From Chapter VII., The Pythagoreans
145. The Numbers Spatial
As the Unlimited is spatial, the Limit must be spatial too, and we should expect to find that the point, the line, and the surface were regarded as forms of the Limit. That was the later doctrine; but the characteristic feature of Pythagoreanism is just that the point was not regarded as a limit, but as the first product of the Limit and the Unlimited, and was identified with the arithmetical unit instead of with zero. According to this view, then, the point has one dimension, the line two, the surface three, and the solid four.61 In, other words, the Pythagorean points have magnitude, their lines breadth, and their surfaces thickness. The whole theory, in short, turns on the definition of the point as a unit "having position" (μονὰς θέσιν ἔχουσα).62 It was out of such elements that it seemed possible to construct a world.
58. Arist. Phys. Γ, 4. 204 a 20 sqq., especially a 26, ἀλλὰ μὴν ὥσπερ ἀέρος ἀὴρ μέρος, οὕτω καὶ ἄπειρον ἀπείρου, εἴ γε οὐσία ἐστὶ καὶ ἀρχή.
59. See Chap. II. § 53.
60. Ar. Phys. Δ, 9. 216 b 25, κυμανεῖ τὸ ὅλον.
61. Cf. Speusippos in the extract preserved in the Theologumena arithmetica, p. 61 (Diels, Vors. 32 A 13) τὸ μὴν γὰρ [α] στιγμή, τὰ δὲ [β] γραμμή, τὰ δὲ [γ] τρίγωνον, τὰ δὲ [δ] πυραμίς. We know that Speusippos is following Philolaos here. Arist. Met. Z, 11. 1036 b 12, καὶ ἀνάγουσι πάντα εἰς τοὺς ἀριθμούς, καὶ γραμμῆς τὸν λόγον τὸν τῶν δύο εἶναι φασιν.. The matter is clearly put by Proclus in Eucl. I. p. 97, 19, τὸ μὲν σημεῖον ἀνάλογον τίθενται μονάδι, τὴν δὲ γραμμήν δυάδι, τὴν δὲ ἐπιφάνειαν τῇ τριάδι καὶ τὸ στερεὸν τῇ τετράδι. καίτοι γε ὡς διαστατὰ λαμβάνοντες μοναδικὴν μὲν εὑρήσομεν τὴν γραμμὴν, δυαδικὴν δὲ τὴν ἐπιφάνειαν, τριαδικὸν δὲ τὸ στερεόν.
62. The identification of the point with the unit is referred to by Aristotle, Phys. B. 227 a 27.
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