## Mister Mathematics: Saunders Mac Lane

**November 20, 2005**

Saunders Mac Lane, whose autobiography provides an excellent description of the life, both professional and personal, of a native 20th-century American mathematician. Photo from Saunders Mac Lane: A Mathematical Autobiography.

**Book ReviewPhilip J. Davis**

**Saunders Mac Lane: A Mathematical Autobiography.** *AK Peters, Wellesley, Massachusetts, 2005, 358 + xvi pages (including many photos and a short preface by David Eisenbud), $39. *

Saunders Mac Lane (1909–2005), a New Englander by heritage, the son and grandson of Congregational ministers, was born in Norwich, Connecticut. He claimed descent on his father's side from Clan MacLean, Scottish Highlanders who were evicted from their homes during the "highland clearances" of the early 1800s, and on his mother's side from William Bradford, a passenger on the Mayflower and a governor of the Plymouth Colony.

My personal acquaintance with Mac Lane was slight. In the spring of 1940, as my freshman adviser at Harvard, he had to agree with and sign my study plan for the next three years. Returning from World War II in 1946, I sat in as a graduate student on his course on Galois theory. I recall raising my hand once to ask a question and his remarking that it was pertinent.

In the years that followed, I would see him at a distance at national meetings, hastening down corridors as though always late, going from talks to committee meetings. Slightly stooped, he was burdened by an old brown briefcase, stuffed to the brim with papers and books. Mac Lane was a very important and influential figure on the U.S. mathematical scene, a president of the American Mathematical Society (1973–74), a vice president of the National Academy of Sciences, an adviser to governmental committees. He seemed to be omnipresent committee-wise, and I began to call him "Mr. Mathematics"--completely to myself, of course. This was in imitation of the sobriquet the newspapers gave to Ohio Senator Robert Taft in the late 1940s: "Mr. Republican."

Over the years Mac Lane was to me simply a mathematical personality. I knew nothing of his personal life. But as I read in the autobiography about the early part of his life--the Flexible Flyer sled, the wanderings of the family in search of employment for his father and grandfather, who earned little as ministers and were involved in sectarian squabbles, the generous uncle who supported Mac Lane in college--all of it was vivid and rang true. As a New Englander, though born a decade and a half later and of a different provenance, I could find parallels with my own early years.

A Yale man (1930), Mac Lane went on to the University of Chicago, where he re-ceived an MA. After what he considered profitless months in Chicago, he chose, in an unusual career departure for those years, to go to Göttingen for a higher degree. He received the D.Phil. from the Georg-August-Universität Göttingen in 1934, with a thesis titled Abbreviated Proofs in the Logical Calculus, written under the supervison of Hermann Weyl and logician Paul Bernays. His stay in Germany spanned the terrible years during which Göttingen, the leading mathematical institution in the world, was destroyed by the fury of Nazi anti-

semitism.

Mac Lane taught at Harvard from 1938 to 1947. During World War II (1943–44), he was part of the Applied Mathematics Group at Columbia University, working on a variety of problems posed by the Air Force, and this, as far as I can tell, was his only foray into applied mathematics. As regards the scattered pages in which Mac Lane describes his mathematical accomplishments, those who know about his work will know what it's all about; those (such as myself) who don't know about categories, functors, cohomology of groups, cross product algebras and group extensions, Eilenberg–Mac Lane spaces will be impressed by the breadth of the areas he worked in.

Mac Lane's great success education-wise was *A Survey of Modern Algebra* (1941), a text he wrote jointly with his contemporary and colleague Garrett Birkhoff. I suspect that many mathematicians know his name only in this connection. This book had been preceded by American texts on what used to be called "higher algebra," Maxime Bôcher's *Introduction to Higher Algebra* (1907, with later editions) and Adrian Albert's *Modern Higher Algebra* (1937) being two examples. But Birkhoff and Mac Lane's *A Survey of Modern Algebra*, inspired to a considerable extent by the work of Emmy Noether and by B.L. van den Waerden's *Moderne Algebra* (1930), hit the right abstract tone at the right time with the right selection of topics and the right manner of presentation. It became an instant undergraduate classic and a path setter for other authors.

