Hermann Weyl

• 1. Introduction

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  Whenever Michael Atiyah turned his attention to a mathematical topic, he kept running into the same name. He found that Hermann Weyl had already been there before him. Freeman Dyson placed Weyl in a category almost no one else could enter. Weyl alone, Dyson wrote, bore comparison with the last great universal mathematicians of the nineteenth century, Henri Poincare and David Hilbert. Hermann Klaus Hugo Weyl was born on the 9th of November 1885 and died on the 8th of December 1955. He was a German mathematician, theoretical physicist, logician and philosopher. How does one person reach across space, time, matter, philosophy, logic, symmetry and the history of mathematics, leaving a mark in each? Why would a man so devoted to his homeland eventually flee it? And how did a mathematician come to write an equation describing particles without mass? The answers run through Gottingen, Zurich and Princeton, and through the people who shaped him along the way.

• 2. The Gottingen Lineage

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  Elmshorn, a small town near Hamburg, was where Weyl began. His father, Ludwig Weyl, was a banker, while his mother, Anna Weyl, came from a wealthy family. He attended the Gymnasium Christianeum in Altona before the larger world of mathematics opened to him. From 1904 to 1908 he studied mathematics and physics at the University of Gottingen and at the Ludwig-Maximilians-Universitat Munchen. At Gottingen his doctorate was awarded under the supervision of David Hilbert, a man Weyl greatly admired. That admiration tied him to a particular tradition of mathematics. The Gottingen line ran back through Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. Weyl would come to be counted among its representatives, even though much of his working life unfolded far from German soil. He was one of the most influential mathematicians of the twentieth century. That influence would later make him an important member of the Institute for Advanced Study during its early years.

• 3. Einstein And The Geometry Of Spacetime

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  In 1913 Weyl left Gottingen for Zurich to take the chair of mathematics at ETH Zurich. There he became 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. Weyl tracked the development of relativity physics in his book Raum, Zeit, Materie, or Space, Time, Matter, first published in 1918 and reaching its fourth edition in 1922. In that same year of 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 attempt to model the electromagnetic field and the gravitational field as geometrical properties of spacetime. The attempt was unsuccessful at the time. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. The Weyl tensor in Riemannian geometry carries his name and is of major importance in understanding the nature of conformal geometry.

• 4. Symmetry And The Quantum

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  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 sit at the foundation of understanding the symmetry structure of quantum mechanics, which Weyl placed on a group-theoretic basis. His treatment included spinors. The mathematical formulation of quantum mechanics owed much to John von Neumann, and together this work produced the treatment that has been familiar since about 1930. Non-compact groups and their representations, particularly the Heisenberg group, were streamlined in his 1927 Weyl quantization, described as the best extant bridge between classical and quantum physics. His book The Classical Groups reconsidered invariant theory. It covered symmetric groups, general linear groups, orthogonal groups and symplectic groups, along with results on their invariants and representations. Helped by Weyl's expositions, Lie groups and Lie algebras became a mainstream part of both pure mathematics and theoretical physics. His own advice captured the method: 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.

• 5. The Crisis Over The Continuum

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  On the 9th of February 1918, at a mathematicians' gathering in Zurich, George Polya and Weyl made a bet about the future direction of mathematics. Weyl predicted that within the next twenty years mathematicians would come to realize the total vagueness of notions such as real numbers, sets and countability. He went further. He held that asking about the truth or falsity of the least upper bound property of the real numbers was as meaningful as asking about the truth of Hegel's basic assertions on the philosophy of nature. In his book The Continuum, Weyl developed the logic of predicative analysis using the lower levels of Bertrand Russell's ramified theory of types. He managed to develop most of classical calculus while using neither the axiom of choice nor proof by contradiction, and while avoiding Georg Cantor's infinite sets. Shortly after publishing The Continuum he shifted his position wholly to the intuitionism of L. E. J. Brouwer. He wrote a controversial article proclaiming, for himself and Brouwer, a revolution. That article proved far more influential in spreading intuitionistic views than Brouwer's own original works. The Crisis article disturbed his formalist teacher Hilbert, a tension Weyl would spend years trying to resolve.

• 6. Phenomenology And A Changing Mind

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  Edmund Husserl's thought reached Weyl through a personal channel. His overall approach in physics was based on Husserl's phenomenological philosophy, specifically Husserl's 1913 work known in English as Ideas of a Pure Phenomenology and Phenomenological Philosophy. Husserl had reacted strongly to Gottlob Frege's criticism of his first work on the philosophy of arithmetic. He was investigating the sense of mathematical and other structures, which Frege had distinguished from empirical reference. Within a few years Weyl decided that Brouwer's intuitionism placed too great a restriction on mathematics, as critics had always said. Later in the 1920s he partially reconciled his position with that of Hilbert. After about 1928 he had apparently decided that mathematical intuitionism was not compatible with his enthusiasm for the phenomenological philosophy of Husserl. In the last decades of his life Weyl emphasized mathematics as symbolic construction and moved toward a position closer to Hilbert and to that of Ernst Cassirer. By 1949 he was thoroughly disillusioned with the ultimate value of intuitionism. He wrote that 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.

• 7. Marriages And Exile

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  In September 1913, in Gottingen, Weyl married Friederike Bertha Helene Joseph, who went by Helene and the nickname Hella. She was a daughter of Dr. Bruno Joseph, a physician who held the position of Sanitatsrat in Ribnitz-Damgarten. Helene was a philosopher and a disciple of Husserl, and she translated Spanish literature into German and English, especially the works of Jose Ortega y Gasset. It was through Helene's close connection with Husserl that Hermann became familiar with his thought. Their two sons, Fritz Joachim Weyl and Michael Weyl, were both born in Zurich. In 1921 Weyl met Erwin Schrodinger, then a professor at the University of Zurich, and they became close friends. Weyl had a childless love affair with Schrodinger's wife Annemarie. Weyl left the University of Zurich in 1930 to become Hilbert's successor at Gottingen. He had been offered one of the first faculty positions at the new Institute for Advanced Study in Princeton, but declined because he did not want to leave his homeland. As the political situation in Germany grew worse he changed his mind, and he left when the Nazis assumed power in 1933, particularly as his wife was Jewish. He remained at Princeton until his retirement in 1951. Helene died in Princeton on the 5th of September 1948, and a memorial service was held there on the 9th of September 1948, with speakers including her son Fritz Joachim Weyl and the mathematicians Oswald Veblen and Richard Courant. In 1950 Hermann married the sculptor Ellen Bar, the widow of professor Richard Josef Bar of Zurich.

• 8. The Massless Particle

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  In 1929 Weyl proposed an equation, now known as the Weyl equation, intended as a replacement for the Dirac equation. The 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. The idea did not stay confined to particle physics. 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. Weyl himself died from a heart attack on the 8th of December 1955 while living in Zurich. He was cremated in Zurich on the 12th of December 1955, and his ashes remained in private hands until 1999, when they were interred in an outdoor columbarium vault in the Princeton Cemetery. The remains of his son Michael Weyl, who died in 2011, are interred right next to Hermann's ashes in the same vault.