“This at least,” said I, “will not be disputed by those who have even a slight acquaintance with geometry, that this science is in direct contradiction with the language employed in it by its adepts. Footnote 1Geometry (and mathematics) is inevitably less abstract than dialectics. But the special purpose of the Platonic education values mathematics chiefly as a discipline in abstraction. Cf. on 523 A, p. 152, note b; and Titchener, A Beginner’s Psychology, pp. 265-266: There are probably a good many of us whose abstract idea of
There have been some attempts to prove (that of Mr. F. M. Cornford in Mind, April , is the most recent) that Plato, if he could not anticipate in detail the modern reduction of mathematics to logic, did postulate something like it as an ideal, the realization of which would abolish his own sharp distinction between mathematics and dialectic. The argument rests on a remote and strained interpretation of two or three texts of the Republic (cf. e.g. 511 and 533 B-D) which, naturally interpreted, merely affirm the general inferiority of the mathematical method and the intermediate position for education of mathematics as a propaedeutic to dialectics. Plato’s purpose throughout is not to exhort mathematicians as such to question their initiatory postulates, but to mark definitely the boundaries between the mathematical and other sciences and pure dialectics or philosophy. The distinction is a true and useful one today. Aristotle often refers to it with no hint that it could not be abolished by a new and different kind of mathematics. And it is uncritical to read that intention into Plato’s words. He may have contributed, and doubtless did contribute, in other ways to the improvement and precision of mathematical logic. But he had no idea of doing away with the fundamental difference that made dialectics and not mathematics the coping-stone of the higher education—science as such does not question its first principles and dialectic does. Cf. 533 B-534 E. ” “How so?” he said. “Their language is most ludicrous, Footnote 2The very etymology of triangle
is simply a mental picture of the little equilateral triangle that stands for the word in text-books of geometry.geometry
implies the absurd practical conception of the science. Cf. Epin. 990 C γελοῖον ὄνομα
. though they cannot help it, Footnote 3Cf. Polit. 302 E, Laws 757 E, 818 B, Phileb. 62 B, Tim. 69 D, and also on 494 A. The word ἀναγκαίως has been variously misunderstood and mistranslated. It simply means that geometers are compelled to use the language of sense perception though they are thinking of abstractions (ideas) of which sense images are only approximations. for they speak as if they were doing something Footnote 4Cf. Aristot. Met. 1051 a 22 εὑρίσκεται δὲ καὶ τὰ διαγράμματα ἐνεργείᾳ· διαιροῦντες γὰρ εὑρίσκουσιν
, geometrical constructions, too, are discovered by an actualization, because it is by dividing that we discover them.
(Loeb tr.) and as if all their words were directed towards action. For all their talk Footnote 5For φθεγγόμενοι cf. on 505 C, p. 89, note g. is of squaring and applying Footnote 6Cf. Thompson on Meno 87 A. and adding and the like, Footnote 7E. Hoffmann, Der gegenwärtige Stand der Platonforschung, p. 1091 (Anhang, Zeller, Plato, 5th ed.), misunderstands the passage when he says: Die Abneigung Platons, dem Ideellen irgendwie einen dynamischen Charakter zuzuschreiben, zeigt sich sogar in terminologischen Andeutungen; so verbietet er Republ. 527 A für die Mathematik jede Anwendung dynamischer Termini wie τετραγωνίζειν, παρατείνειν, προστιθέναι
Plato does not forbid the use of such terms but merely recognizes their inadequacy to express the true nature and purpose of geometry. whereas in fact the real object of the entire study is pure knowledge. Footnote 8Cf. Meyerson, De l’explication dans les sciences, p. 33: En effet, Platon déjà fait ressortir que la géométrie, en dépit de l’apparence, ne poursuit aucun but pratique et n’a tout entière d’autre objet que Ia connaissance.
” “That is absolutely true,” he said. “And must we not agree on a further point?” What?
