Hermann Weyl

Hermann Weyl

Overview
Hermann Klaus Hugo Weyl (9 November 1885 – 8 December 1955) was a German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 and theoretical physicist. Although much of his working life was spent in Zürich
Zürich
Zurich is the largest city in Switzerland and the capital of the canton of Zurich. It is located in central Switzerland at the northwestern tip of Lake Zurich...

, Switzerland
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....

 and then Princeton
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...

, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

 and Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...

.
His research has had major significance for theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

 as well as purely mathematical disciplines including number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...

 during its early years.

Weyl published technical and some general works on space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

, time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

, matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

, philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

, logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

, symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

 and the history of mathematics
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....

.
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Quotations

Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect.

Symmetry
Encyclopedia
Hermann Klaus Hugo Weyl (9 November 1885 – 8 December 1955) was a German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 and theoretical physicist. Although much of his working life was spent in Zürich
Zürich
Zurich is the largest city in Switzerland and the capital of the canton of Zurich. It is located in central Switzerland at the northwestern tip of Lake Zurich...

, Switzerland
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....

 and then Princeton
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...

, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

 and Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...

.
His research has had major significance for theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

 as well as purely mathematical disciplines including number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...

 during its early years.

Weyl published technical and some general works on space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

, time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

, matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

, philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

, logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

, symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

 and the history of mathematics
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....

. He was one of the first to conceive of combining general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

 with the laws of electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

. While no mathematician of his generation aspired to the 'universalism' of Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

 or Hilbert, Weyl came as close as anyone. Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...

, in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him (The Mathematical Intelligencer (1984), vol.6 no.1).

Biography


Weyl was born in Elmshorn
Elmshorn
Elmshorn is a town in the district of Pinneberg in Schleswig-Holstein in Germany. It is located 32 km north of Hamburg at the small river Krückau, close to the Elbe river, is the sixth-largest city in the state of Schleswig-Holstein, Germany...

, a small town near Hamburg
Hamburg
-History:The first historic name for the city was, according to Claudius Ptolemy's reports, Treva.But the city takes its modern name, Hamburg, from the first permanent building on the site, a castle whose construction was ordered by the Emperor Charlemagne in AD 808...

, in Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

, and attended the gymnasium Christianeum in Altona
Altona, Hamburg
Altona is the westernmost urban borough of the German city state of Hamburg, on the right bank of the Elbe river. From 1640 to 1864 Altona was under the administration of the Danish monarchy. Altona was an independent city until 1937...

.

From 1904 to 1908 he studied mathematics and physics in both Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...

 and Munich
Munich
Munich The city's motto is "" . Before 2006, it was "Weltstadt mit Herz" . Its native name, , is derived from the Old High German Munichen, meaning "by the monks' place". The city's name derives from the monks of the Benedictine order who founded the city; hence the monk depicted on the city's coat...

. His doctorate was awarded at the University of Göttingen under the supervision of David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

 whom he greatly admired. After taking a teaching post for a few years, he left Göttingen for Zürich to take the chair of mathematics in the ETH Zurich
ETH Zurich
The Swiss Federal Institute of Technology Zurich or ETH Zürich is an engineering, science, technology, mathematics and management university in the City of Zurich, Switzerland....

, where he was a colleague of Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

, who was working out the details of the theory of general relativity. Einstein had a lasting influence on Weyl who became fascinated by mathematical physics. Weyl met Erwin Schrödinger
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

 in 1921, who was appointed Professor at the University of Zürich
University of Zurich
The University of Zurich , located in the city of Zurich, is the largest university in Switzerland, with over 25,000 students. It was founded in 1833 from the existing colleges of theology, law, medicine and a new faculty of philosophy....

. They were to become close friends over time.

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. The events persuaded him to move to the new Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...

 in Princeton, New Jersey
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...

. He remained there until his retirement in 1951. Together with his wife, he spent his time in Princeton and Zürich, and died in Zürich in 1955.

