While classical accounts of identity (‘classical’ here pertaining to both logics and physics) are generally well understood, the advent of quantum theory, specifically quantum statistics, has cast shadow over these conceptions. Dealing with the consequently surfacing problems is a philosophically rich and interesting enterprise. I begin this thesis by providing an exegesis of the roles played by, and features of, identity in logics, classical physics, and quantum physics. Therein I consider how under a quantal description of reality, classical notions of identity and individuality break down. In the second chapter, I address how this problem has launched an arc of thought in analytic metaphysics and formal philosophy motivating the development of non-standard formal frameworks with which philosophical sense can be made of quantal objects. Among these, I explore and critically evaluate quaset theory, quasi-set theory, and non-reflexive Schr¨odinger logics, identifying some significant problems with quaset theory that arise in defining cardinality and later, pointing out a problem with Schr¨odinger logics in their modelling of the continuity between quantal and classical treatments of the world. The queer character of identity in the quantal regime motivates a turn to vagueness which I introduce in the third chapter, providing a brief outline of vagueness and the sorites paradox. Further, I reflect on the fundamental nature of vagueness, outlining and evaluating the semantic and ontic conceptions thereof. In the final chapter, I proceed to explicate and assess notions that identity and quantal objects can be vague. I shall discuss accounts according to which the vagueness of identity and quantum objects is posited as a feature of nature emerging in quantum systems — the ontic vagueness of identity — finding that these ideas are flawed and/or rely on misinterpretations of vagueness. Finally, I present an argument which suggests how the vagueness of identity can arise as an artifact of the differing treatments of identity in the quantal and classical regimes in which the vagueness involved can be semantic rather than ontic.
While classical accounts of identity (‘classical’ here pertaining to both logics and physics) are generally well understood, the advent of quantum theory, specifically quantum statistics, has cast shadow over these conceptions. Dealing with the consequently surfacing problems is a philosophically rich and interesting enterprise. I begin this thesis by providing an exegesis of the roles played by, and features of, identity in logics, classical physics, and quantum physics. Therein I consider how under a quantal description of reality, classical notions of identity and individuality break down. In the second chapter, I address how this problem has launched an arc of thought in analytic metaphysics and formal philosophy motivating the development of non-standard formal frameworks with which philosophical sense can be made of quantal objects. Among these, I explore and critically evaluate quaset theory, quasi-set theory, and non-reflexive Schr¨odinger logics, identifying some significant problems with quaset theory that arise in defining cardinality and later, pointing out a problem with Schr¨odinger logics in their modelling of the continuity between quantal and classical treatments of the world. The queer character of identity in the quantal regime motivates a turn to vagueness which I introduce in the third chapter, providing a brief outline of vagueness and the sorites paradox. Further, I reflect on the fundamental nature of vagueness, outlining and evaluating the semantic and ontic conceptions thereof. In the final chapter, I proceed to explicate and assess notions that identity and quantal objects can be vague. I shall discuss accounts according to which the vagueness of identity and quantum objects is posited as a feature of nature emerging in quantum systems — the ontic vagueness of identity — finding that these ideas are flawed and/or rely on misinterpretations of vagueness. Finally, I present an argument which suggests how the vagueness of identity can arise as an artifact of the differing treatments of identity in the quantal and classical regimes in which the vagueness involved can be semantic rather than ontic.
In the non-reflexive formulation of quantum physics, attention is given to a metaphysical view of quantum objects (q-objects) as non-individuals, characterised as entities to which the standard notion of identity lacks sense. The reference to the nature of q-objects is usually omitted in other interpretations and the most we can find is some vague reference to “to each [quantum] system we associate a Hilbert space”, and nothing more is said about them; we need to infer from the theory their characteristics, and it gives us several alternatives. One of them, to be justified here in broad aspects, is the non-individual view. It is important to enlighten that we are not trying to just find a way to deal with quantum entities but to defend a possible ontology of non-individuals, which can be applied also to ‘particles’ arising from quantum fields. Our account is based on two assumptions: (i) the ‘classical theory of identity’ does not apply to q-objects, and (ii) their non-individuality must be attributed from the start, that is, as a primitive notion and not ‘made a posteriori’ say by assuming symmetry principles. The underlying logic is called non-reflexive because the reflexive law of identity ∀x(x=x) is supposed not to hold for q-objects. Since all the details demand a lot of space, here we only sketch and justify our view. The references provide a more detailed account.
