Why is gauge symmetry so important in modern physics, given that one must eliminate it when interpreting what the theory represents? In this paper we discuss the sense in which gauge symmetry can be fruitfully applied to constrain the space of possible dynamical models in such a way that forces and charges are appropriately coupled. We review the most well-known application of this kind, known as the 'gauge argument' or 'gauge principle', discuss its difficulties, and then reconstruct the gauge argument (...) as a valid theorem in quantum theory. We then present what we take to be a better and more general gauge argument, based on Noether's second theorem in classical Lagrangian field theory, and argue that this provides a more appropriate framework for understanding how gauge symmetry helps to constrain the dynamics of physical theories. (shrink)

Theoretical equivalence and duality are two closely related notions: but their interconnection has so far not been well understood. In this paper I explicate the contribution of a recent schema for duality to discussions of theoretical equivalence. I argue that duality suggests a construal of theoretical equivalence in the physical sciences. The construal is in terms of the isomorphism of models, as defined by the schema. This construal gives interpretative constraints that should be useful for discussions of theoretical equivalence more (...) generally. I illustrate the construal in various formulations of Maxwell’s electromagnetic theory. (shrink)

We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations \ into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and the structure-semiotics duality. Whereas in classical algebraic geometry (...) a certain kind of rings can be recovered by considering their representations with respect to a unique codomain B, Grothendieck’s theory of schemes permits to reconstruct general rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized—by means of what we shall generally call Gelfand transform—as quantities on a topological space that parameterizes the relevant representations of A. As we shall argue, important dualities in different areas of mathematics can be understood as particular cases of this general pattern. In the wake of Majid’s analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck’s notion of functor of points can be understood as a “relativization of the a priori” that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding domains. (shrink)

Although “theory” has been the prevalent unit of analysis in the meta-study of science throughout most of the twentieth century, the concept remains elusive. I further explore the leitmotiv of several authors in this issue: that we should deal with theorizing (rather than theory) in biology as a cognitive activity that is to be investigated naturalistically. I first contrast how philosophers and biologists have tended to think about theory in the last century or so, and consider recent calls to upgrade (...) the role of theory in the life sciences against the background of the recent “data deluge” in molecular biology, systems biology, etc. I then review thinking about theory in biology in relation to physical theory as a positive or negative exemplar. I conclude by discussing various aspects of a positive program for “naturalizing theorizing.”. (shrink)

We address a recent proposal concerning ‘surplus structure’ due to Nguyen et al.. We argue that the sense of ‘surplus structure’ captured by their formal criterion is importantly different from—and in a sense, opposite to—another sense of ‘surplus structure’ used by philosophers. We argue that minimizing structure in one sense is generally incompatible with minimizing structure in the other sense. We then show how these distinctions bear on Nguyen et al.’s arguments about Yang-Mills theory and on the hole argument.

Las teorías de la referencia puramente descriptivistas o causales fracasan a la hora de dar cuenta del modo en que se fija y puede rastrearse la referencia de los términos teóricos. Psillos propuso dos versiones del descriptivismo causal que recogen argu-mentos presentes en defensas previas de dicha posición. Se trata de una teoría mixta que pretende solucionar el problema y acomodarse a intuiciones presentes en enfo-ques alternativos, como el que apela a oraciones de Ramsey. El artículo se propone mostrar la (...) insuficiencia del descriptivismo causal como solución al problema de la referencia de los términos teóricos. (shrink)

RESUMEN Las teorías de la referencia puramente descriptivistas o causales fracasan a la hora de dar cuenta del modo en que se fija y puede rastrearse la referencia de los términos teóricos. Psillos propuso dos versiones del descriptivismo causal que recogen argumentos presentes en defensas previas de dicha posición. Se trata de una teoría mixta que pretende solucionar el problema y acomodarse a intuiciones presentes en enfoques alternativos, como el que apela a oraciones de Ramsey. El artículo se propone mostrar (...) la insuficiencia del descriptivismo causal como solución al problema de la referencia de los términos teóricos. ABSTRACT Purely descriptivist or purely causal reference theories fail to account for the manner in which the reference of theoretical terms is fixed and can be traced. S. Psillos proposed two versions of causal descriptivism that take up arguments set forth in previous defenses of said position. His is a mixed theory that attempts to solve the problem by adopting insights from alternative approaches, such as those that resort to Ramsey sentences. The objective of the article is to show the insufficiency of causal descriptivism as a solution to the problem of reference of theoretical terms. (shrink)

