Background

Hermann Weyl was born in Elmshorn, German Empire (now Elmshorn, Schleswig-Holstein, Germany), on 9 November 1885. He was the son of Ludwig Weyl, a bank clerk, and Anna (Dieck) Weyl.

Education

The financial standing of his parents enabled Weyl to receive a quality education. From 1895 to 1904 he attended the Gymnasium at Altona, where his performance attracted the attention of his headmaster, a relative of an eminent mathematician of that time, David Hilbert. Weyl soon found himself at the University of Gottingen where Hilbert was an instructor. He remained there for the rest of his student days, with the exception of a semester at the University of Munich. He received his degree under Hilbert in 1908.

Career

At the beginning of his career, Weyl advanced to the ranks of privat docent (unpaid but licensed instructor) in 1910.

In 1913 he accepted a position as professor at the National Technical University (ETH) in Zurich, Switzerland. He declined the offer to be Felix Klein’s successor at Gottingen, despite the university’s central role in the mathematical world.

It is not surprising that Weyl’s early work dealt with topics in which Hilbert, his headmaster, held an interest. His Habilitationsschrift was devoted to boundary conditions of second-order linear differential equations. (The way the German educational system worked, it was necessary to do a substantial piece of original research beyond the doctoral dissertation in order to qualify to teach in the university. This “entitling document” was frequently the launching point of the mathematical career of its author.)

One of the principal areas of Weyl’s research was the topic of Hilbert spaces. The problem was to understand something about the functions that operated on the points of Hilbert space in a way useful for analyzing the result of applying the functions. In particular, Weyl wanted to know where the functions behaved more simply than on the space as a whole, since the behavior of the function on the rest of the space could be represented in terms of its behavior on the simpler regions. Different kinds of functions behaved in radically different ways on a Hilbert space, so Weyl had to restrict his attention to a subclass of functions small enough to be tractable (for example, the functions could not “blow up”) but large enough to be useful. His choice of self-adjoint, compact operators was justified by their subsequent importance in the field of functional analysis.

Among the areas he brought together were geometry and analysis from the nineteenth century and topology, which was largely a creation of the twentieth century. Weyl was able to take advantage of the work of Hilbert and the Dutch mathematician Luitzen Egbertus Jan Brouwer. This effort culminated in his 1913 book on Riemann surfaces, an excellent exposition on how complex analysis and topology could be used together to analyze the behavior of complex functions.

Weyl served briefly in the German army at the outbreak of World War I, but before this military interlude, he did research that led to one of his most important papers. He looked at the way irrational numbers (those that cannot be expressed as a ratio of two whole numbers) were distributed. What he noticed was that the fractional parts of an irrational number and its integral multiples seemed to be evenly distributed in the interval between O and 1. He succeeded in proving this result, and it is known as the Kronecker-Weyl theorem, owing half of its name to an influential number theorist who had an effect on Hilbert. Although the result may seem rather narrow, Weyl was able to generalize it to sequences of much broader application.

During his time in Zurich, Weyl spent a year in collaboration with Albert Einstein and picked up a dose of enthusiasm for the relativity theory. Among the other results of this collaboration was Weyl’s popular account of relativity theory Space, Time, Matter (the original German edition appeared in 1918).

Weyl used the tensor calculus that had been developed by the geometers to come up with neater formulations of general relativity than the original version proposed by Einstein. In later years, he took the evolution of tensors one step further while maintaining a strict mathematical level of rigor.

One of the most visible areas in which Weyl worked after World War I was in the foundations of mathematics. He had used some of the topological results of the Dutch mathematician Brouwer in working on Riemann surfaces. In addition, he had looked at some of Brouwer’s ideas about the philosophy of mathematics and was convinced that they had to be taken seriously.

What especially disturbed Hilbert was Weyl’s support of Brouwer’s concepts, since Hilbert knew the mathematical strength of his former student. In the 1920s, while the argument was being considered, Hilbert was discouraged about the future of mathematics in the hands of the intuitionists.

Weyl subsequently became Professor of Mathematics at Göttingen (1930) and for a brief time Director of its famous Mathematical Institute.

