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1 The interpreters of Plato must allow for his Emersonian habit of hitting each nail in turn as hard as he can. There is no real contradiction between praising mathematics in comparison with mere loose popular thinking, and disparaging it in comparison with dialectics. There is no evidence and no probability that Plato is here proposing a reform of mathematics in the direction of modern mathematical logic, as has been suggested. Cf. on 527 A. It is the nature of mathematics to fall short of dialectics.
2 Cf. Phileb. 20 B and on 520 C, p. 143, note g.
3 Cf. on 531 E.
4 The touch of humor is the expression may be illustrated by Lucian, Hermotimus 74, where it is used to justify Lucian's skepticism even of mathematics, and by Hazlitt's remark on Coleridge, “Excellent talker if you allow him to start from no premises and come to no conclusion.”
5 Or “admission.” Plato thinks of even geometrical reasoning as a Socratic dialogue. Cf. the exaggeration of this idea by the Epicureans in Cic.De fin. i. 21 “quae et a falsis initiis profecta, vera esse non possunt: et si essent vera nihil afferunt quo iucundius, id est, quo melius viveremus.” Dialectic proceeds διὰ συγχωρήσεων, the admission of the interlocutor. Cf. Laws 957 D, Phaedr. 237 C-D, Gorg. 487 E, Lysis 219 C, Prot. 350 E, Phileb. 12 A, Theaet. 162 A, 169 D-E, I 64 C, Rep. 340 B. But such admissions are not valid unless when challenged they are carried back to something satisfactory—ἱκανόν—(not necessarily in any given case to the idea of good). But the mathematician as such peremptorily demands the admission of his postulates and definitions. Cf. 510 B-D, 511 B.
6 Cf. on 519 B, p. 138, note a.
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