Shape space as a conceptual space

TL;DR

This paper explores shape space as a mathematical and conceptual tool in physics, particularly in relational dynamics, questioning whether it is a physical space or a useful conceptual framework.

Contribution

It argues that shape space should be viewed as a conceptual space rather than an actual physical space, clarifying its role in relational physics.

Findings

Shape space is better understood as a conceptual rather than physical space.

Relational dynamics can be framed within shape space as a conceptual tool.

The paper connects shape space to the broader idea of conceptual spaces in science.

Abstract

The notion of shape space was introduced in the second half of the 20th Century as a useful analytical tool for tackling problems related to the intrinsic spatial configuration of material systems. In recent years, the geometrical properties of shape spaces have been investigated and exploited to construct a totally relational description of physics (classical, relativistic, and quantum). The main aim of this relational framework - originally championed by Julian Barbour and Bruno Bertotti - is to cast the dynamical description of material systems in dimensionless and scale-invariant terms only. As such, the Barbour-Bertotti approach to dynamics represents the technical implementation of the famous Leibnizian arguments against the reality of space and time as genuine substances. The question then arises about the status of shape space itself in this picture: Is it an actual physical space in which the fundamental relational dynamics unfolds, or is it just a useful mathematical construction? The present paper argues for the latter answer and, in doing so, explores the possibility that shape space is a peculiar case of a conceptual space.