I argue that a criterion of theoretical equivalence due to Glymour :227–251, 1977) does not capture an important sense in which two theories may be equivalent. I then motivate and state an alternative criterion that does capture the sense of equivalence I have in mind. The principal claim of the paper is that relative to this second criterion, the answer to the question posed in the title is “yes”, at least on one natural understanding of Newtonian gravitation.

I respond to the frequent objection that structural realism fails to sharply state an alternative to the standard predicate-logic, object / property / relation, way of doing metaphysics. The approach I propose is based on what I call a ‘math-first’ approach to physical theories (close to the so-called ‘semantic view of theories') where the content of a physical theory is to be understood primarily in terms of its mathematical structure and the representational relations it bears to physical systems, rather than (...) as a collection of sentences that attempt to make true claims about those systems (a ‘language-first’ approach). I argue that adopting the math-first approach already amounts to a form of structural realism, and that the choice between epistemic and ontic versions of structural realism is then a choice between a language-first and math-first view of metaphysics; I then explore the status of objects (and properties and relations) in fundamental and non-fundamental physics for both versions of math-first structural realism. (shrink)

Using as a starting point recent and apparently incompatible conclusions by Saunders and Knox, I revisit the question of the correct spacetime setting for Newtonian physics. I argue that understood correctly, these two versions of Newtonian physics make the same claims both about the background geometry required to define the theory, and about the inertial structure of the theory. In doing so I illustrate and explore in detail the view—espoused by Knox, and also by Brown —that inertial structure is defined (...) by the dynamics governing subsystems of a larger system. This clarifies some interesting features of Newtonian physics, notably the distinction between using the theory to model subsystems of a larger whole and using it to model complete universes, and the scale-relativity of spacetime structure. _1_ Introduction _2_ Newtonian Mechanics and Galilean Spacetime _3_ Vector Relationism and Maxwellian Spacetime _4_ Recovering the Galilei Group: Dynamics of Subsystems _5_ Knox on Inertial Structure _6_ Connections on Maxwellian Spacetime _7_ Knox on Newtonian Gravity _8_ Vector Relationism and Newton–Cartan Theory _9_ Inertial Structure in Newton–Cartan Gravity _10_ Reconciling Knox and Saunders _11_ Conclusions. (shrink)

Glymour and Quine propose two different formal criteria for theoretical equivalence. In this paper we examine the relationships between these criteria.

We review the dynamical approach to spacetime theories---in particular, its origins in the development of special relativity, its opposition to the contemporary `geometrical' approach, and the manner in which it plays out in general relativity. In addition, we demonstrate that the approach is compatible with the `angle bracket school'.

Logicians and philosophers of science have proposed various formal criteria for theoretical equivalence. In this paper, we examine two such proposals: definitional equivalence and categorical equivalence. In order to show precisely how these two well-known criteria are related to one another, we investigate an intermediate criterion called Morita equivalence.

There is sometimes a sense in which one theory posits ‘less structure’ than another. Philosophers of science have recently appealed to this idea both in the debate about equivalence of theories and in discussions about structural parsimony. But there are a number of different proposals currently on the table for how to compare the ‘amount of structure’ that different theories posit. The aim of this paper is to compare these proposals against one another and evaluate them on their own merits.

There exists a common view that for theories related by a ‘duality’, dual models typically may be taken ab initio to represent the same physical state of affairs, i.e. to correspond to the same possible world. We question this view, by drawing a parallel with the distinction between ‘interpretational’ and ‘motivational’ approaches to symmetries.

