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)

When are two formal theories of broadly logical concepts, such as truth, equivalent? The paper investigates a case study, involving two well-known variants Kripke-Feferman truth. The first, KF+CONS, features a consistent but partial truth predicate. The second, KF+COMP, an inconsistent but complete truth predicate. It is well-known that the two truth predicates are dual to each other. We show that this duality reveals a much stricter correspondence between the two theories: they are intertraslatable. Intertranslatability under natural assumptions coincides with definitional (...) equivalence, and is arguably the strictest notion of theoretical equivalence different from logical equivalence. The case of KF+CONS and KF+COMP raises a puzzle: the two theories can be proved to be strictly related, yet they appear to embody remarkably different conceptions of truth. We discuss the significance of the result for the broader debate on formal criteria of conceptual reducibility for theories of truth. (shrink)

Formal criteria of theoretical equivalence are mathematical mappings between specific sorts of mathematical objects, notably including those objects used in mathematical physics. Proponents of formal criteria claim that results involving these criteria have implications that extend beyond pure mathematics. For instance, they claim that formal criteria bear on the project of using our best mathematical physics as a guide to what the world is like, and also have deflationary implications for various debates in the metaphysics of physics. In this paper, (...) I investigate whether there is a defensible view according to which formal criteria have significant non-mathematical implications, of these sorts or any other, reaching a chiefly negative verdict. Along the way, I discuss various foundational issues concerning how we use mathematical objects to describe the world when doing physics, and how this practice should inform metaphysics. I diagnose the prominence of formal criteria as stemming from contentious views on these foundational issues, and endeavor to motivate some alternative views in their stead. (shrink)

In an earlier article on this journal I argued that the problem of empirical underdetermination can for the largest part be solved by theoretical virtues, and for the remaining part it can be tolerated. Here I confront two further challenges to scientific realism based on underdetermination. First, there are four classes of theories which may seem to be underdetermined even by theoretical virtues. Concerning them I argue that (i) theories produced by trivial permutations and (ii) “equivalent descriptions” are compatible with (...) the truth of standard theories; instead (iii) “as if” versions of standard theories are much worse from the point of view of theoretical virtues; finally (iv) mathematically intertranslatable theories either may become empirically decidable in the future, or can be discriminated by theoretical virtues, or realists may simply plead ignorance about their claims. Secondly, I consider Stanford’s underdetermination with respect unconceived alternatives, arguing that it essentially relies on the pessimistic meta-induction from the falsity of all past theories. Therefore, it can be resisted by (a) considering the radical advancement of present with respect to past science, and (b) arguing with selective realism that past successful theories, even if false, always included some true components. (shrink)

In this article I bring the recent philosophical literature on theoretical equivalence to bear on dualities in physics. Focusing on electromagnetic duality, which is a simple example of S-duality i...

I review the philosophical literature on the question of when two physical theories are equivalent. This includes a discussion of empirical equivalence, which is often taken to be necessary, and sometimes taken to be sufficient, for theoretical equivalence; and "interpretational" equivalence, which is the idea that two theories are equivalent just in case they have the same interpretation. It also includes a discussion of several formal notions of equivalence that have been considered in the recent philosophical literature, including definitional equivalence (...) and categorical equivalence. The article concludes with a brief discussion of the relationship between equivalence and duality. (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.

Cartesian arguments for global skepticism about the external world start from the premise that we cannot know that we are not in a Cartesian scenario such as an evil-demon scenario, and infer that because most of our empirical beliefs are false in such a scenario, these beliefs do not constitute knowledge. Veridicalist responses to global skepticism respond that arguments fail because in Cartesian scenarios, many or most of our empirical beliefs are true. Some veridicalist responses have been motivated using verificationism, (...) externalism, and coherentism. I argue that a more powerful veridicalist response to global skepticism can be motivated by structuralism, on which physical entities are understood as those that play a certain structural role. I develop the structuralist response and address objections. (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)

