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Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
, theoretical physicist and
philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
. Although much of his working life was spent in
Zürich , neighboring_municipalities = Adliswil, Dübendorf, Fällanden, Kilchberg, Maur, Oberengstringen, Opfikon, Regensdorf, Rümlang, Schlieren, Stallikon, Uitikon, Urdorf, Wallisellen, Zollikon , twintowns = Kunming, San Francisco Z ...
,
Switzerland ). Swiss law does not designate a ''capital'' as such, but the federal parliament and government are installed in Bern, while other federal institutions, such as the federal courts, are in other cities (Bellinzona, Lausanne, Luzern, Neuchâtel ...
, and then
Princeton, New Jersey Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of w ...
, he is associated with the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
as well as purely mathematical disciplines such as
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
. He was one of the most influential mathematicians of the twentieth century, and an important member of the
Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...
during its early years. Weyl contributed to an exceptionally wide range of mathematical fields, including works on
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
,
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
,
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
,
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...
,
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
,
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and the history of mathematics. He was one of the first to conceive of combining
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
with the laws of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
. Freeman Dyson wrote that Weyl alone bore comparison with the "last great universal mathematicians of the nineteenth century", Poincaré and 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 Hamburg (, ; nds, label=Hamburg German, Low Saxon, Hamborg ), officially the Free and Hanseatic City of Hamburg (german: Freie und Hansestadt Hamburg; nds, label=Low Saxon, Friee un Hansestadt Hamborg),. is the List of cities in Germany by popul ...
, in
Germany Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwee ...
, 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 Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911. General information The ori ...
and
Munich Munich ( ; german: München ; bar, Minga ) is the capital and most populous city of the German state of Bavaria. With a population of 1,558,395 inhabitants as of 31 July 2020, it is the third-largest city in Germany, after Berlin and ...
. His doctorate was awarded at the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
under the supervision of David Hilbert, whom he greatly admired. In September 1913 in Göttingen, Weyl married Friederike Bertha Helene Joseph (March 30, 1893 – September 5, 1948) 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 Ribnitz-Damgarten () is a town in Mecklenburg-Vorpommern, Germany, situated on Lake Ribnitz (''Ribnitzer See''). Ribnitz-Damgarten is in the west of the district Vorpommern-Rügen. The border between the historical regions of Mecklenburg an ...
, Germany. Helene was a philosopher (she was a disciple of phenomenologist
Edmund Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...
) and a translator of Spanish literature into German and English (especially the works of Spanish philosopher
José Ortega y Gasset José Ortega y Gasset (; 9 May 1883 – 18 October 1955) was a Spanish philosopher and essayist. He worked during the first half of the 20th century, while Spain oscillated between monarchy, republicanism, and dictatorship. His philosoph ...
). 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 – July 20, 1977) and Michael Weyl (September 15, 1917 – March 19, 2011), 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
Oswald Veblen Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this wa ...
and
Richard Courant Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of r ...
. In 1950 Hermann married sculptress Ellen Bär (née Lohnstein) (April 17, 1902 – July 14, 1988), who was the widow of professor Richard Josef Bär (September 11, 1892 – December 15, 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 Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
, who was working out the details of the theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. Einstein had a lasting influence on Weyl, who became fascinated by mathematical physics. In 1921 Weyl met
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
, a theoretical physicist who at the time was a professor at the
University of Zürich The University of Zürich (UZH, german: Universität Zürich) is a public research university located in the city of Zürich, Switzerland. It is the largest university in Switzerland, with its 28,000 enrolled students. It was founded in 1833 f ...
. 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 Oxford () is a city in England. It is the county town and only city of Oxfordshire. In 2020, its population was estimated at 151,584. It is north-west of London, south-east of Birmingham and north-east of Bristol. The city is home to the ...
, England. Weyl was a Plenary Speaker of the
International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rena ...
(ICM) in 1928 at
Bologna Bologna (, , ; egl, label= Emilian, Bulåggna ; lat, Bononia) is the capital and largest city of the Emilia-Romagna region in Northern Italy. It is the seventh most populous city in Italy with about 400,000 inhabitants and 150 different na ...
and an Invited Speaker of the ICM in 1936 at
Oslo Oslo ( , , or ; sma, Oslove) is the capital and most populous city of Norway. It constitutes both a county and a municipality. The municipality of Oslo had a population of in 2022, while the city's greater urban area had a population of ...
. He was elected a fellow of the
American Physical Society The American Physical Society (APS) is a not-for-profit membership organization of professionals in physics and related disciplines, comprising nearly fifty divisions, sections, and other units. Its mission is the advancement and diffusion of k ...
in 1928 and a member of the
National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nat ...
in 1940. For the academic year 1928–1929 he was a visiting professor at
Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
, where he wrote a paper, "On a problem in the theory of groups arising in the foundations of infinitesimal geometry," with
Howard P. Robertson Howard Percy "Bob" Robertson (January 27, 1903 – August 26, 1961) was an American mathematician and physicist known for contributions related to physical cosmology and the uncertainty principle. He was Professor of Mathematical Physics at the C ...
. 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 The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...
in
Princeton, New Jersey Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of w ...
, 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. 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 Pantheism is the belief that reality, the universe and the cosmos are identical with divinity and a supreme supernatural being or entity, pointing to the universe as being an immanent creator deity still expanding and creating, which has e ...
.