In this article we try to justify a way of looking for an alternative approach to quantum mechanics based on a non-classical logic. The main motivation is both to accommodate a metaphysical view according to which quantum objects are non-individuals and to introduce, so to say, these objects within the formalism, for as we know, the usual formalism treats of states of quantum systems and not (directly) of quantum systems properly speaking. We consider two specific questions related to quantum theory, namely, entanglement and the indiscernibility of quanta. Our way of looking to these concepts motivate us to construct a metaphysical view according to which quantum objects do not present identity criterion (they are ‘non-individuals’), and then we try to construct a quantum theory based on such a view, which is grounded on a non-classical logic, termed non-reflexive. From the formal point of view, the main motivation comes from the fact that in using classical logic and classical set theory (such as ZFC), we are necessarily committed to individuals, and non-individuals can be considered only by assuming ‘strong’ symmetry principles, which mask the basic idea involving non-individuality. In the core of the article, due to questions of space, only the main philosophical arguments are presented, but in two appendixes we outline, firstly, quasi-set theory, the non-reflexive mathematical framework that sustain our view, and a bottom-top construction of a quantum mechanics based on quasi-set theory.
The notion of an individual and the related issues on individuation are topics that appear in the philosophical discussion ever since the antiquity. The idea of an individual thing is intuitively clear: an individual is something of a specific kind that is a unity, having its own identity, and being so that it is possible at least in principle to discern it from any other individual, even of similar species. But when we try to leave the intuitive realm and push this idea to a logical analysis, we find a cluster of problems that are difficult to overcome within standard logico-mathematical contexts. In this work, we shall be concerned with some aspects of this intuitive concept of an individual and on some related facts about individuation taken from recent discussions that arose ever since the inception of quantum theory, pushing the discussion to a “logical” view, which in our opinion is still lacking in the usual debates on such issues. In the final part of the paper, we propose a metaphysics where the notion of identity is substituted, for some objects, by a weaker notion of indiscernibility, and we try to justify such a move. In most of the uses of the expression “quantum theory”, we shall not make explicit the distinction between the non-relativistic and the relativistic approaches, although they of course are quite different, for we think that the problems as we shall present them appear in both versions. But, as the text goes, the context will distinguish between them and these questions will become clear to the reader.
Abstract Drawing on philosophical accounts of identity and individuality, as well as the histories of both classical and quantum physics, this book explores two alternative metaphysical approaches to quantum particles. It asks if quantum particles can be regarded as individuals, just like books, tables, and people. Taking the first approach, the book argues that if quantum particles are regarded as individuals, then Leibniz’s famous Principle of the Identity of Indiscernibles is in fact violated. Recent discussions of this conclusion are analysed in detail and the costs involved in saving the Principle are carefully considered. For the second approach, the book considers recent work in non-standard logic and set theory to indicate how we can make sense of the idea that objects can be non-individuals. The concluding chapter suggests how these results might then be extended to quantum field theory.
The topics of identity, individuality and indiscernibility had already entered the domain of the philosophy of physics, mainly in what respects the philosophy of quantum theories. In this paper, I present the main ideas of an underlying metaphysics that considers the basic entities of quantum physics as non-individuals, that is, as entities to which the standard theory of identity of classical logic and standard mathematics (set theory) does not apply. Two things must be emphasized: the first is that we pursue Erwin Schrödinger’s ideas that the notion of identity (or sameness ) does not apply to quantum things; the second is to follow Heinz Post’s claim that the non-individuality of quantum entities ‘must be introduced right at the start’ and not by taking individuals and making them pretend to be non-individuals ‘by hand’, say by confining them to non-rigid structures. Thus, this paper should be regarded as an attempt to justify the metaphysics of non-individuals and the logical systems that cope with them, although these systems are here only referred to. We also answer some criticisms that have been directed to such a view. The paper is directed to an audience of both physicists interested in foundations and philosophers of physics. This article is part of the theme issue ‘Identity, individuality and indistinguishability in physics and mathematics’.
Abstract Traditionally by ‘quantum logic’, one understands the study of the lattice of the subspace of Hilbert space. Here, ‘quantum logic’ is taken to be the underlying logic of quantum physics. Following Schrödinger’s idea that the standard concept of identity cannot be applied to quantum objects, a kind of ‘non-reflexive’ logic called Schrödinger logic is presented, in which the reflexive law of identity is taken not to apply. First order Schrödinger logics are presented, and extended to higher order logics and also to an intensional logic. Classical semantics is presented, and a weak completeness theorem is sketched. The last part of the chapter covers a logic of sortal predicates for which a quasi-set semantics is delineated. This completes the formal framework capable of accommodating quantum non-individuality and thus the initial understanding of Born, Heisenberg, and Schrödinger, as well as the second horn of the above metaphysical underdetermination, can be formally represented.