The problem of theoretical equivalence is traditionally understood as the problem of specifying when superficially dissimilar accounts of the world are reformulations of a single underlying theory. One important strategy for answering this question has been to appeal to formal relations between theoretical structures. This article presents two reasons to think that such an approach will be unsuccessful and suggests an alternative account of theoretical equivalence, based on the notion of interpretive equivalence, in which the problem is merely an instance (...) of a broader problem in the philosophy of physics. I thus conclude that there is no distinctive problem of theoretical equivalence at all. Two difficulties my approach raises for realist replies to the threat of underdetermination are then discussed, with particular emphasis on a recent reply by Norton. 1 Introduction2 Methodological Confusion3 Asymmetrical Equivalence3.1 Maxwell’s field equations3.2 Classical Lagrangian dynamics4 Formal Approaches: Quine and Glymour5 Theoretical Equivalence as Interpretive Equivalence6 Theoretical Equivalence without Interpretive Judgement?7 Lessons for Underdetermination. (shrink)

If two theory formulations are merely different expressions of the same theory, then any problem of choosing between them cannot be due to the underdetermination of theories by data. So one might suspect that we need to be able to tell distinct theories from mere alternate formulations before we can say anything substantive about underdetermination, that we need to solve the problem of identical rivals before addressing the problem of underdetermination. Here I consider two possible solutions: Quine proposes that we (...) call two theories identical if they are equivalent under a reconstrual of predicates, but this would mishandle important cases. Another proposal is to defer to the particular judgements of actual scientists. Consideration of an historical episodethe alleged equivalence of wave and matrix mechanicsshows that this second proposal also fails. Nevertheless, I suggest, the original suspicion is wrong; there are ways to enquire into underdetermination without having solved the problem of identical rivals. (shrink)

The empirical underdetermination of theories is a philosophical problem which until the last century has not seriously troubled actual science. The reason is that confirmation does not depend only on empirical consequences, and theoretical virtues allow to choose among empirically equivalent theories. Moreover, I argue that the theories selected in this way are not just pragmatically or aesthetically better, but more probably true. At present in quantum mechanics not even theoretical virtues allow to choose among many competing theories and interpretations, (...) but this is because none of them possess those virtues to a sufficient degree. However, first, we can hope for some future advancement. Second, even if no further progress came forth, all the most credited competitors agree on a substantial core of theoretical assumptions. Therefore underdetermination does not show that we cannot be realist on unobservable entities in general, but at most that in particular fields our inquiry may encounter some de facto limits. (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)

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.

In this paper, we examine the relationship between general relativity and the theory of Einstein algebras. We show that according to a formal criterion for theoretical equivalence recently proposed by Halvorson and Weatherall, the two are equivalent theories.

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

I review the philosophical literature on the question of when two physical theories are equivalent. This includes a discussion of empirical equivalence, which is often taken to be necessary, and sometimes taken to be sufficient, for theoretical equivalence; and “interpretational” equivalence, which is the idea that two theories are equivalent just in case they have the same interpretation. It also includes a discussion of several formal notions of equivalence that have been considered in the recent philosophical literature, including (generalized) definitional (...) equivalence and categorical equivalence. The article concludes with a brief discussion of the relationship between equivalence and duality. The article is in two parts; this is Part 2, which addresses categorical equivalence, interpretational equivalence, and duality. (shrink)

In recent years, philosophers of science have explored categorical equivalence as a promising criterion for when two theories are equivalent. On the one hand, philosophers have presented several examples of theories whose relationships seem to be clarified using these categorical methods. On the other hand, philosophers and logicians have studied the relationships, particularly in the first order case, between categorical equivalence and other notions of equivalence of theories, including definitional equivalence and generalized definitional equivalence. In this article, I will express (...) some skepticism about categorical equivalence as a criterion of physical equivalence, both on technical grounds and conceptual ones. I will argue that “category structure” likely does not capture the structure of a theory, and discuss some recent work in light of this claim. (shrink)

I review the philosophical literature on the question of when two physical theories are equivalent. This includes a discussion of empirical equivalence, which is often taken to be necessary, and sometimes taken to be sufficient, for theoretical equivalence; and “interpretational” equivalence, which is the idea that two theories are equivalent just in case they have the same interpretation. It also includes a discussion of several formal notions of equivalence that have been considered in the recent philosophical literature, including (generalized) definitional (...) equivalence and categorical equivalence. The article concludes with a brief discussion of the relationship between equivalence and duality. The article is in two parts; this is Part 1, which addresses empirical equivalence and definitional equivalence. (shrink)

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.

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.

