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1 This is one of the chief sources of the fancy that numbers are intermediate entities between ideas and things. Cf. Alexander, Space, Time, and Deity, i. p. 219: “Mathematical particulars are therefore not as Plato thought intermediate between sensible figures and universals. Sensible figures are only less simple mathematical ones.” Cf. on 525 D. Plato here and elsewhere simply means that the educator may distinguish two kinds of numbers—five apples, and the number five as an abstract idea. Cf. Theaet. 19 E: We couldn't err about eleven which we only think, i.e. the abstract number eleven. Cf. also Berkeley, Siris, 288.
2 Cf. Isoc.Antid. 267αὐτοὶ δ᾽ αὑτῶν εὐμαθέστεροι. For the idiom αὐτοὶ αὑτῶν cf. also 411 C. 421 D, 571 D, Prot. 350 A and D, Laws 671 B, Parmen. 141 A, Laches 182 C. “Educators” have actually cited him as authority for the opposite view. On the effect of Mathematical studies cf. also Laws 747 B, 809 C-D, 810 C, Isoc.Antid. 276. Cf. Max Tyr. 37 7ἀλλὰ τοῦτο μὲν εἴη ἄν τι ἐν γεωμετρίᾳ τὸ φαυλότατον. Mill on Hamilton ii. 311 “If the Practice of mathematical reasoning gives nothing else it gives wariness of mind.” Ibid. 312.
3 The translation is, I think, right. Cf. A.J.P. xiii. p. 365, and Adam ad loc.
4 Cf. Burnet, Early Greek Philosophy, p. 111: “Even Plato puts arithmetic before geometry in the Republic in deference to tradition.” For the three branches of higher learning, arithmetic, geometry, and astronomy, Cf. Laws 811 E-818 A, Isoc.Antid. 261-267, Panath. 26, Bus. 226; Max, Tyr. 37 7.
5 Cf. Basilicon Doron(Morley, A Miscellany, p. 144): “I grant it is meete yee have some entrance, specially in the Mathematickes, for the knowledge of the art militarie, in situation of Campes, ordering of battels, making fortifications, placing of batteries, or such like.”
6 This was Xenophon's view, Mem. vi. 7. 2. Whether it was Socrates' nobody knows. Cf. pp. 162-163 on 525 C, Epin. 977 E, Aristoph.Clouds 202.
7 Because it develops the power of abstract thought. Not because numbers are deduced from the idea of good. Cf. on 525, p. 162, note b.
8 Cf. 518 C. Once more we should remember that for the practical and educational application of Plato's main thought this and all similar expressions are rhetorical surplusage or “unction,” which should not be pressed, nor used e.g. to identify the idea of good with god. Cf. Introd. p. xxv.
9 Or “becoming.” Cf. 485 B, 525 B.
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