Une première période où les influences mêlées d’Alessandro Padoa et de Bertrand Russell s’exercent en France culmine avec les essais philosophiques de Jean Nicod. Une seconde période voit fleurir les travaux du mathématicien Jacques Herbrand ; avant de périr, il laisse son nom à un théorème fondamental. Suit une période de débats entre philosophes, mathématiciens et physiciens, stimulés en 1935 et 1937 par la tenue à Paris de deux congrès consacrés, totalement ou en partie, à la philosophie des sciences. Paulette (...) Février y esquisse une logique non classique où l’on postule l’existence de couples de propositions non composables pour ériger en principes les relations de Werner Heisenberg. Jean-Louis Destouches développe cette conception jusqu’à décrire comment édifier une théorie unifiante. La structuration des êtres mathématiques est l’objet d’études philosophiques d’Albert Lautman. Le rôle putatif de la notion de groupe en logique est interrogé. La notion de structure mathématique est l’objet de deux contributions : Marc Krasner généralise les conceptions d’Évariste Galois, attribuées à la logique et étendues à des langages infinitaires ; Nicolas Bourbaki, compte tenu de l’évolution des mathématiques, qualifie de structure ce que nous appelons aujourd’hui un modèle. (shrink)

In his posthumous book from 1914, "New foundations of logic, arithmetic and set theory", Julius Konig develops his philosophy of mathematics. In a previous contribution, we attracted attention on the positive part (his truth and falsehood predicates being excluded) of his "pure logic": his "isology" being assimilated to mutual implication, it constitutes a genuine formalization of positive intuitionistic logic. Konig's intention was to rebuild logic in such a way that the excluded third's principle could no longer be logical. However, his (...) treatment of truth and falsehood (boiling down to negation) is purely classical. We explain here this discrepancy by the choice of the alleged more primitive notions to which the questioned notions of truth and falsehood have been reduced. Finaly, it turns out that the disjunctive and conjunctive forms of the principles of the excluded third and of contradiction have effectively been excluded, but none of their implicative forms. (shrink)

In his posthumous book from 1914, “New foundations of logic, arithmetic andset theory”, Julius König develops his philosophy of mathematics. In a previous contribution, we attracted attention on the positive part of his “pure logic”: his “isology” being assimilated to mutual implication, it constitutes a genuine formalization of positive intuitionistic logic. König’s intention was to rebuild logic in such a way that the excluded third’s principle could no longer be logical. However, his treatment of truth and falsehood is purely classical. (...) We explain here this discrepancy by the choice of the alleged more primitive notions to which the questioned notions of truth and falsehood have been reduced. Finaly, it turns out that the disjunctive and conjunctive forms of the principles of the excluded third and of contradiction have effectively been excluded, but none of their implicative forms. (shrink)

Das obras de Newton da Costa, muitas pessoas na França conhecem apenas o renascimento da paraconsistência. Apresentamos algumas razões em defesa de investigações nessa parte da lógica. Acima de tudo, porém, relembramos uma das maiores contribuições de Newton da Costa: sua demonstração, em 1991, em colaboração com Doria, da indecidibilidade gödeliana do movimento na física matemátcia, um resultado que foi de certa forma previsto, por outras razões, por Duhem em 1906. DOI: 10.5007/1808-1711.2011v15n1p7.

The first step of the construction of Nézondet's models of finite arithmetics which are counter-models to Erdös–Woods conjecture is to add to the natural numbers the non-standard numbers generated by one of them, using addition, multiplication and divisions by a natural factor allowed in an ultrapower construction. After a review of some properties of such a structure, we show that the choice of the ultrafilter can be managed, using just the Chinese remainder's theorem, so that a model as desired is (...) obtained as early as at the first time. (shrink)

The first step of the construction of Nézondet's models of finite arithmetics which are counter-models to Erdös–Woods conjecture is to add to the natural numbers the non-standard numbers generated by one of them, using addition, multiplication and divisions by a natural factor allowed in an ultrapower construction. After a review of some properties of such a structure, we show that the choice of the ultrafilter can be managed, using just the Chinese remainder's theorem, so that a model as desired is (...) obtained as early as at the first time. (shrink)

From Newton da Costa’s works, many people in France know only the revival of paraconsistency. We give some reasons in defence of investigations in this part of logic. But above all we recall one of the major contributions of Newton da Costa: his proof, in 1991, in collaboration with Doria, of the gödelian undecidability of motion in mathematical physics, a result which was somewhat foreseen on other grounds by Duhem in 1906.