Species of structure and physical dimensions

Abstract

This study addresses the often underestimated importance of physical dimensions and units in the formal reconstruction of physical theories, focusing on structuralist approaches that use the concept of ``species of structure" as a meta-mathematical tool. Similar approaches also play a role in current philosophical debates on the metaphysical status of physical quantities. Our approach builds on an earlier proposal by Terence Tao. It involves the representation of fundamental physical quantities by one-dimensional real ordered vector spaces, while derived quantities are formulated using concepts from linear algebra, e.g. tensor products and dual spaces. As an introduction, the theory of Ohm's law is considered. We then formulate a reconstruction of the calculus of physical dimensions, including Buckingham's $\Pi$-theorem. Furthermore, an application of this method to the Newtonian theory of gravitating systems consisting of point particles is demonstrated, emphasizing the role of the automorphism group and its physical interpretations.