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S S symmetry Article The Two Supreme Principles of Plato’s Cosmos—the One and the Indefinite Dyad—the Division of a Straight Line into Extreme and Mean Ratio, and Pingala’s Matr¯ ameru¯ Maria Antonietta Salamone Department of Philosophy and Society, Faculty of Philosophy, Complutense University of Madrid, 28040 Madrid, Spain; salamonema@filos.ucm.es Received: 6 December 2018; Accepted: 8 January 2019; Published: 16 January 2019 Abstract: 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 Matr¯ ameru¯ or The First Analogy of the Double to mould the body of the Cosmos as a whole, to the pint of identifying the two supreme principles of the Cosmos—the One (1) and the Indefinite Dyad (F and1/F)—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 = F) and the Matr¯ ameru¯ sequence. And, this would prove that my mathematical interpretation of the platonic texts is at least plausible. Keywords: Plato; Cosmology; Timaeus; Epinomis; Justice; Equality; Division in Extreme and Mean Ratio; Golden Proportion; Pingala’s Matr¯ ameru;¯ the One and the Indefinite Dyad 1. Introduction: The Idea of Justice and the Paradigm of the Line of the Horizon In my recent research [Salamone, 2017: 27–30] I proposed a cosmological interpretation of the term díke¯ in ancient Greek, in agreement with Aristotle, which establishes the synonymy between justice and equality, through the use of the paradigm of the Divided Line. I analysed the etymological origin of the Greek word díke¯ which derives from the Sanskrit root word * di´s-(dik)whose meaning refers to the «cosmological concept of the Line of the Horizon, that is to say, the apparent boundary line that divides the Cosmos into two equal parts: the Earth and the Sky». Furthermore, I tried to demonstrate the philosophical relationship between the idea of cosmic justice and the paradigm of the Line of the Horizon from the point of view of the founders of Greek scientific thought. In my opinion, the «original conception representing justice as a divine division of the Cosmos into two equal parts, or cosmic dasmós, turns out to be both spatial (the Sky is separated from the Earth) and temporal (Day is split from Night), and has its roots in ancient cosmogony not only of Greek origin, but also Indo-Iranian, Hindu, Old Persian, Egyptian, Babylonian and Chinese ( ... ). The representation is as Symmetry 2019, 11, 98; doi:10.3390/sym11010098 www.mdpi.com/journal/symmetry Symmetry 2019, 11, x 2 of 13 Symmetry 2019, 11, 98 2 of 13 but also Indo-Iranian, Hindu, Old Persian, Egyptian, Babylonian and Chinese (…). The follows: the Cosmosrepresentation began as is an as undifferentiated follows: the Cosmos mass, began without as anyan undifferentiated internal boundaries mass, or without limits any internal (Anaximander’sboundaries boundless or universe). limits (Anaximander’s This mass separated boundless into univ two parts,erse). This Earth mass and separated Sky, which into were two parts, Earth opposed or contrary,and Sky, male which and were female. opposed Finally, or contrary, the male male and theand female female. were Finally, united the bymale Eros, and the the female were contraries wereunited combined, by Eros, and the gave contraries birth to were the individual combined, existence and gave of birth Gods, to orthe of individual things» [1 existence]. It is of Gods, or probable that Platoof things» refers [1]. to theIt is paradigmprobable that of the PlatoLine refers of the to Horizon the paradigmin his Republic of the Line when of the he Horizon speaks in his Republic of the Dividedwhen Line tohe explain speaks his of cosmologicalthe Divided Line doctrine to explain of ideas his [2 ].cosmological In this way, doctri politicalne andof ideas moral [2]. In this way, questions intertwinepolitical with and cosmological moral questions and metaphysicalintertwine with ones, cosmological as prescribed and at metaphysical the beginning ones, of the as prescribed at Book of the Timaeusthe beginningwith reference of the toBookThe of Republic the Timaeus. with reference to The Republic. 