We approach the physics of \emph{minimal coupling} in general relativity, demonstrating that in certain circumstances this leads to violations of the \emph{strong equivalence principle}, which states that, in general relativity, the dynamical laws of special relativity can be recovered at a point. We then assess the consequences of this result for the \emph{dynamical perspective on relativity}, finding that potential difficulties presented by such apparent violations of the strong equivalence principle can be overcome. Next, we draw upon our discussion of the (...) dynamical perspective in order to make explicit two `miracles' in the foundations of relativity theory. We close by arguing that the above results afford us insights into the nature of special relativity, and its relation to general relativity. (shrink)

In this article, I examine whether or not the Hamiltonian and Lagrangian formulations of classical mechanics are equivalent theories. I do so by applying a standard for equivalence that was recently introduced into philosophy of science by Halvorson and Weatherall. This case study yields three general philosophical payoffs. The first concerns what a theory is, while the second and third concern how we should interpret what our physical theories say about the world. 1Introduction 2When Are Two Theories Equivalent? 3Preliminaries on (...) Classical Mechanics 3.1Hamiltonian mechanics 3.2Lagrangian mechanics 4Are Hamiltonian and Lagrangian Mechanics Equivalent Theories? 4.1Tangent bundle versus cotangent bundle 4.2Tangent bundle versus symplectic manifold 4.3Lagrangian vector field versus Hamiltonian vector field 5Conclusion Appendix. (shrink)

Mathematicians, physicists, and philosophers of physics often look to the symmetries of an object for insight into the structure and constitution of the object. My aim in this paper is to explain why this practice is successful. In order to do so, I present a collection of results that are closely related to (and in a sense, generalizations of) Beth’s and Svenonius’ theorems.

In this article I assess the Invariance Principle, which states that only quantities that are invariant under the symmetries of our theories are physically real. I argue, contrary to current orthodoxy, that the variance of a quantity under a theory’s symmetries is not a sufficient basis for interpreting that theory as being uncommitted to the reality of that quantity. Rather, I argue, the variance of a quantity under symmetries only ever serves as a motivation to refrain from any commitment to (...) the quantity in question. (shrink)

I review some recent work on applications of category theory to questions concerning theoretical structure and theoretical equivalence of classical field theories, including Newtonian gravitation, general relativity, and Yang-Mills theories.

It is widely accepted that the notion of an inertial frame is central to Newtonian mechanics and that the correct space-time structure underlying Newton’s methods in Principia is neo-Newtonian or Galilean space-time. I argue to the contrary that inertial frames are not needed in Newton’s theory of motion, and that the right space-time structure for Newton’s Principia requires the notion of parallelism of spatial directions at different times and nothing more. Only relative motions are definable in this framework, never absolute (...) ones. (shrink)

Harvey Brown’s Physical Relativity defends a view, the dynamical perspective, on the nature of spacetime that goes beyond the familiar dichotomy of substantivalist/relationist views. A full defense of this view requires attention to the way that our use of spacetime concepts connect with the physical world. Reflection on such matters, I argue, reveals that the dynamical perspective affords the only possible view about the ontological status of spacetime, in that putative rivals fail to express anything, either true or false. I (...) conclude with remarks aimed at clarifying what is and isn’t in dispute with regards to the explanatory priority of spacetime and dynamics, at countering an objection raised by John Norton to views of this sort, and at clarifying the relation between background and effective spacetime structure. (shrink)

Diffeomorphism invariance is sometimes taken to be a criterion of background independence. This claim is commonly accompanied by a second, that the genuine physical magnitudes (the ``observables'') of background-independent theories and those of background-dependent (non-diffeomorphism-invariant) theories are essentially different in nature. I argue against both claims. Background-dependent theories can be formulated in a diffeomorphism-invariant manner. This suggests that the nature of the physical magnitudes of relevantly analogous theories (one background free, the other background dependent) is essentially the same. The temptation (...) to think otherwise stems from a misunderstanding of the meaning of spacetime coordinates in background-dependent theories. (shrink)

We should see the debate over the existence of spacetime as a debate about the fundamentality of spatiotemporal structure to the physical world. This is a non-traditional conception of the debate, which captures the spirit of the traditional one. At the same time, it clarifies the point of contention between opposing views and offsets worries that the dispute is stagnant or non-substantive. It also unearths a novel argument for substantivalism, given current physics. Even so, that conclusion can be overridden by (...) future physics. I conclude that this debate is a substantive one, which the substantivalist is currently winning. (shrink)

This paper presents a simple pair of first-order theories that are not definitionally (nor Morita) equivalent, yet are mutually conservatively translatable and mutually 'surjectively' translatable. We use these results to clarify the overall geography of standards of equivalence and to show that the structural commitments that theories make behave in a more subtle manner than has been recognized.

