Newton da Costa together with the author of this paper argued in favor of the possibility to consider quantum superpositions in terms of a paraconsistent approach. We claimed that, even though most interpretations of quantum mechanics (QM) attempt to escape contradictions, there are many hints that indicate it could be worth while to engage in a research of this kind. Recently, Arenhart and Krause [1, 2, 3] have raised several arguments against this approach and claimed that —taking into account the square of opposition— quantum superpositions are better understood in terms of contrariety propositions rather than contradictory propositions. In [17] we defended the Paraconsistent Approach to Quantum Superpositions (PAQS) and provided arguments in favor of its development. In the present paper we attempt to analyze the meanings of modality, potentiality and contradiction in QM, and provide further arguments of why the PAQS is better suited, than the Contrariety Approach to Quantum Superpositions (CAQS) proposed by Arenhart and Krause, to face the interpretational questions that quantum technology is forcing us to consider.

La Unesco en su informe sobre la Ciencia hacia el 2030, publicado en 2015, ya había sugerido a los países en vías de desarrollo que tuvieran, como mínimo, un investigador por cada mil habitante; sin embargo, el Observatorio de Ciencia y Tecnología y el Programa de Promoción al Investigador, en funciones hasta el 2009, ubicó el número investigadores en el país en 6.822, una cifra muy lejana de los 30 mil científicos que se requieren. En relación con la producción científica, Scopus, una base de datos de revistas arbitradas e indexadas, muestra que Venezuela pasó de 2.376 artículos científicos en 2009 a 1.476 en 2016. Mientras que SciELO, una biblioteca virtual, indica que las publicaciones pasaron de 2.038, en 2008, a 668,en 2016 (67,23% menos).

In this paper, we show that in two dimensions quantum mechanics can be mapped onto classical mechanics, by transforming the Schr”odinger equation into system of n linear equations known as Kirchhoff equations. These Kirchhoff equations equations satisfy the a poison bracket relationship in phase space which is identical to the Heisenberg uncertainty relationship. Therefore, we conclude that quantum mechanics is consistent with classical mechanics atleast in two dimensions. This allows us to address the wave particle duality in terms of relative phase. As an illustration we show that the equation for optical vortices can be derived as Kirchhoff equation admit a paraxial wave equation in presence of real constant background.

What is the largest number accessible to the human imagination? The question is neither entirely mathematical nor entirely philosophical. Mathematical formulations of the problem fall into two classes: those that fail to fully capture the spirit of the problem, and those that turn it back into a philosophical problem.

Long before current graphic, visualisation and geometric tools were available, John E. Littlewood (1885-1977) wrote in his delightful Miscellany: A heavy warning used to be given [by lecturers] that pictures are not rigorous; this has never had its bluff called and has permanently frightened its victims into playing for safety. Some pictures, of course, are not rigorous, but I should say most are (and I use them whenever possible myself). [34, p. 53] Over the past five to ten years, the role of visual computing in my own research has expanded dramatically. In part this was made possible by the increasing speed and storage capabilities—and the growing ease of programming—of modern multi-core computing environments. But, at least as much, it has been driven by my group’s paying more active attention to the possibilities for graphing, animating or simulating most mathematical research activities.

En este artículo se examina la manera en que Hilbert elabora su primer formalismo al investigar los fundamentos de la geometría. El interés se centra en la forma en que elabora una nueva concepción de las teorías matemáticas. Se contrasta la postura de Hilbert con el constructivismo de Kant, el cual perduró en la filosofía de las matemáticas durante mucho tiempo. Para ello, en la primera parte se examina la manera en que Kant explica la demostración geométrica y se muestra el vínculo entre su explicación y la teoría de esquemas que él mismo sostiene. También se expone la concepción subyacente a los Grundlagen der Geometrie de Hilbert, y se busca reconstruir el camino que siguió hasta alcanzar esa concepción. En particular se examina el lugar que ocupan la geometría proyectiva y el principio de dualidad en sus reflexiones. Por último, se apunta a la idea de que el primer formalismo de Hilbert constituye una generalización necesaria de la filosofía matemática de Kant.

In two articles ([Brisson-Ofman1, 2]), we have analyzed the so-called ‘mathematical passage’ of Plato’s Theaetetus, the first dialogue of a trilogy including the Sophist and the Statesman. In the present article, we study an important point in more detail, the ‘definition’ of ‘powers’ (‘δυνα ´ μϵις ‘). While in [Brisson-Ofman2], it was shown that the different steps to get the definition are mathematically and philosophically incorrect, it is explained why the definition itself is problematic. However, it is the first example, at least in the trilogy, of a definition by division. This point is generally ignored by modern commentators though, as we will try to show, it gives rise, in a mathematical context, to at least three fundamental questions: the meaning(s) of ‘logos’, the connection between ‘elements and compound’ and, of course the question of the ‘power(s)’. One of the main consequences of our works on Theaetetus’ ‘mathematical passage’, including the present one, is to challenge the so-called ‘main standard interpretation’. In particular, following [Ofman2014], we question the claim that Plato praises and glorifies both the mathematician Theodorus and the young Theaetetus. According to our analysis, such a claim, considered as self-evident, entails many errors.

We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or “structural” set theories in the spirit of Lawvere’s “Elementary Theory of the Category of Sets”. We consider weak intuitionistic and predicative theories of pretoposes, and we also propose category-theoretic versions of stronger axioms such as unbounded separation, replacement, and collection. Finally, we compare all of these theories formally to traditional membership-based or “material” set theories, using a version of the classical construction based on internal well-founded relations.

This article argues for the inclusion of the Neapolitan Raimondo di Sangro, il Principe di San Severo (1710-1771) among those thinkers whose ideas, lifestyle, writings, networks and intellectual pursuits have been defined as radical. It explores the ways in which a little known Italian Radical Enlightenment thinker formed his ideas both through contact with the writings of proponents of Radical Enlightenment thought not only in England and Holland, but also in Italy and Switzerland where radical networks have been less visible to scholars. By charting the strategies for the spread and exchange of radical thought from Naples to Lausanne through heretofore unknown paths, new avenues for research are opened while the breadth and depth of the Radical Enlightenment are strengthened.

In [6], Iemhoff introduced the notion of a focused axiom and a focused rule as the building blocks for a certain form of sequent calculus which she calls a focused proof system. She then showed how the existence of a terminating focused system implies the uniform interpolation property for the logic that the calculus captures. In this paper we first generalize her focused rules to semi-analytic rules, a dramatically powerful generalization, and then we will show how the semi-analytic calculi consisting of these rules together with our generalization of her focused axioms, lead to the feasible Craig interpolation property. Using this relationship, we first present a uniform method to prove interpolation for different logics from sub-structural logics FLe, FLec, FLew and IPC to their appropriate classical and modal extensions, including the intuitionistic and classical linear logics.