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. (shrink)

In the present article, we consider the question of the primary elements in Plato’s Timaeus, the components of the whole universe reduced, by an extraordinarily elegant construction, to two right triangles. But how does he reconcile such a model with the infinite diversity of the universe? A large part of this study is devoted to Cornford’s explanation in his commentary of the Timaeus and its shortcomings, in order to finally propose a revised one, which we think to be entirely consistent (...) with Plato’s text. This analysis is an essential step for the understanding of the connection between the sensible world and the mathematical principles that underlies Timaeus’ cosmological account. (shrink)

In this paper, we study the so-called ‘Mathematical part’ of Plato’s Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of integers. As the most ancient text on the subject, and on Greek mathematics and mathematicians as well, its historical importance is enormous. Its interpretation presents a certain degree of difficulty because of the intertwined fields that play a role in it : philosophy, history and mathematics. (...) But conversely, correctly understood, it provides clues on both the question of the origins of the irrationals in Greek mathematics and some points concerning Plato’s thought. Taking into account the historical context and the philosophical background generally forgotten in mathematical analyses, we get a new interpretation of this text which, far from being a tribute to some mathematicians, is a radical criticism of their ways of thinking. And the mathematical lesson, far from being a tribute to some future mathematical achievements, ends on an aporia, in accordance with the whole dialogue. (shrink)

This paper is an updated translation of an article published in French in the Journal Lato Sensu (I, 2014, p. 70-80). We study here the so-called 'Mathematical part' of Plato's Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of integers. As the most ancient text on the subject, and on Greek mathematics and mathematicians as well, its historical importance is enormous. The difficulty to understand it (...) lies in the close intertwining of different fields we found in it: philosophy, history and mathematics. But conversely, correctly understood, it gives some evidences both about the question of the origins of the irrationals in Greek mathematics and some points concerning Plato's thought. Taking into account the historical context and the philosophical background generally forgotten in mathematical analyses, we get a new interpretation of this text, which far from being a tribute to some mathematicians, is a radical criticism of their ways of thinking. And the mathematical lesson, far from being a tribute to some future mathematical achievements, is ending on an aporia, in accordance with the whole dialogue. Résumé. Cet article est une traduction anglaise révisée d'un article paru en français dans la revue Lato Sensu (I, 2014, p. 70-80). (shrink)

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, 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 par 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 numéro. (shrink)