Plato's Sophist [119-120]

The διωρισμένον includes, e.g., ἀριθμός and λόγος the συνεχές includes, e.g., line, surface, body, and, furthermore, χρόνος and τόπος. Insofar as the διωρισμένον consists of parts which are οὐ θέσιν ἔχοντα, whereas the συνεχές consists of parts which are θέσιν ἔχοντα, there is then also a difference in the way the elements of the number series and those of the continuum are connected into unity.

What is the mode of connection of units such as those that belong to the series of numbers? τῶν μὲν γὰρ τοῦ ἀριθμοῦ μορίων οὐδείς ἐστι κοινὸς ὅρος, πρὸς ὃν συνάπτει τὰ μόρια αὐτοῦ· οἷον τὰ πέντε εἰ ἔστι τῶν δέκα μόριον, πρὸς οὐδένα κοινὸν ὅρον συνάπτει τὰ πέντε καὶ τὰ πέντε, ἀλλὰ διώρισται (b25ff.). The parts of a number have no common ὅρος, no common delimitation in the sense that through the ὅρος, which is identical here with the καθόλου, each of the parts would be determined proportionally. For example, in the case of 10, the two μόρια, 5 and 5, have no κοινὸς ὅρος; each is for itself, διωρισμένον, each is distinct. Likewise, 7 + 3 indeed make 10, but 7 does not have a relation, in the sense of the καθόλου or the κοινὸν (b28f.), to 10 or 3. There exists here a peculiar relation, such that the μόρια cannot be connected together, συνάπτεσθαι. οὐδ᾽ ὅλως ἂν ἔχοις ἐπ' ἀριθμοῦ λαβεῖν κοινὸν ὅρον τῶν μορίων, ἀλλ' ἀεὶ διώρισται· ὥστε ὁ μὲν ἀριθμὸς τῶν διωρισμένων ἐστίν (b29ff.). There is therefore for the manifold of numbers no such κοινὸν at all, in relation to which every particular number would be something like an instance, and number itself would be the καθόλου. There is no question here of generalization, to speak in modem terms. Number is not a genus for the particular numbers. This is admittedly only a negative result, but it is still a pressing ahead to the peculiar sort of connection residing in the number series.

Aristotle carries out the same analysis in the case of λόγος the same mode of connectedness resides there. ὡσαύτως δὲ καὶ ὁ λόγος τῶν διωρισμένων ἐστίν· (ὅτι μὲν γὰρ ποσόν ἐστιν ὁ λόγος φανερόν· καταμετρεῖται γὰρ συλλαβῇ μακρᾷ καὶ βραχείᾳ· λέγω δὲ αὐτὸν τὸν μετὰ φωνῆς λόγον γιγνόμενον) πρὸς οὐδένα γὰρ κοινὸν ὅρον αὐτοῦ τὰ μόρια συνάπτει· οὐ γὰρ ἔστι κοινὸς ὅρος πρὸς ὃν αἱ συλλαβαὶ συνάπτουσιν, ἀλλ' ἑκάστη διώρισται αὐτὴ καθ᾽ αὑτήν (b32ff.). Λόγος is taken here as a μετά φωνῆς γιγνόμενος, as vocalization, which is articulated in single syllables as its στοιχεῖα. Aristotle and Plato are fond of the example of λόγος for the question of that peculiar unity of a manifold which is not continuous but in which each part is autonomous instead. Thus λόγος in the sense of vocalization is a ποσόν, whose individual parts are absolutely delimited against one another. Each syllable is autonomously opposed to the others. There is no syllable in general, which would represent what all syllables have in common—however, this does not apply to a point, which is indeed like all other points.