We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we "derive" an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist.

Metaphysicians speak of laws of nature in terms of necessity and universality; scientists, in terms of symmetry and invariance. In this book van Fraassen argues that no metaphysical account of laws can succeed. He analyzes and rejects the arguments that there are laws of nature, or that we must believe there are, and argues that we should disregard the idea of law as an adequate clue to science. After exploring what this means for general epistemology, the author develops the empiricist (...) view of science as a construction of models to represent the phenomena. (shrink)

What knowledge would suffice to yield an interpretation of an arbitrary utterance of a language when such knowledge is based on evidence plausibly available to a nonspeaker of that language? it is argued that it is enough to know a theory of truth for the language and that the theory satisfies tarski's 'convention t' and that it gives an optimal fit to data about sentences held true, Under specified conditions, By native speakers.

I conceptualise the role of dualities in quantum gravity, in terms of their functions for theory construction. I distinguish between two functions of duality in physical practice: namely, discovering and describing ‘equivalent physics’, versus suggesting ‘new physics’. I dub these the ‘theoretical’ versus the ‘heuristic’ functions of dualities. The distinction seems to have gone largely unnoticed in the philosophical literature: and it exists both for dualities, and for the more general relation of theoretical equivalence. The paper develops the heuristic function (...) of dualities: illustrating how they can be used, if one has any luck, to find and formulate new theories. I also point to the different physical commitments about the theories in question that underlie these two functions. I show how a recently developed schema for dualities articulates the differences between the two functions. (shrink)

In this paper we present a schema for describing dualities between physical theories, and illustrate it in detail with the example of bosonization: a boson-fermion duality in two-dimensional quantum field theory. The schema develops proposals in De Haro : these proposals include construals of notions related to duality, like representation, model, symmetry and interpretation. The aim of the schema is to give a more precise criterion for duality than has so far been considered. The bosonization example, or boson-fermion duality, has (...) the feature of being simple, yet rich enough, to illustrate the most relevant aspects of our schema, which also apply to more sophisticated dualities. The richness of the example consists, mainly, in its concern with two non-trivial quantum field theories: including massive Thirring-sine-Gordon duality, and non-abelian bosonization. This prompts two comparisons with the recent philosophical literature on dualities:--- Unlike the standard cases of duality in quantum field theory and string theory, where only specific simplifying limits of the theories are explicitly known, the boson-fermion duality is known to hold {\it exactly}. This exactness can be exhibited explicitly. The bosonization example illustrates both the cases of isomorphic and {\it non-isomorphic} models: which we believe the literature on dualities has not so far discussed. (shrink)

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)

Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions -/- Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in mathematics textbooks: these are aimed squarely at mathematicians; (...) they typically presuppose that the reader has a significant background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are then scattered across disconnected pockets of papers, separated by decades or more. -/- The first aim of Philosophy and Model Theory, then, is to consider the philosophical uses of model theory. On a technical level, we try to show how philosophically significant results connect to one another, and also to state the best version of a result for philosophical purposes. On a philosophical level, we show how similar dialectical situations arise repeatedly, across fragmented debates in varied philosophical areas. -/- The second aim of Philosophy and Model Theory, though, is to consider the philosophy of model theory. Model theory itself is rarely taken as the subject matter of philosophising (contrast this, say, with the philosophy of biology, or the philosophy of set theory). But model theory is a beautiful part of pure mathematics, and worthy of philosophical study in its own right. (shrink)

Numerous approaches to a quantum theory of gravity posit fundamental ontologies that exclude spacetime, either partially or wholly. This situation raises deep questions about how such theories could relate to the empirical realm, since arguably only entities localized in spacetime can ever be observed. Are such entities even possible in a theory without fundamental spacetime? How might they be derived, formally speaking? Moreover, since by assumption the fundamental entities cannot be smaller than the derived and so cannot ‘compose’ them in (...) any ordinary sense, would a formal derivation actually show the physical reality of localized entities? We address these questions via a survey of a range of theories of quantum gravity, and generally sketch how they may be answered positively. (shrink)

We propose that scientific representation is a special case of a more general notion of representation, and that the relatively well worked-out and plausible theories of the latter are directly applicable to thc scientific special case. Construing scientific representation in this way makes the so-called “problem of scientific representation” look much less interesting than it has seerned to many, and suggests that some of the (hotly contested) debates in the literature are concerned with non-issues.

