On Kinds of Indiscernibility in Logic and Metaphysics

Using the Hilbert-Bernays account as a spring-board, we first define four ways in which two objects can be discerned from one another, using the non-logical vocabulary of the language concerned. (These definitions are based on definitions made by Quine and Saunders.) Because of our use of the Hilbert-Bernays account, these definitions are in terms of the syntax of the language. But we also relate our definitions to the idea of permutations on the domain of quantification, and their being symmetries. These relations turn out to be subtle---some natural conjectures about them are false. We will see in particular that the idea of symmetry meshes with a species of indiscernibility that we will call `absolute indiscernibility'. We then report all the logical implications between our four kinds of discernibility. We use these four kinds as a resource for stating four metaphysical theses about identity. Three of these theses articulate two traditional philosophical themes: viz. the principle of the identity of indiscernibles (which will come in two versions), and haecceitism. The fourth is recent. Its most notable feature is that it makes diversity (i.e. non-identity) weaker than what we will call individuality (being an individual): two objects can be distinct but not individuals. For this reason, it has been advocated both for quantum particles and for spacetime points. Finally, we locate this fourth metaphysical thesis in a broader position, which we call structuralism. We conclude with a discussion of the semantics suitable for a structuralist, with particular reference to physical theories as well as elementary model theory.Comment: 55 pages, 21 figures. Forthcoming, after an Appendectomy, in the British Journal for the Philosophy of Scienc

Stationary logic of finitely determinate structures

AbstractIn this part we develop the theory of finitely determinate structures, that is, structures on which the dual quantifiers “stat” and “unreadable” have the same meaning. Among other genera

Dimension, matroids, and dense pairs of first-order structures

A structure M is pregeometric if the algebraic closure is a pregeometry in all M' elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of omega and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique existential matroid. Generalising previous results by van den Dries, we define dense elementary pairs of structures expanding a field and with an existential matroid, and we show that the corresponding theories have natural completions, whose models also have a unique existential matroid. We extend the above result to dense tuples of structures.Comment: Version 2.8. 61 page

Symmetries and Paraparticles as a Motivation for Structuralism

This paper 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 symmetry, in which states (vectors, rays or density operators) are not fixed by all permutations (called `paraparticle states'). Thus Stachel's analogy is revived.Comment: 45 pages, Latex, 3 Figures; forthcoming in British Journal for the Philosophy of Scienc

Symmetries and Paraparticles as a Motivation for Structuralism

This paper 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 symmetry, in which states (vectors, rays or density operators) are not fixed by all permutations (called 'paraparticle states'). Thus Stachel's analogy is revived

Categoricity and covering spaces

This thesis develops some of the basic model theory of covers of algebraic curves. In particular, an equivalence between the good model-theoretic behaviour of the modular j-function, and the openness of certain Galois representations in the Tate-modules of abelian varieties is described.Comment: DPhil Thesi

Vapnik-Chervonenkis density in some theories without the independence property, I

We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and $P$-minimal theories.Comment: 59

Issues of identity and individuality in quantum mechanics

This dissertation is ordered into three Parts. Part I is an investigation into identity, indiscernibility and individuality in logic and metaphysics. In Chapter 2, I investigate identity and discernibility in classical first-order logic. My aim will will be to define four different ways in which objects can be discerned from one another, and to relate these definitions: (i) to the idea of symmetry; and (ii) to the idea of individuality. In Chapter 3, the four kinds of discernibility are put to use in defining four rival metaphysical theses about indiscernibility and individuality. Part II sets up a philosophical framework for the work of Part III. In Chapter 4, I give an account of the rational reconstruction of concepts, inspired chiefly by Carnap and Haslanger. I also offer an account of the interpretation of physical theories. In Chapter 5, I turn to the specific problem of finding candidate concepts of particle. I present five desiderata that any putative explication ought to satisfy, in order that the proposed concept is a concept of particle at all. Part III surveys three rival proposals for the concept of particle in quantum mechanics. In Chapter 6, I define factorism and distinguish it from haecceitism. I then propose an amendment to recent work by Saunders, Muller and Seevinck, which seeks to show that factorist particles are all at least weakly discernible. I then present reasons for rejecting factorism. In Chapter 7, I investigate and build on recent heterodox proposals by Ghirardi, Marinatto and Weber about the most natural concept of entanglement, and by Zanardi about the idea of a natural decomposition of an assembly. In Chapter 8, I appraise the first of my two heterodox proposals for the concept of particle, varietism. I define varietism, and then compare its performance against the desiderata laid out in Chapter 5. I argue that, despite its many merits, varietism suffers a fatal ambiguity problem. In Chapter 9, I present the second heterodox proposal: emergentism. I argue that emergentism provides the best concept of particle, but that it is does so imperfectly; so there may be no concept of particle to be had in quantum mechanics. If emergentism is true, then particles are (higher-order) properties of the assembly, itself treated as the basic bearer of properties.This work was supported by the Arts and Humanities Research Council [grant number 2007/134560]