On the mathematical discovery of new physical phenomena
The objective of this lecture is to try to develop an answer to the following question: how is it possible to physicists to preview new physical phenomena only by looking to some specific terms that lie in the mathematical formalism of some physical theory? Our starting point will be the historical analysis of the prediction of a quantum phenomenon by David Bohm and Yakir Aharonov (Bohm e Aharonov, 1959), the so called Aharonov-Bohm effect. Then we’ll discuss some aspects of the mathematical and physical foundations of quantum theory in opposition to classical theory of electromagnetism. Finally, we are going to show our philosophical point of view about the subject.
As it’s well known, that effect was previewed when Bohm and Aharonov suggested a special kind of interpretation to a vector function called vector potential that entered Schödinger equation. To understand what they had in mind, we need to look very briefly to the way potentials enter in the formulation of classical
electrodynamics. Potentials (e.g., the vector potential) were introduced in the theory of classical electromagnetism only as a mathematical tool to compute the fields whilst the fields (solely) were the responsible for the physical effects. For instance, the electromagnetic field was intended to be obtained by calculating the rotational of the potential, i.e.,- but in quantum mechanics, based on Aharonov and Bohm views, the potential could have a brand new physical interpretation leading to a brand new and non-classical phenomena. That one could be tested empirically. Its existence was empirically tested and rigorously demonstrated by Tonomura (Tonomura, 1989). Our approach will be guided by some of da Silva’s ideas (da Silva, 2010) on the foundations of mathematics we’ve been studying since our PhD work (Grande 2011) and our current studies on the philosophical aspects of the Aharonov-Bohm effect.
BOHM, D. E AHARONOV, Y. Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485-491 (1959).
DA SILVA, J. J.“Structuralism and the applicability of mathematics”, Axiomathes 20 – 229-253 (2010).
GRANDE, R. M. A aplicabilidade da matemática à física. Tese de doutorado apresentada ao instituto de
filosofia e ciências humanas da Unicamp, Campinas (2011).
Restricted normal modal logics and levelled possible worlds semantics
Restricted normal modal logics are defined by imposing conditions on the modal axioms and rules of normal modal systems. The conditions are defined in terms of a depth (associated with the modal connective) and a complexity function. It is proven that the logics obtained are characterized by a subtle adaptation of the possible worlds semantics in which levels are associated with the worlds. Restricted normal modal logics constitute a general framework allowing the definition of a huge variety of modal systems, which can have different applications. For instance, they are useful to define epistemic logics where the logical omniscience problem is partially controlled.
Título: Várias semânticas para umas poucas lógicas
As "semânticas de traduções possíveis" podem dar um significado semântico a vários tipos de sistemas lógicos, embora originalmente projetadas para uma classe de lógicas paraconsistentes chamada "lógicas da inconsistência formal" (que cobre a hierarquia de N. da Costa e diversas outras). Usando tal semântica, várias lógicas complexas podem ser naturalmente decomponíveis (e também componíveis), por meio de combinações adequadas de sistemas lógicos simples. Casos particulares das semânticas de traduções possíveis são as "semânticas da sociedade", as "semânticas diádicas" e as "semânticas não-determinísticas". Pretendo discutir exemplos de semânticas de traduções possíveis e algumas relações entre estas e semânticas conhecidas para LP (Priest), K3 (Kleene) e FDE (Anderson & Belnap), com vistas a um entendimento mais profundo de uma abordagem semântica unificada.