Abstract

This chapter aims to obtain a novel anthyphairetic interpretation of Knowledge as Recollection in Plato’s Meno 80d5-86c3 and 97a9-98b6, in a self-contained manner, in line with the anthyphairetic interpretation I have developed for the whole of Plato’s work.Plato sets out to explain his philosophical notion of Knowledge in the Meno, by explaining what he means by Knowledge in the concrete geometrical case of line a such that a2 = 2b2 for a given line b, in fact of the diameter a of a square with side b. For that purpose, Plato introduces the concepts of True Opinion and Logos/Logismos and formulates his answer as follows:the Knowledge of the diameterKnowledge of the diameter a to the side b of a square, a2 = 2b2 is True Opinion plus Logos/Logismos.Our interpretation of True Opinion is that it consists of the first two anthyphairetic relations of a to ba = b + c1, b = 2c1 + c2, andour interpretation of Logos is the Theaetetus’ Logos Criterion for periodic anthyphairesisAnthyphairesisPeriodic anthyphairesis, assuming for the case in question the form of a continuous proportionb/c1 = c1/c2.Knowledge is then the knowledge of the infinite anthyphairesis of a to ba=b+c1b=2c1+c2c1=2c2+c3…cn‐1=2cn+cn+1…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\displaystyle \begin{array}{l}\mathrm{a}=\mathrm{b}+{\mathrm{c}}_1\\ {}\mathrm{b}=2{\mathrm{c}}_1+{\mathrm{c}}_2\\ {}{\mathrm{c}}_1=2{\mathrm{c}}_2+{\mathrm{c}}_3\\ {}\dots \\ {}{\mathrm{c}}_{\mathrm{n}\hbox{-} 1}=2{\mathrm{c}}_{\mathrm{n}}+{\mathrm{c}}_{\mathrm{n}+1}\\ {}\dots \end{array}} $$\end{document}resulting formally from the two relations above, namely the knowledge of the sequence of the anthyphairetic quotients of a to b [1,2,2,2,…], denoted for short [1, period (2)].Finally, Recollection is a (poetic) reformulation of the repetition c1/c2 of the ratio b/c1, instituting the complete knowledge of the anthyphairesis Anthyphairesisof a to b.The MenoMeno might appear as an isolated work of geometrical Knowledge, of the dyad, certainly related to the Pythagorean discovery of the incommensurability of the diameter to the side of a square, but, for Plato, the Meno is something more, is, in fact, a paradigmatical case for his philosophical theory of Ideas/True Intelligible Beings, and thus the quest for Knowledge in the Meno opens the gate and provides a key for the whole of Plato’s philosophy.The philosophical terms that PlatoPlato uses to describe the geometrical situation, namely True Opinion and Logos, fit in his philosophical theory and strongly suggest that for Plato, a true Intelligible Being is a dyad, to wit either the in the second hypothesis of the Parmenides or the in the Sophist, in the philosophical analogue of anthyphairesisAnthyphairesis, satisfying the philosophical analogue of Theaetetus’ Logos Criterion for anthyphairetic periodicity, as it becomes amply clear in the revealing divisions (of the Angler, Noble Sophistry, and the Sophist) in the Sophist, and in the description of the paradigmatical true Being in the second hypothesis in the ParmenidesParmenidesin terms of the anthyphairetic division of the dyad plus the circularity of the “hapseis”/logoi.Knowledge of True Being, namely the dyad is True Opinion, namely an initial finite segment of the anthyphairesis of the dyad plus Logos, namely the philosophic analogue of Theaetetus’ Logos Criterion. Theaetetus’ proof of the Pythagorean Proposition: if a2 = 2b2, then Anth (a, b) = [1, period (2)].The description of an Intelligible Being and its Knowledge takes various forms in Plato’s dialogues but remains largely equivalent to the model introduced in the Meno.[Part I] The first one and a half steps of the anthyphairesis of a to b (a = b + c1, b = 2c1 + c2) plus.