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 left ZĂŒrich 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 ZĂŒrich, and died from a heart attack on December 8, 1955, while living in ZĂŒrich.
Weyl was cremated in ZĂŒrich on December 12, 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 (1917â2011) are interred right next to Hermann's ashes in the same columbarium vault.
In 1911 Weyl published Ăber die asymptotische Verteilung der Eigenwerte (On the asymptotic distribution of eigenvalues) in which he proved that the eigenvalues of the Laplacian in the 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 domainâasymptotic distribution of eigenvaluesâof modern analysis.
In 1913, Weyl published Die Idee der Riemannschen FlÀche (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 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 PhĂ€nomenologie und phĂ€nomenologischen Philosophie. Erstes Buch: Allgemeine EinfĂŒhrung in die reine PhĂ€nomenologie (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 theoryâ»
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.
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 PĂłlya and Weyl, during a mathematicians' gathering in ZĂŒrich (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/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 âsynthesizedâ from discrete elements, is realized. Although the underlying logic of smooth infinitesimal analysis is intuitionistic â the law of excluded middle not being generally affirmable â mathematics developed within avoids the âunbearable awkwardnessâ to which Weyl refers above."
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.
Quotesâ»
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.
âGesammelte Abhandlungenâas quoted in Year book â 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.
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.
âSymmetry Princeton Univ. Press, p144; 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.
âSpace-Time-Matter â 4th edition (1922), English translation, Dover(1952) p. 10; Weylâs boldfaced highlight.
1925. (publ. 1988 ed. K. Chandrasekharan) Riemann's Geometrische Idee.
1927. Philosophie der Mathematik und Naturwissenschaft, 2d edn. 1949. Philosophy of Mathematics and Natural Science, Princeton 0689702078. With new introduction by Frank Wilczek, Princeton University Press, 2009, ISBN978-0-691-14120-6.
The Theory of Groups and Quantum Mechanics (translated from the second, revised German edition by Howard P. Robertson)1929. "Elektron und Gravitation I", Zeitschrift Physik, 56, pp 330â352. â introduction of the vierbein into GR
^â» Hermann Weyl Collection (AR 3344) (Sys #000195637), Leo Baeck Institute, Center for Jewish History, 15 West 16th Street, New York, NY 10011. The collection includes a typewritten document titled "Hellas letzte Krankheit" ("Hella's Last Illness"); the last sentence on page 2 of the document states: "Hella starb am 5. September â», mittags 12 Uhr." ("Hella died at 12:00 Noon on September 5 â»"). Helene's funeral arrangements were handled by the M. A. Mather Funeral Home (now named the Mather-Hodge Funeral Home), located at 40 Vandeventer Avenue, Princeton, New Jersey. Helene Weyl was cremated on September 6, 1948, at the Ewing Cemetery & Crematory, 78 Scotch Road, Trenton (Mercer County), New Jersey.
^For additional information on Helene Weyl, including a bibliography of her translations, published works, and manuscripts, see the following link: "In Memoriam Helene Weyl"Archived 2020-02-05 at the Wayback Machine by Hermann Weyl. This document, which is one of the items in the Hermann Weyl Collection at the Leo Baeck Institute in New York City, was written by Hermann Weyl at the end of June 1948, about nine weeks before Helene died on September 5, 1948, in Princeton, New Jersey. The first sentence in this document reads as follows: "Eine Skizze, nicht so sehr von Hellas, als von unserem gemeinsamen Leben, niedergeschrieben Ende Juni 1948." ("A sketch, not so much of Hella's life as of our common life, written at the end of June 1948.")
^137: Jung, Pauli, and the Pursuit of a Scientific Obsession (New York and London: W. W. Norton & Company, 2009), by Arthur I. Miller (p. 228).
^Hermann Weyl's cremains (ashes) are interred in an outdoor columbarium vault in the Princeton Cemetery at this location: Section 3, Block 04, Lot C1, Grave B15.
^Hermann Weyl; Peter Pesic (2009-04-20). Peter Pesic (ed.). Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics. Princeton University Press. p. 12. ISBN9780691135458. To use the apt phrase of his son Michael, 'The Open World' (1932) contains "Hermann's dialogues with God" because here the mathematician confronts his ultimate concerns. These do not fall into the traditional religious traditions but are much closer in spirit to Spinoza's rational analysis of what he called "God or nature," so important for Einstein as well. ...In the end, Weyl concludes that this God "cannot and will not be comprehended" by the human mind, even though "mind is freedom within the limitations of existence; it is open toward the infinite." Nevertheless, "neither can God penetrate into man by revelation, nor man penetrate to him by mystical perception."
^Gurevich, Yuri. "Platonism, Constructivism and Computer Proofs vs Proofs by Hand", Bulletin of the European Association of Theoretical Computer Science, 1995. This paper describes a letter discovered by Gurevich in 1995 that documents the bet. It is said that when the friendly bet ended, the individuals gathered cited Pólya as the victor (with Kurt Gödel not in concurrence).
ed. K. Chandrasekharan, Hermann Weyl, 1885â1985, Centenary lectures delivered by C. N. Yang, R. Penrose, A. Borel, at the ETH ZĂŒrich Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo â 1986, published for the Eidgenössische Technische Hochschule, ZĂŒrich.
Deppert, Wolfgang et al., eds., Exact Sciences and their Philosophical Foundations. VortrÀge des Internationalen Hermann-Weyl-Kongresses, Kiel 1985, Bern; New York; Paris: Peter Lang 1988,
Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.
Thomas Hawkins, Emergence of the Theory of Lie Groups, New York: Springer, 2000.
Kilmister, C. W. (October 1980), "Zeno, Aristotle, Weyl and Shuard: two-and-a-half millennia of worries over number", The Mathematical Gazette, 64 (429), The Mathematical Gazette, Vol. 64, No. 429: 149â158, doi:10.2307/3615116, JSTOR3615116, S2CID125725659.
In connection with the WeylâPĂłlya bet, a copy of the original letter together with some background can be found in: PĂłlya, G. (1972). "Eine Erinnerung an Hermann Weyl". Mathematische Zeitschrift. 126 (3): 296â298. doi:10.1007/BF01110732. S2CID118945480.
Erhard Scholz; Robert Coleman; Herbert Korte; Hubert Goenner; Skuli Sigurdsson; Norbert Straumann eds. Hermann Weyl's Raum â Zeit â Materie and a General Introduction to his Scientific Work (Oberwolfach Seminars) (ISBN3-7643-6476-9) Springer-Verlag New York, New York, N.Y.