Republic

No. “Is it, then, music, so far as we have already described it?[*](The ordinary study of music may cultivate and refine feeling. Only the mathematics of music would develop the power of abstract thought.)” “Nay, that,” he said, “was the counterpart of gymnastics, if you remember. It educated the guardians through habits, imparting by the melody a certain harmony of spirit that is not science,[*](Knowledge in the true sense, as contrasted with opinion or habit.) and by the rhythm measure and grace, and also qualities akin to these in the words of tales that are fables and those that are more nearly true. But it included no study that tended to any such good as you are now seeking.” “Your recollection is most exact,” I said; “for in fact it had nothing of the kind. But in heaven’s name, Glaucon, what study could there be of that kind? For all the arts were in our opinion base and mechanical.[*](Cf. ibid, p. 49 note e on 495 E. This idea is the source of much modern prejudice against Plato.)” “Surely; and yet what other study is left apart from music, gymnastics and the arts?” Come, said I, “if we are unable to discover anything outside of these, let us take something that applies to all alike.[*](Cf. Symp. 186 B ἐπὶ πᾶν τείνει.)” What? “Why, for example, this common thing that all arts and forms of thought[*](διάνοιαι is not to be pressed in the special sense of 511 D-E.) and all sciences employ, and which is among the first things that everybody must learn.” What? he said. “This trifling matter,[*](A playful introduction to Plato’s serious treatment of the psychology of number and the value of the study of mathematics.)” I said, “of distinguishing one and two and three. I mean, in sum, number and calculation. Is it not true of them that every art and science must necessarily partake of them?” “Indeed it is,” he said. “The art of war too?” said I. “Most necessarily,” he said. “Certainly, then,” said I, “Palamedes[*](Palamedes, like Prometheus, is a culture hero, who personifies in Greek tragedy the inventions and discoveries that produced civilization. Cf. the speech of Prometheus in Aesch. Prom. 459 ff. and Harvard Studies, xii. p. 208, n. 2.) in the play is always making Agamemnon appear a most ridiculous[*](Quoted by later writers in praise of mathematics. Cf. Theo Smyrn. p. 7 ed. Gelder. For the necessity of mathematics Cf. Laws 818 C.) general. Have you not noticed that he affirms that by the invention of number he marshalled the troops in the army at Troy in ranks and companies and enumerated the ships and everything else as if before that they had not been counted, and Agamemnon apparently did not know how many feet he had if he couldn’t count? And yet what sort of a General do you think he would be in that case?” “A very queer one in my opinion,” he said, “if that was true.” “Shall we not, then,” I said, “set down as a study requisite for a soldier the ability to reckon and number?” “Most certainly, if he is to know anything whatever of the ordering of his troops—or rather if he is to be a man at all.[*](Cf. Laws 819 D.)” “Do you observe then,” said I, “in this study what I do?” What?