Inspired by Ronald Giere’s cognitive approach to scientific models, Cognitive Structural Realism has presented a naturalist account of scientific representation. CSR characterises the structure of theories in terms of cognitive structures. These are informational structures embodied in the brains of scientists. CSR accounts for scientific representation in terms of the dynamical relationship between the organism and its environment. The proposal has been criticised on account of its negligence of social aspects of scientific practice. The present paper aims to chart out (...) a reply to the objection. It shows that cognitive structures do not need to be put inside the brains of single individuals. Cognitive structures are redefined as extended structures in distributed cognitive systems under Free Energy Principle. (shrink)

Free Energy Principle underlies a unifying framework that integrates theories of origins of life, cognition, and action. Recently, FEP has been developed into a Markovian monist perspective. The paper expresses scepticism about the validity of arguments for Markovian monism. The critique is based on the assumption that Markovian models are scientific models, and while we may defend ontological theories about the nature of scientific models, we could not read off metaphysical theses about the nature of target systems from our theories (...) of nature of scientific models. The paper draws attention to different ways of understanding Markovian models, as material entities, fictional entities, and mathematical structures. I argue that none of these interpretations contributes to the defence of a metaphysical stance. This is because scientific representation is a sophisticated process, and properties of Markovian models—such as the property of being neither physical nor mental—could not be projected onto their targets to determine the ontological properties of targets easily. (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.

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.

It has been suggested that we can tell whether two theories are equivalent by comparing the structure that they ascribe to the world. If two theories posit different structures, then they must be i...

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 standard view is that the Lagrangian and Hamiltonian formulations of classical mechanics are theoretically equivalent. Jill North, however, argues that they are not. In particular, she argues that the state-space of Hamiltonian mechanics has less structure than the state-space of Lagrangian mechanics. I will isolate two arguments that North puts forward for this conclusion and argue that neither yet succeeds. 1 Introduction2 Hamiltonian State-space Has less Structure than Lagrangian State-space2.1 Lagrangian state-space is metrical2.2 Hamiltonian state-space is symplectic2.3 Metric > (...) symplectic3 Hamiltonian State-space Does Not Have Less Structure than Lagrangian State-space3.1 Lagrangian state-space has less than metric structure3.1.1 A potential worry3.1.2 General Lagrangians3.2 Hamiltonian state-space has more than symplectic structure3.2.1 A dual structure on the Hamiltonian state-space3.2.2 Simple Hamiltonians3.3 Comparing Lagrangian and Hamiltonian structures3.3.1 Counting mathematical structure3.3.2 Symplectic and metric structure are incomparable4 An Alternative Argument for LS4.1 The argument for P3*4.2 Symplectic manifold vs. tangent bundle structure4.3 Trying to patch up the argument for P3*5 Interpreting Mathematical Structure5.1 State-space realism5.2 Model isomorphism and theoretical equivalence6 Conclusion. (shrink)

In his book Philosophy of Logic, Putnam (1971) presents a short argument which reads like—and indeed, can be reconstructed as—a formal proof that a nominalistic physics is impossible. The aim of this paper is to examine Putnam’s proof and show that it is not compelling. The precise way in which the proof fails yields insight into the relation that a nominalistic physics should bear to standard physics and into Putnam’s indispensability argument.

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.

