ABSTRACT
Lawvere was an assistant professor at Chicago when I taught a version of this course; from listening to my use of differential geometry, he came to think about the justification of older intuitive methods (S. Lie, et al.) in geometry. This, in turn, led him to start to develop the subject synthetic differential geometry, which uses a version of actual infinitesimals to formulate geometric ideas. His current Ph.D. student, Anders Kock, took up the idea, as did others, including my student Eduardo Dubuc: there was a full-blown development of their ideas, which do indeed provide a working system of infinitesimals for geometry, as presented, for example, in a book by Mordijk and Reyes. However, those promising ideas have not yet entered into everyday use by differential geometers.
