Distances between formal theories

TL;DR

This paper introduces methods to measure and analyze the conceptual differences between formal theories, focusing on the minimal number of concepts that distinguish them, with applications to physics theories.

Contribution

It proposes novel measures of distance between formal theories, especially the conceptual distance, and demonstrates their applicability to physics theories like kinematics.

Findings

Conceptual distance can quantify differences between theories.

Relativistic and classical kinematics differ by one concept.

Multiple notions of distance are developed and discussed.

Abstract

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 introduce two main notions for such distances. The first one is that of \textit{axiomatic distance}, but we argue that it might be of limited interest. The more interesting and widely applicable notion is that of \textit{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. We also develop further notions of distance, and we include a number of suggestions for applying and extending our project.