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

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)

The question “what is an interpretation?” is often intertwined with the perhaps even harder question “what is a scientific theory?”. Given this proximity, we try to clarify the first question to acquire some ground for the latter. The quarrel between the syntactic and semantic conceptions of scientific theories occupied a large part of the scenario of the philosophy of science in the 20th century. For many authors, one of the two currents needed to be victorious. We endorse that such debate, (...) at least in the terms commonly expressed, can be misleading. We argue that the traditional notion of “interpretation” within the syntax/semantic debate is not the same as that of the debate concerning the interpretation of quantum mechanics. As much as the term is the same, the term “interpretation” as employed in quantum mechanics has its meaning beyond (pure) logic. Our main focus here lies on the formal aspects of the solutions to the measurement problem. There are many versions of quantum theory, many of them incompatible with each other. In order to encompass a wider variety of approaches to quantum theory, we propose a different one with an emphasis on pure formalism. This perspective has the intent of elucidating the role of each so-called “interpretation” of quantum mechanics, as well as the precise origin of the need to interpret it. (shrink)

Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and Pearce. In Example 2, uncountably many pairs of definitionally inequivalent theories are given such that their model categories are concretely isomorphic via bijections that preserve ultraproducts in the model categories up to isomorphism. Based on these results, we settle several conjectures of Barrett, (...) Glymour and Halvorson. (shrink)

This article concerns the split between syntactic and semantic approaches to scientific theories. It aims at showing that an axiomatic representation of a scientific theory is a precondition of comprehending if the models of contain infinite entities. This result is established on the basis of the proposition that the human mind—which is finitely bounded for all we know—is not capable of directly grasping infinite entities. In view of this cognitive limitation, an indirect and finite representation of possibly infinite components of (...) the models of a scientific theory proves to be indispensable. Sets of axioms and sets of axiom schemes provide such a representation. These considerations will be cast into an argument for an axiomatic conception of scientific theories. The article concludes with a case study of the ideal gas model. (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 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 paper we propose and defend the Synonymy account, a novel account of metaphysical equivalence which draws on the idea (Rayo in The Construction of Logical Space, Oxford University Press, Oxford, 2013) that part of what it is to formulate a theory is to lay down a theoretical hypothesis concerning logical space. Roughly, two theories are synonymous—and so, in our view, equivalent—just in case (i) they take the same propositions to stand in the same entailment relations, and (ii) they (...) are committed to the truth of the same propositions. Furthermore, we put our proposal to work by showing that it affords a better and more nuanced understanding of the debate between Quineans and noneists. Finally we show how the Synonymy account fares better than some of its competitors, specifically, McSweeney’s (Philosophical Perspectives 30(1):270–293, 2016) epistemic account and Miller’s (Philosophical Quarterly 67(269):772–793, 2017) hyperintensional account. (shrink)

Structural realism has been suggested as the best compromise in the debate on scientific realism. It proposes that we should be realist about the relational structure of the world, not its nature. However, it faces an important objection, first raised by Newman against Russell: if relations are not qualified, then the position is either trivial or collapses into empiricism, but if relations are too strongly qualified, then it is no longer SR. A way to overcome this difficulty is to talk (...) about modal, or nomological relations instead of purely extensional relations. I argue that this is insufficient, for then, SR collapses into modal empiricism. I suggest, however, that ME could be the best position in the debate on scientific realism. 1Introduction2Objections to Structural Realism3How to Escape Newman’s Objection4Which Modal Relations Are Retained in Theory Change?5Are Modal Relations Real?6Relativity and Fundamentality7Is Modal Empiricism Really Empiricism?8Could Modal Empiricism Be the Best of Both Worlds? (shrink)

The debate between critics of syntactic and semantic approaches to the formalization of scientific theories has been going on for over 50 years. I structure the debate in light of a recent exchange between Hans Halvorson, Clark Glymour, and Bas van Fraassen and argue that the only remaining disagreement concerns the alleged difference in the dependence of syntactic and semantic approaches on languages of predicate logic. This difference turns out to be illusory.

