Distances Between Formal Theories

Abstract

In the literature, there have been several methods and definitions for working out whether two theories are “equivalent” (essentially the same) or not. In this article, we do something subtler. We provide a means to measure distances (and explore connections) between formal theories. We introduce two natural notions for such distances. The first one is that of axiomatic distance, but we argue that it might be of limited interest. The more interesting and widely applicable notion is that of conceptual distance which measures the minimum number of concepts that distinguish two theories. For instance, we use conceptual distance to show that relativistic and classical kinematics are distinguished by one concept only.