The notion of time reversal has caused some recent controversy in philosophy of physics. The debate has mainly put the focus on how the concept of time reversal should be formally implemented across different physical theories and models, as if time reversal were a single, unified concept that physical theories should capture. In this paper, I shift the focus of the debate and defend that the concept of time reversal involves at least three facets, where each of them gives rise (...) to opposing views. In particular, I submit that any account of time reversal presupposes (explicitly or implicitly) modal, metaphysical, and heuristic facets. The comprehension of this multi-faceted nature of time reversal, I conclude, shows that time reversal can be coherently said in many ways, suggesting a disunified concept. (shrink)

Andrew Wiles' analytic proof of Fermat's Last Theorem FLT, which appeals to geometrical properties of real and complex numbers, leaves two questions unanswered: (i) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? (ii) Why is x^n+y^n=z^n solvable only for n<3? In this inter-disciplinary perspective, we offer insight into, and answers to, both queries; yielding a pre-formal proof of why FLT can be treated as a true (...) arithmetical proposition (one which, moreover, might not be provable formally in the first-order Peano Arithmetic PA), where we admit only elementary (i.e., number-theoretic) reasoning, without appeal to analytic properties of real and complex numbers. We cogently argue, further, that any formal proof of FLT needs---as is implicitly suggested by Wiles' proof---to appeal essentially to formal geometrical properties of formal arithmetical propositions. (shrink)

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

The main intent of this thesis is to defend that the laws of nature are better thought as transcendent universals, such as platonic governism suggests, and that they are metaphysically necessary in a strong way, such as the heterodox version of such platonism defends. With this intention, we sustain that physical symmetries are essential consequences of the laws of nature – what solves the challenge of symmetries – thus being metaphysically necessary, without being governist's necessitation laws. First, we will show (...) what laws of nature are and the reasons to reject other metaphysical theories and to accept platonic governism. Soon after, we will present the challenge of symmetries and the reasons to prefer the platonic governist answer over dispositionalist, regularist, aristotelic (governism), counterfatualist and primitivist ones. At last, we will define what is the metaphysical necessity, argue for the strong metaphysical necessity of laws and their consequences, and show why the reasons for the contingency or weak necessity of laws are bad theoretical paths. (shrink)

This book deals with an internal theme of metaphysics, which is the metaphysics of the laws of nature. The author presents traditional contemporary theories, as well as his own original theory, and evaluates each one at a time. He also addresses the problem of the modality of the laws of nature and makes some criticism of the standard view of necessity as truth in all possible worlds, and shows an application of his discussion to the metaphysics of physics. / Este (...) livro trata de um tema interno à metafísica, que é a metafísica das leis da natureza. O autor apresenta as teorias tradicionais contemporâneas, tal como também a sua própria teoria original, e avalia cada uma delas por vez. Ele também aborda o problema da modalidade das leis da natureza e apresenta uma certa crítica à visão padrão da necessidade como verdade em todos os mundos possíveis, e mostra uma aplicação da sua discussão à metafísica da Física. (shrink)

The aim of this article is to argue that a temporal asymmetry may be established within the framework of quantum field theory, independently of any violation of CP, and thereby T, in weak interactions, and independently of the property of time reversal invariance that its dynamical equations instantiate. Particularly, I shall argue that the temporal asymmetry can be stemmed from assessing the links between the proper group of symmetries of the theory and the ontology of the theory: arguments applied to (...) establish which elements and magnitudes remain invariants under group transformations can also be used to establish a temporal asymmetry. (shrink)

The idea that a serious threat to scientific realism comes from unconceived alternatives has been proposed by van Fraassen, Sklar, Stanford and Wray among others. Peter Lipton's critique of this threat from underconsideration is examined briefly in terms of its logic and its applicability to the case of space-time and particle physics. The example of space-time and particle physics indicates a generic heuristic for quantitative sciences for constructing potentially serious cases of underdetermination, involving one-parameter family of rivals T_m that work (...) as a team rather than as a single rival against default theory T_0. In important examples this new parameter has a physical meaning and makes a crucial _conceptual_ difference, shrinking the symmetry group and in some case putting gauge freedom, formal indeterminism vs. determinism, the presence of the hole argument, etc., at risk. Methodologies akin to eliminative induction or tempered subjective Bayesianism are more demonstrably reliable than the custom of attending only to "our best theory": they can lead either to a serious rivalry or to improved arguments for the favorite theory. The example of General Relativity vs. massive spin 2 gravity, a recent topic in the physics literature, is discussed. Arguably the General Relativity and philosophy literatures have ignored the most serious rival to General Relativity. (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.

