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)

The paper takes up Bell's “Everett theory” and develops it further. The resulting theory is about the system of all particles in the universe, each located in ordinary, 3-dimensional space. This many-particle system as a whole performs random jumps through 3N-dimensional configuration space – hence “Tychistic Bohmian Mechanics”. The distribution of its spontaneous localisations in configuration space is given by the Born Rule probability measure for the universal wavefunction. Contra Bell, the theory is argued to satisfy the minimal desiderata for (...) a Bohmian theory within the Primitive Ontology framework. TBM's formalism is that of ordinary Bohmian Mechanics, without the postulate of continuous particle trajectories and their deterministic dynamics. This “rump formalism” receives, however, a different interpretation. We defend TBM as an empirically adequate and coherent quantum theory. Objections voiced by Bell and Maudlin are rebutted. The “for all practical purposes”-classical, Everettian worlds exist sequentially in TBM. In a temporally coarse-grained sense, they quasi-persist. By contrast, the individual particles themselves cease to persist. (shrink)

Mathematics is often taken to play one of two roles in the empirical sciences: either it represents empirical phenomena or it explains these phenomena by imposing constraints on them. This article identifies a third and distinct role that has not been fully appreciated in the literature on applicability of mathematics and may be pervasive in scientific practice. I call this the “bridging” role of mathematics, according to which mathematics acts as a connecting scheme in our explanatory reasoning about why and (...) how two different descriptions of an empirical phenomenon relate to each other. I discuss two bridging roles appearing in biological and physical explanations. (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)

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)

Dualities are a pervasive phenomenon in contemporary physics, in which two physical theories are empirically equivalent, yet prima facie make different ontological claims about the world (potentially very different claims—differing in e.g. the number and radius of dimensions of the universe). Dualities thus present a particular instantiation of the well-known notion of underdetermination of theory by evidence. Many different philosophical proposals have been made for how such putative underdetermination might be resolved—this continues to be a programme of active research.

A ‘duality’ is a formal mapping between the spaces of solutions of two empirically equivalent theories. In recent times, dualities have been found to be pervasive in string theory and quantum field theory. Naïvely interpreted, duality-related theories appear to make very different ontological claims about the world—differing in e.g. space-time structure, fundamental ontology, and mereological structure. In light of this, duality-related theories raise questions familiar from discussions of underdetermination in the philosophy of science: in the presence of dual theories, what (...) is one to say about the ontology of the world? In this paper, we undertake a comprehensive and non-technical survey of the landscape of possible ontological interpretations of duality-related theories. We provide a significantly enriched and clarified taxonomy of options—several of which are novel to the literature. (shrink)

Orthodoxy has it that only metaphysically elite properties can be invoked in scientifically elite laws. We argue that this claim does not fit scientific practice. An examination of candidate scientifically elite laws like Newton’s F = ma reveals properties invoked that are irreversibly defined and thus metaphysically non-elite by the lights of the surrounding theory: Newtonian acceleration is irreversibly defined as the second derivative of position, and Newtonian resultant force is irreversibly defined as the sum of the component forces. We (...) think that scientists are happy to invoke metaphysically non-elite properties in scientifically elite laws for reasons of convenience, such as to simplify the equations and to make them more modular. On this basis, we draw a deflationary moral about laws themselves, as being merely convenient summaries. 1 Introduction2 Orthodoxy2.1 Orthodoxy stated2.2 Orthodoxy explained2.3 A worrisome commitment: Term objectivism3 The Loose View3.1 The argument from practice3.2 Finding the scientifically elite equations3.3 Diagnosing the metaphysically non-elite properties3.4 Situating our challenge4 Newtonian Mechanics4.1 Newtonian acceleration4.2 Newtonian resultant force4.3 A metaphysical discovery?5 Objections5.1 Metaphysically elite after all?5.2 Scientifically non-elite after all?5.3 But our world is not Newtonian!6 Consequences6.1 Consequences for the motivations for link6.2 Consequences for metaphysically elite properties6.3 Consequences for lawhood. (shrink)

