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[526a] “Suppose now, Glaucon, someone were to ask them, ‘My good friends, what numbers1 are these you are talking about, in which the one is such as you postulate, each unity equal to every other without the slightest difference and admitting no division into parts?’ What do you think would be their answer?” “This, I think—that they are speaking of units which can only be conceived by thought, and which it is not possible to deal with in any other way.” “You see, then, my friend,” said I, “that this branch of study really seems to be [526b] indispensable for us, since it plainly compels the soul to employ pure thought with a view to truth itself.” “It most emphatically does.” “Again, have you ever noticed this, that natural reckoners are by nature quick in virtually all their studies? And the slow, if they are trained and drilled in this, even if no other benefit results, all improve and become quicker than they were2?” “It is so,” he said. [526c] “And, further, as I believe, studies that demand more toil in the learning and practice than this we shall not discover easily nor find many of them.3” “You will not, in fact.” “Then, for all these reasons, we must not neglect this study, but must use it in the education of the best endowed natures.” “I agree,” he said.

“Assuming this one point to be established,” I said, “let us in the second place consider whether the study that comes next4 is suited to our purpose.” “What is that? Do you mean geometry,” he said. “Precisely that,” said I. “So much of it,” he said, [526d] “as applies to the conduct of war5 is obviously suitable. For in dealing with encampments and the occupation of strong places and the bringing of troops into column and line and all the other formations of an army in actual battle and on the march, an officer who had studied geometry would be a very different person from what he would be if he had not.” “But still,” I said, “for such purposes a slight modicum6 of geometry and calculation would suffice. What we have to consider is [526e] whether the greater and more advanced part of it tends to facilitate the apprehension of the idea of good.7 That tendency, we affirm, is to be found in all studies that force the soul to turn its vision round to the region where dwells the most blessed part of reality,8 which it is imperative that it should behold.” “You are right,” he said. “Then if it compels the soul to contemplate essence, it is suitable; if genesis,9 it is not.” “So we affirm.10

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.

10 γε δή is frequent in confirming answers. Cf. 557 B, 517 C, Symp. 172 C, 173 E, Gorg. 449 B, etc.

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