Hermann Weyl
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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, mathematical structure, structure, space, Mathematica ...
, theoretical physicist, logician and
philosopher Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
. Although much of his working life was spent in
Zürich Zurich (; ) is the list of cities in Switzerland, largest city in Switzerland and the capital of the canton of Zurich. It is in north-central Switzerland, at the northwestern tip of Lake Zurich. , the municipality had 448,664 inhabitants. The ...
,
Switzerland Switzerland, officially the Swiss Confederation, is a landlocked country located in west-central Europe. It is bordered by Italy to the south, France to the west, Germany to the north, and Austria and Liechtenstein to the east. Switzerland ...
, and then
Princeton, New Jersey The Municipality of Princeton is a Borough (New Jersey), borough in Mercer County, New Jersey, United States. It was established on January 1, 2013, through the consolidation of the Borough of Princeton, New Jersey, Borough of Princeton and Pri ...
, he is associated with the University of Göttingen 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 List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
as well as purely mathematical disciplines such as
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
. 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) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
during its early years. Weyl contributed to an exceptionally wide range of fields, including works on
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
,
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
,
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 pa ...
,
philosophy Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
,
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
,
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
and the
history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...
. He was one of the first to conceive of combining
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
with the laws of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
. Freeman Dyson wrote that Weyl alone bore comparison with the "last great universal mathematicians of the nineteenth century",
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
and David 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 (, ; ), officially the Free and Hanseatic City of Hamburg,. is the List of cities in Germany by population, second-largest city in Germany after Berlin and List of cities in the European Union by population within city limits, 7th-lar ...
, in
Germany Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...
, 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 (, ; ; ) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. According to the 2022 German census, t ...
and
Munich Munich is the capital and most populous city of Bavaria, Germany. As of 30 November 2024, its population was 1,604,384, making it the third-largest city in Germany after Berlin and Hamburg. Munich is the largest city in Germany that is no ...
. His doctorate was awarded at the University of Göttingen under the supervision of David Hilbert, whom he greatly admired. In September 1913, in Göttingen, Weyl married Friederike Bertha Helene Joseph (30 March 1893 – 5 September 1948) who went by the name Helene (nickname "Hella"). Helene was a daughter of Dr. Bruno Joseph (13 December 1861 – 10 June 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) 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 (19 February 1915 – 20 July 1977) and Michael Weyl (15 September 1917 – 19 March 2011), both of whom were born in Zürich, Switzerland. Helene died in Princeton, New Jersey, on 5 September 1948. A memorial service in her honor was held in Princeton on 9 September 1948. Speakers at her memorial service included her son Fritz Joachim Weyl and mathematicians Oswald Veblen and Richard Courant. In 1950. Hermann married sculptor Ellen Bär (née Lohnstein) (17 April 1902 – 14 July 1988), who was the widow of professor Richard Josef Bär (11 September 1892 – 15 December 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 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
, who was working out the details of the theory of
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. 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 an Austrian-Irish theoretical physicist who developed fundamental results in quantum field theory, quantum theory. In particul ...
, 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 status in the United Kingdom, cathedral city and non-metropolitan district in Oxfordshire, England, of which it is the county town. The city is home to the University of Oxford, the List of oldest universities in continuou ...
, 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 IMU Abacus Medal (known before ...
(ICM) in 1928 at
Bologna Bologna ( , , ; ; ) is the capital and largest city of the Emilia-Romagna region in northern Italy. It is the List of cities in Italy, seventh most populous city in Italy, with about 400,000 inhabitants and 150 different nationalities. Its M ...
and an Invited Speaker of the ICM in 1936 at
Oslo Oslo ( or ; ) 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 1,064,235 in 2022 ...
. He was elected a fellow of the American Physical Society in 1928, a member of the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...
in 1929, a member of the
American Philosophical Society The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...
in 1935, and a member of the
National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, 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 ...
in 1940. For the academic year 1928–1929, he was a visiting professor at
Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...
, 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 Nazism (), formally named National Socialism (NS; , ), is the far-right politics, far-right Totalitarianism, totalitarian socio-political ideology and practices associated with Adolf Hitler and the Nazi Party (NSDAP) in Germany. During H ...
assumed power in 1933, particularly as his wife was
Jewish Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, rel ...
. He had been offered one of the first faculty positions at the new
Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
in
Princeton, New Jersey The Municipality of Princeton is a Borough (New Jersey), borough in Mercer County, New Jersey, United States. It was established on January 1, 2013, through the consolidation of the Borough of Princeton, New Jersey, Borough of Princeton and Pri ...
, 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 8 December 1955, while living in Zürich. Weyl was cremated in Zürich on 12 December 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.