For Knox, ‘spacetime’ is to be defined functionally, as that which picks out a structure of local inertial frames. Assuming that Knox is motivated to construct this functional definition of spacetime on the grounds that it appears to identify that structure which plays theoperationalrole of spacetime—i.e., that structure which is actually surveyed by physical rods and clocks built from matter fields—we identify in this paper important limitations of her approach: these limitations are based upon the fact that there is a (...) gap between inertial frame structure and that which is operationally significant in the above sense. We present five concrete cases in which these two notions come apart, before considering various ways in which Knox’s spacetime functionalism might be amended in light of these issues. (shrink)

I address a question recently raised by Simon Saunders concerning the relationship between the space-time structure of Newton-Cartan theory and that of what I will call “Maxwell-Huygens space-time.” This discussion will also clarify a connection between Saunders’s work and a recent paper by Eleanor Knox.

This article gives an explicit presentation of Newtonian gravitation on the backdrop of Maxwell space-time, giving a sense in which acceleration is relative in gravitational theory. However, caution is needed: assessing whether this is a robust or interesting sense of the relativity of acceleration depends on some subtle technical issues and on substantive philosophical questions over how to identify the space-time structure of a theory.

This paper scrutinises the tenability of a strict conceptual distinction between space and matter via the lens of the debate between modified gravity and dark matter. In particular, we consider Berezhiani and Khoury's novel 'superfluid dark matter theory' as a case study. Two families of criteria for being matter and being spacetime, respectively, are extracted from the literature. Evaluation of the new scalar field postulated by SFDM according to these criteria reveals that it is as much matter as anything could (...) possibly be, but alsoꟷbelow the critical temperature for superfluidityꟷas much spacetime as anything could possibly be. A sequel paper examines possible interpretations of SFDM in light of this result, as well as the consequences for our understanding of the modified gravity/ dark matter distinction and the broader spacetime–matter distinction. (shrink)

Eleanor Knox has argued that our concept of spacetime applies to whichever structure plays a certain functional role in the laws. I raise two objections to this inertial functionalism. First, it depends on a prior assumption about which coordinate systems defined in a theory are reference frames, and hence on assumptions about which geometric structures are spatiotemporal. This makes Knox’s account circular. Second, her account is vulnerable to several counterexamples, giving the wrong result when applied to topological quantum field theories (...) and parity- and time-asymmetric theories. I advance an alternative account on which our spacetime concept is a cluster concept. On this view, the notion of metaphysical fundamentality may feature in the cluster, in which case spacetime functionalism may be uninformative in the absence of answers to fundamental metaphysical questions like the substantivalist/relationist debate. (shrink)

Canonically, ‘classic’ tests of general relativity (GR) include perihelion precession, the bending of light around stars, and gravitational redshift; ‘modern’ tests have to do with, _inter alia_, relativistic time delay, equivalence principle tests, gravitational lensing, strong field gravity, and gravitational waves. The orthodoxy is that both classic and modern tests of GR afford experimental confirmation of that theory _in particular_. In this article, we question this orthodoxy, by showing there are classes of both relativistic theories (with spatiotemporal geometrical properties different (...) from those of GR) and non-relativistic theories (in which the lightcones of a relativistic spacetime are ‘widened’) which would also pass such tests. Thus, (a) issues of underdetermination in the context of GR loom much larger than one might have thought, and (b) given this, one has to think more carefully about what exactly such tests in fact _are_ testing. (shrink)

