View source for Biography:Hermann Weyl

{{Short description|German mathematician (1885–1955)}}
{{Infobox scientist
| birth_name        = Hermann Klaus Hugo Weyl
| image             = Hermann_Weyl_ETH-Bib_Portr_00890.jpg
| image_size        =
| birth_date        = {{Birth date|1885|11|9|df=y}}
| birth_place       = Elmshorn, [[Place:German Empire|German Empire]]
| death_date        = {{Death date and age|1955|12|8|1885|11|9|df=y}}
| death_place       = [[Place:Zürich|Zürich]], {{wipe|Switzerland}}
| residence         = 
| citizenship       = 
| fields            = [[Pure mathematics]], [[Physics:Mathematical physics|Mathematical physics]], Foundations of Mathematics
| workplaces        = Institute for Advanced Study<br />University of Göttingen<br />ETH Zürich
| alma_mater        = University of Munich<br />University of Göttingen
| thesis_title      = Singuläre Integralgleichungen mit besonder Berücksichtigung des Fourierschen Integraltheorems
| thesis_url        = https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/34190/31151014634046.pdf.
| thesis_year       = 1908
| academic_advisors = 
| doctoral_students = {{ublist|Alexander Weinstein|[[Biography:Julius Adams Stratton|Julius Adams Stratton]]}}
| notable_students  = [[Biography:Saunders Mac Lane|Saunders Mac Lane]]
| known_for         = List of topics named after Hermann Weyl<br />Ontic structural realism<ref>[http://plato.stanford.edu/entries/structural-realism/#Rel "Structural Realism"]: entry by James Ladyman in the ''[[Stanford Encyclopedia of Philosophy]]''.</ref><br />[[Physics:Wormhole|Wormhole]]
| signature         = Hermann Weyl signature.svg
| spouses           = Friederike Bertha Helene Joseph (nickname "Hella") (1893–1948)<br />Ellen Bär (née Lohnstein) (1902–1988)
| children          = [[Biography:Fritz Joachim Weyl|Fritz Joachim Weyl]] (1915–1977)<br />Michael Weyl (1917–2011)
| footnotes         =  
}}

'''Hermann Klaus Hugo Weyl''', {{post-nominals|post-noms=ForMemRS}}<ref name="frs"/> ({{IPA-de|vaɪl|lang}}; 9 November 1885 – 8 December 1955) was a German [[Biography:Mathematician|mathematician]], theoretical physicist, logician and [[Biography:Philosopher|philosopher]]. Although much of his working life was spent in [[Place:Zürich|Zürich]], {{wipe|Switzerland}}, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]], [[Biography:David Hilbert|David Hilbert]] and [[Biography:Hermann Minkowski|Hermann Minkowski]].

His research has had major significance for [[Physics:Theoretical physics|theoretical physics]] as well as purely mathematical disciplines such as [[Number theory|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.<ref>{{MacTutor Biography|id=Weyl}}</ref><ref>{{MathGenealogy|id=7373}}</ref>

Weyl contributed to an exceptionally<ref name = "Dyson"/> wide range of mathematical fields, including works on [[Space|space]], [[Time|time]], [[Physics:Matter|matter]], [[Philosophy:Philosophy|philosophy]], [[Logic|logic]], [[Symmetry|symmetry]] and the [[History of mathematics|history of mathematics]]. He was one of the first to conceive of combining [[Physics:General relativity|general relativity]] with the laws of [[Physics:Electromagnetism|electromagnetism]]. [[Biography:Freeman Dyson|Freeman Dyson]] wrote that Weyl alone bore comparison with the "last great universal mathematicians of the nineteenth century", [[Biography:Henri Poincaré|Poincaré]] and [[Biography:David Hilbert|Hilbert]].<ref name = "Dyson">{{cite journal|date=10 March 1956|title=Prof. Hermann Weyl, For.Mem.R.S. |journal=Nature|volume=177|pages= 457–458|doi=10.1038/177457a0|doi-access=free|s2cid=216075495|author=[[Biography:Freeman Dyson|Freeman Dyson]]|issue=4506 |bibcode=1956Natur.177..457D |quote ="He alone could stand comparison with the last great universal mathematicians of the nineteenth century, Hilbert and Poincaré. ... Now he is dead, the contact is broken, and our hopes of comprehending the physical universe by a direct use of creative mathematical imagination are for the time being ended." }}</ref> [[Biography:Michael Atiyah|Michael Atiyah]], in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him.<ref>{{cite journal |last1=Atiyah |first1=Michael |title=An Interview With Michael Atiyah |journal=[[The Mathematical Intelligencer]] |date=1984 |volume=6 |issue=1 |page=19 |doi=10.1007/BF03024202 |s2cid=140298726 |url=https://link.springer.com/article/10.1007/BF03024202}}</ref>