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 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 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 surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
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 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 Gauge ( or ) may refer to: Measurement * Gauge (instrument), any of a variety of measuring instruments * Gauge (firearms) * Wire gauge, a measure of the size of a wire ** American wire gauge, a common measure of nonferrous wire diameter, es ...
, and gave the first example of what is now known as a
gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie grou ...
. Weyl's gauge theory was an unsuccessful attempt to model the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
and the
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational pheno ...
as geometrical properties of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. The Weyl tensor in
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
is of major importance in understanding the nature of conformal geometry. In 1929, Weyl introduced the concept of the
vierbein The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independen ...
into general relativity. His overall approach in physics was based on the phenomenological philosophy of
Edmund Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...
, 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 Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
'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 representation Matrix representation is a method used by a computer language to store matrices of more than one dimension in memory. Fortran and C use different schemes for their native arrays. Fortran uses "Column Major", in which all the elements for a give ...
s. In the compact Lie group case he proved a fundamental character formula. These results are foundational in understanding the symmetry structure of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, which he put on a group-theoretic basis. This included
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s. Together with the mathematical formulation of quantum mechanics, in large measure due to
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
, this gave the treatment familiar since about 1930. Non-compact groups and their representations, particularly the
Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ...
, were also streamlined in that specific context, in his 1927
Weyl quantization Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
, 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 algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s became a mainstream part both of
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
and
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
. His book '' The Classical Groups'' reconsidered invariant theory. It covered
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
s,
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
s,
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
s, 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 In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by r ...
, with his criterion for uniform distribution mod 1, which was a fundamental step in
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...
. This work applied to the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, as well as
additive number theory Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigr ...
. 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 Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
's ramified theory of types. He was able to develop most of classical
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, while using neither the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
nor
proof by contradiction In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known ...
, and avoiding
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
'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 In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...
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 number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, sets, and
countability In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
, and moreover, that asking about the
truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as belief ...
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 Georg Wilhelm Friedrich Hegel (; ; 27 August 1770 – 14 November 1831) was a German philosopher. He is one of the most important figures in German idealism and one of the founding figures of modern Western philosophy. His influence extends a ...
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 Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...
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 In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
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 Weyl equation, for use in a replacement to Dirac equation. This equation describes massless
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s. A normal Dirac fermion could be split into two Weyl fermions or formed from two Weyl fermions.
Neutrino A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s 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.


