zürich. They were to become close friends over time. Weyl had some sort of childless love affair with Schrödinger's wife Annemarie (Anny) Schrödinger (née Bertel), while at the same time Anny was helping raise an illegitimate daughter of Erwin's named Ruth Georgie Erica March, who was born in 1934 in Oxford, England.
Weyl was a Plenary Speaker of the International Congress of Mathematicians (ICM) in 1928 at Bologna and an Invited Speaker of the ICM in 1936 at Oslo. He was elected a fellow of the American Physical Society in 1928, a member of the American Academy of Arts and Sciences in 1929, a member of the American Philosophical Society in 1935, and a member of the National Academy of Sciences in 1940. For the academic year 1928–1929, he was a visiting professor at Princeton University, where he wrote a paper, "On a problem in the theory of groups arising in the foundations of infinitesimal geometry," with Howard P. Robertson.
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. As the political situation in Germany grew worse, he changed his mind and accepted when offered the position again. He remained there until his retirement in 1951. Together with his second wife Ellen, he spent his time in Princeton and Zürich, and died from a heart attack on 8 December 1955, while living in Zürich.
Weyl was cremated in Zürich on 12 December 1955. His ashes remained in private hands until 1999, at which time they were interred in an outdoor columbarium vault in the Princeton Cemetery. The remains of Hermann's son Michael Weyl (1917–2011) are interred right next to Hermann's ashes in the same columbarium vault.
Weyl was a pantheist.

## Contributions
### Distribution of eigenvalues

In 1911 Weyl published Über die asymptotische Verteilung der Eigenwerte (On the asymptotic distribution of eigenvalues) in which he proved that the eigenvalues of the Laplacian in a compact domain are distributed according to the so-called Weyl law. In 1912 he suggested a new proof, based on variational principles. Weyl returned to this topic several times, considered elasticity system and formulated the Weyl conjecture. These works started an important domain—asymptotic distribution of eigenvalues—of modern analysis.

### Geometric foundations of manifolds and physics

In 1913, Weyl published Die Idee der Riemannschen Fläche (The Concept of a Riemann Surface), which gave a unified treatment of Riemann surfaces. In it Weyl utilized point set topology, in order to make Riemann surface theory more rigorous, a model followed in later work on manifolds. He absorbed L. E. J. Brouwer's early work in topology for this purpose.
Weyl, as a major figure in the Göttingen school, was fully apprised of Einstein's work from its early days. He tracked the development of relativity physics in his Raum, Zeit, Materie (Space, Time, Matter) from 1918, reaching a 4th edition in 1922. 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 the 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.
His overall approach in physics was based on the phenomenological philosophy of Edmund Husserl, specifically Husserl's 1913 Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Erstes Buch: Allgemeine Einführung in die reine Phänomenologie (Ideas of a Pure Phenomenology and Phenomenological Philosophy. First Book: General Introduction). Husserl had reacted strongly to Gottlob 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.

### Topological groups, Lie groups and representation theory

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 streamlined in that specific context, in his 1927 Weyl quantization, the best extant bridge between classical and quantum physics to date. 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 reconsidered invariant theory. It covered symmetric groups, general 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 mod 1, which was a 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.

### Foundations of mathematics

In The Continuum Weyl developed the logic of predicative analysis using the lower levels of Bertrand Russell's ramified theory of types. He was able to develop most of classical calculus, while using neither the axiom of choice nor proof by contradiction, and avoiding Georg Cantor's infinite sets. Weyl appealed in this period to the radical constructivism of the German romantic, subjective idealist Fichte.
Shortly after publishing The Continuum Weyl briefly shifted his position wholly to the intuitionism of Brouwer. In The Continuum, the constructible points exist as discrete entities. Weyl wanted a continuum that was not an aggregate of points. He wrote a controversial article proclaiming, for himself and L. E. J. Brouwer, a "revolution." This article was far more influential in propagating intuitionistic views than the original works of Brouwer himself.
George Pólya and Weyl, during a mathematicians' gathering in Zürich (9 February 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 Hegel on the philosophy of nature. Any answer to such a question would be unverifiable, unrelated to experience, and therefore senseless.
However, within a few years Weyl decided that Brouwer's intuitionism did put too great restrictions on mathematics, as critics had always said. The "Crisis" article had disturbed Weyl's formalist teacher Hilbert, but later in the 1920s Weyl partially reconciled his position with that of Hilbert.
After about 1928 Weyl had apparently decided that mathematical intuitionism was not compatible with his enthusiasm for the phenomenological philosophy of Husserl, as he had apparently earlier thought. 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. Weyl however rarely refers to Cassirer, and wrote only brief articles and passages articulating this position.
By 1949, Weyl was thoroughly disillusioned with the ultimate value of intuitionism, and wrote: "Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the contact with intuition much more closely than had been done before. 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."

