Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German

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

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 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.

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

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 ...

, theoretical physicist
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 experi ...

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 t ...

. 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
...

, Switzerland
). Swiss law does not designate a ''capital'' as such, but the federal parliament and government are in Bern, while other federal institutions, such as the federal courts, are in other cities (Bellinzona, Lausanne, Luzern, Neuchâtel, St. Gall ...

, 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 wh ...

, 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 i ...

tradition of mathematics, represented by Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

, David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

and Hermann Minkowski
Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in numbe ...

.
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 experi ...

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, "Mat ...

. 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 con ...

, time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...

, 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 parti ...

, 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
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...

. 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
Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum ...

wrote that Weyl alone bore comparison with the "last great universal mathematicians of the nineteenth century", Poincaré and Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

. 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 inElmshorn
Elmshorn (; nds, Elmshoorn) is a town in the district of Pinneberg in Schleswig-Holstein in Germany. It is 30 km north of Hamburg on the small river Krückau, a tributary of the Elbe, and with about 50,000 inhabitants is the sixth-largest ...

, a small town near Hamburg
(male), (female) en, Hamburger(s),
Hamburgian(s)
, timezone1 = Central (CET)
, utc_offset1 = +1
, timezone1_DST = Central (CEST)
, utc_offset1_DST = +2
, postal ...

, 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 betwe ...

, 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 or ...

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 Ha ...

. 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 i ...

under the supervision of David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

, 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, 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). 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
Fritz Joachim Weyl (February 19, 1915 – July 20, 1977) was born in Zurich, Switzerland. Today Weyl is regarded as a renowned mathematician. During his lifetime he taught at many universities, significantly contributed to research in mathemati ...

(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 was l ...

and Richard Courant. 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. 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 th ...

, 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 ren ...

(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 Nati ...

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.
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 wh ...

, 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 ex ...

.
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 ofRiemann 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 ne ...

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, 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 phen ...

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 diff ...

. 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.
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 phi ...

'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 ofcompact group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...

s, 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
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 slig ...

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 cov ...

, this gave the treatment familiar since about 1930. Non-compact groups and their representations, particularly the Heisenberg group, were also streamlined in that specific context, in his 1927 Weyl quantization, the best extant bridge between
classical and quantum physics to date. From this time, and certainly much helped by Weyl's expositions, Lie groups and Lie 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 ident ...

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 experi ...

.
His book '' The Classical Groups'' reconsidered invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descr ...

. 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 grou ...

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 group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gr ...

s and results on their invariants and representations.
Harmonic analysis and analytic number theory

Weyl also showed how to use exponential sums indiophantine 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 ...

, 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 Diric ...

. 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. It was developed by many others.
Foundations of mathematics

In ''The Continuum'' Weyl developed the logic of predicative analysis using the lower levels ofBertrand 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 mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of ari ...

, 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 o ...

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 set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only se ...

s. 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 ...

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
George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamenta ...

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, 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 belie ...

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 ...

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
, 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 ...

, 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
Ernst Alfred Cassirer ( , ; July 28, 1874 – April 13, 1945) was a German philosopher. Trained within the Neo-Kantian Marburg School, he initially followed his mentor Hermann Cohen in attempting to supply an idealistic philosophy of science.
A ...

. 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 clas ...

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 noncon ...

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 toDirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac pa ...

. 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 an ...

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 m ...

s were once thought to be Weyl fermions, but they are now known to have mass. Weyl fermions are sought after for electronics applications. Quasiparticle
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For exa ...

s 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; 1952Bibliography

* 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
Frank Anthony Wilczek (; born May 15, 1951) is an American theoretical physicist, mathematician and Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology (MIT), Founding Dire ...

, 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 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
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Pete ...

* Schur–Weyl duality
* Weyl algebra
* Weyl basis of the gamma matrices
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(\m ...

* Weyl chamber
* Weyl character formula
* Weyl equation, a relativistic wave equation
* Weyl expansion In physics, the Weyl expansion, also known as the Weyl identity or angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. In a Cartesian coordinate system, it can be denoted as
:\frac=\frac \int_^ ...

* Weyl fermion
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 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
* Weyl–Schouten theorem
* Weyl's criterion
* 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 ( sm ...

* 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 = \nab ...

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, 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