Leslie Saunders Mac Lane

Born: 4 August 1909 in Norwich, Connecticut, USA
Died: 14 April 2005 in San Francisco, California, USA

Saunders Mac Lane came from a Scottish family who fled from Scotland after the Battle of Culloden in 1746. His grandfather, William Ward McLane was born in Lewisville, Pennsylvania in 1846, became a Presbyterian Minister but was forced to leave the Church after preaching the theory of evolution. He became a pastor in the Congregational Church and the eldest son from his second marriage, Donald Bradford McLane, was born 19 January 1882 in Steubenville. Donald married Winifred Saunders, the eldest daughter of George Aretas Saunders, in 1908. Winifred was a graduate of Mount Holyoke College and taught English, Latin and Mathematics at High School before her marriage. After his marriage, Donald became a Congregational Minister in Taftville, Connecticut and his eldest son Leslie Saunders MacLane, the subject of this biography, was born in Norwich, close to Taftville. When he was only one month old his parents decided they did not like the name Leslie and from that time on he was known as Saunders MacLane. The change from MacLane to Mac Lane only came about many years later after his marriage. Saunders had two brothers, Gerald R MacLane, who became a mathematics professor at Rice University and Purdue University, and David Tyler MacLane, born 20 January 1922, in Utica, New York, who became a school teacher and businessman. Mac Lane's early years were spent in several different small towns but he did live in one large city, spending a few years in Boston when aged around seven.

Mac Lane's High School education began in Utica, New York but was interrupted in 1924 when he was 14 years old for, at that time, his father died. After his father's death, Mac Lane moved to Leominster, Massachusetts, to live with his grandfather who, as we noted above, was also a Congregational Minister. Saunders graduated in 1926 from high school and, in that year, he entered the University of Yale. It had been his half-uncle, John Fisher MacLane (born 1878), who had visited him and explained that he and several of his relations had gone to Yale University. He offered to provide Mac Lane with sufficient funds to cover his expenses at Yale but he expected Mac Lane to train for a career in either business or law. Mac Lane's grandfather, William Ward McLane, died on 14 June 1931 a year after Mac Lane graduated from Yale.

In the autumn of 1926, Mac Lane began his studies at Yale but at this time his aim was to specialise in chemistry [3]:-

I took both an honours course in chemistry and the standard freshman mathematics course. I found the chemistry rather dull. I didn't enjoy laboratory work. I had a wonderful teacher in the freshman mathematics course, an instructor working for his Ph.D., named Lester Hill. He gave me lots of encouragement and said, "Mac Lane, why don't you take the Barge Prize examination?" They gave that examination to freshmen every year. So I took the examination, succeeded in winning the prize, and decided that maybe mathematics was a better field than chemistry.

However, he did not take only mathematics courses but, in his second year, he took an accounting course, feeling that he should do that since his uncle was financing his studies. Within weeks he was completely bored by accounting and, after one of his classmates said how much he was enjoying physics, Mac Lane went on to major in both mathematics and physics. He graduated from Yale in 1930 and took up a fellowship at Chicago. At the University of Chicago he was influenced by Eliakim Moore but his first year there, 1930-31, was [33]:-

... a vaguely disappointing year of graduate study.

By this time E H Moore was nearly seventy years old but his advice to Mac Lane to study for a doctorate at Göttingen in Germany certainly persuaded Mac Lane to work at the foremost mathematical research centre in the world at that time. Of course Moore had himself studied in Germany as a young man and had created in Chicago an eminent research school of mathematics based on his experiences of German mathematics at that time.

Mac Lane went to Göttingen in 1931 [33]:-

Hilbert had retired from his professorship, but still lectured once a week on "Introduction to Philosophy on the Basis of Modern Science". His successor, Hermann Weyl, lectured widely on differential geometry, algebraic topology and on the philosophy of mathematics (on which I wrote up lecture notes). From his seminar on group representations, I learned much (e.g., on the use of linear transformations), but I failed to listen to his urging that algebraists should study the structure of Lie algebras. I also was not convinced by his assertion that set theory involved too much "sand". Edmund Landau (professor since 1909) lectured to large audiences with his accustomed polished clarity - and with assistants to wash off used (rolling) blackboards. Richard Courant, administrative head of the Institute, lectured and managed the many assistants working on the manuscript of the Courant-Hilbert book. Gustav Herglotz delivered eloquently his insightful lectures on a wide variety of topics: Lie groups, mechanics, geometrical optics, functions with a positive real part. Felix Bernstein taught statistics, but left in December 1932 before the deluge struck. These were then the 'ordentliche' professors in Göttingen.

