Works in Progress

Studying mathematics in high school or even college rarely gives you a sense of what mathematical research is all about. The strongest impression that often emerges is of a field in which practically all is known—and has been known for a long time.

Courtesy of A K Peters

In a soon-to-be-published autobiography, mathematician Saunders Mac Lane, who spent much of his career at the University of Chicago and helped develop a branch of algebra called category theory, expressed just this sort of sentiment about his time as a student at Yale University in the 1920s.

“At the start of my active interest in a mathematics career, I do not believe I was aware of research that leads to new results,” Mac Lane wrote. “Mathematics was fascinating, and much of it new to me. Calculus, for example, was exciting, but it seemed as though it had long since been entirely worked out.”

Born in 1909, Mac Lane died earlier this month at the age of 95. His autobiography will be published in May by A K Peters.

Mac Lane, at 17, had gone off to Yale in the fall of 1926. He had intended to major in chemistry but found laboratory work a bit dull and unappealing. In calculus, however, he discovered a “remarkable world of derivation and limits, as well as Newton’s wonderful use of calculus to describe the planetary orbits.”

By the following spring, after winning a mathematics prize, Mac Lane was pondering the possibility of a career in mathematics—becoming an actuary with a job in an insurance company. He became a math major.

Mac Lane relished his encounters with mathematics, but there was a dearth of new ideas and open questions.

“I remember well my course on the beautiful subject of theoretical mechanics with E.W. Brown, an expert on the motions of the moon,” Mac Lane wrote. “He lectured to us from dog-eared notes on classical mechanics and the deep ideas of Hamiltonian mechanics. This subject, apparently, had all been worked out.”

Mac Lane reacted in much the same way to advanced calculus, Cantor’s set theory, Hausdorff’s work on topological spaces, and other areas of mathematics that he encountered. Even in newer areas, the work seemed largely finished. It was simply a matter of filling in some details.

“It seemed as if all of our attention was directed toward knowledge that was already known; therefore, during the first years of my undergraduate education, I put my own emphasis on acquiring universal knowledge—the assimilation and organization of everything known,” Mac Lane recalled.

Mac Lane proved very good at accumulating this knowledge—not just in mathematics but also in all the other courses that he took. His grades were high. Indeed, Mac Lane was awarded a prize for achieving the “highest-ever” overall average in his studies during his first two-and-a-half years at Yale.

It wasn’t until his senior year that Mac Lane got his first, tantalizing glimpse of mathematics as a work in progress. Norwegian mathematician Oystein Ore (1899–1968) had come to Yale and was teaching two graduate courses, one in group theory and the other in Galois theory.

“I came to listen, and discovered the developing ideas of modern abstract algebra,” Mac Lane said.

“The work of Emmy Noether [1882–1932] and her successors indicated to me that there were brand-new ideas to be found in mathematics,” he noted. “With this indication, my focus shifted from the accumulation of knowledge to the hope of discovering new knowledge.”

Mac Lane’s experience provided a valuable lesson. As he put it, “effective teaching that motivates and stimulates students into creativity grows out of research and the incorporation of new ideas into the curriculum.”

Several generations earlier, English mathematician G.H. Hardy (1877–1947) described his own early mathematical training and interests as being focused on examinations and scholarships.

“I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble,” Hardy wrote in 1940 in A Mathematician’s Apology. “I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.”

Even at Trinity College, Cambridge, which he entered in 1896, Hardy spent 2 years focused on cramming for the infamous Mathematical Tripos examination. There was nothing of new mathematics. It was a matter of learning the tricks of the trade to achieve the maximum possible score on the examination.

“. . . I was really quite ignorant, even when I took the Tripos, of the subjects on which I have spent the rest of my life; and I still thought of mathematics as essentially a ‘competitive’ subject,” Hardy wrote.

Luckily, with the help of Cambridge professor A.E.H. Love (1863–1940), Hardy developed his first serious concept of the area of mathematics known as analysis. And from reading the influential, path-breaking book Cours d’analyse by Camille Jordan (1838–1938), he learned for the first time “what mathematics really meant.”

Nowadays, the situation may be a little better—at least for some students. There are undergraduate research opportunities at many colleges, and even high school students, with proper guidance and mentoring, can undertake sophisticated mathematics projects that venture into new mathematics. Nonetheless, a continuing emphasis on standardized tests, mathematics competitions, and simply learning the “tricks of the trade” pushes such endeavors out of the reach of many students.

Mathematics is full of unanswered questions, which far outnumber known theorems and results. Indeed, it’s the nature of mathematics to pose more problems that it can solve. Part of a mathematical education should include some sense of what is known and what is not yet known (and may never be known) and what progress is being made in creating new mathematics.

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