Abstract

We are used to seeing foundations linked to the mainstream mathematics of the late nineteenth century: the arithmetization of analysis, non-Euclidean geometry, and the rise of abstract structures in algebra. And a growing number of case studies bring a more philosophy-of-science viewpoint to the latest mathematics, as in [Carter, 2005; Corfield, 2006; Krieger, 2003; Leng, 2002]. Mac Lane's autobiography is a valuable bridge between these, recounting his experience of how the mid- and late-twentieth-century mainstream grew especially through Hilbert's school.An autobiography at age 94 obviously has a lot of ground to cover. Mac Lane entered Yale in 1926 to study chemistry as a good practical career field but by the end of his first year he had won a $50 prize in mathematics and learned that you could make a living with it—as an actuary. He decided to do that. The next year he was excited to learn from his philosophy professor F.S.C. Northrop that you can also pursue new mathematical discoveries! Northrop had studied with A.N. Whitehead and sold Mac Lane on the excitement of Principia Mathematica. In a pattern that foreshadowed his career Mac Lane bought and annotated a copy of the first volume of Principia but his planned tutorial study of the book turned into a study of [Hausdorff, 1914] applying set theory to topology and analysis. As a graduate student at Chicago he was powerfully influenced by E.H. Moore, who taught the new axiomatic methods as tools for unifying and advancing the most classical analysis . Then he went to Göttingen. He heard Hilbert lecture on philosophy. He studied mathematics and its philosophy with Weyl. He followed Noether's lectures on abstract algebra for number theory. He had gone there to study logic and he wrote a …