**Title: **Reconciling two ways to generalize counting

**Abstract:** In this talk, we will look at two ways to generalize counting from finite sets to structures that allow subtraction and division, respectively, in a topologically meaningful way. At first glance they seem to be orthogonal generalizations, but some intuitions involving divergent sums suggest they can be combined. This will indeed be the case given some restrictions, as we shall see. One application is to a question that can be stated in group theory. Furthermore, some accounts lead to structures from chromatic homotopy theory e.g. generalizations of character theory. This is intended to be a fun introductory talk, and in particular very little knowledge of anything in homotopy theory is assumed until towards the end.