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Hermann Weyl
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Biography
Weyl was born in Elmshorn, a small town near Hamburg, in Germany, and attended the gymnasium Christianeum in Altona.JOURNAL, Bernd, Elsner, Die Abiturarbeit Hermann Weyls, Christianeum, 63, 1, 2008, 3â€“15, From 1904 to 1908 he studied mathematics and physics in both GÃ¶ttingen and Munich. His doctorate was awarded at the University of GÃ¶ttingen under the supervision of David Hilbert whom he greatly admired.In September 1913 in GÃ¶ttingen, Weyl married Friederike Bertha Helene Joseph (March 30, 1893UniversitÃ¤t ZÅ©rich Matrikeledition â€“ September 5, 1948weblink 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 [1948], mittags 12 Uhr." ("Hella died at 12:00 Noon on September 5 [1948]"). 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.) who went by the name Helene (nickname "Hella"). Helene was a daughter of Dr. Bruno Joseph (December 13, 1861 â€“ June 10, 1934), a physician who held the position of SanitÃ¤tsrat 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 JosÃ© Ortega y Gasset).For additional information on Helene Weyl, including a bibliography of her translations, published works, and manuscripts, see the following link: "In Memoriam Helene Weyl" 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.") 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 (February 19, 1915 â€“ July 20, 1977) and Michael Weyl (September 15, 1917 â€“ March 19, 2011),WashingtonPost.com both of whom were born in ZÃ¼rich, Switzerland. Helene died in Princeton, New Jersey on September 5, 1948. A memorial service in her honor was held in Princeton on September 9, 1948. Speakers at her memorial service included her son Fritz Joachim Weyl and mathematicians Oswald Veblen and Richard Courant.In Memoriam Helene Weyl (1948) by Fritz Joachim Weyl. See: (i)weblink and (ii)weblink In 1950 Hermann married sculptress Ellen BÃ¤r (nÃ©e Lohnstein) (April 17, 1902 â€“ July 14, 1988),artist-finder.com who was the widow of professor Richard Josef BÃ¤r (September 11, 1892 â€“ December 15, 1940)Ellen Lohnstein and Richard Josef BÃ¤r were married on September 14, 1922 in ZÃ¼rich, Switzerland. of ZÃ¼rich.After taking a teaching post for a few years, Weyl left GÃ¶ttingen in 1913 for ZÃ¼rich to take the chair of mathematicsWeyl went to ETH ZÃ¼rich in 1913 to fill the professorial chair vacated by the retirement of Carl Friedrich Geiser. at the ETH Zurich, 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 SchrÃ¶dinger, a theoretical physicist who at the time was a professor at the University of ZÃ¼rich. They were to become close friends over time. Weyl had some sort of childless love affair with SchrÃ¶dinger's wife Annemarie (Anny) SchrÃ¶dinger (nÃ©e 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.BOOK, Walter, Moore, SchrÃ¶dinger: Life and Thought, Cambridge University Press, 1989, 0-521-43767-9, {{Google books, yes, m-YF1g1KWLoC, 175, |pages=175â€“176 }}weblink Ruth Georgie Erica March was born on May 30, 1934 in Oxford, England, butâ€”according to the records presented hereâ€”it appears that her birth wasn't "registered" with the British authorities until the 3rd registration quarter (the Julyâ€“Augustâ€“September quarter) of the year 1934. Ruth's actual, biological father was Erwin SchrÃ¶dinger (1887â€“1961), and her mother was Hildegunde March (nÃ©e Holzhammer) (born 1900), wife of Austrian physicist Arthur March (February 23, 1891 â€“ April 17, 1957). Hildegunde's friends often called her "Hilde" or "Hilda" rather than Hildegunde. Arthur March was Erwin SchrÃ¶dinger's assistant at the time of Ruth's birth. The reason Ruth's surname is March (instead of SchrÃ¶dinger) is because Arthur had agreed to be named as Ruth's father on her birth certificate, even though he wasn't her biological father. Ruth married the engineer Arnulf Braunizer in May 1956, and they have lived in Alpbach, Austria for many years. Ruth has been very active as the sole administrator of the intellectual (and other) property of her father Erwin's estate, which she manages from Alpbach.Weyl was a Plenary Speaker of the International Congress of Mathematicians (ICM) in 1928 at BolognaBOOK, Atti del Congresso internazionale dei Matematici, Bologna, 1928, Tomo I, 233â€“246, Bologna, N. Zanichelli, 1929, Kontinuierliche Gruppen und ihre Darstellung durch lineare Transformationen von H. Weyl,weblink and an Invited Speaker of the ICM in 1936 at Oslo. For the academic year 1928â€“1929 he was a visiting