==Biography==
Hermann Weyl was born in Elmshorn, a small town near [[Place:Hamburg|Hamburg]], in {{wipe|Germany}}, and attended the Gymnasium Christianeum in Altona.<ref>{{cite journal |first=Bernd |last=Elsner |title=Die Abiturarbeit Hermann Weyls |journal=Christianeum |volume=63 |issue=1 |year=2008 |pages=3–15 }}</ref> His father, Ludwig Weyl, was a banker; whereas his mother, Anna Weyl (née Dieck), came from a wealthy family.<ref>{{cite book |last=James |first=Ioan |date=2002 |title=Remarkable Mathematicians |url= |location= |publisher=Cambridge University Press |page= 345 |isbn=978-0-521-52094-2}}</ref>

From 1904 to 1908, he studied mathematics and physics in both Göttingen and [[Place:Munich|Munich]]. His doctorate was awarded at the University of Göttingen under the supervision of [[Biography:David Hilbert|David Hilbert]], whom he greatly admired.

In September 1913, in Göttingen, Weyl married Friederike Bertha Helene Joseph (March 30, 1893<ref>[http://www.matrikel.uzh.ch/active/static/28236.htm Universität Zũrich Matrikeledition]</ref> – September 5, 1948<ref>[https://archive.org/details/hermannweylf001] 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.</ref>) who went by the name Helene (nickname "Hella"). Helene was a daughter of Dr. Bruno Joseph (December 13, 1861 – June 10, 1934), a physician who held the position of Sanitätsrat in Ribnitz-Damgarten, Germany. Helene was a philosopher (she was a disciple of phenomenologist [[Biography:Edmund Husserl|Edmund Husserl]]) and a translator of Spanish literature into German and English (especially the works of Spanish philosopher [[Biography:José Ortega y Gasset|José Ortega y Gasset]]).<ref>For additional information on Helene Weyl, including a bibliography of her translations, published works, and manuscripts, see the following link: [http://digital.cjh.org/R/JVSF47K3SQLELY84UGG7YPFA37DG678XF613V1EL1lAVK8UXA2-03415?func=dbin-jump-full&object_id=418384&local_base=GEN01&pds_handle=GUEST "In Memoriam Helene Weyl"] {{Webarchive|url=https://web.archive.org/web/20200205130420/http://digital.cjh.org/R/JVSF47K3SQLELY84UGG7YPFA37DG678XF613V1EL1lAVK8UXA2-03415?func=dbin-jump-full&object_id=418384&local_base=GEN01&pds_handle=GUEST |date=2020-02-05 }} 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.")</ref>  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, [[Biography:Fritz Joachim Weyl|Fritz Joachim Weyl]] (February 19, 1915 – July 20, 1977) and Michael Weyl (September 15, 1917 – March 19, 2011),<ref>[https://www.washingtonpost.com/local/obituaries/michael-weyl-foreign-service-officer/2011/04/04/AFRUXBlC_story.html WashingtonPost.com]</ref> both of whom were born in Zürich, 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 [[Biography:Oswald Veblen|Oswald Veblen]] and [[Biography:Richard Courant|Richard Courant]].<ref>''In Memoriam Helene Weyl'' (1948) by Fritz Joachim Weyl. See: (i) http://www.worldcat.org/oclc/724142550 and (ii) http://d-nb.info/993224164</ref> In 1950. Hermann married sculptor Ellen Bär (née Lohnstein) (April 17, 1902 – July 14, 1988),<ref>[http://www.artist-finder.com/stuff/database/info_popup.php?menu=artist&info=80111 artist-finder.com]</ref> who was the widow of professor Richard Josef Bär (September 11, 1892 – December 15, 1940)<ref>[http://www.tribalpages.com/tribe/familytree?uid=jeffreymarx&surname=Baer Ellen Lohnstein and Richard Josef Bär were married on September 14, 1922 in Zürich, Switzerland.]</ref> of Zürich.

