William Lawvere

(English version of Italian: Saunders Mac Lane, un matematico, Lettera Matematica PRISTEM 55, p. 13 )

Saunders Mac Lane, a mathematician

Saunders Mac Lane, who was for many years Professor of Mathematics at the University of Chicago, died on April 14, 2005, in San Francisco at the age of 95. As president of the Mathematical Association of America in the early 1950s, as president of the American Mathematical Society in the early 1970s, and later as vice-president of the National Academy of Sciences, he labored tirelessly to raise, through organizational means, the level of support for scientific research and for science education in the United States. But his lasting legacy is in the vast variety and depth of the mathematical advances made possible by his own direct contributions to mathematical research, and in the legions of students who have learned over the past 65 years from his popular textbooks on algebra and its applications.

In those textbooks Mac Lane produced original expositions of the most recent results: the “modern” algebra of Artin and Noether, which van der Waerden had diffused only at an advanced level, Mac Lane & Birkhoff treated for beginners; homological algebra and category theory have still received no better expositions than Mac Lane’s; and the still-developing field of topos theory is provided a “first introduction” in his 1991 book with Moerdijk.

Saunders Mac Lane’s doctoral dissertation, at the Goettingen of Hilbert, Noether, Weyl, and Bernays, was in logic. But his best-known contributions were in algebraic topology (after initial work in pure algebra, which resulted, for example, in the still frequently used “Mac Lane-Steinitz exchange relations”).

Algebraic topology became necessary in the epoch of Betti and Volterra, who noted that the behavior of electromagnetic fields and elastic fluids may depend on qualitative features of the form of bodies and containers, and that those qualitative features require measurement by quantities of a new type. The subsequent study of the Poincare’ groupoid and the Poincare’ problem was followed by Brouwer’s gathering Hopf, Noether, and Vietoris for intensive research in the 1920s; Hurewicz, de Rham, and Steenrod continued the advance in the 1930s and beyond, while Whitney distinguished one important class of these qualitative quantities as “cohomology”.

Mac Lane entered algebraic topology through his friend Samuel Eilenberg. Together they constructed the famous Eilenberg-Mac Lane spaces, which “represent cohomology”. That seemingly technical result of geometry and algebra, required, in fact, several striking methodological advances: (a) cohomology is a “functor”, a specific kind of dependence on change of domain space; (b) the category where these functors are defined has as maps not the ordinary continuous ones, but rather equivalence classes of such maps, where arbitrary continuous deformations of maps serve to establish the equivalences; and (c) although in any category any fixed object K determines a special “representable” functor that assigns, to any X, the set [X,K] of maps from X to K, most functors are not of that form and thus it is remarkable that the particular cohomological functors of interest turned out to be isomorphic to H*(X) = [X,K]  but only for the Hurewicz category (b) and only for the spaces K of the kind constructed for H* by Eilenberg and Mac Lane. All those advances depended on the concepts of category and functor, invented likewise in 1942 by the collaborators! Even as the notion of category itself was being made explicit, this result made apparent that “concrete” categories, in which maps are determined by their values on points, do not suffice.

It is sometimes said that Mac Lane moved away from his initial preoccupation with logic. But why then is his last book entitled “Sheaves in Geometry and Logic”? Before going to Goettingen, he had studied with E.H. Moore (the author of “General Analysis”). That same fierce quest for understanding urged him inexorably to later fundamental work that enabled a qualitative broadening and deepening of logic. Logic, as the serious study of the general aspects of all exact thinking, could not remain confined to the narrow “symbolic” tradition that emphasized presentations (“languages”) above the algebras themselves, nor to the narrow Frege tradition that maintained that concepts are mere properties. Taking due account of the achievements of those traditions, the Eilenberg-Mac Lane tradition has made possible during the past 60 years the systematic construction of actual geometric concepts for which properties can then be defined (and whose algebraic aspect sometimes needs presentations). In that way the hopes for a conscious guiding vision of the development of unified mathematics, which Moore a hundred years ago could only fragmentarily realize, have now become possible.

Mac Lane’s philosophy is broadly based on geometry, but a geometry grown so broad and multi-layered that it encompasses the earlier logic. With recognition of the unity of the principles of logic and geometry, and with the conceptual tools we inherit, we can hope to fulfill the central dream of Mac Lane (and Moore), to convey deeper understanding of modern mathematics to students and mathematical scientists.