professor at Princeton University,JOURNAL, Shenstone, Allen G., Allen Shenstone, Princeton & Physics, 24 February 1961, Princeton Alumni Weekly, 61, 7â€“8 of article on pp. 6â€“13 & p. 20,weblink where he wrote a paper with Howard P. Robertson.JOURNAL, On a problem in the theory of groups arising in the foundations of infinitesimal geometry, Robertson, H. P., Weyl, H., Bull. Amer. Math. Soc., 35, 5, 1929, 686â€“690, 10.1090/S0002-9904-1929-04801-8, Weyl left ZÃ¼rich in 1930 to become Hilbert's successor at GÃ¶ttingen, 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 ZÃ¼rich, and died from a heart attack on December 8, 1955 while living in ZÃ¼rich.Weyl was cremated in Zurich on December 12, 1955.137: Jung, Pauli, and the Pursuit of a Scientific Obsession (New York and London: W. W. Norton & Company, 2009), by Arthur I. Miller (p. 228). His cremains remained in private hands{{Unreliable source?|date=March 2019}} until 1999, at which time they were interred in an outdoor columbarium vault in the Princeton Cemetery.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. The remains of Hermann's son Michael Weyl (1917â€“2011) are interred right next to Hermann's ashes in the same columbarium vault.Weyl was a pantheist.BOOK, Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics, Princeton University Press, 9780691135458, Hermann Weyl, Peter Pesic, Peter Pesic, 12, 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.",Contributions
{{refimprove section|date=February 2017}}missing image!
- Hermann Weyl, Ernst Peschl.jpg -
right|Hermann Weyl (left) and Ernst Peschl (right).
- Hermann Weyl, Ernst Peschl.jpg -
right|Hermann Weyl (left) and Ernst Peschl (right).
Distribution of eigenvalues
{{further|Weyl law|Weyl law#Weyl conjecture}}In 1911 Weyl published Ãœber 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 domainâ€”asymptotic distribution of eigenvaluesâ€”of modern analysis.Geometric foundations of manifolds and physics
{{further|Weyl transformation|Weyl tensor}}In 1913, Weyl published Die Idee der Riemannschen FlÃ¤che (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 GÃ¶ttingen 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. In 1929, Weyl introduced the concept of the vierbein into general relativity.1929. "Elektron und Gravitation I", Zeitschrift Physik, 56, pp 330â€“352.His overall approach in physics was based on the phenomenological philosophy of Edmund Husserl, specifically Husserl's 1913 Ideen zu einer reinen PhÃ¤nomenologie und phÃ¤nomenologischen Philosophie. Erstes Buch: Allgemeine EinfÃ¼hrung in die reine PhÃ¤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.{{citation needed|date=January 2016}}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 betweenclassical 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
{{details|Weyl's criterion}}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 that, for himself and L. E. J. Brouwer, "We are the revolution." This article was far more influential in propagating intuitionistic views than the original works of Brouwer himself.George PÃ³lya and Weyl, during a mathematicians' gathering in ZÃ¼rich (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.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 PÃ³lya as the victor (with Kurt GÃ¶del not in concurrence). 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."Weyl fermions
In 1929, Weyl proposed a fermion for use in a replacement theory for relativity. This fermion would be a massless quasiparticle and carry electric charge. An electron 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 to solve some problems that electrons present. Such quasiparticles were discovered in 2015, in a form of crystals known as Weyl semimetals, a type of topological material.NEWS,weblink Weyl Fermions Found, a Quasiparticle That Acts Like a Massless Electron, Charles Q. Choi, 16 July 2015, IEEE Spectrum, IEEE, WEB,weblink After 85-year search, massless particle with promise for next-generation electronics found, 16 July 2015, Science Daily, JOURNAL, Discovery of a Weyl Fermion semimetal and topological Fermi arcs, 10.1126/science.aaa9297, Su-Yang Xu, Ilya Belopolski, Nasser Alidoust, Madhab Neupane, Guang Bian, Chenglong Zhang, Raman Sankar, Guoqing Chang, Zhujun Yuan, Chi-Cheng Lee, Shin-Ming Huang, Hao Zheng, Jie Ma, Daniel S. Sanchez, BaoKai Wang, Arun Bansil, Fangcheng Chou, Pavel P. Shibayev, Hsin Lin, Shuang Jia, M. Zahid Hasan, M. Zahid Hasan, Science, 1502.03807, 2015Sci...349..613X, 349, 6248, 613â€“617, 2015,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.
