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)

TL;DR

This paper draws a philosophical and mathematical analogy between Plato's method of understanding Forms through Division and Collection and the ancient geometric theory of anthyphairesis, highlighting their shared structure of self-similarity and periodicity.

Contribution

It establishes a novel analogy between Plato's epistemological method and the ancient mathematical theory of anthyphairesis, linking philosophical and mathematical concepts.

Findings

Shows that Plato's Division and Collection mirror periodic anthyphairesis.

Identifies the self-similar structure of Forms with periodic continued fractions.

Provides a new interdisciplinary perspective on Plato's theory of knowledge.

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.