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1 See crit. note and Adam ad loc.
2 This is the problem of the one and the many with which Plato often plays, which he exhaustively and consciously illustrates in the Parmenides, and which the introduction to the Philebus treats as a metaphysical nuisance to be disregarded in practical logic. We have not yet got rid of it, but have merely transferred it to psychology.
3 Cf. Gorg. 450 D, 451 B-C.
4 Cf. my review of Jowett, A.J.P. xiii. p. 365. My view there is adopted by Adam ad loc., and Apelt translates in the same way.
5 It is not true as Adam says that “the nature of numbers cannot be fully seen except in their connection with the Good.” Plato never says that and never really meant it, though he might possibly have affirmed it on a challenge. Numbers are typical abstractions and educate the mind for the apprehension of abstractions if studied in their nature, in themselves, and not in the concrete form of five apples. There is no common sense nor natural connection between numbers and the good, except the point made in the Timaeus 53 B, and which is not relevant here, that God used numbers and forms to make a cosmos out of a chaos.
6 Instead of remarking on Plato's scorn for the realities of experience we should note that he is marking the distinctive quality of the mind of the Greeks in contrast with the Egyptians and orientals from whom they learned and the Romans whom they taught. Cf. 525 Dκαπηλεύειν, and Horace, Ars Poetica 323-332, Cic.Tusc. i. 2. 5. Per contraXen. Mem. iv. 7, and Libby, Introduction to History of Science, p. 49: “In this the writer did not aim at the mental discipline of the students, but sought to confine himself to what is easiest and most useful in calculation, ‘such as men constantly require in cases of inheritance, legacies, partition, law-suits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned.’”
7 Cf. on 521 D, p. 147, note e.
8 Cf. Aristot. Met. 982 a 15τοῦ εἰδέναι χάριν, and Laws 741 C. Montesquieu apud Arnold, Culture and Anarchy, p. 6: “The first motive which ought to impel us to study is the desire to augment the excellence of our nature and to render an intelligent being more intelligent.”
9 Lit. “numbers (in) themselves,” i.e. ideal numbers or the ideas of numbers. For this and the following as one of the sources of the silly notion that mathematical numbers are intermediate between ideal and concrete numbers, cf. my De Platonis Idearum Doctrina, p. 33, Unity of Plato's Thought, pp. 83-84, Class. Phil. xxii. (1927) pp. 213-218.
10 Cf. Meno 79 Cκατακερματίζῃς, Aristot.Met. 1041 a 19ἀδιαίρετον πρὸς αὑτὸ ἕκαστον: τοῦτο δ᾽ ἦν τὸ ἑνὶ εἶναι, Met. 1052 b a ff., 15 ff. and 1053 a 1τὴν γὰρ μονάδα τιθέασι πάντῃ ἀδιαίρετον. κερματίζειν is also the word used of breaking money into small change.
11 Numbers are the aptest illustration of the principle of the Philebus and the Parmenides that thought has to postulate unities which sensation (sense perception) and also dialectics are constantly disintegrating into pluralities. Cf. my Ideas of Good in Plato's Republic, p. 222. Stenzel, Dialektik, p. 32, says this dismisses the problem of the one and the many “das ihn (Plato) später so lebhaft beschäftigen sollte.” But that is refuted by Parmen. 159 Cοὐδὲ μὴν μόριά γε ἔχειν φαμὲν τὸ ὡς ἀληθῶς ἕν. The “problem” was always in Plato's mind. He played with it when it suited his purpose and dismissed it when he wished to go on to something else. Cf. on 525 A, Phaedr. 266 B, Meno 12 C, Laws 964 A, Soph. 251.
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