'The arrow of time' refers to the curious asymmetry that distinguishes the future from the past. Reversing the Arrow of Time argues that there is an intimate link between the symmetries of 'time itself' and time reversal symmetry in physical theories, which has wide-ranging implications for both physics and its philosophy. This link helps to clarify how we can learn about the symmetries of our world, how to understand the relationship between symmetries and what is real, and how to overcome (...) pervasive illusions about the direction of time. Roberts explains the significance of time reversal in a way that intertwines physics and philosophy, to establish what the arrow of time means and how we can come to know it. This book is both mathematically and philosophically rigorous yet remains accessible to advanced undergraduates in physics and the philosophy of physics. This title is also available as Open Access on Cambridge Core. (shrink)

This dissertation is divided into two parts. In the first part I defend substantivalism. I do this by offering, in chapter 1, a counterpart-theoretic defense of substantivalism from Leibniz’ shift arguments. Then, in chapter 2, I defend substantivalism from the hole argument and argue, against the consensus, that the question of haecceitism is irrelevant to substantivalism in the context of general relativity. In the second part of the dissertation I defend supersubstantivalism. I do this by offering, in chapter 3, an (...) argument against dualistic substantivalism. The argument appeals to plausible principles of modal plenitude to show that the dualist is committed to a range of problematic possibilities. Then, in chapter 4, I consider a range of supersubstantivalist positions. I conclude by arguing for a version of supersubstantivalism I call compresence supersubstantivalism. (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)

Science continually contributes new models and rethinks old ones. The way inferences are made is constantly being re-evaluated. The practice and achievements of science are both shaped by this process, so it is important to understand how models and inferences are made. But, despite the relevance of models and inference in scientific practice, these concepts still remain contro-versial in many respects. The attempt to understand the ways models and infer-ences are made basically opens two roads. The first one is to (...) produce an analy-sis of the role that models and inferences play in science. The second one is to produce an analysis of the way models and inferences are constructed, especial-ly in the light of what science tells us about our cognitive abilities. The papers collected in this volume go both ways. (shrink)

The semantic view of theories is normally considered to be an ac-count of theories congenial to Scientific Realism. Recently, it has been argued that Ontic Structural Realism could be fruitfully applied, in combination with the semantic view, to some of the philosophical issues peculiarly related to bi-ology. Given the central role that models have in the semantic view, and the relevance that mathematics has in the definition of the concept of model, the fo-cus will be on population genetics, which is (...) one of the most mathematized areas in biology. We will analyse some of the difficulties which arise when trying to use Ontic Structural Realism to account for evolutionary biology. (shrink)

Since the beginning of the 20th century, philosophers of science have asked, "what kind of thing is a scientific theory?" The logical positivists answered: a scientific theory is a mathematical theory, plus an empirical interpretation of that theory. Moreover, they assumed that a mathematical theory is specified by a set of axioms in a formal language. Later 20th century philosophers questioned this account, arguing instead that a scientific theory need not include a mathematical component; or that the mathematical component need (...) not be specified by a set of axioms in a formal language. We survey various accounts of scientific theories entertained in the 20th century -- removing some misconceptions, and clearing a path for future research. (shrink)

Olaf Müller’s book develops a new case for underdetermination, and, as he is focusing on theories of a ‘limited domain’, this assumes the containability of the theories. First, the paper argues that Müller’s theory of darkness is fundamentally Newtonian, but for Newton’s optical theory the type of theoretical structure Müller adopts is problematic. Second, the paper discusses seventeenth-century challenges to Newton, changes in the proof-structure of Newton’s optical theory, and how these affect Müller’s reconstruction. Müller’s book provides empirically equivalent theories, (...) yet the historical theories were not empirically equivalent, and the same experiments were used to extract different bodies of evidence to rebut the opponent. Third, Goethe’s multi-layered critique of Newton’s experimental proof is investigated, including his developmental account of prismatic colours, the role of experimental series in rejecting Newton’s observations, and his incorporation of the ‘limited domain’ of prismatic colours in a broader framework. Two key elements of Goethe’s method, polarity and strengthening are discussed in contrast to Müller, who only utilises polarity in his account. Finally Neurath’s attempts to come to grips with the optical controversies and the prism-experiments with ‘blurred edges’ are recalled. Müller also discusses in detail some of these experiments and heavily draws on Quine. Neurath developed Duhem’s and Poincaré’s conventionalist insights and had good reasons to be pessimistic about theory-containment. Their differences provide some additions to the history of the Duhem–Quine thesis. (shrink)

Karim Thébault has argued that for ontic structural realism to be a viable ontology it should accommodate two principles: physico-mathematical structures it deploys must be firstly consistent and secondly substantial. He then contends that in geometric quantization, a transitional machinery from classical to quantum mechanics, the two principles are followed, showing that it is a guide to ontic structure. In this article, I will argue that geometric quantization violates the consistency principle. To compensate for this shortcoming, the deformation quantization procedure (...) will be offered. (shrink)

_ doi:10.1093/analys/anx072 _, published: 27 June 2017.