Eleanor Knox has argued that our concept of spacetime applies to whichever structure plays a certain functional role in the laws (the role of determining local inertial structure). I raise two complications for this approach. First, our spacetime concept seems to have the structure of a cluster concept, which means that Knox's inertial criteria for spacetime cannot succeed with complete generality. Second, the notion of metaphysical fundamentality may feature in the spacetime concept, in which case spacetime functionalism may be uninformative (...) in the absence of answers to fundamental metaphysical questions like the substantivalist/relationist debate. (shrink)

For Knox, ‘spacetime’ is to be defined functionally, as that which picks out a structure of local inertial frames. Assuming that Knox is motivated to construct this functional definition of spacetime on the grounds that it appears to identify that structure which plays the operational role of spacetime—i.e., that structure which is actually surveyed by physical rods and clocks built from matter fields—we identify in this paper important limitations of her approach: these limitations are based upon the fact that there (...) is a gap between inertial frame structure and that which is operationally significant in the above sense. We present five concrete cases in which these two notions come apart, before considering various ways in which Knox’s spacetime functionalism might be amended in light of these issues. (shrink)

In this paper, I consider the role of exact symmetries in theories of physics, working throughout with the example of gravitation set in Newtonian spacetime. First, I spend some time setting up a means of thinking about symmetries in this context; second, I consider arguments from the seeming undetectability of absolute velocities to an anti-realism about velocities; and finally, I claim that the structure of the theory licences us to interpret models which differ only with regards to the absolute velocities (...) of objects as depicting the same physical state of affairs. In defending this last claim, I consider how ideas and resources from the philosophy of language may usefully be brought to bear on this topic. (shrink)

I point out a radical indeterminism in potential-based formulations of Newtonian gravity once we drop the condition that the potential vanishes at infinity. This indeterminism, which is well known in theoretical cosmology but has received little attention in foundational discussions, can be removed only by specifying boundary conditions at all instants of time, which undermines the theory's claim to be fully cosmological, i.e., to apply to the Universe as a whole. A recent alternative formulation of Newtonian gravity due to Saunders (...) pp.22-48) provides a conceptually satisfactory cosmology but fails to reproduce the Newtonian limit of general relativity in homogenous but anisotropic universes. I conclude that Newtonian gravity lacks a fully satisfactory cosmological formulation. (shrink)

This article uncovers a foundational relationship between the ‘gauge symmetry’ of a Newton-Cartan theory and the celebrated Trautman Recovery Theorem and explores its implications for recent philosophical work on Newton-Cartan gravitation.

This is an essay about general covariance, and what it says about spacetime structure. After outlining a version of the dynamical approach to spacetime theories, and how it struggles to deal with generally covariant theories, I argue that we should think about the symmetry structure of spacetime rather differently in generally-covariant theories compared to non-generally-covariant theories: namely, as a form of internal rather than external symmetry structure.

Absolutism about mass claims that mass ratios obtain in virtue of absolute masses. Comparativism denies this. Dasgupta, Oxford studies in metaphysics, Oxford University Press, Oxford, 2013) argues for comparativism about mass, in the context of Newtonian Gravity. Such an argument requires proving that comparativism is empirically adequate. Dasgupta equates this to showing that absolute masses are undetectable, and attempts to do so. This paper develops an argument by Baker to the contrary: absolute masses are in fact empirically meaningful, that is (...) detectable. Additionally, it is argued that the requirement of empirical adequacy should not be cashed out in terms of undetectability in the first place. The paper closes by sketching the possible strategies that remain for the comparativist. Along the way a framework is developed that is useful for thinking about these issues: Ozma games—how could one explain to an alien civilisation what an absolute mass is? (shrink)