After taking a teaching post for a few years, Weyl left Göttingen in 1913 for Zürich to take the chair of mathematics<ref>Weyl went to ETH Zürich in 1913 to fill the professorial chair vacated by the retirement of Carl Friedrich Geiser.</ref> at the ETH Zürich, where he was a colleague of [[Biography:Albert Einstein|Albert Einstein]], who was working out the details of the theory of [[Physics:General relativity|general relativity]]. Einstein had a lasting influence on Weyl, who became fascinated by mathematical physics. In 1921, Weyl met [[Biography:Erwin Schrödinger|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.<ref>{{cite book |first=Walter |last=Moore |title=Schrödinger: Life and Thought |publisher=Cambridge University Press |year=1989 |isbn=0-521-43767-9 |url={{Google books |plainurl=yes |id=m-YF1g1KWLoC |page=175 }} |pages=175–176 }}</ref><ref>[https://familysearch.org/pal:/MM9.1.1/QVQC-RNQH] Ruth Georgie Erica March was born on May 30, 1934 in Oxford, England, but—according to the records presented here—it appears that her birth wasn't "registered" with the British authorities until the 3rd registration quarter (the July–August–September quarter) of the year 1934. Ruth's actual, biological father was Erwin Schrödinger (1887–1961), and her mother was Hildegunde March (née Holzhammer) (born 1900), wife of Austrian physicist Arthur March (February 23, 1891 – April 17, 1957). Hildegunde's friends often called her "Hilde" or "Hilda" rather than Hildegunde. Arthur March was Erwin Schrödinger's assistant at the time of Ruth's birth. The reason Ruth's surname is March (instead of Schrödinger) 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.</ref>

Weyl was a Plenary Speaker of the International Congress of Mathematicians (ICM) in 1928 at Bologna<ref>{{cite book|title=Atti del Congresso internazionale dei Matematici, Bologna, 1928|volume=Tomo I|pages=233–246|location=Bologna|publisher=N. Zanichelli|year=1929|chapter=''Kontinuierliche Gruppen und ihre Darstellung durch lineare Transformationen'' von H. Weyl|isbn=9783540043881|chapter-url=https://books.google.com/books?id=vzMKd3IlqswC&pg=PA189}}</ref> and an Invited Speaker of the ICM in 1936 at [[Place:Oslo|Oslo]]. He was elected a fellow of the [[Organization:American Physical Society|American Physical Society]] in 1928,<ref>{{cite web|title=APS Fellow Archive|url=https://www.aps.org/programs/honors/fellowships/archive-all.cfm?initial=&year=1928&unit_id=&institution=Princeton+University}}</ref> a member of the American Academy of Arts and Sciences in 1929,<ref>{{Cite web |date=2023-02-09 |title=Hermann Weyl |url=https://www.amacad.org/person/hermann-weyl |access-date=2023-06-05 |website=American Academy of Arts & Sciences |language=en}}</ref> a member of the American Philosophical Society in 1935,<ref>{{Cite web |title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=Hermann+Weyl&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=2023-06-05 |website=search.amphilsoc.org}}</ref> and a member of the National Academy of Sciences in 1940.<ref>{{cite web|title=Hermann Weyl|website=National Academy of Sciences|url=http://www.nasonline.org/member-directory/deceased-members/20000916.html}}</ref> For the academic year 1928–1929, he was a visiting professor at [[Organization:Princeton University|Princeton University]],<ref>{{cite journal|author=Shenstone, Allen G.|title=Princeton & Physics|date=24 February 1961|journal=Princeton Alumni Weekly|volume=61|pages= 7–8 of article on pp. 6–13 & p. 20|url=https://books.google.com/books?id=8hJbAAAAYAAJ&pg=PA363}}</ref> where he wrote a paper, "On a problem in the theory of groups arising in the foundations of infinitesimal geometry," with [[Biography:Howard P. Robertson|Howard P. Robertson]].<ref>{{cite journal|title=On a problem in the theory of groups arising in the foundations of infinitesimal geometry|author=Robertson, H. P.|author2=Weyl, H.|journal=Bull. Amer. Math. Soc.|volume=35|issue=5|year=1929|pages=686–690|doi=10.1090/S0002-9904-1929-04801-8|doi-access=free}}</ref>

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 December 8, 1955, while living in Zürich.

