Powers and division in the 'mathematical part' of Plato's Theaetetus

Abstract: 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' ('$\delta\upsilon\nu\acute\alpha\mu\epsilon\iota\varsigma$'). 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. Conversely, our analysis of Theaetetus' mathematical mistakes highlights the main cause of some generally overlooked failures in the dialogue: the forgetting of the 'logos', first in the 'mathematical part', then in the following discussion, and finally the failure of the four successive tries of its definition at the end of the dialogue. Namely, as we will show, the passage is closely connected with the problems studied at the end of the dialogue, but also to the two other parts of the trilogy through the method of 'definition by division'. Finally, if our conclusions are different from the usual ones, it is probably because the passage is analyzed, maybe for the first time, simultaneously from the philosophical, historical and mathematical points of view. It had been considered usually either as an excursus by historians of philosophy (for instance [Burnyeat1978]), or as an isolated text separated from the rest of the dialogue by historians of mathematics (for instance [Knorr1975]), or lastly as a pretext to discuss some astute developments in modern mathematics by mathematicians (for instance [Kahane1985]).[Brisson-Ofman1]: Luc Brisson-Salomon Ofman, Theodorus' lesson in Plato's Theaetetus(147d3-d6) Revisited-A New Perspective', to appear[Brisson-Ofman2]: Luc Brisson-Salomon Ofman,The Philosophical Interpretation of Plato'sTheaetetus and the Final Part of the Mathematical Lesson (147d7-148b)', to appear[Burnyeat 1978]: Myles Burnyeat, The Philosophical Sense of Theaetetus' Mathematics',Isis, 69, 1978, 489-514[Kahane1985]: Jean-Pierre Kahane,la th{\'e}orie de Th{\'e}odore des corps quadratiques r{\'e}els',L'enseignement math{\'e}matique, 31, 1985, p. 85-92[Knorr1975]: Wilbur Knorr, The evolution of the Euclidean elements, Reidel, 1975[Ofman2014]: Salomon Ofman, Comprendre les math{\'e}matiques pour comprendre Platon-Th{\'e}{\'e}t{\e}te (147d-148b)', Lato Sensu, I, 2014, p. 70-80