Plato's theory of knowledge of Forms by Division and Collection in the Sophistes is a philosophic analogue of periodic anthyphairesis (and modern continued fractions)

Abstract

The aim of this paper is to show that Plato's theory of knowledge of Forms (intelligible Beings, Ideas) in the Sophistes, obtained by Division and Collection, is a close philosophic analogue of the geometric theory of periodic anthyphairesis, an ancient theory of incommensurability (developed by the Pythagoreans, Theodorus and Theaetetus) having its modern counterpart in the theory of continued fractions. Division corresponds to infinite anthyphairetic division, Collection to the Logos Criterion, resulting in periodicity and a self-similar One, precisely the One of a Platonic Form.