â€”Gesammelte Abhandlungenâ€”as 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. {{harvtxt|Weyl|1939b|loc=p. 500}}
Bibliography
- 1911. Ãœber die asymptotische Verteilung der Eigenwerte, Nachrichten der KÃ¶niglichen Gesellschaft der Wissenschaften zu GÃ¶ttingen, 110â€“117 (1911).
- 1913. Die Idee der Riemannschen FlÄche,JOURNAL, Moulton, F. R., Forest Ray Moulton, Bull. Amer. Math. Soc., 1914, 20, 7, 384â€“387, Review: Die Idee der Riemannschen FlÃ¤che by Hermann Weyl,weblink 10.1090/s0002-9904-1914-02505-4, 2d 1955. The Concept of a Riemann Surface. Addisonâ€“Wesley.
- 1918. Das Kontinuum, trans. 1987 The Continuum : A Critical Examination of the Foundation of Analysis. {{isbn|0-486-67982-9}}
- 1918. Raum, Zeit, Materie. 5 edns. to 1922 ed. with notes by JÅ«rgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922 Space Time Matter, Methuen, rept. 1952 Dover. {{isbn|0-486-60267-2}}.
- 1923. Mathematische Analyse des Raumproblems.
- 1924. Was ist Materie?
- 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, {{isbn|978-0-691-14120-6}}.
- 1928. Gruppentheorie und Quantenmechanik. transl. by H. P. Robertson, The Theory of Groups and Quantum Mechanics, 1931, rept. 1950 Dover. {{isbn|0-486-60269-9}}
- 1929. "Elektron und Gravitation I", Zeitschrift Physik, 56, pp 330â€“352. â€“ introduction of the vierbein into GR
- 1933. The Open World Yale, rept. 1989 Oxbow Press {{isbn|0-918024-70-6}}
- 1934. Mind and Nature U. of Pennsylvania Press.
- 1934. "On generalized Riemann matrices," Ann. Math. 35: 400â€“415.
- 1935. Elementary Theory of Invariants.
- 1935. The structure and representation of continuous groups: Lectures at Princeton university during 1933â€“34.
- {{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=The Classical Groups. Their Invariants and Representations | url=https://books.google.com/?id=2twDDAAAQBAJ&printsec=frontcover&dq=The+Classical+Groups:+Their+Invariants+and+Representationsv=onepage&q=The%20Classical%20Groups%3A%20Their%20Invariants%20and%20Representations&f=false | publisher=Princeton University Press | isbn=978-0-691-05756-9 | year=1939 | mr=0000255}}JOURNAL, Jacobson, N., Nathan Jacobson, Review: The Classical Groups by Hermann Weyl, Bull. Amer. Math. Soc., 1940, 46, 7, 592â€“595,weblink 10.1090/s0002-9904-1940-07236-2,
- {{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=Invariants | url=http://projecteuclid.org/euclid.dmj/1077491405 | year=1939b | journal=Duke Mathematical Journal | issn=0012-7094 | volume=5 | pages=489â€“502 | mr=0000030 | doi=10.1215/S0012-7094-39-00540-5 | issue=3}}
- 1940. Algebraic Theory of Numbers rept. 1998 Princeton U. Press. {{isbn|0-691-05917-9}}
- {{Citation | title=Ramifications, old and new, of the eigenvalue problem | last1 = Weyl |first1 = Hermann | journal=Bull. Amer. Math. Soc. | volume=56 | issue = 2 | year=1950 | pages=115â€“139 | doi=10.1090/S0002-9904-1950-09369-0 | url=http://dml.cz/bitstream/handle/10338.dmlcz/103099/AplMat_12-1967-4_2.pdf }} (text of 1948 Josiah Wilard Gibbs Lecture)
- 1952. Symmetry. Princeton University Press. {{isbn|0-691-02374-3}}
- 1968. in K. Chandrasekharan ed, Gesammelte Abhandlungen. Vol IV. Springer.
