This is a general philosophical paper where I overview some ideas concerning the non-reflexive foundations of quantum mechanics (NRFQM). By NRFQM I mean formalism and an interpretation of QM that considers an involved ontology of non-individuals as explained in the text. Thus, I do not endorse a purely instrumentalist view of QM, but believe that it speaks of something, and then I try to show that one of the plausible views of this `something' is as entities devoid of identity conditions. Warning note: This is a revised version of a paper with the same name that was written by invitation to be published in a book titled \textit{The Mammoth Book on Quantum Mechanics Interpretations}, edited by Open Academic Press, Berlin, and having as editor a certain Ulf Edvinsson, who has invited me. The book was announced in the page of OAP and should appear by 2016. This never happened. Later I discovered that OAP is in a list of predatory editorial houses and that "Ulf Edvinsson" is (apparently) a fake name. Furthermore, I couldn't contact anyone responding by OAP to retire my name from the announcement of the book and for impeding them to publish the paper. I strongly apologize for such a fault, which is completely mine. Since the subject presented here has been among my preoccupations ever since I met Franscicso Antonio Doria for the first time (in 1987), it is a pleasure to dedicate the stuff to him. And of course I thank the editors for accepting this version of the paper for this book.
This is a general philosophical paper where I overview some ideas concerning the non-reflexive foundations of quantum mechanics (NRFQM). By NRFQM I mean formalism and an interpretation of QM that considers an involved ontology of non-individuals as explained in the text. Thus, I do not endorse a purely instrumentalist view of QM, but believe that it speaks of something, and then I try to show that one of the plausible views of this `something' is as entities devoid of identity conditions. Warning note: This is a revised version of a paper with the same name that was written by invitation to be published in a book titled \textit{The Mammoth Book on Quantum Mechanics Interpretations}, edited by Open Academic Press, Berlin, and having as editor a certain Ulf Edvinsson, who has invited me. The book was announced in the page of OAP and should appear by 2016. This never happened. Later I discovered that OAP is in a list of predatory editorial houses and that "Ulf Edvinsson" is (apparently) a fake name. Furthermore, I couldn't contact anyone responding by OAP to retire my name from the announcement of the book and for impeding them to publish the paper. I strongly apologize for such a fault, which is completely mine. Since the subject presented here has been among my preoccupations ever since I met Franscicso Antonio Doria for the first time (in 1987), it is a pleasure to dedicate the stuff to him. And of course I thank the editors for accepting this version of the paper for this book.
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.
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.
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.
Several experiments have suggested that the complete indistinguishability of quantum systems is to be considered a real phenomenon. Experiments such as the HOM effect, the existence of BECs, the path identity cases, and many others are physical realities of today. Physicists `explain’ these phenomena using different models, such as Bohmian mechanics, relational quantum mechanics or the QBism approach. Most of these models are constructed within a standard mathematical framework, meaning standard mathematics, where all things can, even if only in principle, be distinguished from one another, so complete indistinguishability can only be mimicked. These approaches may work for physical purposes but are debatable regarding logical and metaphysical questions. If we accept that such questions are at least part of the philosophical aspect of quantum physics, then we are authorised to analyse them and try to offer an alternative. Looking for an alternative, we consider Alain Aspect’s and Anton Zeilinger’s views about quantum realism, opting for Aspect’s acceptance of some kind of reality behind the wave-function, while Zeilinger suggests that everything comes from information. In our opinion, the case is the opposite: it is information that comes from the existence of things. The aim of this paper is to justify why we think that the underlying reality (in Aspect’s sense) can be seen as a quantum ontology of non-individuals.
When investigating various interpretations of quantum mechanics one notices that each interpretation contains an element which escapes a complete and full description. This element is always associated with the stochasticity of the individual event in the quantum measurement process. It appears that the implications of this limit to any description of the world has not been sufficiently appreciated with notable exceptions of, for example, Heisenberg, Pauli and Wheeler. If we assume that a deeper foundation of quantum mechanics is possible, the question arises which features such a philosophical foundation might have. It is suggested that the objective randomness of the individual quantum event is a necessity of a description of the world in view of the significant influence the observer in quantum mechanics has. It is also suggested that the austerity of the Copenhagen interpretation should serve as a guiding principle in a search for deeper understanding. In recent years, a significant increase of interest in the foundations of quantum mechanics can be observed. This increase is certainly related to the immense progress made during the last two decades in experiments with individual quantum systems. This technological progress has made it possible to really perform more and more of the famous gedanken experiments which were so important in the early days of the theory. Not only did these experiments confirm the theory in every detail, they also opened up avenues for new experimental directions which might even lead some day to new technology. While such applications are certainly still far away, areas like quantum cryptography[1], quantum computation[2] and interaction-free measurement[3], just to name a few examples, certainly deserve the attention they receive. All this activity has also drawn more attention to the problems of understanding and interpreting quantum mechanics. It seems that there, a definitive consensus has not yet been reached. As witnesses quote authorities like Feynman as saying[4]: I think can safely say that nobody today understands quantum mechanics, whereby he apparently included himself, or like Roger Penrose[5], who describes his opinion remarking that while the theory agrees incredibly well with experiment and while it is of profound mathematical beauty, it makes absolutely no sense. Why did even physicists who have contributed so significantly to quantum theory Feynman was even rewarded the Nobel Prize for one of its mathematical formulations choose such emphatic and strong formulations? Why is it that the average physicist hears very little about such problems during his/her education? Why is the comprehension of the theory very often focused on formalism while questions that probe for deeper meaning are usually not tackled? To take a closer look at the situation, let us turn to a brief analysis of the interpretation of quantum mechanics. We note that there are at least two levels of interpreting a formalism, a physical theory. The first, basic, level supplies the rules that determine which element of the formalism corresponds to which measurable quantity or to which observable fact in a concrete experimental situation. Theses rules are a large, mostly not explicated but only implicit, set of instructions. These instructions concern the manner of working with the statements of the theory in order to obtain predictions for future experiments or events, and they concern the instructions on how to proceed in experiment in order to demonstrate or test a theoretical