“That it is the knowledge of that which always is, Footnote 9i.e. mathematical ideas are (Platonic) ideas like other concepts. Cf. on 525 D, p. 164, note a. and not of a something which at some time comes into being and passes away.” “That is readily admitted,” he said, “for geometry is the knowledge of the eternally existent.” “Then, my good friend, it would tend to draw the soul to truth, and would be productive of a philosophic attitude of mind, directing upward the faculties that now wrongly are turned earthward.” “Nothing is surer,” he said. “Then nothing is surer,” said I, “than that we must require that the men of your Fair City Footnote 10καλλιπόλει: Plato smiles at his own Utopia. There were cities named Callipolis, e.g. in the Thracian Chersonese and in Calabria on the Gulf of Tarentum. Cf. also Herod. vii. 154. fanciful is the attempt of some scholars to distinguish the Callipolis as a separate section of the Republic, or to take it as the title of the Republic. shall never neglect geometry, for even the by-products of such study are not slight.” “What are they?” said he. “What you mentioned,” said I, “its uses in war, and also we are aware that for the better reception of all studies Footnote 11Plato briefly anticipates much modern literature on the value of the study of mathematics. Cf. on 526 B, p. 166, note a. Olympiodorus says that when geometry deigns to enter into matter she creates mechanics which is highly esteemed. there will be an immeasurable Footnote 12For ὅλῳ καὶ παντί cf. 469 C. Laws 779 B, 734 E, Phaedo 79 E, Crat. 434 A. difference between the student who has been imbued with geometry and the one who has not.” “Immense indeed, by Zeus,” he said. “Shall we, then, lay this down as a second branch of study for our lads?” “Let us do so,” he said.
“Shall we set down astronomy as a third, or do you dissent?” “I certainly agree,” he said; “for quickness of perception about the seasons and the courses of the months and the years is serviceable, Footnote 13Xen. Mem. iv. 7. 3 ff. attributes to Socrates a similar utilitarian view of science. not only to agriculture and navigation, but still more to the military art.” “I am amused, Footnote 14For ἡδὺς εἶ cf. 337 D, Euthydem. 300 A, Gorg. 491 E ἥδιστε
, Rep. 348 C γλυκὺς εἶ
, Hipp. Maj. 288 B. ” said I, “at your apparent fear lest the multitude Footnote 15Cf. on 499 D-E, p. 66, note a. may suppose you to be recommending useless studies. Footnote 16Again Plato anticipates much modern controversy. It is indeed no trifling task, but very difficult to realize that there is in every soul an organ or instrument of knowledge that is purified Footnote 17Cf. Xen. Symp. 1. 4 ἐκκεκαθαρμένοις τὰς ψυχάς
, and Phaedo 67 B-C. and kindled afresh by such studies when it has been destroyed and blinded by our ordinary pursuits, a faculty whose preservation outweighs ten thousand eyes Footnote 18Another instance of Plato’s unction.
Cf. Tim. 47 A-B, Eurip. Orest. 806 μυρίων κρείσσων
, and Stallbaum ad loc. for imitations of this passage in antiquity. ; for by it only is reality beheld. Those who share this faith will think your words superlatively Footnote 19For ἀμηχάνως ὡς Cf. Charm. 155 D ἀμήχανόν τι οἷον
. Cf. 588 A, Phaedo 80 C, 95 C, Laws 782 A, also Rep. 331 A θαυμάστος ὡς
, Hipp. Maj. 282 C, Epin. 982 C-E, Aristoph. Birds 427, Lysist. 198, 1148. true. But those who have and have had no inkling of it will naturally think them all moonshine. Footnote 20This is the thought more technically expressed in the earlier
work, Crito 49 D. Despite his faith in dialectics Plato recognizes that the primary assumptions on which argument necessarily proceeds are irreducible choices of personality. Cf. What Plato Said, p. 478, Class. Phil. ix. () p. 352. For they can see no other benefit from such pursuits worth mentioning.
Book 7