Weyl was cremated in Zürich on December 12, 1955.<ref>''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).</ref> His ashes 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.<ref>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.</ref> 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.<ref>{{cite book|title=Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics|publisher=Princeton University Press|isbn=9780691135458|author1=Hermann Weyl |author2=Peter Pesic |editor=Peter Pesic|page=12|quote=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."|date=2009-04-20}}</ref>

==Contributions==
[[Image:Hermann Weyl, Ernst Peschl.jpg|thumb|right|Hermann Weyl (left) and [[Biography:Ernst Peschl|Ernst Peschl]] (right)]]

===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 and eigenvectors|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 law#Weyl conjecture|Weyl conjecture]]. These works started an important domain—[[Asymptote|asymptotic]] distribution of eigenvalues—of modern analysis.

===Geometric foundations of manifolds and physics===
{{further|Physics:Weyl transformation|Weyl tensor}}

In 1913, Weyl published ''Die Idee der Riemannschen Fläche'' (''The Concept of a Riemann Surface''), which gave a unified treatment of [[Riemann surface]]s. In it Weyl utilized point set topology, in order to make Riemann surface theory more rigorous, a model followed in later work on [[Manifold|manifold]]s. 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 [[Physics:General relativity|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 [[Physics:Gauge theory|gauge theory]]. Weyl's gauge theory was an unsuccessful attempt to model the [[Physics:Electromagnetic field|electromagnetic field]] and the [[Physics:Gravitational field|gravitational field]] as geometrical properties of [[Physics:Spacetime|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.<ref>1929. "Elektron und Gravitation I", ''Zeitschrift Physik'', 56, pp 330–352.</ref>

His overall approach in physics was based on the [[Philosophy:Phenomenology|phenomenological]] philosophy of [[Biography:Edmund Husserl|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 [[Biography:Gottlob Frege|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 group|compact group]]s, in terms of [[Representation theory|matrix representation]]s. In the compact Lie group case he proved a fundamental character formula.

These results are foundational in understanding the symmetry structure of [[Physics:Quantum mechanics|quantum mechanics]], which he put on a group-theoretic basis. This included [[Physics:Spinor|spinor]]s. Together with the [[Physics:Mathematical formulation of quantum mechanics|mathematical formulation of quantum mechanics]], in large measure due to [[Biography:John von Neumann|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 group]]s and [[Lie algebra]]s became a mainstream part both of [[Pure mathematics|pure mathematics]] and [[Physics:Theoretical physics|theoretical physics]].

His book ''[[The Classical Groups]]'' reconsidered [[Invariant theory|invariant theory]]. It covered [[Symmetric group|symmetric group]]s, [[General linear group|general linear group]]s, [[Orthogonal group|orthogonal group]]s, and [[Symplectic group|symplectic group]]s and results on their [[Invariant (mathematics)|invariants]] and [[Group representation|representations]].

===Harmonic analysis and analytic number theory===
Weyl also showed how to use [[Exponential sum|exponential sum]]s in [[Diophantine approximation|diophantine approximation]], with his criterion for uniform distribution mod 1, which was a fundamental step in [[Analytic number theory|analytic number theory]]. This work applied to the [[Riemann zeta function]], as well as [[Additive number theory|additive number theory]]. It was developed by many others.

