In Plato’s eponymous dialogue, Timaeus, the main character presents the universe as an (almost) perfect sphere filled by tiny, invisible particles having the form of four regular polyhedrons. At first glance, such a construction may seem close to an atomistic theory. However, one does not find any text in Antiquity that links Timaeus’ cosmology to the atomists, while Aristotle opposes clearly Plato to the latter. Nevertheless, Plato is commonly presented in contemporary literature as some sort of atomist, sometimes as supporting a form of so-called ‘mathematical atomism’. However, the term ‘atomism’ is rarely defined when applied to Plato. Since it covers many different theories, it seems that this term has almost as different meanings as different authors. The purpose of this article is to consider whether it is correct to connect Timaeus’ cosmology to some kind of ‘atomism’, however this term may be understood. Its purpose is double: to obtain a better understanding of the cosmology of the Timaeus, and to consider the different modern ‘atomistic’ interpretations of this cosmology. In short, we would like to show that such a claim, in any form whatsoever, is misleading, an impediment to the understanding of the dialogue, and more generally of Plato’s philosophy.
Pour rendre compte de la première démonstration d'existence d'une grandeur irrationnelle, les historiens des sciences et les commentateurs d'Aristote se réfèrent aux textes sur l'incommensurabilité de la diagonale qui se trouvent dans les Premiers Analytiques, en tant qu'ils sont les plus anciens sur la question. Les preuves usuelles proposées dérivent d'un même modèle qui se trouve à la fin du livre X des Éléments d'Euclide. Le problème est que ses conclusions, passant par la représentation des fractions comme rapport de deux entiers premiers entre eux i.e. la proposition VII.22 des Éléments, ne correspondent pas aux écrits aristotéliciens. Dans cet article, nous proposons une nouvelle démonstration, conforme aux textes des Analytiques, fondés sur des résultats très anciens de la théorie du pair et de l'impair. Ne passant pas par la proposition VII.22, ni aucune autre propriété établie par l'absurde, cette irrationalité apparaît comme le premier résultat que l'on ne pouvait établir par une autre méthode. L'importance de ce résultat, révélant un nouveau domaine mathématique, celui des grandeurs irrationnelles, rend compte de la centralité que cette forme de raisonnement acquiert alors, d'abord en mathématique, puis dans tout type de discours rationnel. À partir des conséquences qui suivent de cette nouvelle démonstration, on peut interpréter très simplement la leçon sur les irrationnels du passage mathématique figurant dans le Théétète de Platon (147d-148b), ce que nous ferons dans un article à paraître dans un prochain volume. Abstract.To account for the first proof of existence of an irrational magnitude, historians of science as well as commentators of Aristotle refer to the texts on the incommensurability of the diagonal in Prior Analytics, since they are the most ancient on the subject. The usual proofs suggested by the historians of science derive from a proposition found at the end of Book X of Euclid's Elements. But its conclusions, using the representation of fractions as a ratio of two integers relatively prime i.e. the proposition VII.22 of the Elements, do not match the Aristotelian texts. In this article, we propose a new demonstration conformed to these texts. They are based on very old results of the odd/even theory. Since they use neither the proposition VII.22, nor any other result proved by a reductio ad absurdum, it seems to be the first result which was impossible to prove in another way. The significance of this result, revealing a complete new territory in Mathematics, the field of irrational magnitudes, accounts for the centrality gained afterwards by this kind of reasoning, firstly in Mathematics, then in all forms of rational discourse. From the consequences of this new proof, we can construe very simply the lecture on the irrationals in the mathematical text in . It will be done in an article to appear in a forthcoming volume.
d d et d-cohomologies d'une variété compacte privée d'un point. Application à l'intégration sur les cycles Bulletin de la S. M. F., tome 113 (1985), p. 241-254
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