# Hermann Klaus Hugo Weyl

# Hermann Klaus Hugo Weyl

**1885-1955**

**German-American Mathematician**

Hermann Weyl served as a link between an older generation of mathematicians, dominated by David Hilbert (1862-1943) and others at the University of Göttingen, and the world of refugees from Nazism who fled to America and there helped establish the world of the future. His mathematical work concerned a variety of topics, ranging from the most purely philosophical pursuits in the foundations of the discipline to highly practical applications in physics. Many scholars consider Weyl among the immortals of twentieth-century mathematics.

Born on November 9, 1885, in the town of Elmshorn near Hamburg, Weyl was the son of a bank clerk, Ludwig, whose wife Anna Dieck had come from a wealthy family. While studying in the gymnasium at Altona, Weyl's abilities caught the attention of a headmaster who was related to Hilbert. He went on to study with the latter at Göttingen, where he earned his doctorate in 1908.

In 1913, the same year he married Helene Joseph, Weyl accepted a professorship at the National Technical University (ETH) in Zurich, where sons Fritz and Michael were raised. There he became acquainted with Albert Einstein (1879-1955), about whose relativity theory he would write in *Space, Time, Matter* (1918), a book comprehensible to non-technically educated readers. He also examined tensor calculus, involving functions on a number of vectors that take a number as their value, and his findings in this area helped clarify some of the still-untidy mathematical under-pinnings of Einstein's theory.

During this extremely fruitful period, Weyl investigated the boundary conditions of second-order linear differential equations—that is, the behavior of functions within a given region in which the behavior at the boundary has been defined. He also became interested in Hilbert spaces, another realm in which he considered the behavior of different functions at different points. In addition, Weyl brought together classical methods of geometry and analysis with the relatively new discipline of topology, and in 1913 published a highly readable exposition on Riemann surfaces.

In the period leading up to World War I, during which he served briefly in the German army, Weyl examined the properties of irrational numbers—numbers, such as the square root of a negative, that cannot be expressed as a ratio of two whole numbers. He discovered that the fractional parts of an irrational number and its integral multiples appeared to be evenly distributed in the interval between 0 to 1.

In the 1920s, Weyl became involved in a battle over the foundations of mathematics. In this he found himself torn between the formalism of his mentor Hilbert and the intuitionism of Luitzen Jan Brouwer (1881-1966), which he found appealing. The saga underlying this conflict, and the arguments for both sides, are intriguing; however, they are academic, because the incompleteness theorem of Kurt Gödel (1906-1978) in 1931 would render moot any attempt to provide an all-encompassing explanation of mathematics.

Weyl, whose wife was Jewish, left newly Nazified Germany for the United States in 1933, and became involved with Princeton's Institute for Advanced Study. He became an American citizen, and remained in America for the rest of his life, though he traveled widely and frequently. In 1948, his first wife died, and in 1950 he married Ellen Bär. During the last three decades of his life, Weyl continued to investigate a wide array of mathematical questions, and remained active up to the very end. He died of a heart attack on December 8, 1955, a month after his 70th birthday, in Zurich.

**JUDSON KNIGHT**

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**Hermann Klaus Hugo Weyl**