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. *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.
Über die asymptotische Verteilung der Eigenwerte
', Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 110–117 (1911). * 1913. ''Die Idee der Riemannschen Flāche'', 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''. * 1918.
Raum, Zeit, Materie
'. 5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922
Space Time Matter
', Methuen, rept. 1952 Dover. . * 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, . * 1928. ''Gruppentheorie und Quantenmechanik''. transl. by H. P. Robertson,
The Theory of Groups and Quantum Mechanics
', 1931, rept. 1950 Dover. * 1929. "Elektron und Gravitation I", ''Zeitschrift Physik'', 56, pp 330–352. – introduction of the
vierbein The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independen ...
into GR * 1933. ''The Open World'' Yale, rept. 1989 Oxbow Press * 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''. * * * 1940. ''Algebraic Theory of Numbers'' rept. 1998 Princeton U. Press. * (text of 1948 Josiah Wilard Gibbs Lecture) * 1952. ''Symmetry''. Princeton University Press. * 1968. in K. Chandrasekharan ''ed'', ''Gesammelte Abhandlungen''. Vol IV. Springer.


See also


Topics named after Hermann Weyl

* Majorana–Weyl spinor * Peter–Weyl theorem * Schur–Weyl duality * Weyl algebra *
Weyl basis In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
of the gamma matrices * Weyl chamber * Weyl character formula * Weyl equation, a relativistic wave equation * Weyl expansion * Weyl fermion * Weyl gauge * Weyl gravity * Weyl notation *
Weyl quantization Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
* Weyl spinor *
Weyl sum In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function :e(x) = \exp(2\pi ix).\, Therefore, a typi ...
, a type of exponential sum *
Weyl symmetry :''See also Wigner–Weyl transform, for another definition of the Weyl transform.'' In theoretical physics, the Weyl transformation, named after Hermann Weyl, is a local rescaling of the metric tensor: :g_\rightarrow e^g_ which produces anoth ...
: see Weyl transformation *
Weyl tensor In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tida ...
* Weyl transform *
Weyl transformation :''See also Wigner–Weyl transform, for another definition of the Weyl transform.'' In theoretical physics, the Weyl transformation, named after Hermann Weyl, is a local rescaling of the metric tensor: :g_\rightarrow e^g_ which produces anoth ...
*
Weyl–Schouten theorem In the mathematical field of differential geometry, the existence of isothermal coordinates for a (pseudo-)Riemannian metric is often of interest. In the case of a metric on a two-dimensional space, the existence of isothermal coordinates is uncond ...
*
Weyl's criterion In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequenc ...
* Weyl's lemma on
hypoellipticity In the theory of partial differential equations, a partial differential operator P defined on an open subset :U \subset^n is called hypoelliptic if for every distribution u defined on an open subset V \subset U such that Pu is C^\infty (smooth ...
* Weyl's lemma on the "very weak" form of the
Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \n ...


References


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, *
Ivor Grattan-Guinness Ivor Owen Grattan-Guinness (23 June 1941 – 12 December 2014) was a historian of mathematics and logic. Life Grattan-Guinness was born in Bakewell, England; his father was a mathematics teacher and educational administrator. He gained his b ...
, 2000. ''The Search for Mathematical Roots 1870-1940''. Princeton Uni. Press. *Thomas Hawkins, ''Emergence of the Theory of Lie Groups'', New York: Springer, 2000. * *In connection with the Weyl–Pólya bet, a copy of the original letter together with some background can be found in: *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) () 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. 48–70. *


External links


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
* * {{DEFAULTSORT:Weyl, Hermann Klaus Hugo 1885 births 1955 deaths Burials at Princeton Cemetery Differential geometers ETH Zurich faculty Institute for Advanced Study faculty Fellows of the American Physical Society Foreign Members of the Royal Society German male writers 20th-century German mathematicians Members of the United States National Academy of Sciences Number theorists Linear algebraists Pantheists People from Elmshorn People from the Province of Schleswig-Holstein German relativity theorists University of Göttingen alumni People educated at the Gymnasium Christianeum 20th-century German philosophers