### Weyl equation

In 1929, Weyl proposed an equation, known as the Weyl equation, for use in a replacement to the Dirac equation. This equation describes massless fermions. A normal Dirac fermion could be split into two Weyl fermions or formed from two Weyl fermions. Neutrinos were once thought to be Weyl fermions, but they are now known to have mass. Weyl fermions are sought after for electronics applications. Quasiparticles that behave as Weyl fermions were discovered in 2015, in a form of crystals known as Weyl semimetals, a type of topological material.

### Philosophy

Weyl had been interested in philosophy since his youth, when he read Immanuel Kant's "Critique of Pure Reason" with space and time as a priori concepts of knowledge (even if he later disliked Kant's too close ties to Euclidean geometry). From 1912 onwards he was strongly influenced by Edmund Husserl and his phenomenology, which was also reflected in some passages in his book “Space, Time, Matter”. In 1927 his contribution Philosophy of Mathematics and Natural Sciences to the Handbook of Philosophy was published by Oldenbourg Verlag, which was later published separately and revised as a book. In an attempt to reconstruct the origins of Hermann Weyl's philosophy and to integrate them into the main currents of philosophy, Norman Sieroka pointed to intensive, long-term discussions between Weyl and his Zurich philosopher colleague Fritz Medicus, a specialist on Johann Gottlieb Fichte. Fichte's Wissenschaftslehre and philosophy, according to which "being" results from the interaction of the "absolute ego" with its material neighbourhood (Umgebung), is also of great influence on Weyl and is reflected in Weyl's use of the neighbourhood concept of topology (continuum) and in Weyl's conception of the general theory of relativity, alongside the influences of Edmund Husserl's phenomenology known directly from Weyl's writings. According to Sieroka, Weyl also finds influences from Gottfried Wilhelm Leibniz's theory of matter (the theory of monads, etc.) 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). He understands the intra-mathematical debate about intuitionism and formalism along the lines of a debate between Husserlian phenomenology and Fichtean constructivism. In the 1920s, before the development of quantum mechanics and inspired by the statistical nature of quantum theory, which was becoming increasingly clear at the time, Weyl turned away from the field-theoretical description of matter towards a theory of active (agens) matter, which was achieved by including the spatial environment in the field theoretical description expressed. He had previously described the general theory of relativity and his own extensions of it, which led to the origin of today's concept of gauge field theories, using differential geometric methods. Under the influence of quantum theory, he turned away from this “geometric field theory”. According to Sieroka, Fichte and Ernst Cassirer were also an important influence in Weyl's late philosophy (science as a “symbolic construction”). Weyl's involvement with Martin Heidegger was less well known. Although Weyl did not agree with Heidegger's opinion about death, his conception of neighbourhood (Umgebung) was influenced by Heidegger's Existentialism.

## 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.
    —Gesammelte Abhandlungen—as quoted in Year book – The American Philosophical Society, 1943, p. 392
* In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.
    —Weyl (1939b, p. 500)
* Whenever you have to do with a structure-endowed entity S try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of S in this way.
    —Symmetry, Princeton Univ. Press, p. 144; 1952
* Beyond the knowledge gained from the individual sciences, there remains the task of comprehending. In spite of the fact that the views of philosophy sway from one system to another, we cannot dispense with it unless we are to convert knowledge into a meaningless chaos.
    —Space-Time-Matter — 4th edition (1922), English translation, Dover (1952) p. 10; Weyl’s boldfaced highlight.