Early Greek Philosophy by John Burnet, with Burnet's notes
143. Aristotle on the Numbers 145. The Numbers Spatial

From Chapter VII., The Pythagoreans

144. The Elements of Numbers
Aristotle speaks of certain "elements" (στοιχεῖα) of numbers, which were also the elements of things. That is clearly the key to the problem, if we can discover what it means. Primarily, the "elements of number" are the Odd and the Even, but that does not seem to help us much. We find, however, that the Odd and Even were identified with the Limit and the Unlimited, which we have seen reason to regard as the original principles of the Pythagorean cosmology (§ 53). Aristotle tells us that it is the Even which gives things their unlimited character when it is contained in them and limited by the Odd,53 and the commentators are at one in understanding this to mean that the Even is in some way the cause of infinite divisibility. They get into difficulties, however, when they try to show how this can be. Simplicius has preserved an explanation, in all probability Alexander's, to the effect that they called the even number unlimited "because every even is divided into equal parts, and what is divided into equal parts is unlimited in respect of bipartition; for division into equals and halves goes on ad infinitum. But, when the odd is added, it limits it; for it prevents its division into equal parts."54 Now it is plain that we must not impute to the Pythagoreans the view that even numbers can be halved indefinitely. They must have known that the even numbers 6 and 10 can only be halved once. The explanation is rather to be found in a fragment of Aristoxenos, where we read that "even numbers are those which are divided, into equal parts, while odd numbers are divided into unequal parts and have a middle term."55 This is still further elucidated by a passage which is quoted in Stobaios and ultimately goes back to Poseidonios. It runs: "When the odd is divided into two equal parts, a unit is left over in the middle; but when the even is so divided, an empty field is left, without a master and without a number, showing that it is defective and incomplete."56 Again, Plutarch says: "In the division of numbers, the even, when parted in any direction, leaves as it were within itself . . . a field; but, when the same thing is done to the odd, there is always a middle left over from the division."57 It is clear that all these passages refer to the same thing, and that can hardly be anything else than the "terms" or dots with which we are already familiar (§ 47). The division must fall between these; for, if it meets with an indivisible unit, it is at once arrested.

Burnet's Notes


53. Met. A, 5. 986 a 17 (R. P. 66); Phys. Γ, 4. 203 a 10 (R. P. 66 a).

54. Simpl. Phys. p. 455, 20 (R. P. 66 a). I owe the passages which I have used in illustration of this subject to W. A. Heidel, "Πέρας and ἄπειρον in the Pythagorean Philosophy" (Arch. xiv. pp. 384 sqq.). The general principle of my interpretation is the same as his, though I think that, by bringing the passage into connexion with the numerical figures, I have avoided the necessity of regarding the words ἡ γὰρ εἰς ἴσα καὶ ἡμίση διαίρεσις ἐπ' ἄπειρον as "an attempted elucidation added by Simplicius."

55. Aristoxenos, fr. 81, ap. Stob. i. p. 20, ἐκ τῶν Ἀριστοξένου Περὶ ἀριθμητικῆς. . . τῶν δὲ ἀριθμῶν ἄρτιοι μέν εἰσιν οἱ εἰς ἴσα διαιρούμενοι, περισσοὶ δὲ οἱ εἰς ἄνισα καὶ μέσον ἔχοντες.

56. [Plut.] ap. Stob. i. p. 22, 19, καὶ μὴν εἰς δύο διαιρουμένων ἴσα τοῦ μὲν περισσοῦ μονὰς ἐν μέσῳ περίεστι, τοῦ δὲ ἀρτίου κενὴ λείπεται χώρα καὶ ἀδέσποτος καὶ ἀνάριθμος, ὡς ἂν ἐνδεοῦς καὶ ἀτελοῦς ὄντος.

57. Plut. De E apud Delphos, 388 a, ταῖς γὰρ εἰς ἴσα τομαῖς τῶν ἀριθμῶν, ὁ μὲν ἄρτιος πάντῃ διϊστάμενος ὑπολείπει τινὰ δεκτικὴν ἀρχὴν οἶον ἐν ἑαυτῷ καὶ χώραν, ἐν δὲ τῷ περιττῷ ταὐτὸ παθόντι μέσον ἀεὶ περίεστι τῆς νεμήσεως γόνιμον. The words which I have omitted in translating refer to the further identification of Odd and Even with Male and Female. The passages quoted by Heidel might be added to. Cf., for instance, what Nikomachos says (p. 13, 10, Hoche), ἔστι δὲ ἄρτιον μὲν ὃ οἶόν τε εἰς δύο ἴσα διαιρεθῆναι μονάδος μέσον μὴ παρεμπιπτούσης, περιττὸν δὲ τὸ μὴ δυνάμενον εἰς δύο ἴσα μερισθῆναι διὰ τὴν προειρημένην τῆς μονάδος μεσιτείαν. He significantly adds that this definition is ἐκ τῆς δημώδους ὑπολήψεως.

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