Une nouvelle d\'emonstration d'irrationalit\'e de racine carr\'ee de 2 d'apr\`es les Analytiques d'Aristote

TL;DR

This paper presents a novel proof of the irrationality of the square root of 2 based on ancient results and texts of Aristotle, differing from Euclid's traditional methods, and highlighting its significance in the history of mathematics.

Contribution

It offers a new demonstration aligned with Aristotelian texts, avoiding Euclidean propositions and reductio ad absurdum, pioneering a different approach in understanding irrational magnitudes.

Findings

New proof based on old odd/even theory results

Avoids Euclidean proposition VII.22 and reductio ad absurdum

Highlights historical significance of irrational magnitudes

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 Plato's Theaetetus (147d-148b). It will be done in an article to appear in a forthcoming volume.