Early Greek Philosophy by John Burnet, with Burnet's notes
142. The Problem 144. The Elements of Numbers

From Chapter VII., The Pythagoreans

143. Aristotle on the Numbers
In the first place, Aristotle is quite clear that Pythagoreanism was intended to be a cosmological system like the others. "Though the Pythagoreans," he tells us, "made use of less obvious first principles and elements than the rest, seeing that they did not derive them from sensible objects, yet all their discussions and studies had reference to nature alone. They describe the origin of the heavens, and they observe the phenomena of its parts, all that happens to it and all it does."41 They apply their first principles entirely to these things, "agreeing apparently with the other natural philosophers in holding that reality was just what could be perceived by the senses, and is contained within the compass of the heavens,"42 though "the first principles and causes they made use of were really adequate to explain realities of a higher order than the sensible."43

The doctrine is more precisely stated by Aristotle to be that the elements of numbers are the elements of things, and that therefore things are numbers .44 He is equally positive that these "things" are sensible things,45 and indeed that they are bodies,46 the bodies of which the world is constructed.47 This construction of the world out of numbers was a real process in time, which the Pythagoreans described in detail.48

Further, the numbers were intended to be mathematical numbers, though they were not separated from the things of sense.49 On the other hand, they were not mere predicates of something else, but had an independent reality of their own. "They did not hold that the limited and the unlimited and the one were certain other substances, such as fire, water, or anything else of that sort; but that the unlimited itself and the one itself were the reality of the things of which they are predicated, and that is why they said that number was the reality of everything."50 Accordingly the numbers are, in Aristotle's own language, not only the formal, but also the material, cause of things.51

Lastly, Aristotle notes that the point in which the Pythagoreans agreed with Plato was in giving numbers an independent reality of their own; while Plato differed from the Pythagoreans in holding that this reality was distinguishable from that of sensible things.52 Let us consider these statements in detail.

Burnet's Notes


41. Arist. Met. A, 8. 989 b 29 (R. P. 92 a).

42. Arist. Met. A, 8. 990 a 3,ὁμολογοῦντες τοῖς ἄλλοις φυσιολόγοις ὅτι τό γ' ὂν τοῦτ' ἐστὶν ὅσον αἰσθητόν ἐστι καὶ περιείληφεν ὁ καλούμενος οὐρανός.

43. Arist. Met. ib., 8. 990 a 5,τὰς δ' αἰτίας καὶ τὰς ἀρχάς, ὥσπερ εἴπομεν, ἰκανὰς λέγουσιν ἐπαναβῆναι καὶ ἐπὶ τὰ ἀνωτέρω τῶν ὄντων, καὶ μᾶλλον ἢ τοῖς περὶ φύσεως λόγοις ἁρμοττούσας.

44. Met. A, 5. 986 a 1; τὰ τῶν ἀριθμῶν στοιχεῖα τῶν ὄντων στοιχεῖα πάντων ὑπέλαβον εἶναι N, 3. 1090 a 22 εἶναι μὲν ἀριθμοὺς ἐποίησαν τὰ ὄντα, οὐ χωριστοὺς δέ, ἀλλ' ἐξ ἀριθμῶν τὰ ὄντα.

45. Met. M, 6. 1080 b 2, ὡς ἐκ τῶν ἀριθμῶν ἐνυπαρχόντων ὄντα τὰ αἰσθητά; ib. 1080 b 17, ἐκ τούτου (τοῦ μαθηματικοῦ ἀριθμοῦ) τὰς αἰσθητὰς οὐσίας συνεστάναι φασίν.

46. Met. M, 8. 1083 b 11, τὰ σώματα ἐξ ἀριθμῶν εἶναι συγκείμενα; ib. b 17, ἐκεῖνοι δὲ τὸν ἀριθμὸν τὰ ὄντα λέγουσιν· τὰ γοῦν θεωρήματα προσάπτουσι τοῖς σώμασιν ὡς ἐξ ἐκείνων ὄντων τῶν ἀριθμῶν; N. 3. 1090 a 32 κατὰ μέντοι τὸ ποιεῖν ἐξ ἀριθμῶν τὰ φυσικὰ σώματα, ἐκ μὴ ἐχόντων βάρος μηδὲ κουφότητα ἔχοντα κουφότητα καὶ βάρος.

47. Met. A, 5. 986 a 2, τὸν ὅλον οὐρανὸν ἁρμονίαν εἶναι καὶ ἀριθμόν; A, 8. 990 a 21 τὸν ἀριθμὸν τοῦτον ἐξ οὗ συνέστηκεν ὁ κόσμος; M. 6. 1080 b 18 τὸν γὰρ ὅλον οὐρανὸν κατασκευάζουσιν ἐξ ἀριθμῶν; De caelo Γ. 1. 300 a 15, τοῖς ἐξ ἀριθμῶν συνιστᾶσι τὸν οὐρανόν· ἔνιοι γὰρ τὴν φύσιν ἐξ ἀριθμῶν συνιστᾶσιν, ὥσπερ τῶν Πυθαγορείων τινές.

48. Met. N, 3. 1091 a 18, κοσμοποιοῦσι καὶ φυσικῶς βούλονται λέγειν.

49. Met. M, 6. 1080 b l6; N, 3. 1090 a 20.

50. Arist. Met. A, 5. 987 a 15.

51. Met. ib. 986 a 15 (R. P. 66).

52. Met. A, 6. 987 b 27, ὁ μὲν (Πλάτων) τοὺς ἀριθμοὺς παρὰ τὰ αἰσθητά, οἱ δ' (οἱ Πυθαγόρειοι) ἀριθμοὺς εἶναί φασιν αὐτὰ τὰ αἰσθητά.

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