Scientific properties come in degrees: elephants are more massive than mice. Are facts like these fundamental or can they be explained in other terms? This article argues that the structure of physical quantities like mass reduces to facts about the role that mass plays in the laws of nature. On this view elephants are more massive than mice partly in virtue of the fact that elephants are harder to throw around.

Perhaps the hottest topic in the philosophy of chemistry is that of the relationship between chemistry and physics. The problem finds one of its main manifestations in the debate about the nature of molecular structure, given by the spatial arrangement of the nuclei in a molecule. The traditional strategy to address the problem is to consider chemical cases that challenge the definition of molecular structure in quantum–mechanical terms. Instead of taking that top-down strategy, in this paper we face the problem (...) of the reduction of molecular structure to quantum mechanics from a bottom-up perspective: our aim is to show how the theoretical peculiarities of quantum mechanics stand against the possibility of molecular structure, defined in terms of the spatial relations of the nuclei conceived as individual localized objects. We will argue that, according to the theory, quantum “particles” are not individuals that can be identified as different from others and that can be reidentified through time; therefore, they do not have the ontological stability necessary to maintain the relations that can lead to a spatially definite system with an identifiable shape. On the other hand, although quantum chemists use the resources supplied by quantum mechanics with successful results, this does no mean reduction: their “approximations” add certain assumptions that are not justified in the context of quantum mechanics or are even inconsistent with the very formal structure of the theory. (shrink)

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)

Path-dependence offers a promising way of understanding the role historicity plays in explanation, namely, how the past states of a process can matter in the explanation of a given outcome. The two main existing accounts of path-dependence have sought to present it either in terms of dynamic landscapes or branching trees. However, the notions of landscape and tree both have serious limitations and have been criticized. The framework of causal networks is both more fundamental and more general that that of (...) landscapes and trees. Within this framework, I propose that historicity in networks should be understood as symmetry breaking. History matters when an asymmetric bias towards an outcome emerges in a causal network. This permits a quantitative measure for how path-dependence can occur in degrees, and offers suggestive insights into how historicity is intertwined both with causal structure and complexity. (shrink)

This paper is concerned with the relation between two notions: that of two solutions or models of a theory being related by a symmetry of the theory and that of solutions or models being physically equivalent. A number of authors have recently discussed this relation, some taking an optimistic view, on which there is a suitable concept of the symmetry of a theory relative to which these two notions coincide, others taking a pessimistic view, on which there is no such (...) concept. The present paper arrives at a cautiously pessimistic conclusion. (shrink)

In this paper, we put forward an alternative interpretation of time-reversal symmetry in philosophy of physics: Humean time-reversal symmetry. According to it, time-reversal symmetry is understood as a heuristic, epistemic virtue of the best system, not as a property of the Humean mosaic. One of the consequences of this view is that one of the main arguments against a primitive direction of time is rendered harmless, which paves the way for primitivism about the direction of time.

A widespread view in physics holds that the implementation of time reversal in standard quantum mechanics must be given by an anti-unitary operator. In foundations and philosophy of physics, however, there has been some discussion about the conceptual grounds of this orthodoxy, largely relying on either its obviousness or its mathematical-physical virtues. My aim in this paper is to substantively change the traditional structure of the debate by highlighting the philosophical commitments underlying the orthodoxy. I argue that the persuasive force (...) of the orthodoxy can benefit from a relational metaphysics of time and a by-stipulation view on symmetries. Within such philosophical background, I submit, the orthodoxy of time reversal in standard quantum mechanics could find a fertile terrain to lay the groundwork for a more thorough conceptual justification. (shrink)