However, political events would soon disrupt Göttingen. Mac Lane began to work for his doctorate under Paul Bernays' supervision but in 1933 the Nazis came to power. They began to remove the top mathematicians from Göttingen, and other universities, who had Jewish connections. Mac Lane wrote to his mother on 3 May 1933 (see [33]):-

So many professors and instructors have been fired or have left that the mathematics department is pretty thoroughly emasculated. It is rather hard on mathematics, and we have but the cold comfort that it is the best thing for the Volk.

Mac Lane had seen that he had to work quickly for his doctorate and leave Germany as soon as possible before things deteriorated further. He defended his thesis Abbreviated Proofs in the Logical Calculus, with Weyl as examiner, on 19 July 1933. At the end of his thesis he thanked his advisor Paul Bernays "for his criticism", "and most of all Professor Hermann Weyl for his advice and for the inspiration of his lectures". Mac Lane was rather disappointed to learn that Weyl had only rated his thesis "sufficient". A couple of days after getting his degree, he married Dorothy Jones who he had met in Chicago and who had joined him in Göttingen (in fact she had typed his thesis). It was a small ceremony followed by a wedding dinner with a couple of friends in the Rathaus Keller. The newly married couple quickly returned to the United States. The article [33] by Mac Lane gives an interesting account of the events at Göttingen in 1933.

On returning to the United States, Mac Lane spent the session 1933-34 at Yale while he tried to get a university position for the following year. He went to the American Mathematical Society meeting in December 1933 and talked with George Birkhoff, Marshall Stone and Joseph Ritt about possible positions. A couple of weeks after this he received a letter from Harvard offering him an appointment for a year as Benjamin Peirce Instructor. He was also asked to give an advanced course. He gladly accepted. Up to this time he had worked on mathematical logic but that was not a topic that was attractive to those making appointments to mathematics departments. Offered the option of giving the advanced course at Harvard on logic or algebra, he opted for algebra and he began to move in that direction. He spent two years at Harvard and left to take up a post of instructor at Cornell for session 1936-37. He spent the following session back at Chicago, working in algebra and getting much help from Adrian Albert, before accepting an appointment as an assistant professor at Harvard which he took up in 1938.

It was during the years at Harvard (1938-47) that he wrote his famous text A survey of modern algebra with Garrett Birkhoff which was published in 1941. Kaplansky writes in [24] about this text:-

"A Survey of Modern Algebra" opened to American undergraduates what had until then been largely reserved for mathematicians in van der Waerden's "Moderne Algebra", published a decade earlier. The impact of Birkhoff and Mac Lane on the content and teaching of algebra in colleges and universities was immediate and long sustained. What we recognise in undergraduate courses in algebra today took much of its start with the abstract algebra which they made both accessible and attractive.

During World War II Mac Lane worked in the Applied Mathematics Group at Columbia [3]:-

We were doing some immediately practical problems of calculating curves for fire control and such. It was elementary differential equations and I learned to understand more about them. But the work at Columbia was not in really profitable directions of applied mathematical research. Also it was partly administrative and I decided that I didn't especially like administration. I remember making a conscious decision at that time: if you want to go into business and make that your career, now is the time to do it. I didn't. I went back to Harvard, happy to go back to mathematics.

Then in 1947 he was appointed professor of mathematics at Chicago. The research centre there had Marshall Stone, Abraham Albert, Irving Kaplansky, Otto Schilling and André Weil on the staff and was led by Stone. In 1952, five years after being appointed, Mac Lane took over the chairmanship of the department from Stone who stepped down as chairmen but remained on the staff for another sixteen years. Wojciech Komornicki attended Mac Lane's algebra course in 1968-69 and describes the experience in [27]:-

... it was nothing like any mathematics course I had ever taken. Though teaching from his own book, it seemed that Mac Lane would go out of his way to present the material differently than what was in the text. Sometimes he would give more than one proof of the same theorem. He made sure that we were aware of this, explaining that the more ways we understood something the better our understanding. Though I once again saw the same definition of a group that I remembered from my NSF program, I heard that a group was a category with one object in which all the arrows were invertible. The product of two groups was a universal object. I remember struggling with these new concepts especially since Mac Lane's style was to give us the big picture leaving most, but not all, of the details to the book and to the problems he assigned. However, the understanding of the mathematics was paramount in Mac Lane's presentations. If he thought that someone did not understand a proof, he would provide an alternative proof. And there was never any hint that a concept or a proof was too complex for someone to understand. Another aspect of his teaching that struck me was that he never came to class with notes. He would every once in a while pull out an index card on which I assume he had the subject matter of the day's class, look at it, put it back in his shirt or jacket pocket and continue the lecture. Though his lectures were meticulous in presenting the global picture of what we were learning he took great care to answer questions. No question was too small and he never blew off a question nor showed any irritation by student asking questions.

