Saunders Mac Lane (1909–2005) Outstanding Mathematician and Mentor Who Invented Category Theory

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NEWS & VIEWS NATURE|Vol 435|19 May 2005

OBITUARY Saunders Mac Lane (1909–2005) Outstanding mathematician and mentor who invented category theory.

Saunders Mac Lane, who died on 14 April, and that abstraction is a tool, not a goal. This was one of the few mathematicians to create belief is reflected in the books that he wrote. a major new concept that has had a lasting A Survey of Modern Algebra, written 50 years effect on the subject. In 1945, he and Sammy ago with his co-instructor at Harvard, Garrett Eilenberg introduced the basic ideas of Birkhoff, is still used as a text; and his classic category theory, often described as a monograph, Homology, in which every language of mathematics that allows the general concept is presented starting from a description of transformations from one specific mathematical situation, still serves as area of the subject into another by distilling an introduction and reference to that subject. their common properties. More specifically, Mac Lane’s impact on the mathematical one can look at common properties of community went beyond his contributions mathematical objects such as algebraic as a researcher. According to the online structures (groups, fields or rings) or ‘Mathematics Genealogy Project’, he had topological spaces by studying structure- 39 students and 1,028 ‘descendants’. He preserving mappings — ‘morphisms’ — recognized talent and let it flourish with between such objects. For example, in the modest interference. Among his protégés case of vector spaces as mathematical objects, were Irving Kaplansky, his first PhD the structure-preserving mappings are the student at Harvard, who later headed the linear transformations, and one can study Mathematical Sciences Research Institute the ‘category’ of vector spaces using linear at the University of California, Berkeley; and mappings as the morphisms between objects John Thompson, winner in 1970 of the of the category. Concentrating on common highest mathematical honour, the Fields properties of these structures by stripping medal, for his work on the so-called ‘odd- away the non-essential aspects of a problem order’ problem, which solved the Burnside In his later career, Mac Lane took his clears the path to new results. conjecture that every finite simple group responsibilities to the wider scientific Mac Lane was born in Norwich, of non-prime order must have even order. community very seriously, always devoting

Connecticut, and studied at Yale University Recalling his work with Thompson, his full energies to the task at hand. He was CHICAGO UNIV. and the University of Chicago, where Mac Lane wrote: “By the end of the second elected to the National Academy of Sciences he obtained a master’s degree in 1931. quarter, in which I had tended to emphasize in 1949, and served as vice-president of the After doctoral studies at the University of infinite groups, I had essentially come to academy and of the American Philosophical Göttingen in Germany, and appointments the end of my knowledge on group theory. Society, and as president of the American first as an assistant and then as a full professor Thompson, who was in the course, came Mathematical Society. During his presidency at Harvard University, he returned to the to me to say that he wished to write a thesis of the Mathematical Association of America University of Chicago in 1947. He remained on group theory. I encouraged him, but did in the 1950s, he began the association’s first there for the rest of his career, being not trouble to say that my own knowledge efforts to improve the teaching of modern appointed emeritus professor in 1982. of the subject was somewhat limited. But not mathematics. He was also a member of the Perhaps responding to some early to worry — I arranged for eminent theorists, National Science Board (1974–80), which comments on category theory, Mac Lane such as Richard Brauer, Reinhold Baer, and provided science policy advice to the US wrote in his autobiography: “At the time, Marshall Hall, to visit Chicago. Each Saturday government. In 1976, he led a delegation we sometimes called our subject ‘general morning, I listened to Thompson tell me of mathematicians to the People’s Republic abstract nonsense’. We didn’t really mean what he had been up to with groups; the of China to examine the conditions affecting the nonsense part, and we were proud of subject fitted his interest, and he chose the development of mathematics there. its generality.” Later, Mac Lane wrote his own problems.” In 1989, he received the highest US award Categories for the Working Mathematician, This spirit of guidance and selfless for scientific achievement, the National the title of which reflects his ability to encouragement probably grew out of Mac Medal of Science. convey humorously a serious intention Lane’s time in Göttingen, then the Mecca During the past few years, Mac Lane was — in this case, to make an abstract of mathematics, in the early 1930s, where engaged in writing his autobiography, and idea relevant. he became immersed in an atmosphere I have been privileged to be a part of that Mac Lane’s innovative contributions of intellectual challenge and excitement process. In the book, he wanted to convey to to mathematics had two roots. One was that formed his views. It was there that a new generation of mathematicians the joy his interest in what he called ‘universal he received his PhD with a thesis on of the creative process and the excitement knowledge’, which he recognized as a student “Abbreviated proofs in logic calculus”. that he experienced in being part of history. ■ at Yale. With a wry awareness of the problems Following ideas from Alfred North Klaus Peters this posed, he later wrote: “I believed that a Whitehead and Bertrand Russell, he worked Klaus Peters is at A K Peters Ltd, 888 Worcester proper academic ought to know everything, on formalizing mathematical proof. The Street, Suite 230, Wellesley, Massachusetts but of course, I never did succeed in subject did not impress his advisers Hermann 02482–3717, USA. He is the publisher of mastering universality, nor do I know exactly Weyl and Paul Bernays, and in Mac Lane’s Saunders Mac Lane — A Mathematical what this might have meant.” The other root own words: “My thesis did not attract any Autobiography, which will appear at the was his firm belief that mathematics has to be following, nor did it have any influence on end of May. developed from the specific to the general subsequent studies of mechanized proof.” e-mail: [email protected]