Plato Republic 7.525

ἡ περὶ τὸ αὐτὸ ὄψις . I formerly read αὐτό instead of τὸ αὐτό with Ξ and a few inferior MSS. αὐτό, which Bekker, Schneider and Stallbaum adopt, is easier, but lacking in authority; and τὸ αὐτό is in reality more elegant. The marked antithesis between ἡ περὶ τὸ ἓν μάθησις (‘the intellectual apprehension of the one’) and ἡ περὶ τὸ αὐτὸ ὄψις (‘the visual apprehension of the same’) makes it clear that τὸ αὐτό means ‘the same’ as that with which ἡ μάθησις was concerned (viz. τὸ ἕν), and not (as Hermann imagined) ‘one and the same object of vision’ (like ταὐτόν presently). Plato may have deliberately employed the two forms τὸ αὐτό and ταὐτόν in order to dissociate them from one another.

καὶ ξύμπας ἀριθμὸς κτλ. Because ἀριθμός is τὸ ἐκ μονάδων συγκείμενον πλῆθος (Euclid VII def. 2), or in other words a σύστημα μονάδων (Theo Smyrn. p. 18 ed. Hiller), and thus for example a visible three (i.e. three visible things) presents us with three separate cases of the contrast between ἕν and πολλά.

τοῦτο (see cr. n.) is preferable to τούτῳ, which appears in no MS except A, and would be superfluous after εἴπερ τὸ ἕν. Two MSS do in point of fact omit the word altogether.

λογιστική τε καὶ ἀριθμητική . Greek mathematicians distinguished between ἀριθμητική ‘the science of numbers’ and λογιστική ‘the art of calculation’ (Gow Greek Math. p. 22). It has been doubted whether Plato also held this distinction; but a comparison of Gorg. 451 B , 453 E, Theaet. 198 A (on ἀριθμητική) with Gorg. 451 C , Charm. 166 A , Pol. 259 E (on λογιστική) proves that he did (Rothlauf, l. c. pp. 19—21). Plato does not insist on the distinction here, but we may reasonably suppose that his pupils would begin with λογισμοί ( λογιστική) and rise from thence to ἀριθμητική: cf. C, D and Laws 817 E, 819 A ff. See also on λογιστικῷ in B.

ταῦτα : i.e. τὰ τοῦ ἀριθμοῦ.

ἀλήθειαν : viz. the Ideas, and ultimately the Idea of Good (517 B).

γενέσεως . See on 519 A.

ἤ =‘alioquin’ (V 463 D note). Liebhold absurdly adds ἔστιν after γενέσθαι.

λογιστικῷ : ‘a reasoning proficient in the art of calculation,’ with a play on λογιστικός in its deeper sense, as Shorey points out ( Chicago Studies I p. 222 note 4), comparing the double meaning of παρανομία in IV 424 D. λογισμόν in 524 B prepared the way for this; and the same ambiguity partly explains why Plato puts λογιστική rather than ἀριθμητική in the forefront of this discussion (cf. λογιστικήν below and λογισμούς in C). We readily feel that λογιστική will arouse τὸ λογιστικόν. Cf. also X 602 E note

καὶ πείθειν . προσῆκον ἂν εἴη is carried on: cf. I 334 B note and infra 530 B. J. and C.'s explanation, that “ μάθημα (or αὐτό) is to be repeated in the accusative after νομοθετῆσαι and πείθειν ἐπὶ λογιστικὴν ἰέναι,” is untenable.

τῶν μεγίστων is idiomatically used of government: cf. 534 D and Apol. 22 D with my note ad loc.