The philosopher of science faces overwhelming disagreement in the literature on the definition, nature, structure, ontology, and content of scientific theories. These disagreements are at least partly responsible for disagreements in many of the debates in the discipline which put weight on the concept scientific theory. I argue that available theories of theories and conceptual analyses of theory are ineffectual options for addressing this difficulty: they do not move debates forward in a significant way. Directing my attention to debates about (...) the properties of particular, named theories, I introduce ‘theory eliminativism’ as a certain type of debate-reformulation. As a methodological tool it has the potential to be a highly effective way to make progress in the face of the noted problem: post-reformulation disagreements about theory cannot compromise the debate, and the questions that really matter can still be asked and answered. In addition the reformulation process demands that philosophers engage with science and the history of science in a more serious way than is usual in order to answer important questions about the justification for targeting a particular set of propositions (say) in a given context. All things considered, we should expect the benefits of a theory-eliminating debate-reformulation to heavily outweigh the costs for a highly significant number of debates of the relevant type. (shrink)

I provide an explicit formulation of empirical adequacy, the central concept of constructive empiricism, and point out a number of problems. Based on one of the inspirations for empirical adequacy, I generalize the notion of a theory to avoid implausible presumptions about the relation of theoretical concepts and observations, and generalize empirical adequacy with the help of approximation sets to allow for lack of knowledge, approximations, and successive gain of knowledge and precision. As a test case, I provide an application (...) of these generalizations to a simple interference phenomenon. (shrink)

I argue that instead of a rather narrow focus on N. Bohr's account of complementarity as a particular and perhaps obscure metaphysical or epistemological concept (or as being motivated by such a concept), we should consider it to result from pursuing a particular method of studying physical phenomena. More precisely, I identify a strong undercurrent of Baconian method of induction in Bohr's work that likely emerged during his experimental training and practice. When its development is analyzed in light of Baconian (...) induction, complementarity emerges as a levelheaded rather than a controversial account, carefully elicited from a comprehensive grasp of the available experimental basis, shunning hasty metaphysically motivated generalizations based on partial experimental evidence. In fact, Bohr's insistence on the “classical” nature of observations in experiments, as well as the counterintuitive synthesis of wave and particle concepts that have puzzled scholars, seem a natural outcome (an updated instance) of the inductive method. Such analysis clarifies the intricacies of early Schrödinger's critique of the account as well as Bohr's response, which have been misinterpreted in the literature. If adequate, the analysis may lend considerable support to the view that Bacon explicated the general terms of an experimentally minded strand of the scientific method, developed and refined by scientists in the following three centuries. (shrink)

A recent rethinking of the early history of Quantum Mechanics deemed the late 1920s agreement on the equivalence of Matrix Mechanics and Wave Mechanics, prompted by Schrödinger's 1926 proof, a myth. Schrödinger supposedly failed to prove isomorphism, or even a weaker equivalence (“Schrödinger-equivalence”) of the mathematical structures of the two theories; developments in the early 1930s, especially the work of mathematician von Neumann provided sound proof of mathematical equivalence. The alleged agreement about the Copenhagen Interpretation, predicated to a large extent (...) on this equivalence, was deemed a myth as well. In response, I argue that Schrödinger's proof concerned primarily a domain-specific ontological equivalence, rather than the isomorphism or a weaker mathematical equivalence. It stemmed initially from the agreement of the eigenvalues of Wave Mechanics and energy-states of Bohr's Model that was discovered and published by Schrödinger in his first and second communications of 1926. Schrödinger demonstrated in this proof that the laws of motion arrived at by the method of Matrix Mechanics are satisfied by assigning the auxiliary role to eigenfunctions in the derivation of matrices (while he only outlined the reversed derivation of eigenfunctions from Matrix Mechanics, which was necessary for the proof of both isomorphism and Schrödinger-equivalence of the two theories). This result was intended to demonstrate the domain-specific ontological equivalence of Matrix Mechanics and Wave Mechanics, with respect to the domain of Bohr's atom. And although the mathematical equivalence of the theories did not seem out of the reach of existing theories and methods, Schrödinger never intended to fully explore such a possibility in his proof paper. In a further development of Quantum Mechanics, Bohr's complementarity and Copenhagen Interpretation captured a more substantial convergence of the subsequently revised (in light of the experimental results) Wave and Matrix Mechanics. -/- I argue that both the equivalence and Copenhagen Interpretation can be deemed myths if one predicates the philosophical and historical analysis on a narrow model of physical theory which disregards its historical context, and focuses exclusively on its formal aspects and the exploration of the logical models supposedly implicit in it. -/- . (shrink)