Mac Lane's work covered a wide range of mathematics. He worked on and off throughout his career on mathematical logic, no surprise for a student of Bernays, and he did some early work on planar graphs. He studied valuations and their extensions to polynomial rings. In the 1940s he worked on cohomology and introduced the basic notions of category theory. Peter May, Professor in Mathematics at the University of Chicago, said (see [28]):-

[In his research] he was extraordinarily perceptive and original, and he was especially strong as a philosopher of mathematics. With Sammy Eilenberg he created a new way of thinking about mathematics. In a landmark 1945 paper, they introduced and named the concepts of 'categories,' 'functors' and 'natural transformations.' The language they introduced there transformed modern mathematics. In fact, a very great deal of mathematics since then would quite literally have been unthinkable without that language.

Mac Lane was the author of seven books: (with Garrett Birkhoff) A Survey of Modern Algebra (1941); Homology (1963); (with Garrett Birkhoff) Algebra (1967); Categories for the Working Mathematician (1971); Mathematics, Form and Function (1985); (with Ieke Moerdijk) Sheaves in Geometry and Logic: A First Introduction to Topos Theory (1992); and Saunders Mac Lane: A Mathematical Autobiography (2005).

Among the many prizes and honours Mac Lane received we mention the Chauvenet Prize (1941):-

... for his writing 'Modular Fields' and 'Some Recent Advances in Algebra' ...

and the Leroy P Steele Prize (1986):-

... for his many contributions to algebra and algebraic topology, and in particular for his pioneering work in homological and categorical algebra.

He received the National Medal of Science in 1989:-

For revolutionizing the language and content of modern mathematics by his collaboration in the creation and development of the fields of homological algebra and category theory, for outstanding contributions to mathematics education, and for incisive leadership of the mathematical and scientific communities.

For his contributions to the Mathematical Association of America, Mac Lane received the Association's Award for Distinguished Service for 1975. He was a Governor of the Association (1943-45), a Vice-President (1948-49), and President for 1951-52 [10]:-

His presidency made an enduring impression on those who were active in the Association at that time. He saw the office less as an honour than as an opportunity, attacked the problems of collegiate mathematics with characteristic imagination and energy, and set the Association on the active course that it has followed ever since.

He was elected to the National Academy of Sciences in 1949 and served as its vice-president from 1973 to 1981. He was also elected to the American Philosophical Society in 1949, and served as its vice-president from 1968 to 1971. He was president of the American Mathematical Society in 1973-74 [10]:-

As President of the American Mathematical Society, Mac Lane has given that Society the dynamic leadership that one would have anticipated, initiating new activities and vigorously pursuing old ones. Many members of the Society especially valued the openness of his administration and his new lines of communication with the membership. He handled difficult problems with exemplary skill which many of us have enjoyed observing at first hand. His presidency of the Society, like his presidency of the Association, will be remembered with admiration.

Among his other honours we note that he was elected an Honorary Fellow of the Royal Society of Edinburgh in 1972. He received honorary degrees from many universities including Purdue University, Yale University and the University of Glasgow.

Saunders and Dorothy Mac Lane had two daughters, Gretchen and Cynthia. Dorothy died in 1985 and Mac Lane married the artist Osa Skotting Segal (who was divorced from Irving Segal in 1977) in the following year. Mac Lane's hobbies included skiing, hiking and writing poetry. He would read aloud the works of poets such as Byron, Shelley, Keats and Wordsworth. Osa said [55]:-

And so I was reading poetry to him - those same poets - when he was lying there at the end of his life.

Kelly, in [25], writes:-

No man could so stimulate others unless, alongside an incisive intellect, he was possessed of enthusiasm and warmth, a deep interest in his fellow man, and a sympathy the more real for being unsentimental. Those who proudly call themselves his friends know these things: others will infer them in reading [his works].

Article by: J J O'Connor and E F Robertson

October 2015

MacTutor History of Mathematics