===Foundations of mathematics===
In ''The Continuum'' Weyl developed the logic of [[Impredicativity|predicative analysis]] using the lower levels of [[Biography:Bertrand Russell|Bertrand Russell]]'s [[Ramified theory of types|ramified theory of types]]. He was able to develop most of classical calculus, while using neither the [[Axiom of choice|axiom of choice]] nor [[Proof by contradiction|proof by contradiction]], and avoiding [[Biography:Georg Cantor|Georg Cantor]]'s [[Infinite set|infinite set]]s. Weyl appealed{{clarify|date=August 2021}} 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|intuitionism]] of Brouwer. In ''The Continuum'', the constructible points exist as discrete entities. Weyl wanted a [[Continuum (set theory)|continuum]] that was not an aggregate of points. He wrote a controversial article proclaiming, for himself and L. E. J. Brouwer, a "revolution."<ref>[https://gdz.sub.uni-goettingen.de/id/PPN266833020_0010 "Über die neue Grundlagenkrise der Mathematik" ](About the new foundational crisis of mathematics), H. Weyl, Springer Mathematische Zeitschrift 1921 Vol. 10, p.45 (22 pages)</ref> This article was far more influential in propagating intuitionistic views than the original works of Brouwer himself.

[[Biography:George Pólya|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 number|real number]]s, [[Set (mathematics)|sets]], and countability, and moreover, that asking about the [[Truth|truth]] or falsity of the [[Completeness of the real numbers|least upper bound property]] of the real numbers was as meaningful as asking about truth of the basic assertions of [[Biography:Georg Wilhelm Friedrich Hegel|Hegel]] on the philosophy of nature.<ref>Gurevich, Yuri. [http://research.microsoft.com/~gurevich/Opera/123.pdf "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 [[Biography:Kurt Gödel|Kurt Gödel]] not in concurrence).</ref> 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 [[Philosophy:Phenomenology|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 [[Biography:Ernst Cassirer|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 [[Philosophy:Classical logic|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|law of excluded middle]] not being generally affirmable — mathematics developed within avoids the “unbearable awkwardness” to which Weyl refers above."

===Weyl equation===
{{main|Physics:Weyl equation}}
In 1929, Weyl proposed an equation, known as [[Physics:Weyl equation|Weyl equation]], for use in a replacement to [[Physics:Dirac equation|Dirac equation]]. This equation describes massless [[Physics:Fermion|fermion]]s. A normal Dirac fermion could be split into two Weyl fermions or formed from two Weyl fermions. [[Physics:Neutrino|Neutrino]]s were once thought to be Weyl fermions, but they are now known to have mass. Weyl fermions are sought after for electronics applications. [[Physics:Quasiparticle|Quasiparticle]]s that behave as Weyl fermions were discovered in 2015, in a form of crystals known as [[Physics:Weyl semimetal|Weyl semimetal]]s, a type of topological material.<ref name=I3Espectrum-2015-07-16>{{cite news |url= https://spectrum.ieee.org/tech-talk/semiconductors/materials/exotic-particles-could-lead-to-faster-electronics |title= Weyl Fermions Found, a Quasiparticle That Acts Like a Massless Electron |author= Charles Q. Choi |date= 16 July 2015 |journal= IEEE Spectrum |publisher= IEEE }}</ref><ref name=ScienceDaily-2015-07-16>{{cite web |url= https://www.sciencedaily.com/releases/2015/07/150716160325.htm |title= After 85-year search, massless particle with promise for next-generation electronics found |date= 16 July 2015 |website= Science Daily }}</ref><ref name="10.1126/science.aaa9297"> {{cite journal |title= Discovery of a Weyl Fermion semimetal and topological Fermi arcs |doi= 10.1126/science.aaa9297 |author1=Su-Yang Xu |author2=Ilya Belopolski |author3=Nasser Alidoust |author4=Madhab Neupane |author5=Guang Bian |author6=Chenglong Zhang |author7=Raman Sankar |author8=Guoqing Chang |author9=Zhujun Yuan |author10=Chi-Cheng Lee |author11=Shin-Ming Huang |author12=Hao Zheng |author13=Jie Ma |author14=Daniel S. Sanchez |author15=BaoKai Wang |author16=Arun Bansil |author17=Fangcheng Chou |author18=Pavel P. Shibayev |author19=Hsin Lin |author20=Shuang Jia |author21=M. Zahid Hasan |journal= Science |arxiv = 1502.03807 |bibcode = 2015Sci...349..613X |volume=349 |issue= 6248 |pages=613–617|year= 2015 |pmid=26184916 |s2cid= 206636457 }}</ref>

