Stelios Negrepontis Negrepontis

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.

Plato, most unexpectedly, in the middle of the Timaeus (48e2-49a7) declares that the sensible bodies cannot be explained solely by their participation in the intelligible, as we were led to believe by reading the long succession of all his previous dialogues, but that it is now necessary to introduce, beside the intelligibles and the sensibles, a Third Kind, the Receptacle.We must, however, in beginning our fresh account of the Universe make more distinctions than we did before; for whereas then we distinguished two Kinds (τότε μὲν γὰρ δύο εἴδη διειλόμεθα), we must now declare another Third Kind (τρίτον ἄλλο γένος). For our former exposition, those two were sufficient, one of them being assumed as a Paradigm (παραδείγματος εἶδος), intelligible, and always self-similar Beings (νοητὸν καὶ ἀεὶ κατὰ ταὐτὰ ὄν), and the second as the Paradigm’s Image (μίμημα δὲ παραδείγματος), subject to generation and visible (γένεσιν ἔχον καὶ ὁρατόν). A Third Kind we did not at that time distinguish, considering that those two were sufficient, but now the argument seems to compel us (ἔοικεν εἰσαναγκάζειν) to try to reveal by words a Kind that is baffling and obscure (χαλεπὸν καὶ ἀμυδρὸν εἶδος). What essential property, then, are we to conceive it to possess? This in particular – that it should be the receptacle (ὑποδοχὴν), and as it were the nurse of all Generation (πάσης εἶναι γενέσεως αὐτὴν οἷον τιθήνην).Yet true though this statement is, we need to describe it more plainly 48e2-49a7.(Emphasis by italics or bold here and in future occurences is ours)It is fair to state that both the reason that Plato felt compelled to introduce the Third Kind and its nature have so far eluded understanding and remained a mystery, and this is certainly not because of any lack in efforts.In the present work, we aim to obtain a new and, we believe, definitive understanding of the nature of the Receptacle and of the reason that Plato felt obliged to introduce it. This resolution is possible only because we have uncovered, under Plato’s fascinating account, the well-hidden deep Mathematics of incommensurability and of infinite anthyphairesis, some going back to the Pythagoreans, some discovered recently by Theaetetus, the great resident mathematician of the Academy, and which are fundamental, not only for Plato’s philosophical system of the unchanging intelligible Beings, but also for his ambitious description of the incessantly changing sensible, physical World, as well. Without reaching an understanding of the underlying Mathematics, it is impossible to gain any true understanding of the philosophy based on it; it is very much like trying to understand Newtonian physics without calculus or quantum physics without Hilbert space operators.The starting point of our work is the interpretation of the intelligible Being as the philosophical analogue of a dyad in periodic anthyphairesis, and the pre-Timaeus interpretation of a sensible body participating in the intelligible, in terms of an initial finite segment of the infinite intelligible anthyphairesis. The serious problem that results from the built-in intelligible anthyphairetic infinityAnthyphairetic infinity is that this infinity leads to the formation of infinite kosmoi, a problem akin to the intelligible Third Man Argument; it has not been realized that the sole reason for the necessity of introducing the Third Kind in the Timaeus is the necessity to avoid infinite kosmoi. At this point, it must be emphasized that the intelligibility is neutral on this problem, its only necessary condition being that the intelligible must retain the control of the sensibles, whether these sensibles live in infinitely many kosmoi or in one kosmos.Plato attempts to deal with the problem, by eliminating the infinite multitude of anthyphairetic remainders, replacing them by the mathematically equivalent tight double inequalitiesDouble inequalities and generalized side and diameter numbers (the “convergents” of modern continued fractions), which are vividly and aptly described, in order to represent every sensible body, as a dyad consisting of Content & Receptacle, but he realizes that such arithmetical methods, while they work perfectly well for the geometric anthyphairetic remainders, fail to make sense for the intelligible philosophical analoguesPhilosophical analogue of the geometric anthyphairetic remainders.It is precisely because of this differentiation that Plato finds it necessary to introduce the geometric Third Kind; it consists of the four primary bodiesFour primary bodies, each identified with one of the surfaces of the four (minus the 12hedron) canonical solids, primary earth PE/cube, primary water PW/20hedron, primary air PA/8hedron, primary fire PF/pyramid, studied in the Theaetetean Book XIII of Euclid’s ElementsElements. The comprehension of the canonical solids in the sphere, a central theme in Book XIII, in fact the world sphere in Plato’s mind, provides several crucial advantages for this choice, to wit:It makes possible the canonical comparisonCanonical comparison by which the primary earth PE turns to be incommensurable, in fact in periodic anthyphairesis to each of the other three, mutually commensurable, primary bodies, and thus allows for the substitution of the intelligible dyad by the Third Kind dyad in its role of generating and being the cause of the sensible bodies and their motions.(“intelligence”) It makes possible the participation, via the Soul of the World, of the Third Kind into the intelligible, thus turning the Third Kind into a “bastard intelligible” and salvaging Plato’s absolutely essential philosophical requirement that the intelligible governs over the sensibles (by means of True Opinion).(“necessity”) It entraps and imprisons the Third Kind in the World sphere, not allowing the generation of infinitely many remainders and Kosmoi above and beyond the initial one, but turning, equivalently, the participation into sensible dyads, vividly and accurately described as having the form Content & Receptacle, underlying the tight remainder-free double inequalities and anthyphairetic “convergents,” thus saving the One Kosmos with the traditional four (not elements any more, but) derived kinds.

In the present chapter, we provide a novel interpretation of the concept of Plato’s and Xenocrates’ indivisible line, in fact, we show that indivisibility is just another description of the Platonic intelligible true Being. Our claim and arguments are based on our earlier interpretation of Plato’s intelligible Being as the philosophic analogue of a dyad of opposite kinds in periodic anthyphairesis (as revealed primarily in the Meno, the second hypothesis of the One in the Parmenides, and the Sophist), and take into account the ancient sources on the indivisible by Plato (Sophist, Phaedrus, Republic, Philebus), Aristotle (Metaphysics), Proclus, and Simplicius. Additional arguments, based on Zeno (Dichotomy Paradox), and Aristotle (Physics, and the pseudo-Aristotelian On Indivisible Lines), establish that Zeno’s true Being is also indivisible. Earlier attempts by modern Platonists were not successful, mainly because these have not taken into account the geometric/anthyphairetic nature of a true Being.