In 1933 Weyl decided that he could no longer remain in Nazi Germany, and he took up a permanent position at the Institute for Advanced Study, newly founded in Princeton, New Jersey. He remained there until 1951.

There may have been the additional attraction of the wealth of intellectual company available at the institute, between its visitors from all over the globe and permanent residents such as Einstein. Weyl took his official duties as a faculty member seriously, although his reputation could be terrifying to younger mathematicians unaware of the poet within.

Weyl’s work continued to bridge the gap between physics and mathematics. As long ago as 1929, he developed a mathematical theory for the subatomic particle the neutrino. The theory was internally consistent but failed to preserve left-right symmetry and so was abandoned. Subsequent experimentation revealed that symmetry need not be conserved, with the result that Weyl’s theory reentered the mathematical physics mainstream all the more forcefully. Another area for the interaction of mathematics and physics was the study of spinors, a kind of tensor that has proven to be of immense use in quantum mechanics.

In a 1923 paper Weyl had come up with a suitable definition for congruence in relativistic spacetime. Even more influential was his book on group theory and quantum mechanics, which imposed a model that would have been welcome to Klein due to the previously rather disjointed results assembled by quantum physicists. Weyl was an artist in the use of group theory and could accomplish wonders with modest mathematical structure.

After the end of the World War II, Weyl divided his time between Zurich and Princeton.

He took a serious view of the history of mathematics and arranged with Princeton to give a course on the subject. One of his magisterial works was a survey of the previous half-century of mathematics that appeared in the American Mathematical Monthly in 1951. Although he never became as fluent in English as he had in German, he retained a strong commitment to the public’s right to be informed about scientific developments.

Achievements

• Hermann Weyl was one of the twentieth century's most important mathematicians, as well as a seminal figure in the development of quantum physics and general relativity. He was one of the first to conceive of combining general relativity with the laws of electromagnetism.
  
  Although spinors had been known before Weyl, he was the first to give a full treatment of them. Perhaps it was this work that led Roger Penrose, one of the most insightful mathematical physicists of the second half of the twentieth century, to label Weyl “the greatest mathematician of this century.”
  
  

Works

More photos

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  • Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics

    (Mind and Nature is a collection of Weyl's most important ...)

  • Philosophy of Mathematics and Natural Science

    (When mathematician Hermann Weyl decided to write a book o...)

  • Die Idee der Riemannschen Fläche

    (Weyl created a new branch of mathematics by uniting funct...)

  • Symmetry (Princeton Science Library)

    (Symmetry is a classic study of symmetry in mathematics, t...)

  • The Concept of a Riemann Surface (Dover Books on Mathematics)

    (This classic on the general history of functions was writ...)

  • The Theory of Groups and Quantum Mechanics (Dover Books on Mathematics)

    (This book is devoted to the consistent and systematic app...)

  • Levels of Infinity: Selected Writings on Mathematics and Philosophy (Dover Books on Mathematics)

    (This original anthology assembles ten accessible essays b...)

  • Space, Time, Matter

    ("The standard treatise on the general theory of relativit...)

  • The Continuum: A Critical Examination of the Foundation of Analysis (Dover Books on Mathematics)

    ("The hard won power ... to assess correctly the continuum...)

  • Raum, Zeit, Materie
  • Gruppentheorie und Quantenmechanik

Views

Quotations: "I could not bear to live under the rule of that demon [Hitler] who had dishonoured the name of Germany, and although the wrench was hard and the mental agony so cruel that I suffered a severe breakdown, I shook the dust of the fatherland from my feet."

Membership

Weyl was a fellow of the Royal Society.

Connections

Weyl married Helene Joseph (known as Hella to the family) in 1913. His first wife died in 1948, and two years later he married Ellen Bair. He had two children from the first marriage - Fritz Joachim and Michael.

Father:

Ludwig Weyl

Mother:

Anna (Dieck) Weyl

1-st wife:

Helene Joseph

2-nd wife:

Ellen Bair

Son:

Fritz Joachim Weyl

Son:

Michael Weyl