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διάνοιαι scarcely means the mathematical sciences here, as J. and C. suggest, but simply ‘modes of thought’: cf. VI 511 D note Plato does not, I believe, use the plural of διάνοια in its peculiar technical sense, and in any case such a meaning is here unsuitable. τὸ ἕν τε κτλ. Cf. Laws 818 C μήτε ἓν μήτε δύο μήτε τρία μήθ᾽ ὅλως ἄρτια καὶ περιττὰ δυνάμενος γιγνώσκειν, μηδὲ ἀριθμεῖν τὸ παράπαν εἰδώς and [Epin.] 977 C. ἀριθμόν τε καὶ λογισμόν. See on λογιστική τε καὶ ἀριθμητική 525 A.
The Republic of Plato. James Adam. 1902. Cambridge. Cambridge University Press.
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