Early Greek Philosophy by John Burnet, with Burnet's notes
144. The Elements of Numbers 146. The Numbers as Magnitudes

From Chapter VII., The Pythagoreans

145. The Numbers Spatial
Now there can be no doubt that by his Unlimited Pythagoras meant something spatially extended; for he identified it with air, night, or the void. We are prepared, then, to find that his followers also thought of the Unlimited as extended. Aristotle certainly regarded it so. He argues that, if the Unlimited is itself a reality, and not merely the predicate of some other reality, then every part of it must be unlimited too, just as every part of air is air.58 The same thing is implied in his statement that the Pythagorean Unlimited was outside the heavens.59 Further than this, it is not safe to go. Philolaos and his followers cannot have regarded the Unlimited as Air; for, as we shall see, they adopted the theory of Empedokles as to that "element," and accounted for it otherwise. One of them, Xouthos, argued that rarefaction and condensation implied the void; without it the universe would overflow.60 We do not know, however, whether he was earlier than the Atomists or not. It is enough to say that the Pythagoreans meant by the Unlimited the res extensa.

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.

Burnet's Notes


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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