Distances between formal theories

In the literature, there have been several methods and definitions for working out if two theories are "equivalent" (essentially the same) or not. In this article, we do something subtler. We provide means to measure distances (and explore connections) between formal theories. We define two main notions for such distances. A natural definition is that of axiomatic distance, but we argue that it might be of limited interest. The more interesting and widely applicable definition is that of conceptual distance which measures the minimum number of concepts that separate two theories. For instance, we use conceptual distance to show that relativistic and classical kinematics are distinguished by one concept only. We also develop further notions of distance, and we include a number of suggestions for applying and extending our project. We end with a philosophical discussion of the significance of these approaches.