==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. {{harvtxt|Weyl|1939b|loc=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, p144; 1952

==Bibliography==
* 1911. ''[http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PID=GDZPPN002502526 Über die asymptotische Verteilung der Eigenwerte]'', Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 110–117 (1911).
* 1913.  ''Die Idee der Riemannschen Flāche'',<ref>{{cite journal|author=Moulton, F. R.|journal=Bull. Amer. Math. Soc.|year=1914|volume=20|issue=7|pages=384–387|title=Review: ''Die Idee der Riemannschen Fläche'' by Hermann Weyl|url=https://www.ams.org/journals/bull/1914-20-07/S0002-9904-1914-02505-4/S0002-9904-1914-02505-4.pdf|doi=10.1090/s0002-9904-1914-02505-4|doi-access=free}}</ref> 2d 1955. ''The Concept of a Riemann Surface''. Addison–Wesley.
* 1918. ''Das Kontinuum'', trans. 1987 ''The Continuum : A Critical Examination of the Foundation of Analysis''. {{isbn|0-486-67982-9}}
* 1918. ''[https://archive.org/details/raumzeitmateriev00weyl Raum, Zeit, Materie]''. 5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922 ''[https://archive.org/details/spacetimematter00weyluoft 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 [[Biography:Frank Wilczek|Frank Wilczek]], Princeton University Press, 2009, {{isbn|978-0-691-14120-6}}.
* 1928. ''Gruppentheorie und Quantenmechanik''. transl. by H. P. Robertson, ''[https://books.google.com/books?isbn=0486602699 The Theory of Groups and Quantum Mechanics]'', 1931, rept. 1950 Dover. {{isbn|0-486-60269-9}}
* [[File:Weyl-2.jpg|alt=The Theory of Groups and Quantum Mechanics (translated from the second, revised German edition by Howard P. Robertson|thumb|234x234px|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 330–352. – introduction of the vierbein into [[Physics:General relativity|GR]]
* 1931: ''[https://libserv.aip.org/ipac20/ipac.jsp?session=167NT8P289225.1867993&profile=rev-nbl&source=~!horizon&view=subscriptionsummary&uri=full=3100006~!44625~!2&ri=5&aspect=power&menu=search&ipp=20&spp=20&staffonly=&term=%22The+Theory+of+Groups+and+Quantum+Mechanics%22&index=.GW&uindex=&aspect=power&menu=search&ri=5 The Theory of Groups and Quantum Mechanics]''," London, : Methuen & Co., [1931]
* 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'': 400–415.
* 1935. ''Elementary Theory of Invariants''.
* 1935. ''The structure and representation of continuous groups: Lectures at Princeton university during 1933–34''.
*{{Citation | last1=Weyl | first1=Hermann |  title=The Classical Groups. Their Invariants and Representations | url=https://books.google.com/books?id=2twDDAAAQBAJ&q=The+Classical+Groups%3A+Their+Invariants+and+Representations | publisher=Princeton University Press | isbn=978-0-691-05756-9 | year=1939 | mr=0000255}}<ref>{{cite journal|author=Jacobson, N.|title=Review: ''The Classical Groups'' by Hermann Weyl|journal=Bull. Amer. Math. Soc.|year=1940|volume=46|issue=7|pages=592–595|url=https://www.ams.org/journals/bull/1940-46-07/S0002-9904-1940-07236-2/S0002-9904-1940-07236-2.pdf|doi=10.1090/s0002-9904-1940-07236-2|doi-access=free}}</ref>
*{{Citation | last1=Weyl | first1=Hermann |  title=Invariants | year=1939b | journal=Duke Mathematical Journal | issn=0012-7094 | volume=5 | pages=489–502 | 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=115–139 | doi=10.1090/S0002-9904-1950-09369-0 | url=https://projecteuclid.org/download/pdf_1/euclid.bams/1183514553| doi-access=free }} (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.
<gallery>
File:Weyl, Hermann – Raum, Zeit, Materie, 1922 – BEIC 3898041.jpg|''Temps, espace, matière'' (French, 1922)
File:Weyl-17.jpg|''Space, Time, Matter'' (English, 1922: translated from German from Henry L. Brose)
File:Weyl-9.jpg|''Raum - Zeit - Materie'' (German, 1918)
</gallery>

