George David Birkhoff

George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician and a leading figure in his field during his era. His work spanned differential equations, dynamical systems, the four-color problem, the three-body problem, and general relativity. He is most famous today for the ergodic theorem. His home in Cambridge, Massachusetts, is a National Historic Landmark.

Birkhoff was born in Overisel Township, Michigan, to Dutch immigrant parents. His father, David Birkhoff, arrived in the U.S. in 1870 and worked as a physician in Chicago. As a teenager, Birkhoff attended the Lewis Institute from 1896 to 1902.

He belonged to the first generation of American mathematicians to be trained entirely in the United States. After the Lewis Institute, he spent a year at the University of Chicago, then earned his A.B. and A.M. from Harvard. He returned to Chicago for his Ph.D., graduating summa cum laude in 1907 at age twenty-three. Though E. H. Moore was his advisor, Henri Poincaré’s writings influenced him most. He taught at the University of Wisconsin–Madison (1907–1909) and Princeton (1909–1912) before joining Harvard in 1912, where he remained until his death. As the only American familiar with all three major U.S. mathematical institutions—Chicago, Harvard, and Princeton—he was highly respected by his peers.

Within the American Mathematical Society, Birkhoff served as vice president in 1919 and edited its *Transactions* from 1920 to 1924. He was president from 1925 to 1926, during which he pushed for a lectureship program to travel the U.S. promoting mathematics. In 1926, he traveled to Europe as an unofficial representative of the Rockefeller Foundation’s International Education Board, working to build ties between American and French institutions, partly due to his fondness for Paris. In 1937, he became president of the American Association for the Advancement of Science, an unusual honor for a mathematician that reflected his standing in the scientific community.

In 1912, while working on the four-color problem, Birkhoff introduced the chromatic polynomial. Though it didn’t solve the problem, it became important in algebraic graph theory. In 1913, he proved Poincaré’s “Last Geometric Theorem,” a special case of the three-body problem, which brought him worldwide fame and boosted the international reputation of American mathematics. He also contributed to general relativity, writing on its foundations and quantum mechanics, and co-authoring *Relativity and Modern Physics* (1923) with R. E. Langer. That same year, he proved that the Schwarzschild geometry is the only spherically symmetric solution to Einstein’s field equations, showing that black holes could arise from any sufficiently massive spherical star—a result later used in the Oppenheimer–Snyder model. He published *Dynamical Systems* in 1927.

His most lasting work came in 1931 with the ergodic theorem, which combined insights from physics on the ergodic hypothesis with measure theory to address a core problem in statistical mechanics. This theorem also influenced dynamics, probability, group theory, and functional analysis. He worked on number theory, the Riemann–Hilbert problem, and the four-color problem, and proposed an axiomatization of Euclidean geometry different from Hilbert’s (Birkhoff’s axioms), culminating in *Basic Geometry* (1941). In 1933, he published *Aesthetic Measure*, a mathematical theory of aesthetics, for which he studied art, music, and poetry from various cultures. His 1938 book *Electricity as a Fluid* blended philosophy and science. His 1943 theory of gravitation is puzzling because, despite knowing it, Birkhoff accepted that it allowed only matter that is a perfect fluid where the speed of sound equals the speed of light.

Birkhoff believed foreign fellowships could raise U.S. mathematical standards, but he wanted stricter qualifications, arguing that standards in less-developed countries differed and that access should be limited. He insisted foreign fellows meet language requirements. Albert Einstein and Norbert Wiener accused Birkhoff of promoting antisemitic selection processes. During the 1930s, as many Jewish mathematicians fled Europe for the U.S., Birkhoff allegedly influenced hiring to exclude Jews. Saunders Mac Lane, then at Harvard, dismissed Einstein’s claims as “worthless,” saying Einstein was unfamiliar with American processes and had competing ideas about general relativity. Mac Lane also rebutted reports of antisemitism toward Wiener, noting that Birkhoff clearly listened to Wiener’s ideas.