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Species of structure and physical dimensions

Heinz-Jürgen Schmidt

TL;DR

This work constructs a structuralist reconstruction of physical theories that foregrounds dimensional analysis by representing physical quantities as one-dimensional ordered real vector spaces and building derived quantities via tensor products and duals. It embeds this dimensional calculus within the broader meta-theory of species of structure, enabling a formal treatment of Buckingham's $\Pi$-theorem and offering a unified framework for Ohm's law and Newtonian gravitation. The paper analyzes automorphisms and their physical interpretations (active, passive, and Leibniz), shows how dilations constrain symmetries, and demonstrates that constants like the gravitational constant $\Gamma$ can be uniquely determined within a model yet transform under isomorphisms. It argues that integrating dimensional structure into structuralist reconstructions clarifies debates about the metaphysical status of physical quantities and paves the way for extensions to relativistic settings and natural-unit analyses.

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 $Π$-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.

Species of structure and physical dimensions

TL;DR

This work constructs a structuralist reconstruction of physical theories that foregrounds dimensional analysis by representing physical quantities as one-dimensional ordered real vector spaces and building derived quantities via tensor products and duals. It embeds this dimensional calculus within the broader meta-theory of species of structure, enabling a formal treatment of Buckingham's -theorem and offering a unified framework for Ohm's law and Newtonian gravitation. The paper analyzes automorphisms and their physical interpretations (active, passive, and Leibniz), shows how dilations constrain symmetries, and demonstrates that constants like the gravitational constant can be uniquely determined within a model yet transform under isomorphisms. It argues that integrating dimensional structure into structuralist reconstructions clarifies debates about the metaphysical status of physical quantities and paves the way for extensions to relativistic settings and natural-unit analyses.

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 -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.
Paper Structure (17 sections, 4 theorems, 78 equations, 3 figures)

This paper contains 17 sections, 4 theorems, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Ohm's law $U=R\,I$
  3. Calculus of physical quantities and Buckingham's $\Pi$-theorem
  4. Scalar quantities
  5. Buckingham's $\Pi$-theorem
  6. $3$-dimensional vector quantities
  7. Species of structure
  8. General account
  9. Species of structure and $\Pi$-theorem
  10. Theory of $N$ gravitating point particles
  11. General account
  12. Isomorphisms and automorphisms
  13. Active and passive interpretation of automorphisms
  14. Summary and Outlook
  15. Proof of Theorem \ref{['TPi']}
  16. ...and 2 more sections

Key Result

Theorem 1

With the preceding notation assume that a physical law of the form $f(Q_1,\ldots, Q_L)=0$ is dilationally invariant in the domain $\widetilde{\mathcal{Q}}\subset {\mathcal{Q}}_+$. Then there exist $K$ dimensionless quantities with values of the form and a function $F:\Pi_+\to {\mathbb R}$ such that

Figures (3)

  • Figure 1: Schematic sketch of an electrical circuit for carrying out measurements to confirm Ohm's law.
  • Figure 2: Schematic sketch of the affine geometry of the $\Pi$ theorem. The three-dimensional affine space shown in the figure represents ${\mathcal{Q}}_+$. A dilationally invariant physical law is represented by a subset ${\mathcal{S}}\subset {\mathcal{Q}}_+$ which is invariant under translations from a certain subspace, here represented as translations in $3$-direction. This subspace generates an equivalence relation $\sim$ on ${\mathcal{Q}}_+$. The lines parallel to the $3$-direction in the figure represent certain $\sim$ equivalence classes contained in ${\mathcal{S}}$, which are generally $R$-dimensional planes in ${\mathcal{Q}}_+$. Alternatively, the law can also be represented by a subset ${\mathcal{S}}"$ of the $K$-dimensional affine space space $\Pi_+$, which is isomorphic to the quotient space ${\mathcal{Q}}_+/_\sim$.
  • Figure 3: Sketch of a structural element ${\mathcal{G}}$ of $TN$. The $x$-coordinate represents the norm of the total gravitational force upon a particle divided by its mass and by the gravitational constant $\Gamma$ (physical dimension $\hbox{mass}\times \hbox{length}^{-2}$) denoted by $r_i(t)$ in (\ref{['eomnorm']}). The $y$-coordinate represents the norm of its acceleration (physical dimension $\hbox{length}\times \hbox{time}^{-2}$) denoted by $\ell_i(t)$ in (\ref{['eomnorm']}). Possible gravitational experiments (some of which are indicated in this Figure) give rise to points that lie on a "universal line" ${\mathcal{G}}$ with slope $\Gamma$. For special cases the total gravitational force upon a particle may vanish which is indicated by the point with coordinates $(0,0)$.

Theorems & Definitions (4)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • Proposition 2