θέαν ‐‐ αὐτῇ . The ‘nature of numbers’ cannot be fully seen except in their connexion with the Good and with all other νοητά (VI 511 B—D notes). Plato does not of course imply that ἀριθμητική by itself will achieve this result (although it may be doubted whether some of his successors did not exalt the science to something like this dignity: see e.g. the Epinomis): neither ἀριθμητική nor all the propaedeutic studies taken together will ever carry us so far. He only means that the student, having once set foot on the ladder, must not redescend until he reaches the Good. Then and then only will he understand the ‘nature of numbers’ i.e. the Ideas of 1, 2, etc., because only then will he know Numbers dialectically (VI 511 B). On the use of φύσις see X 597 B note

τῇ νοήσει αὐτῇ : ‘by thought alone.’ αὐτῇ is ‘by itself’ i.e. (in this case) unadulterated with αἴσθησις: cf. 525 D note and supra IV 437 E, 438 B, VI 510 B, D notes

ῥᾳστώνης . A few inferior MSS add καί after this word: A alone has ῥᾳστώνης τε. I agree with Schneider in holding that the conjunctions are interpolated to avoid the concurrence of genitives, in which there is, however, no difficulty at all: cf. V 449 A note

νῦν καὶ ἐννοῶ . Cf. (with J. and C.) II 370 A ἐννοῶ γὰρ καὶ αὐτὸς εἰπόντος σοῦ .

λογισμούς : see on λογιστικῷ in B.

αὐτῶν τῶν ἀριθμῶν : ‘numbers themselves,’ e.g. 1, 2, 3, 4 etc., in other words individual mathematical numbers and nothing more. αὐτῶν means ‘by themselves,’ ‘alone,’ i.e. with nothing αἰσθητόν about them, such as is present in the ὁρατὰ ἢ ἁπτὰ σώματα ἔχοντας ἀριθμούς (=Aristotle's αἰσθητικοὶ or σωματικοὶ ἀριθμοί: v. Bonitz Ind. Arist. s. v. ἀριθμός), e.g. one man, two men etc. These mathematical numbers are not Ideas, but (like τὰ μαθηματικά generally) a half-way house between sensibles and Ideas, and for this reason valuable as a προπαιδεία to Dialectic: cf. 526 A note and see on VI 510 D and App. I. For αὐτῶν in this sense cf. αὐτὸ τὸ ἕν in E, αὐτῇ τῇ νοήσει 526 B and ἀριθμῶν αὐτῶν ἀλλ’ οὐ σώματα ἐχόντων [ Epin.] 990 C.

δεινούς . The word δύο, which was originally written after δεινούς (see cr. n.) in Α and Π, is probably due to a marginal adscript on the words ἐάν τις αὐτὸ τὸ ἓν ἐπιχειρῇ—τέμνειν. Burnet neatly conjectures δεινοὺς αὖ, but αὖ is inappropriate here.

ἐάν τις κτλ. αὐτὸ τὸ ἕν means ‘the unit itself’ i.e. the mathematical number ‘one’ which is ex hypothesi and by definition ἀμέριστον καὶ ἀδιαίρετον (Theo Smyrn. 18). If any one maintains that the mathematical unit is divisible, the mathematicians καταγελῶσί τε καὶ οὐκ ἀποδέχονται. Quâ mathematicians, they never condescend to justify either this or any other mathematical definition ( οὐδένα λόγον οὔτε αὑτοῖς οὔτε ἄλλοις ἔτι ἀξιοῦσι—διδόναι VI 510 C), and think it ridiculous that any one should question the foundations of their science. The moment they begin to render an account of their ὑποθέσεις they cease to be mathematicians and become διαλεκτικοί. See also on VI 510 C and App. III.

ἐὰν σὺ κερματίζῃς κτλ. : ‘if you mince it, they multiply it.’ If you insist on dividing their unit, they insist on multiplying it (viz. by your divisor), and so defeat your purpose and keep the unit one and indivisible as before. ‘I cut that unit up!’ you exclaim. ‘I multiply it!’ is their reply; and you are checkmated. They have just as much right to multiply it as you to divide it; for the mathematical unit is only a ὑπόθεσις when all is said and done. Plato is humorously describing a passage-atarms between mathematicians and some obstinate fellow who will not admit the indivisibility of their unit. The words ‘back again’ in D. and V.'s translation “they multiply it back again” correspond to nothing in the Greek and suggest an erroneous idea; nor can the Greek mean “that division is regarded by them as a process of multiplication, for the fractions of one continue to be units” (as Jowett suggests). Each of these explanations misses the humour of the original. The word μόρια is doubtless genuine, though its rejection (proposed by Herwerden) would improve the antithesis. Cf. μόριόν τε ἔχον ἐν ἑαυτῷ οὐδέν (526 A), for which μόρια here prepares the way.