Every Thing Must Go aruges that the only kind of metaphysics that can contribute to objective knowledge is one based specifically on contemporary science as it ...

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.

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)

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)

Available for the first time in 20 years, here are two important works from the 1920s by the best-known representative of the Vienna Circle.

SummaryenThe main argument for scientific realism is that our present theories in science are so successful empirically that they can't have got that way by chance - instead they must somehow have latched onto the blueprint of the universe. The main argument against scientific realism is that there have been enormously successful theories which were once accepted but are now regarded as false. The central question addressed in this paper is whether there is some reasonable way to have the best (...) of both worlds: to give the argument from scientific revolutions its full weight and yet still adopt some sort of realist attitude towards presently accepted theories in physics and elsewhere. I argue that there is such a way - through structural realism, a position adopted by Poincare, and here elaborated and defended. (shrink)

I argue that the Hole Argument is based on a misleading use of the mathematical formalism of general relativity. If one is attentive to mathematical practice, I will argue, the Hole Argument is blocked.

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.

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)

In a number of publications, John Earman has advocated a tertium quid to the usual dichotomy between substantivalism and relationism concerning the nature of spacetime. The idea is that the structure common to the members of an equivalence class of substantival models is captured by a Leibniz algebra which can then be taken to directly characterize the intrinsic reality only indirectly represented by the substantival models. An alleged virtue of this is that, while a substantival interpretation of spacetime theories falls (...) prey to radical local indeterminism, the Leibniz algebras do not. I argue that the program of Leibniz algebras is subject to radical local indeterminism to the same extent as substantivalism. In fact, for the category of topological spaces of interest in spacetime physics, the program is equivalent to the original spacetime approach. Moreover, the motivation for the program--that isomorphic substantival models should be regarded as representing the same physical situation--is misguided. (shrink)

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.

I argue that the best spacetime setting for Newtonian gravitation (NG) is the curved spacetime setting associated with geometrized Newtonian gravitation (GNG). Appreciation of the ‘Newtonian equivalence principle’ leads us to conclude that the gravitational field in NG itself is a gauge quantity, and that the freely falling frames are naturally identified with inertial frames. In this context, the spacetime structure of NG is represented not by the flat neo-Newtonian connection usually made explicit in formulations, but by the sum of (...) the flat connection and the gravitational field. 1 Introduction2 Newtonian Gravity: The Orthodox Approach3 Newtonian Gravity: Additional Symmetries4 Cosmological Considerations5 A Newtonian Equivalence Principle: Inertial Frames in Newtonian Gravitation6 Theory Equivalence?7 Conclusion. (shrink)

Classically, a gauge potential was merely a convenient device for generating a corresponding gauge field. Quantum-mechanically, a gauge potential lays claim to independent status as a further feature of the physical situation. But whether this is a local or a global feature is not made any clearer by the variety of mathematical structures used to represent it. I argue that in the theory of electromagnetism (or a non-Abelian generalization) that describes quantum particles subject to a classical interaction, the gauge potential (...) is best understood as a feature of the physical situation whose global character is most naturally represented by the holonomies of closed curves in space-time. (shrink)

Those looking for holism in contemporary physics have focused their attention primarily on quantum entanglement. But some gauge theories arguably also manifest the related phenomenon of nonseparability. While the argument is strong for the classical gauge theory describing electromagnetic interactions with quantum “particles”, it fails in the case of general relativity even though that theory may also be formulated in terms of a connection on a principal fiber bundle. Anandan has highlighted the key difference in his analysis of a supposed (...) gravitational analog to the Aharonov-Bohm effect. By contrast with electromagnetism in the original Aharonov-Bohm effect, gravitation is separable and exhibits no novel holism in this case. Whether the nonseparability of classical gauge theories of non-gravitational interactions is associated with holism depends on what counts as the relevant part-whole relation. Loop representations of quantized gauge theories of non- gravitational interactions suggest that these conclusions about holism and nonseparability may extend also to quantum theories of the associated fields. (shrink)

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)

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Stephen Hawking, among others, has proposed that the topological stability of a property of space-time is a necessary condition for it to be physically significant. What counts as stable, however, depends crucially on the choice of topology. Some physicists have thus suggested that one should find a canonical topology, a single ‘right’ topology for every inquiry. While certain such choices might be initially motivated, some little-discussed examples of Robert Geroch and some propositions of my own show that the main candidates—and (...) each possible choice, to some extent—faces the horns of a no-go result. I suggest that instead of trying to decide what the ‘right’ topology is for all problems, one should let the details of particular types of problems guide the choice of an appropriate topology. (shrink)

Spacetime substantivalism leads to a radical form of indeterminism within a very broad class of spacetime theories which include our best spacetime theory, general relativity. Extending an argument from Einstein, we show that spacetime substantivalists are committed to very many more distinct physical states than these theories' equations can determine, even with the most extensive boundary conditions.

