Abstract This chapter presents the theory of quasi-sets, and argues that this offers an appropriate set-theoretic formalism for non-individual quantum objects. The basic idea is that in a quasi-set, there may exist elements for which the traditional concept of identity does not apply — these elements are called the ‘m-atoms’. Thus, a quasi-set may have a cardinal but not an associated ordinal. The other elements can be regarded as standard elements of a set. Hence, the theory encompasses a copy of Zermelo-Frankel set theory with Urelement. A number of alternatives are also presented with a view to possible physical applications, such as allowing the cardinality of a quasi-set to vary in time and thereby accommodating aspects of relativistic quantum theory. The formal framework is also applied to the development of quantum statistics, which can be naturally captured by the theory.
Abstract This chapter presents the theory of quasi-sets, and argues that this offers an appropriate set-theoretic formalism for non-individual quantum objects. The basic idea is that in a quasi-set, there may exist elements for which the traditional concept of identity does not apply — these elements are called the ‘m-atoms’. Thus, a quasi-set may have a cardinal but not an associated ordinal. The other elements can be regarded as standard elements of a set. Hence, the theory encompasses a copy of Zermelo-Frankel set theory with Urelement. A number of alternatives are also presented with a view to possible physical applications, such as allowing the cardinality of a quasi-set to vary in time and thereby accommodating aspects of relativistic quantum theory. The formal framework is also applied to the development of quantum statistics, which can be naturally captured by the theory.
In this article, we discuss some questions about the nature of elementary particles treated by quantum mechanics, in particular related to the concepts of identity and individuality of these particles. We started briefly exposing the philosophical and formal concepts of the identity and individuality, and then show how these notions are problematic when applied to elementary particles such as electrons, protons and neutrons. In particular, we emphasize that both philosophy, logic and set theory (and thus the mathematics) admit the usual assumption that things have a ‘type’ of identity and individuality (i.e., are individuals), in the sense that objects which have all the same properties are the same object (are equal). Nevertheless, we show that in the quantum universe is possible to find objects that share all their properties, but are not just one, constituting in something like “nonindividuals” (thesis defended by many physicists and philosophers of science). In sequence, we show how classical mathematics — which, as we said, assumes an individuality to your entities — handle this situation, in particular admitting assumptions external to the physical theories. To avoid this procedure, and look for a formalism more ‘natural’ and appropriate to work with these quantum characteristics, we suggest the use of a non-classical set theory called the quasi-set theory, based on a non-reflective logics, which admits objects devoid of identity and individuality ab initio. Finally, we show another application of this theory: the search to a quasi-set mathematical structure that describes the ‘behaviour’ of non-relativistic quantum theory, and discuss some advantages of this ‘quasi-structure’ to a classical one.
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.
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.
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.
Quantum theory is the land of indiscernible things, of things that in some situations cannot be put apart. This contrasts in much with standard mathematics and classical logic, which were elaborated to deal with \textit{individuals}, discernible things. Many thinkers, among them the mathematician Yuri Manin, have proposed that a more general theory of `sets' than standard set theories (where sets are collections of discernible elements) should be elaborated to cope with quantum physics. Quasi-set theory was proposed with such an aim. In this paper, we consider some aspects of this theory and discuss the way we can use such a theory as a framework for constructing deformable (not rigid) structures where indiscernible things can be considered but with one additional and fundamental detail: these structures cannot be extended to rigid structures as it happens in the standard frameworks. Really, the way of dealing with indiscernible (or indistinguishable) elements within a `standard' framework such as a `classical' set theory (like the ZFC system) requires the confinement of the discussion to deformable (non-rigid) structures. But in ZFC every structure can be extended to a rigid structure (encompassing only the trivial automorphism -- the identity function), so that in the extended structure (or in the whole universe) one can realize that the supposed indiscernible objects are not indiscernible at all. Quasi-set theory is such that the universe of quasi-sets is deformable and in such a theory one can construct deformable structures which cannot be extended to rigid ones. The aim of this paper is to put these things in a clear way.