There are several theorems in algebra where one purposely forgets certain information about the coefficients of a polynomial and then sees whether certain properties of the roots can still be determined. A prototypical example is Descartes’ Rule of Signs, where we forget everything about a polynomial P except for the signs of its coefficients and then ask for information about the signs of the real roots of P. I will explain a novel algebraic framework for systematically understanding results of this type. I will then discuss connections to matroid theory, including the construction of a "moduli space of matroids" and a generalization of a theorem of Lafforgue about “rigid” matroids. This is joint work with Oliver Lorscheid.