Quantum Mechanics, Ontology, and Non-Reflexive Logics

This is a general philosophical paper where I overview some ideas concerning the non-reflexive foundations of quan- tum mechanics (NRFQM). By NRFQM I mean formalism and an interpretation of QM that considers an involved on- tology 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 en- tities devoid of identity conditions

Do `classical' space and time provide identity to quantum particles?

Non-relativistic quantum mechanics is grounded on ‘classical’ (Newtonian) space and time (NST). The mathematical description of these concepts entails that any two spatially separated objects are necessarily different, which implies that they are discernible (in classical logic, identity is defined by means of indiscernibility) — we say that the space is T2, or "Hausdorff". But quantum systems, in the most interesting cases, some- times need to be taken as indiscernible, so that there is no way to tell which system is which, and this holds even in the case of fermions. But in the NST setting, it seems that we can always give an identity to them, which seems to be contra the physical situation. In this paper we discuss this topic for a case study (that of two potentially infinite wells) and con- clude that, taking into account the quantum case, that is, when physics enter the discussion, even NST cannot be used to say that the systems do have identity. This case study seems to be relevant for a more detailed discussion on the interplay between physical theories (such as quantum theory) and their underlying mathematics (and logic), in a way never considered before

Potentiality and Contradiction in Quantum Mechanics

Following J.-Y.B\'eziau in his pioneer work on non-standard interpretations of the traditional square of opposition, we have applied the abstract structure of the square to study the relation of opposition between states in superposition in orthodox quantum mechanics in \cite{are14}. Our conclusion was that such states are \ita{contraries} (\ita{i.e.} both can be false, but both cannot be true), contradicting previous analyzes that have led to different results, such as those claiming that those states represent \ita{contradictory} properties (\ita{i. e.} they must have opposite truth values). In this chapter we bring the issue once again into the center of the stage, but now discussing the metaphysical presuppositions which underlie each kind of analysis and which lead to each kind of result, discussing in particular the idea that superpositions represent potential contradictions. We shall argue that the analysis according to which states in superposition are contrary rather than contradictory is still more plausible

Axiomatization and Models of Scientific Theories

In this paper we discuss two approaches to the axiomatization of scien- tific theories in the context of the so called semantic approach, according to which (roughly) a theory can be seen as a class of models. The two approaches are associated respectively to Suppes’ and to da Costa and Chuaqui’s works. We argue that theories can be developed both in a way more akin to the usual mathematical practice (Suppes), in an informal set theoretical environment, writing the set theoretical predicate in the language of set theory itself or, more rigorously (da Costa and Chuaqui), by employing formal languages that help us in writing the postulates to define a class of structures. Both approaches are called internal, for we work within a mathematical framework, here taken to be first-order ZFC. We contrast these approaches with an external one, here discussed briefly. We argue that each one has its strong and weak points, whose discussion is relevant for the philosophical foundations of science

Aspectos da indiscernibilidade quântica e da teoria de quase-conjuntos

A indiscernibilidade das entidades quânticas é considerada como uma das noções centrais das teorias quânticas. Neste artigo expositivo, são apontadas algumas das razões para essa alegada importância, bem como as ideias básicas de uma teoria matemática que permite tratar de coleções dessas entidades. O artigo inicia com uma posição geral do autor sobre a filosofia da ciência

Memorial de Atividades Acadêmicas

Memorial da trajetória acadêmica.Memorial de Atividades Acadêmicas apresentado para progressão para a Classe de Professor Titular do Departamento de Filosofia da UFSC em 2015

On Otavio Bueno on identity and quantification (v.2)

Otavio Bueno has pointed out an interesting and relevant topic about quantification. He stresses that the meaning of quantifiers can have a sense if and only if the entities being quantified have well defined identity conditions. In this paper we discuss his view and show that this is not the case in all situations, being quite reasonable to quantify also over entities to which identity conditions fail to hold

Estruturas em Ciência

Este trabalho destaca o papel das estruturas matemáticas no estudo das teorias científicas, em especial das teorias da física. Identificamos a contraparte matemática de uma teoria científica com uma determinada estrutura abstrata, que descreve abstratamente os modelos da teoria, ou seja, as estruturas “concretas” às quais a teoria se aplica. Estas fornecem a contraparte empírica das teorias, e a noção de verdade assumida é a de quase-verdade. A tese é a de que a atividade científica, sendo uma atividade conceitual, se vale da noção de estrutura para organizar os conceitos que o cientista assume como primitivos (ainda que incons- cientemente), e que elas servem de instrumento epistemológico para se alcançar o conhecimento em certos domínios. Destaca-se então uma nova forma de realismo estrutural, que denominamos de realismo es- trutural gnosiológico. A base matemática na qual essas estruturas são erigidas é levada em consideração, o que não é comum nas discussões filosóficas usuais.Fundação para a Ciência e a Tecnologia, Faculdade de Letras da Universidade de Lisboa,LanCog Grou

Quantifying over indiscernibles

One of the main criticisms of the theory of collections of in- discernible objects is that once we quantify over one of them, we are quantifying over all of them since they cannot be dis- cerned from one another. In this way, we would call the collapse of quantifiers: ‘There exists one x such as P ’ would entail ‘All x are P’. In this paper we argue that there are situations (quantum theory is the sample case) where we do refer to a certain quantum entity, saying that it has a certain property, even without committing all other indistinguishable entities with the considered property. Mathematically, within the realm of the the- ory of quasi-sets Q, we can give sense to this claim. We show that the above-mentioned ‘collapse of quantifiers’ depends on the interpretation of the quantifiers and on the mathematical background where they are ranging. In this way, we hope to strengthen the idea that quantification over indiscernibles, in particular in the quantum domain, does not conform with quantification in the standard sense of classical logic