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 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 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
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

s. In it Weyl utilized point set topology, in order to make Riemann surface theory more rigorous, a model followed in later work on manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s. He absorbed L. E. J. Brouwer's
Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer FRS , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.-Biography:Early in his career,...

 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
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

 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
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

. Weyl's gauge theory was an unsuccessful attempt to model the electromagnetic field
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

 and the gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

 as geometrical properties of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

. The Weyl tensor in Riemannian geometry
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 which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

 is of major importance in understanding the nature of conformal geometry. In 1929, Weyl introduced the concept of the vierbein
Cartan connection applications
The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. This section is an approach to tetrads, but written in general terms...

 into general relativity.

His overall approach in physics was based on the phenomenological philosophy of Edmund Husserl
Edmund Husserl
Edmund Gustav Albrecht Husserl was a philosopher and mathematician and the founder of the 20th century philosophical school of phenomenology. He broke with the positivist orientation of the science and philosophy of his day, yet he elaborated critiques of historicism and of psychologism in logic...

, 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). Apparently this was Weyl's way of dealing with Einstein's controversial dependence on the phenomenological physics of Ernst Mach.

Husserl had reacted strongly to Gottlob Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...

'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. Hence there is good reason for viewing gauge theory as it developed from Weyl's ideas as a formalism of physical measurement and not a theory of anything physical, i.e. as scientific formalism
Scientific formalism
Scientific formalism is a broad term for a family of approaches to the presentation of science. It is viewed as an important part of the scientific method, especially in the physical sciences.-Levels of formalism:...

.

Topological groups, Lie groups and representation theory


From 1923 to 1938, Weyl developed the theory of compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

s, in terms of matrix representation
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

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
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, which he put on a group-theoretic basis. This included spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

s. Together with the mathematical formulation of quantum mechanics
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...

, in large measure due to John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

, 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
Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...

, 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
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

s became a mainstream part both of pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

 and theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

.

His book The Classical Groups, a seminal if difficult text, reconsidered invariant theory
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...

. It covered symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

s, general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...

s, orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

s, and symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...

s and results on their invariants
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

 and representations
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

.

Harmonic analysis and analytic number theory



Weyl also showed how to use exponential sum
Exponential sum
In mathematics, an exponential sum may be a finite Fourier series , or other finite sum formed using the exponential function, usually expressed by means of the functione = \exp.\,...

s in diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....

, with his criterion for uniform distribution mod 1, which was a fundamental step in analytic number theory
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 Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

. This work applied to the Riemann zeta function, as well as additive number theory
Additive number theory
In number theory, the specialty additive number theory studies subsets of integers and their behavior under addition. More abstractly, the field of "additive number theory" includes the study of Abelian groups and commutative semigroups with an operation of addition. Additive number theory has...

. 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 Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

's ramified theory of types. He was able to develop most of classical calculus, while using neither the axiom of choice nor proof by contradiction, and avoiding Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...

'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
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...

 of Brouwer. In The Continuum, the constructible points exist as discrete entities. Weyl wanted a continuum that was not an aggregate of points. He wrote a controversial article proclaiming that, for himself and L. E. J. Brouwer, "We are the revolution." This article was far more influential in propagating intuitionistic views than the original works of Brouwer himself.

George Pólya
George Pólya
George Pólya 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 fundamental contributions to combinatorics, number theory, numerical analysis and probability theory...

 and Weyl, during a mathematicians' gathering in Zürich (9 February 1918), made a bet concerning the future direction of mathematics. Weyl predicted that in the subsequent 20 years, mathematicians would come to realize the total vagueness of notions such as real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, sets, and countability, and moreover, that asking about the truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...

 or falsity of the least upper bound property
Property (philosophy)
In modern philosophy, logic, and mathematics a property is an attribute of an object; a red object is said to have the property of redness. The property may be considered a form of object in its own right, able to possess other properties. A property however differs from individual objects in that...

 of the real numbers was as meaningful as asking about truth of the basic assertions of Hegel
Georg Wilhelm Friedrich Hegel
Georg Wilhelm Friedrich Hegel was a German philosopher, one of the creators of German Idealism. His historicist and idealist account of reality as a whole revolutionized European philosophy and was an important precursor to Continental philosophy and Marxism.Hegel developed a comprehensive...