==See also==
===Topics named after Hermann Weyl===<!--Please respect alphabetical order-->
{{Div col|colwidth=25em}}
* Majorana–Weyl spinor
* [[Peter–Weyl theorem]]
* [[Schur–Weyl duality]]
* [[Weyl algebra]]
* Weyl basis of the [[Gamma matrices|gamma matrices]]
* Weyl chamber
* [[Weyl character formula]]
* [[Physics:Weyl equation|Weyl equation]], a relativistic wave equation
* [[Physics:Weyl expansion|Weyl expansion]]
* Weyl fermion
* Weyl gauge
* [[Physics:Conformal gravity|Weyl gravity]]
* Weyl notation
* Weyl quantization
* Weyl spinor
* Weyl sum, a type of [[Exponential sum|exponential sum]]
* Weyl symmetry: see Weyl transformation
* [[Weyl tensor]]
* Weyl transform
* [[Physics:Weyl transformation|Weyl transformation]]
* [[Weyl–Schouten theorem]]
* Weyl's criterion
* Weyl's lemma on hypoellipticity
* [[Weyl's lemma (Laplace equation)|Weyl's lemma]] on the "very weak" form of the Laplace equation
{{div col end}}

==References==
{{Reflist}}

==Further reading==
* ed. K. Chandrasekharan, ''Hermann Weyl, 1885–1985, Centenary lectures delivered by C. N. Yang, R. Penrose, A. Borel, at the ETH Zürich''  Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo – 1986,  published for the Eidgenössische Technische Hochschule, Zürich.
*Deppert, Wolfgang et al., eds., ''Exact Sciences and their Philosophical Foundations. Vorträge des Internationalen Hermann-Weyl-Kongresses, Kiel 1985'', Bern; New York; Paris: Peter Lang 1988,
*[[Biography:Ivor Grattan-Guinness|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=149–158| doi=10.2307/3615116| publisher=The Mathematical Gazette, Vol. 64, No. 429| jstor=3615116|s2cid=125725659 | postscript=. }}
*In connection with the Weyl–Pólya bet, a copy of the original letter together with some background can be found in: {{Cite journal | last1 = Pólya | first1 = G. | title = Eine Erinnerung an Hermann Weyl | journal = Mathematische Zeitschrift | volume = 126 | issue = 3 | pages = 296–298 | year = 1972 | doi = 10.1007/BF01110732| s2cid = 118945480 }}
*Erhard Scholz; Robert Coleman; Herbert Korte; Hubert Goenner; Skuli Sigurdsson; Norbert Straumann eds. ''Hermann Weyl's Raum – Zeit – 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, Schrödinger, and Weyl in Exile." In Mitchell G. Ash, and Alfons Söllner, eds., ''Forced Migration and Scientific Change: Emigré German-Speaking Scientists and Scholars after 1933'' (Washington, D.C.: German Historical Institute and New York: Cambridge University Press, 1996), pp.&nbsp;48–70.
*{{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==
* [http://www.nap.edu/readingroom/books/biomems/hweyl.html National Academy of Sciences biography]
* [[Biography:John Lane Bell|Bell, John L.]] ''[http://publish.uwo.ca/~jbell/Hermann%20Weyl.pdf Hermann Weyl on intuition and the continuum]''
* Feferman, Solomon. [http://math.stanford.edu/~feferman/papers/DasKontinuum.pdf "Significance of Hermann Weyl's das Kontinuum"]
* Straub, William O. [http://www.weylmann.com Hermann Weyl Website]
* {{Gutenberg author | id=41982| name=Hermann Weyl}}
* {{Internet Archive author |sname=Hermann Weyl}}

{{relativity}}

{{DEFAULTSORT:Weyl, Hermann Klaus Hugo}}
[[Category:Differential geometers]]
[[Category:Number theorists]]
[[Category:Linear algebraists]]

{{Sourceattribution|Hermann Weyl}}