Inferences about Structure

This chapter locates a few rules that govern our inferences about structure in physics, three in particular: inferring structure from the laws, minimizing structure, and matching structure. The chapter illustrates these rules by means of familiar inferences that rely on them. These inferences concern a variety of physical theories, from Aristotle’s physics and Newton’s laws to time reversal invariant laws and special relativity. The discussion contrasts these epistemic rules with other guiding principles in the literature. Along the way, the general idea of the structure presupposed by the laws, a theory’s dynamical structure, is elucidated. The chapter ends with a discussion of the extent to which coordinate systems and the form of an equation can tell us about the nature of physical reality.

What is Structure? Why Care about It?

This chapter explains the notion of structure that will be the focus of the book and illustrates it by means of examples drawn from mathematics and physics. The discussion begins with a simple example of the structure of the Euclidean plane, and goes on to explain how similar ideas apply to physical theories such as Newtonian physics and special relativity. Taken together, the examples illustrate that this notion is implicit in many aspects of our theorizing in physics and mathematics. The chapter also discusses the idea of allowable coordinate systems and reference frames; contrasts the relevant notion of structure with other related notions, including invariance, symmetry, and objectivity; and explains how to compare different types and amounts of structure.

On the Equivalence of Physical Theories

This chapter argues against formal accounts of theoretical equivalence in physics. It defends the importance of a theory’s picture of the world and its explanations of the phenomena, drawing on examples from classical physics, Newtonian gravitation, classical electromagnetism, special relativity, and quantum mechanics. The discussion draws a distinction between metaphysical equivalence and informational equivalence and argues that these are equally important to the equivalence of physical theories. The chapter concludes that there are fewer cases of wholly equivalent theories in physics than usually thought. However, this is not a problem, for it is still possible to talk about the various respects in which physical theories are, or are not, equivalent to one another.

Spatiotemporal Structure

This chapter applies our reasoning about structure in physics to the debate about spacetime ontology. The discussion starts by explaining why the traditional debate about the existence of space(time) may well be at a stalemate: each side has plausible responses to the usual cases aimed against it. It then argues that reconstruing the debate as being about the fundamentality of spatiotemporal structure yields a substantive dispute that is relevant to current and future physics. It argues that the epistemic principles governing structure support substantivalism over relationalism, given current formulations of physics, and challenges the relationalist to come up with reformulations that meet certain criteria. The idea of direct formulations of physical theories and why they are preferable is also discussed.

Classical Mechanics

This chapter applies our reasoning about structure in physics to two formulations of classical mechanics, Lagrangian and Newtonian mechanics, that are generally taken to be completely equivalent. It argues that these two formulations differ in both the type and amount of structure presupposed by their dynamical laws, as revealed by the invariances of the equations representing the laws as well as the theories’ statespace structures. This suggests that these are not fully equivalent theories: they differ in dynamical structure. There are also various metaphysical differences between them. The chapter goes on to argue that the minimize-structure rule tells us to choose one over the other. Along the way, the idea that preferred or natural coordinate systems indicate underlying structure is discussed.

Realism about Structure

This chapter rebuts objections to the account, dispelling concerns about taking the mathematical structures of our best physical theories seriously. It outlines further aspects of the book’s realism about structure and explains what this realism is, and what it is not, committed to. It discusses the ideas of reasonably straightforward interpretations of physics; of perspicuous formulations of physics; and of direct and indirect representations. It argues against a different kind of realism about structure in physics, which is at once both stronger and weaker than the realism about structure defended in the book. It ends by arguing against focusing on the structure of a theory’s models in our physical theorizing in general, and the model isomorphism criterion for theoretical equivalence in particular.

Introduction

This chapter provides an overview of the book: its aims, scope, and key themes. The chief project is to unearth how we figure out what our mathematically formulated, best physical theories tell us about the nature of the world, and how we figure out what these theories tell us when they can be mathematically formulated in different ways. The chapter outlines the core theme of taking the mathematics in which our best physical theories are formulated seriously; notes the background assumption of scientific realism; outlines the idea that direct formulations of physical theories are preferable; explains the idea of taking a theory’s metaphysical aspects seriously; and notes that one upshot of the account will be fewer cases of theoretical equivalence than usually thought.