We discuss some aspects of the relation between dualities and gauge symmetries. Both of these ideas are of course multi-faceted, and we confine ourselves to making two points. Both points are about dualities in string theory, and both have the ‘flavour’ that two dual theories are ‘closer in content’ than you might think. For both points, we adopt a simple conception of a duality as an ‘isomorphism’ between theories: more precisely, as appropriate bijections between the two theories’ sets of states (...) and sets of quantities. The first point is that this conception of duality meshes with two dual theories being ‘gauge related’ in the general philosophical sense of being physically equivalent. For a string duality, such as T-duality and gauge/gravity duality, this means taking such features as the radius of a compact dimension, and the dimensionality of spacetime, to be ‘gauge’. The second point is much more specific. We give a result about gauge/gravity duality that shows its relation to gauge symmetries to be subtler than you might expect. For gauge theories, you might expect that the duality bijections relate only gauge-invariant quantities and states, in the sense that gauge symmetries in one theory will be unrelated to any symmetries in the other theory. This may be so in general; and indeed, it is suggested by discussions of Polchinski and Horowitz. But we show that in gauge/gravity duality, each of a certain class of gauge symmetries in the gravity/bulk theory, viz. diffeomorphisms, is related by the duality to a position-dependent symmetry of the gauge/boundary theory. (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 I develop a framework for relating dualities and emergence: two notions that are close to each other but also exclude one another. I adopt the conception of duality as 'isomorphism', from the physics literature, cashing it out in terms of three conditions. These three conditions prompt two conceptually different ways in which a duality can be modified to make room for emergence; and I argue that this exhausts the possibilities for combining dualities and emergence. I apply this (...) framework to gauge/gravity dualities, considering in detail three examples: AdS/CFT, Verlinde's scheme, and black holes. My main point about gauge/gravity dualities is that the theories involved, qua theories of gravity, must be background-independent. I distinguish two senses of background-independence: minimalistic and extended. I argue that the former is sufficiently strong to allow for a consistent theory of quantum gravity; and that AdS/CFT is background-independent on this account; while Verlinde's scheme best fits the extended sense of background-independence. I argue that this extended sense should be applied with some caution: on pain of throwing the baby out with the bath-water. Nevertheless, it is an interesting and potentially fruitful heuristic principle for quantum gravity theory construction. It suggests some directions for possible generalisations of gauge/gravity dualities. The interpretation of dualities is discussed; and the so-called 'internal' vs. 'external' viewpoints are articulated in terms of: epistemic and metaphysical commitments; parts vs. wholes. I then analyse the emergence of gravity in gauge/gravity dualities in terms of the two available conceptualisations of emergence; and I show how emergence in AdS/CFT and in Verlinde's scenario differ from each other. Finally, I give a novel derivation of the Bekenstein-Hawking black hole entropy formula based on Verlinde's scheme; the derivation sheds light on several aspects of Verlinde's scheme and how it compares to Bekenstein's original calculation. (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.

According to the semantic view of scientific theories, theories are classes of models. I show that this view -- if taken seriously as a formal explication -- leads to absurdities. In particular, this view equates theories that are truly distinct, and it distinguishes theories that are truly equivalent. Furthermore, the semantic view lacks the resources to explicate interesting theoretical relations, such as embeddability of one theory into another. The untenability of the semantic view -- as currently formulated -- threatens to (...) undermine scientific structuralism. (shrink)

The dominance of string theory in the research landscape of quantum gravity physics (despite any direct experimental evidence) can, I think, be justified in a variety of ways. Here I focus on an argument from mathematical fertility, broadly similar to Hilary Putnam’s ‘no miracles argument’ that, I argue, many string theorists in fact espouse in some form or other. String theory has generated many surprising, useful, and well-confirmed mathematical ‘predictions’—here I focus on mirror symmetry and the mirror theorem. These predictions (...) were made on the basis of general physical principles entering into string theory. The success of the mathematical predictions are then seen as evidence for the framework that generated them. I shall attempt to defend this argument, but there are nonetheless some serious objections to be faced. These objections can only be evaded at a considerably high (philosophical) price. (shrink)