Anti-exceptionalism about logic takes logical theories to be continuous with scientific theories. Scientific theories are subject to criteria of theoretical equivalence. This article compares two types of theoretical equivalence – one syntactic and one semantic – in the context of logical anti-exceptionalism, and argues that the syntactic approach leads to undesirable consequences. The anti-exceptionalist should therefore take a semantic approach when evaluating whether logical theories, understood as scientific theories, are equivalent. This article argues for a particular semantic approach, in terms (...) of categorical equivalence, to determine whether logical theories are equivalent. (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 2, which addresses categorical equivalence, interpretational equivalence, and duality. (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 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)

The structuralist approach represents the relation between a model and physical system as a relation between two mathematical structures. However, since a physical system is _prima facie_ _not_ a mathematical structure, the structuralist approach seemingly fails to represent the fact that science is about concrete, physical reality. In this paper, I take up this _problem of lost reality_ and suggest how it may be solved in a purely structuralist fashion. I start by briefly introducing both the structuralist approach and the (...) problem of lost reality and discussing the various (non-structuralist) solutions that have been proposed in the literature. Following this, I decompose the problem into the _ontological mismatch_ and _specification_ problems. In response to the former, I present a _metascientific dissolution argument_, according to which the difference in kind between mathematical structures and physical systems poses no deep obstacle to the structuralist approach, and consider some upshots of this argument for our views on representation. By way of conclusion, I argue that the metascientific dissolution argument paves the way for a solution to the specification problem as well. (shrink)

In recent papers Hans Halvorson has offered a critique of the semantic view of theories, showing that theories may be the same although the corresponding sets of models are different and, conversely, that theories may be different although the corresponding sets of models are the same. This critique will be assessed, first, as it pertains to issues concerning scientific models in the empirical sciences and, second, independent of any concern with empirical science.

Structuralist foundations of mathematics aim for an ‘invariant’ conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. I argue in favor of the former over the latter. First, I explain why to choose between them we need to ask the question of what is the correct ‘categorified’ version of a set. Second, I argue in favor of groupoids over categories as ‘categorified’ sets by introducing a pre-formal understanding of groupoids as (...) abstract shapes. This conclusion lends further support to the perspective taken by the Univalent Foundations of mathematics. (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 this paper, I discuss one form of the idea that spacetime and gravity might ‘emerge’ from quantum theory, i.e. via a holographic duality, and in particular via AdS/CFT duality. I begin by giving a survey of the general notion of duality, as well as its connection to emergence. I then review the AdS/CFT duality and proceed to discuss emergence in this context. We will see that it is difficult to find compelling arguments for the emergence of full quantum gravity (...) from gauge theory via AdS/CFT, i.e. for the boundary theory's being metaphysically more fundamental than the bulk theory. (shrink)

As a prolegomenon to understanding the sense in which dualities are theoretical equivalences, we investigate the intuitive `equivalence' of hyper-regular Lagrangian and Hamiltonian classical mechanics. We show that the symplectification of these theories provides a sense in which they are isomorphic, and mutually and canonically definable through an analog of `common definitional extension'.

This paper argues for a representational semantic conception of scientific theories, which respects the bare claim of any semantic view, namely that theories can be characterised as sets of models. RSC must be sharply distinguished from structural versions that assume a further identity of ‘models’ and ‘structures’, which we reject. The practice-turn in the recent philosophical literature suggests instead that modelling must be understood in a deflationary spirit, in terms of the diverse representational practices in the sciences. These insights are (...) applied to some mathematical models, thus showing that the mathematical sciences are not in principle counterexamples to RSC. (shrink)