In this article I shall defend, against the conventional understanding of the matter, that two coherent and tenable approaches to time reversal can be suitably introduced in standard quantum mechanics: an “orthodox” approach that demands time reversal to be represented in terms of an anti-unitary and anti-linear time-reversal operator, and a “heterodox” approach that represents time reversal in terms of a unitary, linear time-reversal operator. The rationale shall be that the orthodox approach in quantum theories assumes a relationalist metaphysics of (...) time, according to which time reversal is nothing but motion reversal. But, when one shifts gears and turn to a substantivalist metaphysics of time the heterodox approach to time reversal in quantum mechanics comes up in a more natural way. (shrink)

In this paper it will be argued that any realist interpretation of quantum mechanics intending to preserve the objectivity of the set of the definite-valued observables should require such a set to be invariant under the symmetry group of the theory. In particular, it will be shown that the natural way to reach this goal is to appeal to the Casimir operators of the Galilean group. Additionally, this idea will be generalized in two ways: by selecting the definite-valued observables of (...) a system in quantum field theory as those represented by the Casimir operators of the Poincaré group, and by extending the strategy to gauge symmetries. (shrink)

Fundamental physics contains an important link between properties of elementary particles and continuous symmetries of particle systems. For example, properties such as mass and spin are said to be 'associated' with specific continuous symmetries. -/- These 'associations' have played a key role in the discovery of various new particle kinds, but more importantly: they are thought to provide a deep insight into the nature of physical reality. The link between properties and symmetries has been said to call for a radical (...) revision of perceived metaphysical orthodoxy. However, if we are to use claims about an 'association' between properties and symmetries in the articulation of metaphysical views, we first need to develop a sufficiently precise understanding of the content of these claims. The goal of this paper is to do just that. (shrink)

The aim of this paper is to consider in what sense the modal-Hamiltonian interpretation of quantum mechanics satisfies the physical constraints imposed by the Galilean group. In particular, we show that the only apparent conflict, which follows from boost-transformations, can be overcome when the definition of quantum systems and subsystems is taken into account. On this basis, we apply the interpretation to different well-known models, in order to obtain concrete examples of the previous conceptual conclusions. Finally, we consider the role (...) played by the Casimir operators of the Galilean group in the interpretation. (shrink)

This article develops an analogy proposed by Stachel between general relativity (GR) and quantum mechanics (QM) as regards permutation invariance. Our main idea is to overcome Pooley's criticism of the analogy by appeal to paraparticles. In GR, the equations are (the solution space is) invariant under diffeomorphisms permuting spacetime points. Similarly, in QM the equations are invariant under particle permutations. Stachel argued that this feature—a theory's ‘not caring which point, or particle, is which’—supported a structuralist ontology. Pooley criticizes this analogy: (...) in QM the (anti-)symmetrization of fermions and bosons implies that each individual state (solution) is fixed by each permutation, while in GR a diffeomorphism yields in general a distinct, albeit isomorphic, solution. We define various versions of structuralism, and go on to formulate Stachel's and Pooley's positions, admittedly in our own terms. We then reply to Pooley. Though he is right about fermions and bosons, QM equally allows more general types of particle symmetry, in which states (vectors, rays, or density operators) are not fixed by all permutations (called ‘paraparticle states’). Thus Stachel's analogy is revived. (shrink)

Merely approximate symmetry is mundane enough in physics that one rarely finds any explication of it. Among philosophers it has also received scant attention compared to exact symmetries. Herein I invite further consideration of this concept that is so essential to the practice of physics and interpretation of physical theory. After motivating why it deserves such scrutiny, I propose a minimal definition of approximate symmetry—that is, one that presupposes as little structure on a physical theory to which it is applied (...) as seems needed. Then I apply this definition to three topics: first, accounting for or explaining the symmetries of a theory emeritus in intertheoretic reduction; second, explicating and evaluating the Curie-Post principle; and third, a new account of accidental symmetry. (shrink)

Over the past few decades the notion of symmetry has played a major role in physics and in the philosophy of physics. Philosophers have used symmetry to discuss the ontology and seeming objectivity of the laws of physics. We introduce several notions of symmetry in mathematics and explain how they can also be used in resolving different problems in the philosophy of mathematics. We use symmetry to discuss the objectivity of mathematics, the role of mathematical objects, the unreasonable effectiveness of (...) mathematics and the relationship of mathematics to physics. (shrink)