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Plato’s view that mathematics paves the way for his philosophy of forms is well known. This book attempts to flesh out the relationship between mathematics and philosophy as Plato conceived them by proposing that in his view, although it is philosophy that came up with the concept of beings, which he calls forms, and highlighted their importance, first to natural philosophy and then to ethics, the things that do qualify as beings are inchoately revealed by mathematics as the raw materials that must be further processed by philosophy (mathematicians, to use Plato’s simile in the Euthedemus, do not invent the theorems they prove but discover beings and, like hunters who must hand over what they catch to chefs if it is going to turn into something useful, they must hand over their discoveries to philosophers). Even those forms that do not bear names of mathematical objects, such as the famous forms of beauty and goodness, are in fact forms of mathematical objects. The first chapter is an attempt to defend this thesis. The second argues that for Plato philosophy’s crucial task of investigating the exfoliation of the forms into the sensible world, including the sphere of human private and public life, is already foreshadowed in one of its branches, astronomy.
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Theokritos Kouremenos Plato’s forms, mathematics and astronomy Trends in Classics – Supplementary Volumes Edited by Franco Montanari and Antonios Rengakos Associate Editors Evangelos Karakasis · Fausto Montana · Lara Pagani Serena Perrone · Evina Sistakou · Christos Tsagalis Scientific Committee Alberto Bernabé · Margarethe Billerbeck Claude Calame · Jonas Grethlein · Philip R. Hardie Stephen J. Harrison · Richard Hunter · Christina Kraus Giuseppe Mastromarco · Gregory Nagy Theodore D. Papanghelis · Giusto Picone Tim Whitmarsh · Bernhard Zimmermann Volume 67 Theokritos Kouremenos Plato’s forms, mathematics and astronomy ISBN 978-3-11-060143-5 e-ISBN (PDF) 978-3-11-060186-2 e-ISBN (EPUB) 978-3-11-060148-0 ISSN 1868-4785 Library of Congress Control Number: 20189402984 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter Inc., Boston/Berlin Printing and binding: CPI books GmbH, Leck www.degruyter.com Contents Introduction | 1 Platonic Forms as Forms only of Mathematical Objects | 8 The identification of forms with forms of mathematical objects in the cave simile | 8 1.1.1 The cave simile | 8 1.1.2 The forms | 8 1.1.3 The forms of mathematical objects | 16 1.1.4 Forms and forms of mathematical objects in the cave simile | 20 1.2 The problem of the range of forms | 21 1.2.1 The evidence | 21 1.2.2 The forms of negations, of similarity and dissimilarity and of opposites | 22 1.2.3 The forms of artifacts | 24 1.2.4 The form of number | 25 1.2.4.1 The Good | 25 1.2.4.2 The Good-like form of number | 26 1.2.4.3 The ‘generation’ of numbers in the Parmenides | 27 1.2.4.4 Aristotle’s testimony in EE A 8, 1218a15–28 | 30 1.2.4.5 Aristotle’s testimony in EN A 4, 1096a10–18 | 32 1.2.4.6 Aristotle and Plato’s conception of the forms of mathematical objects | 33 1.2.4.7 The philosophical arithmetic of the Philebus | 36 1.2.4.8 Numbers in Republic 7 | 37 1.2.4.9 Plato and Aristotle on mathematics | 41 1.2.4.10 Forms galore | 47 1.3 Are all forms only forms of mathematical objects? | 50 1.3.1 The forms of oneness and multitude(s) | 50 1.3.2 The forms of half and double, third and triple etc., small and large | 51 1.3.3 The forms of equality and inequality | 53 1.3.4 The forms of the elements: fire, air, water, and earth | 53 1.3.5 The forms of compound material substances | 57 1.3.6 The fo