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Reyes G.E. A mathematical analysis of Masaccio’s Trinity. Preliminary version (February 2004). Published in Categories and Types in Logic, Language, and Physics. Editors C. Casadio et al. Springer LNCS 8222. 2014.
The aim of this note is to study several questions of a mathematical nature suggested by this fresco: (1) How accurate is the use of perspective? (2) What are the dimensions of the chapel? (3) What are the dimensions of the coffers of the vaulted ceiling of the chapel? (4) Where is the point of view situated with respect to the fresco? (5) Where are the different characters situated inside the chapel? (6) What are the “real” heights of the characters portrayed? Questions (1)-(4) admit answers that may be computed starting from the data of the fresco, by using some rules of perspective and simple mathematical facts. This is not true for the others. Nevertheless, we will show that under some reasonable hypotheses estimates may be made. A pictorial reproduction of the Trinity may be unloaded by clicking the next document in Varia: “Masaccio Trinity in the WEB”.
Cette pièce se passe dans un studio où a lieu un vernissage. Les personnages papillonnent de l’un à l’autre comme dans un party. Par la suite, ils se retrouvent dans un wagon de métro et s’enfoncent dans une réalité plus étrange que la fiction
Une mise à jour du procès de Socrate montrant sa pertinence pour la démocratie d’aujourd’hui. LA VRAIE NATURE DE SOCRATE OU LE PROCÈS INACHEVÉ (2003) Creative Commons Attribution
Galli, A., Reyes G.E. and M. Sagastume (2003). Strong amalgamation, Beck-Chevalley for equivalence relations and interpolation in Algebraic Logic. Fuzzy sets and systems 138, 3-23.
We extend Makkai’s proof of strong amalgamation (push-outs of monos along arbitrary maps are monos) from the category of Heyting algebras to a class which includes the categories of symmetric bounded distributive lattices, symmetric Heyting algebras, Heyting modal S4-algebras, Heyting modal bi-S4-algebras, and Lukasiewicz n-valued algebras. We also extend and improve Pitt’s proof that strong amalgamation implies Beck-Chevalley for filters of Heyting algebras to exact categories with certain push-outs. As a consequence, a form of the Interpolation Lemma for some non-classical calculi is proved.
Kock A. and G.E. Reyes
Theory and Applications of Categories, Vol. 11, No. 14, 2003, pp. 321–336.
ABSTRACT. We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation.
Cette pièce a pour sujet les dernières heures de la vie du Général Jean-Victor Poncelet, né à Metz en 1788 et mort à Paris en 1867. Sur son lit de mort, le général se remémore l’une des périodes les plus heureuses de sa vie, lorsque après une longue marche dans les steppes glacées de la Russie, à l’hiver 1812, il arriva enfin dans les prisons de Saratoff, où reprenant goût à la vie, il découvrit les fondements de la géométrie projective. La pièce jette un regard sur les horreurs de la campagne de Russie par Napoléon en écoutant les réflexions de quelques personnages en butte à la faim, au froid, à la mort.
LE JEUNE HOMME ET LA MORT (LES DERNIÈRES HEURES DU GÉNÉRAL PONCELET) (2002)
Cette pièce a pour sujet les trois dernières années de la courte vie mouvementée d’Évariste Galois (1811-1832), génial mathématicien français et républicain exalté, contemporain d’Alexandre Dumas, de François Vincent Raspail, de Stendhal et de Victor Hugo. Du suicide de son père, juillet 1829, à sa propre mort en duel, fin mai 1832, alors qu’il n’avait pas vingt et un ans, Évariste Galois a vécu la frustration de l’action politique manquée, l’exaspération devant l’incompréhension des autres face à ses découvertes mathématiques et les tiraillements d’amours malheureux.
Galli, A., Reyes G.E. and M. Sagastume (2000). Completeness theorems via the double dual functor. Studia Logica 64, pp 61-81.
The aim of this paper is to apply properties of the double dual endofunctor on the category of bounded distributive lattices and some extensions thereof to obtain completeness of certain non-classical propositional logics in a unified way. In particulart, we obtain completeness theorems for Moisil calculus, n-valued Lukasiewicz calculus and Nelson calculus. Furthermore we show some conservativeness results by these methods.
Anders Kock et Gonzalo Reyes
Cahiers de topologie et géométrie différentielle catégoriques, tome 40, no 2 (1999), p. 127-140.
Dans le contexte de la théorie constructive des locales ou des cadres (c’est-à-dire de la théorie de locales sur un locale de base), nous étudions quelques aspects des “distributions sur les cadres” , i.e., des applications sur un cadre à valeurs dans un cadre de base préservant les suprema arbitraires. Nous obtenons une relation entre certains résultats dus à Jibladze- Johnstone et d’autres dus à Bunge-Funk. De plus, nous donnons des descriptions de l’opérateur “intérieur de fermeture” défini sur les parties ouvertes d’un locale en termes des distributions sur les cadres ainsi qu’en termes des opérations de double négation généralisée.
La Palme Reyes M., Macnamara J., Reyes G. E. and H. Zolfaghari (1999). Models for Non-Boolean Negations in Natural Languages based on Aspect Analysis. In Gabbay D. and H. Wansing (eds.). What is negation? Kluwer Academic Publishers. Dordrecht, Boston, London. pp 241-260.
Since antiquity two different negations in natural languages have been noted: predicate negation (`not honest’) and predicate term negation (`dishonest’). Aristotle tried to formalize them in his system of oppositions, distinguishing between affirmation and negation (`honest’ and `not honest’) and contraries (`honest’ and `dishonest’). The Stoics replaced Aristotle’s logic of terms by their logic of propositions. Although they considered three types of negation, none of them corresponded to Aristotle’s predicate term negation. Frege and modern logic have followed the Stoics in either identifying predicate term negation with predicate negation or in casting predicate term negation out of logic into the realm of pragmatics. Although an extensive literature has arisen on these issues, we have not found mathematical models. We propose category-theoretic models with two distinct negation operations, neither of them in general Boolean. We study combinations of the two (`not dishonest’) and sentential counterparts of each. We touch briefly on quantifiers and modalities. The models are based on an analysis of aspects. For instance, to give an overall, global judgement of John’s honesty we must agree on what aspects of John are relevant for that judgement: John qua person (global aspect), John qua social being (social aspect), John qua family man, John qua professional man, etc. We conceptualize this `Aristotelian’ analysis by means of a category of `aspects’. A model (for the negations) is obtained from the category of presheaves on this category. Although neither of the negations is Boolean, predicate negation turns out to be Boolean at the `global’ aspect (the aspect of the overall judgement) which may help to explain the persistent belief that logic is naturally Boolean.
Models for Non-Boolean Negations in Natural Languages based on Aspect Analysis
Gonzalo E. Reyes - Marie La Palme Reyes 