Various forms of underdetermination that might threaten the realist stance are examined. That which holds between different 'formulations' of a theory (such as the Hamiltonian and Lagrangian formulations of classical mechanics) is considered in some detail, as is the 'metaphysical' underdetermination invoked to support 'ontic structural realism'. The problematic roles of heuristic fruitfulness and surplus structure in attempts to break these forms of underdetermination are discussed and an approach emphasizing the relevant structural commonalities is defended.

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.

The aim of the famous Born and Jordan 1925 paper was to put Heisenberg’s matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg’s theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger’s wave mechanics cannot be equivalent to Heisenberg’s more physically motivated matrix mechanics unless its observables are quantized using this rule, and not (...) the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born–Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born–Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments. (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)

Quine’s thesis of underdetermination is significantly weaker than it has been taken to be in the recent literature, for the following reasons: (i) it does not hold for all theories, but only for some global theories, (ii) it does not require the existence of empirically equivalent yet logically incompatible theories, (iii) it does not rule out the possibility that all perceived rivalry between empirically equivalent theories might be merely apparent and eliminable through translation, (iv) it is not a fundamental thesis (...) within Quine’s philosophy, and (v) it does not carry with it the anti-realistic consequences often associated with the thesis in recent debates. The paper analyzes Quine’s views on the matter and the changes they underwent over the years. A conjecture is put forth about why Quine’s thesis has been so widely misrepresented: Quine’s writings up to 1975 tackled primarily the formulation and justification of the thesis, but afterwards were concerned mostly with the question whether empirically equivalent rivals to the theory we hold are to be considered true also. When this latter discussion is read without bearing in mind Quine’s earlier formulation and justification of the thesis, his thesis seems to have stronger epistemic consequences than it actually does. A careful reading of his later writings shows, however, that the formulation of the thesis remained unchanged after 1975, and that his mature and considered views supported only a very mitigated version of the thesis. (shrink)

Scientific change has two important dimensions: conceptual change and structural change. In this paper, I argue that the existence of conceptual change brings serious difficulties for scientific realism, and the existence of structural change makes structural realism look quite implausible. I then sketch an alternative account of scientific change, in terms of partial structures, that accommodates both conceptual and structural changes. The proposal, however, is not realist, and supports a structuralist version of van Fraassen’s constructive empiricism (structural empiricism).

