The Anthyphairetic Revolutions of the Platonic Ideas

TL;DR

This paper explores the nature of Platonic Ideas through the lens of anthyphairetic periodicity, revealing their self-similar, incommensurable, and unified structure, and links ancient mathematical methods to philosophical concepts.

Contribution

It introduces a novel interpretation of Platonic Ideas as incommensurable pairs analyzed via anthyphairesis, connecting ancient mathematics with philosophical theory.

Findings

Platonic Ideas are analogous to incommensurable line pairs.

Division and Collection mirror anthyphairesis processes.

Theaetetus proved the periodicity of anthyphairesis for incommensurable lines.

Abstract

In the present work it is shown, by an examination of the Platonic dialogues Theaetetus, Sophistes, Politicus, and Philebus, that (a) a Platonic Idea is the philosophic analogue of a pair of lines incommensurable in length only, (b) the Division and Collection, the method by which humans obtain knowledge of a Platonic Idea, is the philosophic analogue of the palindromically periodic anthyphairesis of this pair, and (c) a Platonic Idea is One in the sense of the self-similarity induced by periodic anthyphairesis. A byproduct of the above analysis is that (d) Theaetetus had obtained a proof of the Proposition: The anthyphairesis of a dyad of lines incommensurable in length only is palindromically periodic. It is further verified that the concepts and tools contained in the Theaetetean Book X of the Elements suffice for the proof of the Proposition.