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[510a] first, shadows, and then reflections in water and on surfaces of dense, smooth and bright texture, and everything of that kind, if you apprehend.” “I do.” “As the second section assume that of which this is a likeness or an image, that is, the animals about us and all plants and the whole class of objects made by man.” “I so assume it,” he said. “Would you be willing to say,” said I, “that the division in respect of reality and truth or the opposite is expressed by the proportion:1 as is the opinable to the knowable so is the likeness to that [510b] of which it is a likeness?” “I certainly would.” “Consider then again the way in which we are to make the division of the intelligible section.” “In what way?” “By the distinction that there is one section of it which the soul is compelled to investigate by treating as images the things imitated in the former division, and by means of assumptions from which it proceeds not up to a first principle but down to a conclusion, while there is another section in which it advances from its assumption to a beginning or principle that transcends assumption,2 and in which it makes no use of the images employed by the other section, relying on ideas3 only and progressing systematically through ideas.” “I don't fully understand4 what you mean by this,” he said. “Well, I will try again,” [510c] said I,” for you will better understand after this preamble. For I think you are aware that students of geometry and reckoning and such subjects first postulate the odd and the even and the various figures and three kinds of angles and other things akin to these in each branch of science, regard them as known, and, treating them as absolute assumptions, do not deign to render any further account of them5 to themselves or others, taking it for granted that they are obvious to everybody. They take their start [510d] from these, and pursuing the inquiry from this point on consistently, conclude with that for the investigation of which they set out.” “Certainly,” he said, “I know that.” “And do you not also know that they further make use of the visible forms and talk about them, though they are not thinking of them but of those things of which they are a likeness, pursuing their inquiry for the sake of the square as such and the diagonal as such, and not for the sake of the image of it which they draw6? [510e] And so in all cases. The very things which they mould and draw, which have shadows and images of themselves in water, these things they treat in their turn7 as only images, but what they really seek is to get sight of those realities which can be seen

1 Cf. on 508 C, p. 103. note b.

2 Cf. my Idea of good in Plato's republic, pp. 230-234, for the ἀνυπόθετον. Ultimately, the ἀνυπόθετον is the Idea of Good so far as we assume that idea to be attainable either in ethics or in physics. But it is the Idea of Good, not as a transcendental ontological mystery, but in the ethical sense already explained. The ideal dialectician is the man who can, if challenged, run his reasons for any given proposition back, not to some assumed axioma medium, but to its relation to ultimate Good, To call the ἀνυπόθετον the Unconditioned or Absolute introduces metaphysical associations foreign to the passage. Cf. also Introd. pp. xxxiii-xxxiv.

3 The practical meaning of this is independent of the disputed metaphysics. Cf. Introd. pp. xvi-xviii.

4 Cf. Vol. I. p. 79, note c on 347 A and p. 47, not f on 338 D; What Plato Said, p. 503 on Gorg. 463 D.

5 100 b 2-3οὐ δεῖ γὰρ ἐν ταῖς ἐπιστημονικαῖς ἀρχαῖς ἐπιζητεῖσθαι τὸ διὰ τί, exactly expresses Plato's thought and the truth, though Aristotle may have meant it mainly for the principle of non-contradiction and other first principles of logic. Cf. the mediaeval “contra principium negantem non est disputandum.” A teacher of geometry will refuse to discuss the psychology of the idea of space, a teacher of chemistry will not permit the class to ask whether matter is “real.”

6 Cf. 527 A-B. This explanation of mathematical reasoning does not differ at all from that of Aristotle and Berkely and the moderns who praise Aristotle, except that the metaphysical doctrine of ideas is in the background to be asserted if challenged.

7 i.e. a bronze sphere would be the original of its imitative reflection in water, but it is in turn only the imperfect imitation of the mathematical idea of a sphere.

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