 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
Formalism (mathematics)
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....

 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
Ernst Cassirer
Ernst Cassirer was a German philosopher. He was one of the major figures in the development of philosophical idealism in the first half of the 20th century...

. Weyl however rarely refers to Cassirer, and wrote only brief articles and passages articulating this position.

Quotes


Weyl's comment, although half a joke, sums up his personality:
My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.

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

The problems of mathematics are not problems in a vacuum....

[Impredicative definition's] vicious circle, which has crept into analysis through the foggy nature of the usual set and function concepts, is not a minor, easily avoided form of error in analysis.

In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.

Primary

  • 1911. Über die asymptotische Verteilung der Eigenwerte, Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 110–117 (1911).
  • 1913. Idee der Riemannflāche, 2d 1955. The Concept of a Riemann Surface. Addison–Wesley.
  • 1918. Das Kontinuum, trans. 1987 The Continuum : A Critical Examination of the Foundation of Analysis. ISBN 0-486-67982-9
  • 1918. 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. ISBN 0-486-60267-2.
  • 1923. Mathematische Analyse des Raumproblems.
  • 1924. Was ist Materie?
  • 1925. (publ. 1988 ed. K. Chandrasekharan) Riemann's Geometrische Idee.
  • 1927. Philosophie der Mathematik und Naturwissenschaft, 2d edn. 1949. Philosophy of Mathematics and Natural Science, Princeton 0689702078. With new introduction by Frank Wilczek
    Frank Wilczek
    Frank Anthony Wilczek is a theoretical physicist from the United States and a Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology ....

    , Princeton University Press, 2009, ISBN 978-0691141206.
  • 1928. Gruppentheorie und Quantenmechanik. transl. by H. P. Robertson, The Theory of Groups and Quantum Mechanics, 1931, rept. 1950 Dover. ISBN 0-486-60269-9
  • 1929. "Elektron und Gravitation I", Zeitschrift Physik, 56, pp 330–352. – introduction of the vierbein
    Cartan connection applications
    The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. This section is an approach to tetrads, but written in general terms...

     into GR
    General relativity
    General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

  • 1933. The Open World Yale, rept. 1989 Oxbow Press ISBN 0-918024-70-6
  • 1934. Mind and Nature U. of Pennsylvania Press.
  • 1934. "On generalized Riemann matrices," Ann. Math. 35: 400–415.
  • 1935. Elementary Theory of Invariants.
  • 1935. The structure and representation of continuous groups: Lectures at Princeton university during 1933–34.
  • 1940. Algebraic Theory of Numbers rept. 1998 Princeton U. Press. ISBN 0-691-05917-9
  • 1952. Symmetry. Princeton University Press. ISBN 0-691-02374-3
  • 1968. in K. Chandrasekharan ed, Gesammelte Abhandlungen. Vol IV. Springer.

Secondary

  • 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 Herman-Weyl-Kongresses, Kiel 1985, Bern; New York; Paris: Peter Lang 1988,
  • Ivor Grattan-Guinness
    Ivor Grattan-Guinness
    Ivor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...

    , 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.
  • Erhard Scholz; Robert Coleman; Herbert Korte; Hubert Goenner; Skuli Sigurdsson; Norbert Straumann eds. Hermann Weyl's Raum – Zeit – Materie and a General Introduction to his Scientific Work (Oberwolfach Seminars) (ISBN 3-7643-6476-9) Springer-Verlag New York, New York, N.Y.
  • Thomas Hawkins, Emergence of the Theory of Lie Groups, New York: Springer, 2000.

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