Something New about the Line (Plato, Republic, Book VI -VII)

It is no exaggeration to state that the critical debate surrounding the metaphor of the divided line, featured in Book  of the Republic, has spawned an endless secondary literature on the subject. Fortunately, we can here draw upon a significant and valuable bibliographical resource, which provides an analytical survey of the one hundred and eighty years of research on the topic (-).¹ Given that I cannot review even just a fair share of these contributions in the present article, I shall refer readers searching for a broader overview of the various suggested interpretations to the book in question. Here I shall only mention and discuss some of the most recent studies. As a preliminary assumption for our investigation, let us consider a vertical line divided into four segments, conventionally referred to as , ,  and , and corresponding to the four epistemological levels of eikasia (), pistis (), dianoia () and noesis ()-each associated to one of the four ontological levels: A corresponds to shadows and reflections,  to sense objects; the counterparts to  and  are among the objects of the present enquiry. .     - Let us start by examining b-, the passage in which Socrates briefly describes segments  and , at lines - and - respectively: Socrates b ἧι τὸ μὲν αὐτοῦ τοῖς τότε μιμηθεῖσιν ὡς εἰκόσιν χρωμένη b ψυχὴ ζητεῖν ἀναγκάζεται ἐξ ὑποθέσεων, οὐκ ἐπ᾽ ἀρχὴν b πορευομένη ἀλλ᾽ ἐπὶ τελευτήν, τὸ δ᾽ αὖ ἕτερον [τὸ] ἐπ᾽ b ἀρχὴν ἀνυπόθετον ἐξ ὑποθέσεως ἰοῦσα καὶ ἄνευ τῶν περὶ b ἐκεῖνο εἰκόνων, αὐτοῖς εἴδεσι δι᾽ αὐτῶν τὴν μέθοδον ποιου-b μένη. ¹ Lafrance ().

Journal of the History of Philosophy, 2018

Elaborating the analogy between the sun and the good, Plato's Socrates tells Glaucon to divide a line αβ into two unequal segments at γ. The result is that αγ represents what is intelligible and γβ what is visible. 1 Then Glaucon is to divide each of the two segments by the same ratio as he used in the original division (Republic 509d6-8). 2 Whatever proportion he used to make the cuts γ, δ, and ε in the divided line, generating its four segments, the geometrical implication is that the two middle segments must be equal in length. As both Nicholas D. Smith and Richard Foley have emphasized, when Socrates reiterates the characteristics of the line at 534a3-5, transposing δγ and γε, there should be no doubt that Plato knew the two middle segments were equal. 3 Their equality constitutes half the mystery of the divided line that my interpretation attempts to solve. But let us follow the text awhile. "[D]ifferences in relative clarity and obscurity" (d9) 4 determine what is assigned to each part of the line, its contents, now labeled with uppercase letters. The lower part of what is visible, Δ, represents what is most obscure: images, shadows and reflections; the next part, Γ, represents spatiotemporal objects, sensible particulars, such as animals, plants, and artifacts. The lower part of what is intelligible, Β, represents hypotheses such as those used in geometry and arithmetic; and the upper intelligible part, Α, represents "what reason itself grasps through the power of dialectic" (511b3), namely, Platonic forms. Corresponding to each sort of content is a cognitive state (511d8-e1): A for the contents of highest segment, αδ, νόησις or ἐπιστήμη; Β for δγ, διάνοια; Γ for γε, πίστις; and Δ for εβ, εἰκασία, the lowest of the four segments.

2020

A novel interpretation of a single sentence in Proclus' Commentary to Plato's "Republic" suggests a virtually algebraic rigorous derivation of an infinite sequence of pairs of side and diameter numbers.

The Line is a diagram, a geometrical device, working as a recursive model, by which Plato suggests that, as geometry is involved in the search of proportional ratios (logistikê), it is only an analogical representation of reality

We raise two questions on Euclid's Elements: How to explain that Propositions 16 and 27 in his first book do not follow, strictly speaking, from his postulates (or are perhaps meaningless)? and: What are the mathematical consequences of the meanings of the term eutheia, which we today often prefer to consider as different?

International Journal of Mathematical Education in Science and Technology, 2016

ABSTRACT Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs are given under the headings: standard proof, elegant proof, and the proof without words. The solutions were obtained through a combination of mathematical tools and by dynamic investigation of the geometrical properties.

2015

In 2007—possibly earlier—I started to write a paper entitled Euclid’s straight lines. During the academic year 2010-2011 I studied Classical Greek for Ove Strid in order to improve my understanding of Euclid (but not only). I was aware that publication might be a problem, since the paper contained too much of Classical Greek for a mathematics journal, and too much mathematics for a language journal. I submitted it three times without success. The fourth time the paper was accepted. This note tells the story of my attempts . . . maybe somebody could learn something from my experience . . . perhaps even beyond the ever-present

Symmetry, 11(1) 98, 2019

The objective of this paper is to propose a mathematical interpretation of the continuous geometric proportion (Timaeus, 32a) with which Plato accomplishes the goal to unify, harmonically and symmetrically, the Two Opposite Elements of Timaeus Cosmos—Fire and Earth—through the Mean Ratio. As we know, from the algebraic point of view, it is possible to compose a continuous geometric proportion just starting from two different quantities a (Fire) and b (Earth); their sum would be the third term, so that we would obtain the continuous geometric proportion par excellence, which carries out the agreement of opposites most perfectly: (a + b)/a = a/b. This equal proportion, applied to linear geometry, corresponds to what Euclid called the Division into Extreme and Mean Ratio (DEMR) or The Golden Proportion. In fact, according to my mathematical interpretation, in the Timaeus 32b and in the Epinomis 991 a–b, Plato uses Pingala’s Mātrāmeru or The First Analogy of the Double to mould the body of the Cosmos as a whole, to the point of identifying the two supreme principles of the Cosmos—the One (1) and the Indefinite Dyad (Φ and1/Φ)—with the DEMR. In effect, Fire and Earth are joined not by a single Mean Ratio but by two (namely, Air and Water). Moreover, using the Platonic approach to analyse the geometric properties of the shape of the Cosmos as a whole, I think that Timaeus constructed the 12 pentagonal faces of Dodecahedron by means of elementary Golden Triangles (a/b = Φ) and the Mātrāmeru sequence. And, this would prove that my mathematical interpretation of the platonic texts is at least plausible.

Apeiron, 1988

The main aim of this article is to defend the thesis that Plato apprehended the structure of incommensurable magnitudes in a way that these magnitudes correspond in a unique and well defined manner to the modern concept of the "Dedekind cut". Thus, the notion of convergence is consistent with Plato's apprehension of mathematical concepts, and in particular these of "density" of magnitudes and the complete continuum in the sense that they include incommensurable cuts. For this purpose I discuss and interpret, in a new perspective, the mathematical framework and the logic of the Third Man Argument (TMA) that appears in Plato's "Parmenides" as well as mathematical concepts from other Platonic dialogues. I claim that in this perspective the apparent infinite sequence of F-Forms, that it is generated by repetitive applications of the TMA, converges (in a mathematical sense) to a unique F-Form for the particular predicate. I also claim and prove that ...