Quine often argued for a simple, untyped system of logic rather than the typed systems that were championed by Russell and Carnap, among others. He claimed that nothing important would be lost by eliminating sorts, and the result would be additional simplicity and elegance. In support of this claim, Quine conjectured that every many-sorted theory is equivalent to a single-sorted theory. We make this conjecture precise, and prove that it is true, at least according to one reasonable notion of theoretical (...) equivalence. Our clarification of Quine’s conjecture, however, exposes the shortcomings of his argument against many-sorted logic. (shrink)

ABSTRACT It has recently been argued that Harvey Brown and Oliver Pooley’s ‘dynamical approach’ to special relativity should be understood as what might be called an ontologically and ideologically relationalist approach to Minkowski geometry, according to which Minkowski geometrical structure supervenes upon the symmetries of the best-systems dynamical laws for a material world with primitive topological or differentiable structure. Fleshing out the details of some such primitive structure, and a conception of laws according to which Minkowski geometry could so supervene, (...) has been referred to by some as the ‘constructivist project’. Here, it is explained that Nick Huggett’s work on ‘regularity relationalism’ provides a framework for such an approach, and a relativistic version of Huggett’s regularity relationalism is outlined for that purpose. Finally, by way of examples, it is shown that this approach fails in the simplest cases. Still, reasons are given for which this should not necessarily discourage an advocate of this interpretation of the dynamical approach. 1 Introduction to the Dynamical Approach 2 Huggett’s Regularity Relationalism 3 The Dynamical Approach as a Form of Regularity Relationalism 4 A Simple Way to Set the Stage 5 A First Attempt 5.1 Complex-valued planewave solutions 5.2 Real-valued planewave solutions 6 More General Solutions 7 Summary and Reflections. (shrink)

As Harvey Brown emphasizes in his book Physical Relativity, inertial motion in general relativity is best understood as a theorem, and not a postulate. Here I discuss the status of the "conservation condition", which states that the energy-momentum tensor associated with non-interacting matter is covariantly divergence-free, in connection with such theorems. I argue that the conservation condition is best understood as a consequence of the differential equations governing the evolution of matter in general relativity and many other theories. I conclude (...) by discussing what it means to posit a certain spacetime geometry and the relationship between that geometry and the dynamical properties of matter. (shrink)

I discuss several issues related to "classical" spacetime structure. I review Galilean, Newtonian, and Leibnizian spacetimes, and briefly describe more recent developments. The target audience is undergraduates and early graduate students in philosophy; the presentation avoids mathematical formalism as much as possible.

We explore the significance of physical theories set in Euclidean spacetimes. In particular, we explore the use of these theories in contemporary physics at large, and the sense in which there can be a notion of temporal evolution in these theories. Having achieved these tasks, we proceed to reflect on the lessons that one can take from such theories for Knox’s ‘inertial frame’ version of spacetime functionalism, which seems to issue incorrect verdicts in the case of theories with Euclidean metrical (...) structure. (shrink)

There is a venerable position in the philosophy of space and time that holds that the geometry of spacetime is conventional, provided one is willing to postulate a “universal force field.” Here we ask a more focused question, inspired by this literature: in the context of our best classical theories of space and time, if one understands “force” in the standard way, can one accommodate different geometries by postulating a new force field? We argue that the answer depends on one’s (...) theory. In Newtonian gravitation the answer is yes; in relativity theory, it is no. (shrink)

The equivalence principle has constituted one of the cornerstones of discussions in the foundations of spacetime theories over the past century. However, up to this point the principle has been considered overwhelmingly only within the context of relativistic physics. In this article, we demonstrate that the principle has much broader, super-theoretic significance: to do so, we present a unified framework for understanding the principle in its various guises, applicable to both relativistic and Newtonian contexts. We thereby deepen significantly our understanding (...) of the role played by the equivalence principle in a broad class of spacetime theories. (shrink)

Metaphysics of Quantum Gravity The metaphysics of quantum gravity explores metaphysical issues related to research programs in theoretical physics clustered under the term quantum gravity. These research programs aim at the formulation of a theory that reconciles the theory of general relativity with quantum theory. The goal is not necessarily to come up with a … Continue reading Metaphysics of Quantum Gravity →.