Symmetries in physics are a guide to reality. That much is well known. But what is less well known is why symmetry is a guide to reality. What justifies inferences that draw conclusions about reality from premises about symmetries? I argue that answering this question reveals that symmetry is an epistemic notion twice over. First, these inferences must proceed via epistemic lemmas: premises about symmetries in the first instance justify epistemic lemmas about our powers of detection, and only from those (...) epistemic lemmas can we draw conclusions about reality. Second, in order to justify those epistemic lemmas, the notion of symmetry must be defined partly in epistemic terms. 1 Symmetry-to-Reality Reasoning1.1 A rough introduction to symmetry1.2 The symmetry-to-reality inference1.3 Two questions1.4 Two answers1.5 Preliminary clarifications2 Against Redundancy2.1 Redundancy2.2 Is absolute velocity redundant?2.3 Some redundancies3 Against Objectivity4 From Symmetry to Detection4.1 The epistemic approach4.2 The Occamist norm4.3 From symmetry to detection5 The Meaning of ‘Symmetry’5.1 A framework5.2 Formal definitions5.3 Ontic definitions6 Epistemic Definitions6.1 Taking observation seriously6.2 How things look6.3 Observation sentences6.4 Observational equivalence7 Symmetry as an Epistemic Notion 7.1 Observational equivalence and metaphysics7.2 The Occamist norm revisted7.3 Consequences8 Conclusion. (shrink)

One can (for the most part) formulate a model of a classical system in either the Lagrangian or the Hamiltonian framework. Though it is often thought that those two formulations are equivalent in all important ways, this is not true: the underlying geometrical structures one uses to formulate each theory are not isomorphic. This raises the question of whether one of the two is a more natural framework for the representation of classical systems. In the event, the answer is yes: (...) I state and sketch proofs of two technical results—inspired by simple physical arguments about the generic properties of classical systems—to the effect that, in a precise sense, classical systems evince exactly the geometric structure Lagrangian mechanics provides for the representation of systems, and none provided by Hamiltonian. The argument not only clarifies the conceptual structure of the two systems of mechanics, but also their relations to each other and their respective mechanisms for representing physical systems. It also shows why naïvely structural approaches to the representational content of physical theories cannot work. [Lagrange] grasped that he had gained a method of stating dynamical truths in a way, which is perfectly indifferent to the particular methods of measurement employed in fixing the positions of the various parts of the system. Accordingly, he went on to deduce equations of motion, which are equally applicable whatever quantitative measurements have been made, provided that they are adequate to fix positions. The beauty and almost divine simplicity of these equations is such that these formulae are worthy to rank with those mysterious symbols which in ancient times were held directly to indicate the Supreme Reason at the base of all things. (Whitehead [1948], p. 63)1. Introduction2.Classical Systems3. The Possible Interactions of a Classical System and the Structure of Its Space of States4. Classical Systems Are Lagrangian5. Classical Systems Are Not Hamiltonian6. How Lagrangian and Hamiltonian Mechanics Represent Classical Systems7. The Conceptual Structure of Classical Mechanics. (shrink)

It is often said that the Aharonov-Bohm effect shows that the vector potential enjoys more ontological significance than we previously realized. But how can a quantum-mechanical effect teach us something about the interpretation of Maxwell's theory—let alone about the ontological structure of the world—when both theories are false? I present a rational reconstruction of the interpretative repercussions of the Aharonov-Bohm effect, and suggest some morals for our conception of the interpretative enterprise.

The classical field theories that underlie the quantum treatments of the electromagnetic, weak, and strong forces share a peculiar feature: specifying the initial state of the field determines the evolution of some degrees of freedom of the theory while leaving the evolution of some others wholly arbitrary. This strongly suggests that some of the variables of the standard state space lack physical content-intuitively, the space of states of such a theory is of higher dimension than the corresponding space of genuine (...) physical possibilities. The structure of such theories can helpfully be characterized in terms of the action of symmetry groups on their space of states; and the conceptual problems surrounding their strange behavior can be sharpened in light of the observation that it is usually possible to eliminate the redundant variables associated with these symmetries-which turn out to be precisely those variables whose evolution is unconstrained by the dynamical laws of the theory. This paper discusses this approach, uses it to frame questions about the interpretation of classical gauge theories, and to reflect (pessimistically) on our prospects of reaching satisfactory answers to these questions. (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)

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.