In his later years, Mac Lane became a bit philosophical and wrote a number of papers that I (mistakenly) thought were edging close to my own views. He describes his 1986 book *Mathematics, Form and Function* as a philosophical work, but I see it as an overview of the whole of mathematics according to his lights--and, nonetheless, a considerable tour de force. Yes, there is in it the occasional philosophical remark. He gave a puff to one of my books with Ruben Hersh, *The Mathematical Experience*, in his introduction, so I conclude that at least he knew about philosophies other than the standard platonism, formalism, and intuitionism. I believe he was au fond a formalist. He called himself a "formal functionalist," but as far as the functions of mathematics in the sense of contemporary applications--forget it. I can only conjecture how he must have regarded the quip of his thesis adviser Hermann Weyl: "In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." Looking around in recent times, Mac Lane might have reformulated Weyl to have the angel of abstract algebra and the devil of computation fighting for the soul of mathematics.

While in his eighties, Mac Lane submitted an initial manuscript of his autobiography to AK Peters, but it was very rough, tentative, and incomplete. Janet Beissinger of the University of Illinois, Chicago, was called in as manuscript doctor. By working with Mac Lane personally, in weekly all-day sessions for three months or so, Beissinger was able to produce a publishable version. Indeed, divided into 64 short chapters and studded with many photos, the book is an easy read. The pages reveal Mac Lane as an open, liberal, sympathetic individual, an excellent and influential teacher, long on family and friends, and a man wholly devoted to mathematics and his own contributions to it.

Mac Lane had a light side: He liked a good joke, the occasional parody, and he wrote verse that he was not averse to declaim. He was quite capable of the sharp comment: Although soft on Bourbaki, he quipped, "The only thing that Bourbaki never understood was the foundations of mathematics."

Speaking of his move (in 1947) from Harvard to the University of Chicago, where he spent the rest of his career, he gave as one reason that "Dorothy (his wife) never fit in well with the traditions or autocracy in New England."

"(William F.) ‘Foggy' Osgood [author of undergraduate texts and of the very influential *Lehrbuch der Funktionentheorie* (1907, with later editions)] thought that the formal discussion of limits was a bit much for Harvard freshmen so he used the older notion of infinitesimals."

Anecdotes pour out of Mac Lane: Senior faculty systematically assessed the classroom work of the junior faculty. Professor Julian Lowell Coolidge bawled out Mac Lane for talking too fast--a sin that Coolidge himself admitted to. (What would Coolidge have said about today's PowerPoint presentations, whose information overload would blow anyone's circuits?) A decade later, I got bawled out by G.D. Birkhoff for not wearing a jacket when I lectured on a hot summer's day. (What would Birkhoff have said about today's sartorial sloppiness in academia?)

At a Harvard lecture given by Bertrand Russell, Mac Lane, in an impish mood, tried to upstage Russell by asking him about recent developments in mathematical logic, even though he knew that Russell had long left the field. Russell wisely did not respond, and the moderator moved on with other questions.

One misses a bit of pain in the autobiography. Mac Lane does write at some length about his vexation that Hermann Weyl gave his doctoral thesis a grade of "genügend" (sufficient) and that the ideas in it, which in euphoric letters to his mother Mac Lane hoped would set the world of logic on its ear, didn't pan out. He mentions his wife Dorothy's long-term disability. But all in all, I suspect that Mac Lane's long life was more serene than the average.

His autobiography provides an excellent description of the life, both professional and personal, of a native 20th-century American mathematician. The personal part is absolutely representative of one fine American tradition and sensibility. In my mind's eye, I can easily imagine him posing for *American Gothic*, Grant Wood's famous painting. The professional part is paradigmatic: the research; the constant jetting around to conferences; the endless committees, local, national, international. The students, the courses and curricula; the personality squabbles and acts of one-upmanship; the never-ending applications to foundations for contracts and grants; the visiting lectureships; the honors. These were the ingredients that made up the life of a 20th-century mathematician. None of the cheek-slashing duels he had found on his arrival in Germany.

It would be revealing and amusing if someone were to write the life of a mathematician of the deep past about which a fair amount is known--Cardano, say--and embed it in today's rituals. "Saint Andrew's Colloquium: August 25, 1552. Prof. G. Cardano of the University of Pavia, mathematician, physician, and papal astrologer, will describe his joint work with N. Tartaglia on the applications of the cubic equation to the casting of horoscopes. This research was supported in part by grant 51– 002317 from the Sforza Discretionary Fund of Milan."

All mathematicians are creatures of the age in which they live.

*Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at **[email protected]**.*