This paper investigates the significance of T-duality in string theory: the indistinguisha- bility with respect to all observables, of models attributing radically different radii to space – larger than the observable universe, or far smaller than the Planck length, say. Two interpretational branch points are identified and discussed. First, whether duals are physically equivalent or not: by considering a duality of the familiar simple harmonic oscillator, I argue that they are. Unlike the oscillator, there are no measurements ‘outside’ string theory (...) that could distinguish the duals. Second, whether duals agree or disagree on the radius of ‘target space’, the space in which strings evolve according to string theory. I argue for the latter position, because the alternative leaves it unknown what the radius is. Since duals are physically equivalent yet disagree on the radius of target space, it follows that the radius is indeterminate between them. Using an analysis of Brandenberger and Vafa (1989), I explain why – even so – space is observed to have a determinate, large radius. The conclusion is that observed, ‘phenomenal’ space is not target space, since a space cannot have both a determinate and indeterminate radius: instead phenomenal space must be a higher-level phenomenon, not fundamental. (shrink)

1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (...) . . . . . . . . . . . 7 1.3 Vector Fields, Integral Curves, and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Tensors and Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 The Action of Smooth Maps on Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.7 Derivative Operators and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.8 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.9 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.10 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.11 Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.. (shrink)

Several philosophers of science construe models of scientific theories as set-theoretic structures. Some of them moreover claim that models should not be construed as structures in the sense of model theory because the latter are language-dependent. I argue that if we are ready to construe models as set-theoretic structures (strict semantic view), we could equally well construe them as model-theoretic structures of higher-order logic (liberal semantic view). I show that every family of set-theoretic structures has an associated language of higher-order (...) logic and an up to signature isomorphism unique model-theoretic counterpart, which is able to serve the same purposes. This allows to carry over every syntactic criterion of equivalence for theories in the sense of the liberal semantic view to theories in the sense of the strict semantic view. Taken together, these results suggest that the recent dispute about the semantic view and its relation to the syntactic view can be resolved. (shrink)

We discuss ways in which category theory might be useful in philosophy of science, in particular for articulating the structure of scientific theories. We argue, moreover, that a categorical approach transcends the syntax-semantics dichotomy in 20th century analytic philosophy of science.

String theory has played a highly influential role in theoretical physics for nearly three decades and has substantially altered our view of the elementary building principles of the Universe. However, the theory remains empirically unconfirmed, and is expected to remain so for the foreseeable future. So why do string theorists have such a strong belief in their theory? This book explores this question, offering a novel insight into the nature of theory assessment itself. Dawid approaches the topic from a unique (...) position, having extensive experience in both philosophy and high-energy physics. He argues that string theory is just the most conspicuous example of a number of theories in high-energy physics where non-empirical theory assessment has an important part to play. Aimed at physicists and philosophers of science, the book does not use mathematical formalism and explains most technical terms. (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)

The AdS/CFT duality has been a source of several strong conceptual claims in the physics literature that have yet to be explored by philosophers. In this paper I focus on one of these: the extent to which spacetime geometry and locality can be said to emerge from this duality, so that neither is fundamental. I argue: that the kind of emergence in question is relatively weak, involving one kind of spacetime emerging from another kind of spacetime; inasmuch as there is (...) something conceptually interesting to say about the emergence of spacetime and locality , it is no different from that already well known to those within canonical quantum gravity; that at the core of AdS/CFT is an issue of representation and redundancy in representation. (shrink)