I provide an alternative characterization of a "standard of rotation" in the context of classical spacetime structure that does not refer to any covariant derivative operator.

Background independence begins life as an informal property that a physical theory might have, often glossed as 'doesn't posit a fixed spacetime background'. Interest in trying to offer a precise account of background independence has been sparked by the pronouncements of several theorists working on quantum gravity that background independence embodies in some sense an essential discovery of the General Theory of Relativity, and a feature we should strive to carry forward to future physical theories. This paper has two goals. (...) The first is to investigate what a world must be like in order to be truly described by a background independent theory given extant accounts of background independence. The second is to argue that there are no non-empirical reasons to be more confident in theories that satisfy extant accounts of background independence than in theories that don't. The paper concludes by drawing a general moral about a way in which focussing primarily on mathematical formulations of our physical theories can adversely affect debates in the metaphysics of physics. (shrink)

Laplace wondered about the minimal choice of initial variables and parameters corresponding to a well-posed initial value problem. Discussions of Laplace’s problem in the literature have focused on choosing between spatiotemporal variables relative to absolute space or merely relative to other material bodies and between absolute masses or merely mass ratios. This paper extends these discussions of Laplace’s problem, in the context of Newtonian Gravity, by asking whether mass needs to be included in the initial state at all, or whether (...) a purely spatiotemporal initial state suffices. It is argued that mass indeed needs to be included; removing mass from the initial state drastically reduces the predictive and explanatory power of Newtonian Gravity. (shrink)

I discuss singular space-times in the context of the geometrized formulation of Newtonian gravitation. I argue first that geodesic incompleteness is a natural criterion for when a model of geometrized Newtonian gravitation is singular, and then I show that singularities in this sense arise naturally in classical physics by stating and proving a classical version of the Raychaudhuri-Komar singularity theorem.

A popular method for comparing the structures of mathematical objects, which I call the ‘subset approach’, says that X has more structure than Y just in case X’s automorphisms form a proper subset of Y’s automorphisms. This approach is attractive, in part, because it seems to yield the right results in some comparisons of spacetime structure. But as I show, it yields the wrong results in a number of other cases. The problem is that the subset approach compares structure using (...) automorphism sets. So a different approach is needed: structure should be compared using automorphism groups. (shrink)

We study a long-recognised but under-appreciated symmetry called dynamical similarity and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a system where the unit of Hamilton’s principal function is rescaled, and therefore represent a kind of dynamical scaling symmetry with formal properties that differ from many standard symmetries. To study this symmetry, we develop a general framework for symmetries that distinguishes the observable and surplus structures of a theory by using the (...) minimal freely specifiable initial data for the theory that is necessary to achieve empirical adequacy. This framework is then applied to well-studied examples including Galilean invariance and the symmetries of the Kepler problem. We find that our framework gives a precise dynamical criterion for identifying the observables of those systems, and that those observables agree with epistemic expectations. We then apply our framework to dynamical similarity. First we give a general definition of dynamical similarity. Then we show, with the help of some previous results, how the dynamics of our observables leads to singularity resolution and the emergence of an arrow of time in cosmology. (shrink)

There is a long tradition of preferring local theories to ones that posit lawful or causal influence at a spacetime distance. In this paper, we argue against this preference. We argue that nonlocality is scientifically unobjectionable and that nonlocal theories can be known. Scientists can gather evidence for them and confirm them in much the same way that they do for local theories. We think these observations point to a deeper constraint on scientific theorizing and experimentation: the (quasi‐) isolation of (...) causal or lawful influence. We argue that this requirement ought to replace the locality desideratum in science. We then explore the possibility that the order of explanation has been reversed: perhaps it is isolatable influence that determines what counts as local in the first place. (shrink)