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 mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
in a 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 In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...
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 surfaces. In it Weyl utilized point set topology, in order to make Riemann surface theory more rigorous, a model followed in later work on manifolds. He absorbed L. E. J. Brouwer's early work in topology for this purpose. Weyl, as a major figure in the Göttingen school, was fully apprised of Einstein's work from its early days. He tracked the development of relativity physics in his ''Raum, Zeit, Materie'' (''Space, Time, Matter'') from 1918, reaching a 4th edition in 1922. In 1918, he introduced the notion of gauge, and gave the first example of what is now known as a gauge theory. Weyl's gauge theory was an unsuccessful attempt to model the electromagnetic field and the
gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
as geometrical properties of
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. The Weyl tensor in
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
is of major importance in understanding the nature of conformal geometry. His overall approach in physics was based on the phenomenological philosophy of Edmund Husserl, specifically Husserl's 1913 ''Ideen zu einer reinen 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 philos ...
'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 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 the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, which he put on a group-theoretic basis. This included spinors. Together with the mathematical formulation of quantum mechanics, in large measure due to
John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
, 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 In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
s 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 List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
. 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 grou ...
s,
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
s,
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
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 ...
, 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 Dir ...
. This work applied to the Riemann zeta function, 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 semigro ...
. 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 philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
's ramified theory of types. He was able to develop most of classical
calculus Calculus 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 generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
, while using neither the axiom of choice nor
proof by contradiction In logic, 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. Although it is quite freely used in mathematical pr ...
, and avoiding
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
's infinite sets. Weyl appealed in this period to the radical constructivism of the German romantic, subjective idealist Fichte. Shortly after publishing ''The Continuum'' Weyl briefly shifted his position wholly to the intuitionism of Brouwer. In ''The Continuum'', the constructible points exist as discrete entities. Weyl wanted a continuum that was not an aggregate of points. He wrote a controversial article proclaiming, 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, sets, and countability, and moreover, that asking about the
truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...
or falsity of the least upper bound property of the real numbers was as meaningful as asking about truth of the basic assertions of Hegel on the philosophy of nature. 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 c ...
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 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 the Weyl equation, for use in a replacement to the Dirac equation. This equation describes massless
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s. A normal Dirac fermion could be split into two Weyl fermions or formed from two Weyl fermions. Neutrinos were once thought to be Weyl fermions, but they are now known to have mass. Weyl fermions are sought after for electronics applications. Quasiparticles that behave as Weyl fermions were discovered in 2015, in a form of crystals known as Weyl semimetals, a type of topological material.


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, p. 144; 1952 *Beyond the knowledge gained from the individual sciences, there remains the task of comprehending. In spite of the fact that the views of philosophy sway from one system to another, we cannot dispense with it unless we are to convert knowledge into a meaningless chaos. :—''Space-Time-Matter'', 4th edition (1922), English translation, Dover (1952) p. 10; Weyl’s boldfaced highlight.


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 into GR * 1931:
The Theory of Groups and Quantum Mechanics
'," London, : Methuen & Co., 931* 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. 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)


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 commutation relation, anticommutation relations that ensure they generating set, generate a matrix representat ...
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 * 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 * Weyl's lemma on the "very weak" form of the Laplace equation


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. * * Clary, David C. (2012)
''Schrödinger in Oxford''
World Scientific Publishing, .


External links



* 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 Academic staff of ETH Zurich 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 German 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 Members of the American Philosophical Society Presidents of the German Mathematical Society