2. Plato’s Cosmology2. Plato’s and Cosmology the Division and into the ExtremeDivision and into Mean Extreme Ratio and (DEMR) Mean Ratio (DEMR) Against the innumerableAgainst the coexistinginnumerable universes coexisting of theuniverses Atomists, of the and Atomists, the succession and the of singlesuccession of single universes whichuniverses had figured which in had some figured Ionian in systems some Ionian and in syst Empedocles,ems and in Plato Empedocles, contemplated Plato thecontemplated the Cosmos as a singleCosmos and as unique a single living and creature:unique living «Heaven creature: has come «Heaven to be has and come is and to shallbe and be is hereafter and shall be hereafter one and unique»one [3 and]. According unique» [3]. to the According philosopher’s to the views,philosopher’s the creator views, made the onlycreator one made copy only in order one copy in order that the Cosmosthat should the Cosmos resemble should its model resemble in respect its model of its in uniqueness. respect of its Uniqueness uniqueness. is perfection,Uniqueness and is perfection, and the Cosmos isthe marvellous Cosmos foris marvellous possessing for it. Inpossessing fact, if it it. were In fact, not unique,if it were there not wouldunique, be there body would left be body left outside it, whoseoutside strong it, powerswhose strong might powers impair itsmight life andimpair even its destroylife and it. even That destroy is why it. the That Cosmos is why the Cosmos must be (1) wholemust and be complete,(1) whole consistingand complete, of parts consisting each of whichof parts is each whole of and which complete; is whole (2) and single complete; or (2) single unique (not oneor of unique many coexistent(not one of worlds); many (3)coexistent everlasting worlds); (not destroyed (3) everlasting and superseded (not destroyed by another and superseded by world), which itanother could hardlyworld), be which if it were it could exposed hardly to assaultsbe if it were from exposed outside. to Also, assaults Aristotle from explained outside. Also, Aristotle that there is noexplained body, place, that or there void is or no time body, outside place, the or Cosmos void or [time4]. outside the Cosmos [4]. Furthermore, PlatoFurthermore, reduced thePlato four reduced material the Elements four material of Empedocles Elements of to Empedocles two from start—as to two from did start—as did Parmenides, whoParmenides, postulated who Fire postulated and Earth—and Fire and made Earth— “theand middle” made a “the blend middle” of these. a blend of these. In fact, for Empedocles,In fact, for who Empedocles, postulated who four postulated elements: four elements: Fire and Earth, moreover,Fire and Earth, are extremes moreover, and are purest: extremes Water and and purest: Air, onWa theter contrary,and Air, on are the intermediates contrary, are intermediates and more like blends.and more The like same blends. course The is same followed course by is those followed who by advocate those who three advocate (we may three compare (we may compare what what Plato doesPlato in the does Divisions: in the forDivisions: he makes for “the he middle”makes “the a blend). middle” Indeed a blend). there is Indeed practically there no is practically no difference betweendifference those who between postulate those two who and postulate those who two postulate and those three, who exceptpostulate that three, the former except split that the former split the middle elementthe intomiddle two, element while theinto latter two, treatwhile it the as latter only onetreat[De it as Generatione only one [De et Generatione Corruptione, et Corruptione, 330b] [5]. 330b] [5]. Plato deducesPlato the need deduces of two the Primary need of Elementstwo Primary by theElements following by the argument. following (1) argument. There must (1) There must be be two (not onetwo primary (not one form primary of matter, form asof thematter, Ionian as the monists Ionian had monists held) had because held) Firebecause is needed Fire is to needed to make make the world’sthe world’s body visible, body visible, and Earth and to Earth make toit make tangible. it tangible. Fire and Fire Earth and Earth had been had been commonly commonly regarded regarded as theas two the extreme two extreme and contrary and contrary elements elements (or end elements), (or end elements), since Fire belongssince Fire to the belongs Sky in to the Sky in opposition to theopposition Earth; (2) to But, the twoEarth; cannot (2) But, hold two together cannot withouthold together a middle without element a middle which element serves as which serves as bond, becausebond, the Cosmos because is athe three-dimensional Cosmos is a three-dimensional space.
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