The widely held picture of dynamical symmetry as surplus structure in a physical theory has many metaphysical applications. Here, I focus on its relevance to the question of which quantities in a theory represent fundamental natural properties.

Steven French articulates and defends the bold claim that there are no objects in the world. He draws on metaphysics and philosophy of science to argue for structural realism--the position that we live in a world of structures--and defends a form of eliminativism about objects that sets laws and symmetry principles at the heart of ontology.

Philosophers of quantum mechanics have generally addressed exceedingly simple systems. Laura Ruetsche offers a much-needed study of the interpretation of more complicated systems, and an underexplored family of physical theories, such as quantum field theory and quantum statistical mechanics, showing why they repay philosophical attention. She guides those familiar with the philosophy of ordinary QM into the philosophy of 'QM infinity', by presenting accessible introductions to relevant technical notions and the foundational questions they frame--and then develops and defends answers to (...) some of those questions. Finally, Ruetsche highlights ties between the foundational investigation of QM infinity and philosophy more broadly construed, in particular by using the interpretive problems discussed to motivate new ways to think about the nature of physical possibility and the problem of scientific realism. (shrink)

This is a prize-winning study of an area of physics not previously explored by philosophy: gauge theory. Gauge theories have provided our most successful representations of the fundamental forces of nature. But how do such representations work? Healey defends an original answer to this question.

We are used to talking about the “structure” posited by a given theory of physics, such as the spacetime structure of relativity. What is “structure”? What does the mathematical structure used to formulate a theory tell us about the physical world according to the theory? What if there are different mathematical formulations of a given theory? Do different formulations posit different structures, or are they merely notational variants? I consider the case of Lagrangian and Hamiltonian classical mechanics. I argue that, (...) contrary to standard wisdom, these are not genuinely equivalent theories: they differ in statespace structure. I suggest that we should be realists about statespace structure. (shrink)

A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30, ], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to (...) the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation. (shrink)

I give an informal outline of the hole argument which shows that spacetime substantivalism leads to an undesirable indeterminism in a broad class of spacetime theories. This form of the argument depends on the selection of differentiable manifolds within a spacetime theory as representing spacetime. I consider the conditions under which the argument can be extended to address versions of spacetime substantivalism which select these differentiable manifolds plus some further structure to represent spacetime. Finally, I respond to the criticisms of (...) Tim Maudlin and Jeremy Butterfield. (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.

I explain the difficulty of making various concepts of and relating to probability precise, rigorous and physically significant when attempting to apply them in reasoning about objects living in infinite-dimensional spaces, working through many examples from cosmology. I focus on the relation of topological to measure-theoretic notions of and relating to probability, how they diverge in unpleasant ways in the infinite-dimensional case, and are even difficult to work with on their own. Even in cases where an appropriate family of spacetimes (...) is finite-dimensional, and so admits a measure of the relevant sort, however, it is always the case that the family is not a compact topological space, and so does not admit a physically significant, well behaved probability measure. Problems of a different but still deeply troubling sort plague arguments about likelihood in that context, which I also discuss. I conclude that most standard forms of argument used in cosmology to estimate the likelihood of the occurrence of various properties or behaviors of spacetimes have serious mathematical, physical and conceptual problems. (shrink)

Leibniz vertrat auf der einen Seite die Überzeugung, es gebe Relationen weder als abstrakte Universalien noch als konkrete Akzidenzen. Auf der anderen Seite war er überzeugt, daß es relationale Eigenschaften von physischen Gegenständen, die nicht auf nicht-relationale Eigenschaften dieser Objekte reduziert werden können, gebe. Die wirklichen Einzeldinge haben jedoch keine nicht-formalen relationalen Eigenschaften. Sie stehen zwar in Beziehung oder sind miteinander verknüpft, aber nur durch Perzeptionen, so daß der Begriff Beziehung hier ein Begriff der zweiten Ordnung ist. Die physische Welt (...) mit ihren relationalen Eigenschaften ist fundiert in den Monaden. Die Fundierung ist eine Reduktion, aber es ist ein Mißverständnis, wenn man diese Reduktion so beschreibt, als impliziere sie eine Reduktion von relationalen Eigenschaften physischer Gegenstände; denn ihre nicht relationalen Eigenschaften werden in gleicher Weise reduziert. (shrink)