In this essay I begin to lay out a conceptual scheme for: analysing dualities as cases of theoretical equivalence; assessing when cases of theoretical equivalence are also cases of physical equivalence. The scheme is applied to gauge/gravity dualities. I expound what I argue to be their contribution to questions about: the nature of spacetime in quantum gravity; broader philosophical and physical discussions of spacetime. - proceed by analysing duality through four contrasts. A duality will be a suitable isomorphism between models: (...) and the four relevant contrasts are as follows: Bare theory: a triple of states, quantities, and dynamics endowed with appropriate structures and symmetries; vs. interpreted theory: which is endowed with, in addition, a suitable pair of interpretative maps. Extendable vs. unextendable theories: which can, respectively cannot, be extended as regards their domains of application. External vs. internal intepretations: which are constructed, respectively, by coupling the theory to another interpreted theory vs. from within the theory itself. Theoretical vs. physical equivalence: which contrasts formal equivalence with the equivalence of fully interpreted theories. I will apply this scheme to answering questions - for gauge/gravity dualities. I will argue that the things that are physically relevant are those that stand in a bijective correspondence under duality: the common core of the two models. I therefore conclude that most of the mathematical and physical structures that we are familiar with, in these models, are largely, though crucially never entirely, not part of that common core. Thus, the interpretation of dualities for theories of quantum gravity compels us to rethink the roles that spacetime, and many other tools in theoretical physics, play in theories of spacetime. (shrink)

This article proposes to explicate theoretical equivalence by supplementing formal equivalence criteria with preservation conditions concerning interpretation. I argue that both the internal structure of models and choices of morphisms are aspects of formalisms that are relevant when it comes to their interpretation. Hence, a formal criterion suitable for being supplemented with preservation conditions concerning interpretation should take these two aspects into account. The two currently most important criteria—gener-alized definitional equivalence (Morita equivalence) and categorical equivalence—are not optimal in this respect. (...) I put forward a criterion that takes both aspects into account: the criterion of definable categorical equivalence. (shrink)

The purported fact that geometric theories formulated in terms of points and geometric theories formulated in terms of lines are “equally correct” is often invoked in arguments for conceptual relativity, in particular by Putnam and Goodman. We discuss a few notions of equivalence between first-order theories, and we then demonstrate a precise sense in which this purported fact is true. We argue, however, that this fact does not undermine metaphysical realism.

'Holographic' relations between theories have become a main theme in quantum gravity research. These relations entail that a theory without gravity is equivalent to a gravitational theory with an extra spatial dimension. The idea of holography was first proposed in 1993 by Gerard 't Hooft on the basis of his studies of evaporating black holes. Soon afterwards the holographic 'AdS/CFT' duality was introduced, which since has been heavily studied in the string theory community and beyond. Recently, Erik Verlinde has proposed (...) that even Newton's law of gravitation can be related holographically to the thermodynamics of information on screens. We discuss inter-theoretical relations in these scenarios: what is the status of the holographic relation in them and in what sense is gravity, or spacetime, emergent? (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)

String theory promises to be able to provide us with a working theory of quantum gravity and a unified description of all fundamental forces. In string theory there are so called ‘dualities’; i.e. different theoretical formulations that are physically equivalent. In this article these dualities are investigated from a philosophical point of view. Semantic and epistemic questions relating to the problem of underdetermination of theories by data and the debate on realism concerning scientific theories are discussed. Depending on ones views (...) on semantic issues and realism different interpretations are possible of the dualities. (shrink)

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

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)

I extract some philosophical morals from some aspects of Lagrangian mechanics. One main moral concerns methodology: Lagrangian mechanics provides a level of description of phenomena which has been largely ignored by philosophers, since it falls between their accustomed levels---``laws of nature'' and ``models''. Another main moral concerns ontology: the ontology of Lagrangian mechanics is both more subtle and more problematic than philosophers often realize. The treatment of Lagrangian mechanics provides an introduction to the subject for philosophers, and is technically elementary. (...) In particular, it is confined to systems with a finite number of degrees of freedom, and for the most part eschews modern geometry. (shrink)

We analyse the possibility that string-theoretic dualities present a genuine case of strong underdetermination of theory by evidence. Drawing on the parallel discussion of the hole argument, we assess the possible interpretations of dualities. We conclude that there exist at least two defensible interpretations on which dualities do not present a worrying case of underdetermination per se.

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'.