From Chapter VII., The Pythagoreans
146. The Numbers as Magnitudes
Zeller, moreover, allows, and indeed insists, that in the Pythagorean cosmology the numbers were spatial, but he raises difficulties about the other parts of the system. There are other things, such as the Soul and Justice and Opportunity, which are said to be numbers, and which cannot be regarded as constructed of points, lines, and surfaces.66 Now it appears to me that this is just the meaning of a passage in which Aristotle criticises the Pythagoreans. They held, he says, that in one part of the world Opinion prevailed, while a little above it or below it were to be found Injustice or Separation or Mixture, each of which was, according to them, a number. But in the very same regions of the heavens were to be found things having magnitude which were also numbers. How can this be, since justice has no magnitude?67 This means surely that the Pythagoreans had failed to give any clear account of the relation between these more or less fanciful analogies and their geometrical construction of the universe.
63. Arist. Met. M, 6. 1080 b 18 sqq., 1083 b 8 sqq. ; De caelo, Γ, 1. 300 a 16 (R. P. 76 a).
64. Zeller, p. 381.
65. Zeno in his fourth argument about motion, which, we shall see (§ 163), was directed against the Pythagoreans, used ὄγκοι for points. Aetios, i. 3, 19 (R. P. 76 b), says that Ekphantos of Syracuse was the first of the Pythagoreans to say that their units were corporeal. Cf. also the use ofὄγκοι in Plato, Parm. 164 d, and Galen, Hist. Phil. 18 (Dox. p. 610), Ἡρακλείδης δὲ ὁ Ποντικὸς καὶ Ἀσκληπιάδης ὁ βιθυνὸς ἀνάρμους ὄγκους τὰς ἀρχὰς ὑποτίθενται τῶν ὅλων.
66. Zeller, p. 381.
67. Arist. Met. A, 8.,990 a 22 (R. P. 81 e). I read and interpret thus "For, seeing that, according to them, Opinion and Opportunity are in a given part of the world, and a little above or below them Injustice and Separation and Mixture,—in proof of which they allege that each of these is a number,—and seeing that it is also the case (reading συμβαίνῃ with Bonitz) that there is already in that part of the world a number of composite magnitudes (i.e. composed of the Limit and the Unlimited), because those affections (of number) are attached to their respective regions (seeing that they hold these two things), the question arises whether the number which we are to understand each of these things (Opinion, etc.) to be is the same as the number in the world (i.e. the cosmological number) or a different one." I cannot doubt that these are the extended numbers which are composed (συνίσταται) of the elements of number, the limited and the unlimited, or, as Aristotle here says, the "affections of number," the odd and the even. Zeller's view that "celestial bodies" are meant comes near this, but the application is too narrow. Nor is it the number (πλῆθος) of those bodies that is in question, but their magnitude (μέγεθος). For other views of the passage see Zeller, p. 391, n. 1.
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