In this article I connect two debates in the philosophy of science: the questions of scientific representation and both model and theoretical equivalence. I argue that by paying attention to how a model is used to draw inferences about its target system, we can define a notion of theoretical equivalence that turns on whether models license the same claims about the same target systems. I briefly consider the implications of this for two questions that have recently been discussed in the (...) context of the formal philosophy of science. (shrink)

Einstein considered general covariance to characterize the novelty of his General Theory of Relativity (GTR), but Kretschmann thought it merely a formal feature that any theory could have. The claim that GTR is ``already parametrized'' suggests analyzing substantive general covariance as formal general covariance achieved without hiding preferred coordinates as scalar ``clock fields,'' much as Einstein construed general covariance as the lack of preferred coordinates. Physicists often install gauge symmetries artificially with additional fields, as in the transition from Proca's to (...) Stueckelberg's electromagnetism. Some post-positivist philosophers, due to realist sympathies, are committed to judging Stueckelberg's electromagnetism distinct from and inferior to Proca's. By contrast, physicists identify them, the differences being gauge-dependent and hence unreal. It is often useful to install gauge freedom in theories with broken gauge symmetries (second-class constraints) using a modified Batalin-Fradkin-Tyutin (BFT) procedure. Massive GTR, for which parametrization and a Lagrangian BFT-like procedure appear to coincide, mimics GTR's general covariance apart from telltale clock fields. A generalized procedure for installing artificial gauge freedom subsumes parametrization and BFT, while being more Lagrangian-friendly than BFT, leaving any primary constraints unchanged and using a non-BFT boundary condition. Artificial gauge freedom licenses a generalized Kretschmann objection. However, features of paradigm cases of artificial gauge freedom might help to demonstrate a principled distinction between substantive and merely formal gauge symmetry. (shrink)

Classical and quantum field theory provide not only realistic examples of extant notions of empirical equivalence, but also new notions of empirical equivalence, both modal and occurrent. A simple but modern gravitational case goes back to the 1890s, but there has been apparently total neglect of the simplest relativistic analog, with the result that an erroneous claim has taken root that Special Relativity could not have accommodated gravity even if there were no bending of light. The fairly recent acceptance of (...) nonzero neutrino masses shows that widely neglected possibilities for nonzero particle masses have sometimes been vindicated. In the electromagnetic case, there is permanent underdetermination at the classical and quantum levels between Maxwell's theory and the one-parameter family of Proca's electromagnetisms with massive photons, which approximate Maxwell's theory in the limit of zero photon mass. While Yang–Mills theories display similar approximate equivalence classically, quantization typically breaks this equivalence. A possible exception, including unified electroweak theory, might permit a mass term for the photons but not the Yang–Mills vector bosons. Underdetermination between massive and massless (Einstein) gravity even at the classical level is subject to contemporary controversy. (shrink)

Although logical empiricism is now mostly decried, their naturalist claim that the content of a theory can be read off from its structure, without any philosophical considerations needed, still supports traditional strategies to escape cases of underdetermination. The appeal to theoretical equivalence or to theoretical virtues, for instance, both assume that there is a neutral standpoint from which the structure of the theories can be analyzed, the physically relevant from the superfluous separated, and a comparison made between their theoretical content (...) and virtues. In my dissertation, I examine the presuppositions upon which such strategies depend. I argue that the methodological principle underlying them, according to which theories with no superfluous structure should be preferred, is unpractical, for what constitutes relevant structure is determined by epistemic considerations about the aim of scientific theories. In chapter 1, I analyze the claim that theories with ordinary bosons and fermions are theoretically equivalent to theories with exotic `paraparticles'. I argue that this claim does not do justice to the latter, as the proof is formulated in a vocabulary parochial to the former and thus favors one of the theory while giving an impoverished version of the second. In chapter 2, I examine recent arguments to the effects that any interpretation of Quantum Mechanics that can offer a no-go theorem against paraparticles possesses an explanatory advantage over other interpretations and should, as such, be favored over others. Given that most physicists consider paraparticles as surplus structure whose non-observation does not require an explanation, I evaluate arguments of both sides and suggest a third way to approach the question. Finally, in chapter 3, I turn my attention to methods for excluding another kind of unphysical structure, mathematical artifacts, from rival dark matter models. Simulations are our only window into what rival models predict about the universe’s structure. But for them to play a useful role in generating knowledge, we need to distinguish reliably between real predictions and artifacts. I argue that robustness analysis fails to fulfill this task. I propose in its place another methodology, that of crucial simulations. (shrink)