A recent rethinking of the early history of Quantum Mechanics deemed the late 1920s agreement on the equivalence of Matrix Mechanics and Wave Mechanics, prompted by Schrödinger’s 1926 proof, a myth. Schrödinger supposedly failed to achieve the goal of proving isomorphism of the mathematical structures of the two theories, while only later developments in the early 1930s, especially the work of mathematician John von Neumman (1932) provided sound proof of equivalence. The alleged agreement about the Copenhagen Interpretation, predicated to a (...) large extent on this equivalence, was deemed a myth as well. If such analysis is correct, it provides considerable evidence that, in its critical moments, the foundations of scientific practice might not live up to the minimal standards of rigor, as such standards are established in the practice of logic, mathematics, and mathematical physics, thereby prompting one to question the rationality of the practice of physics. In response, I argue that Schrödinger’s proof concerned primarily a domain-specific ontological equivalence, rather than the isomorphism. It stemmed initially from the agreement of the eigenvalues of Wave Mechanics and energy-states of Bohr’s Model that was discovered and published by Schrödinger in his First and Second Communications of 1926. Schrödinger demonstrated in this proof that the laws of motion arrived at by the method of Matrix Mechanics could be derived successfully from eigenfunctions as well (while he only outlined the reversed derivation of eigenfunctions from Matrix Mechanics, which was necessary for the proof of isomorphism of the two theories). This result was intended to demonstrate the domain-specific ontological equivalence of Matrix Mechanics and Wave Mechanics, with respect to the domain of Bohr’s atom. And although the full-fledged mathematico-logical equivalence of the theories did not seem out of the reach of existing theories and methods, Schrödinger never intended to fully explore such a possibility in his proof paper. In a further development of Quantum Mechanics, Bohr’s complementarity and Copenhagen Interpretation captured a more substantial convergence of the subsequently revised (in light of the experimental results) Wave and Matrix Mechanics. I argue that both the equivalence and Copenhagen Interpretation can be deemed myths if one predicates the philosophical and historical analysis on a narrow model of physical theory which disregards its historical context, and focuses exclusively on its formal aspects and the exploration of the logical models supposedly implicit in it. (shrink)

The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is (...) Heisenberg's `quantum-theoretical Umdeutung (reinterpretation) of classical observables', which lies at the basis of quantization theory. Similarly, Bohr's correspondence principle (in somewhat revised form) and Schroedinger's wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely in the limit h -> 0 of small Planck's constant (in a finite system), in the limit N goes to infinity of a large system with $N$ degrees of freedom (at fixed h), and through decoherence and consistent histories. The first limit is closely related to modern quantization theory and microlocal analysis, whereas the second involves methods of C*-algebras and the concepts of superselection sectors and macroscopic observables. In these limits, the classical world does not emerge as a sharply defined objective reality, but rather as an approximate appearance relative to certain ``classical" states and observables. Decoherence subsequently clarifies the role of such states, in that they are ``einselected", i.e. robust against coupling to the environment. Furthermore, the nature of classical observables is elucidated by the fact that they typically define (approximately) consistent sets of histories. This combination of ideas and techniques does not quite resolve the measurement problem, but it does make the point that classicality results from the elimination of certain states and observables from quantum theory. Thus the classical world is not created by observation (as Heisenberg once claimed), but rather by the lack of it. (shrink)

This paper re-evaluates the formative year of quantum mechanics—from Heisenberg’s first paper on matrix mechanics to Schrödinger’s equivalence proof—by focusing on the role of radiation in the emerging theory. We argue that the radiation problem played a key role in early quantum mechanics, a role that has not been taken into account in the standard histories. Radiation was perceived by the main protagonists of matrix and wave mechanics as a central lacuna in these emerging theories and continued to contribute to (...) the theoretical development and conceptual clarification of quantum mechanics. Studying the interplay between quantum mechanics and radiation, the paper provides an account of (a) how quantum mechanics was able to connect to its empirical basis in spectroscopy and (b) how Schrödinger’s equivalence proof emerged from his explorative calculations on the emission of radiation. (shrink)

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

The ontic structural realist stance is motivated by a desire to do philosophical justice to the success of science, whilst withstanding the metaphysical undermining generated by the various species of ontological underdetermination. We are, however, as yet in want of general principles to provide a scaffold for the explicit construction of structural ontologies. Here we will attempt to bridge this gap by utilizing the formal procedure of quantization as a guide to ontic structure of modern physical theory. The example of (...) non-relativistic particle mechanics will be considered and, for that case, it will be shown that a viable candidate for an ontic structural realism framework can be constituted in terms of the combination of a state-space with Poisson bracket structure, and a set of observables, with Lie algebra structure. 1 Introduction2 Formulation Underdetermination and Structural Ontologies3 Quantization and Structural Ontologies4 The Case of Non-relativistic Particle Mechanics4.1 Formulation underdetermination4.2 The classical ontology4.3 Quantum theory and the generalized structural ontology4.4 Interpretation of results5 Conclusion and Prospects. (shrink)