See also
Topics named after Hermann Weyl
{{Div col|colwidth=25em}}- Majoranaâ€“Weyl spinor
- Peterâ€“Weyl theorem
- Schurâ€“Weyl duality
- Weyl algebra
- Weyl basis of the gamma matrices
- Weyl chamber
- Weyl character formula
- Weyl equation, a relativistic wave equation
- Weyl fermion
- Weyl gauge
- Weyl gravity
- Weyl notation
- Weyl quantization
- Weyl spinor
- Weyl sum, a type of exponential sum
- Weyl symmetry: see Weyl transformation
- Weyl tensor
- Weyl transform
- Weyl transformation
- Weylâ€“Schouten theorem
- Weyl's criterion
- Weyl's lemma on hypoellipticity
- Weyl's lemma on the "very weak" form of the Laplace equation
References
{{Reflist}}Further reading
- ed. K. Chandrasekharan,Hermann Weyl, 1885â€“1985, Centenary lectures delivered by C. N. Yang, R. Penrose, A. Borel, at the ETH ZÃ¼rich Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo â€“ 1986, published for the EidgenÃ¶ssische Technische Hochschule, ZÃ¼rich.
- Deppert, Wolfgang et al., eds., Exact Sciences and their Philosophical Foundations. VortrÃ¤ge 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.
- {{Citation| last=Kilmister |first=C. W. |date=October 1980 |title=Zeno, Aristotle, Weyl and Shuard: two-and-a-half millennia of worries over number |journal=The Mathematical Gazette |volume=64 |issue=429 |pages=149â€“158| doi=10.2307/3615116| publisher=The Mathematical Gazette, Vol. 64, No. 429| jstor=3615116| postscript=. }}
- In connection with the Weylâ€“PÃ³lya bet, a copy of the original letter together with some background can be found in: JOURNAL, PÃ³lya, G., Eine Erinnerung an Hermann Weyl, Mathematische Zeitschrift, 126, 3, 296â€“298, 1972, 10.1007/BF01110732,
- 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) ({{isbn|3-7643-6476-9}}) Springer-Verlag New York, New York, N.Y.
- Skuli Sigurdsson. "Physics, Life, and Contingency: Born, SchrÃ¶dinger, and Weyl in Exile." In Mitchell G. Ash, and Alfons SÃ¶llner, eds., Forced Migration and Scientific Change: EmigrÃ© German-Speaking Scientists and Scholars after 1933 (Washington, D.C.: German Historical Institute and New York: Cambridge University Press, 1996), pp. 48â€“70.
- {{citation|first=Hermann|last=Weyl|editor=Peter Pesic|title=Levels of Infinity / Selected Writings on Mathematics and Philosophy|year=2012|publisher=Dover|isbn=978-0-486-48903-2}}
External links
{{commons category}}- National Academy of Sciences biography
- Bell, John L. Hermann Weyl on intuition and the continuum
- Feferman, Solomon. "Significance of Hermann Weyl's das Kontinuum"
- Straub, William O. Hermann Weyl Website
- {{Gutenberg author | id=Weyl,+Hermann | name=Hermann Weyl}}
- {{Internet Archive author |sname=Hermann Weyl}}
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