“It seems likely that it is one of those studies which we are seeking that naturally conduce to the awakening of thought, but that no one makes the right use[*](Plato’s point of view here, as he will explain, is precisely the opposite of that of modern educators who would teach mathematics concretely and not puzzle the children with abstract logic. But in the Laws where he is speaking of primary and secondary education for the entire population he anticipates the modern kindergarten ideas (819 B-C).) of it, though it really does tend to draw the mind to essence and reality.” “What do you mean?” he said. “I will try,” I said, “to show you at least my opinion. Do you keep watch and observe the things I distinguish in my mind as being or not being conducive to our purpose, and either concur or dissent, in order that here too we may see more clearly[*](For σαφέστερον cf. 523 C. Cf. Vol. I. p. 47, note f, on 338 D, and What Plato Said, p. 503, on Gorg. 463 D.) whether my surmise is right.” “Point them out,” he said. “I do point them out,” I said, “if you can discern that some reports of our perceptions do not provoke thought to reconsideration because the judgement[*](Cf. Phileb. 38 C. Unity of Plato’s Thought, n. 337.) of them by sensation seems adequate,[*](ἱκανῶς is not to be pressed here.) while others always invite the intellect to reflection because the sensation yields nothing that can be trusted.[*](For οὐδὲν ὑγιές cf. 496 C, 584 A, 589 C, Phaedo 69 B, 89 E, 90 E, Gorg. 524 E, Laws 776 E, Theaet. 173 B, Eurip. Phoen. 201, Bacch. 262, Hel.. 746, etc.)” “You obviously mean distant[*](The most obvious cause of errors of judgement. Cf. Laws 663 B.) appearances,” he said, “and shadow-painting.[*](Cf. Vol. I. p. 137 on 365 C.)” “You have quite missed my meaning,[*](The dramatic misapprehension by the interlocutor is one of Plato’s methods for enforcing his meaning. Cf. on 529 A, p. 180, note a, Laws 792 B-C.)” said I. “What do you mean?” he said. “The experiences that do not provoke thought are those that do not at the same time issue in a contradictory perception.[*](Cf. Jacks, Alchemy of Thought, p. 29: The purpose of the world, then, being to attain consciousness of itself as a rational or consistent whole, is it not a little strange that the first step, so to speak, taken by the world for the attainment of this end is that of presenting itself in the form of contradictory experience? αἴσθησις is not to be pressed. Adam’s condescending apology for the primitive character of Plato’s psychology here is as uncalled-for as all such apologies. Plato varies the expression, but his meaning is clear. Cf. 524 D. No modern psychologists are able to use sensation, perception, judgement, and similar terms with perfect consistency.) Those that do have that effect I set down as provocatives, when the perception no more manifests one thing than its contrary, alike whether its impact[*](For προσπίπτουσα Cf. Tim. 33 A, 44 A, 66 A, Rep. 515 A, 561 C, Laws 791 C, 632 A, 637 A, Phileb. 21 C; also accidere in Lucretius, e.g. iv. 882, ii. 1024-1025, iv. 236 and iii. 841, and Goethe’s Das Blenden der Erscheinung, die sich an unsere Sinne drängt.) comes from nearby or afar. An illustration will make my meaning plain. Here, we say, are three fingers, the little finger, the second and the middle.” “Quite so,” he said. “Assume that I speak of them as seen near at hand. But this is the point that you are to consider.” What? “Each one of them appears to be equally a finger,[*](This anticipates Aristotle’s doctrine that substances do not, as qualities do, admit of more or less.) and in this respect it makes no difference whether it is observed as intermediate or at either extreme, whether it is white or black, thick or thin, or of any other quality of this kind. For in none of these cases is the soul of most men impelled to question the reason and to ask what in the world is a finger, since the faculty of sight never signifies to it at the same time that the finger is the opposite of a finger.” “Why, no, it does not,” he said. Then, said I, “it is to be expected that such a perception will not provoke or awaken[*](We should never press synonyms which Plato employs for ποικιλία of style or to avoid falling into a rut of terminology.) reflection and thought.” “It is.” “But now, what about the bigness and the smallness of these objects? Is our vision’s view of them adequate, and does it make no difference to it whether one of them is situated[*](κεῖσθαι perhaps anticipates the Aristotelian category.) outside or in the middle; and similarly of the relation of touch, to thickness and thinness, softness and hardness? And are not the other senses also defective in their reports of such things?

Or is the operation of each of them as follows? In the first place, the sensation that is set over the hard is of necessity related also to the soft,[*](Cf. Theaet. 186 ff., Tim. 62 B, Taylor, Timaeus, p. 233 on 63 D-E, Unity of Plato’s Thought, nn. 222 and 225, Diels, Dialex. 5 (ii.3 p. 341). Protag. 331 D anticipates this thought, but Protagoras cannot follow it out. Cf. also Phileb. 13 A-B. Stallbaum also compares Phileb. 57 D and 56 C f.) and it reports to the soul that the same thing is both hard and soft to its perception.” “It is so,” he said. Then, said I, “is not this again a case where the soul must be at a loss[*](Plato gives a very modern psychological explanation. Thought is provoked by the contradictions in perceptions that suggest problems. The very notion of unity is contradictory of uninterpreted experience. This use of ἀπορεῖν (Cf. 515 D) anticipates much modern psychology supposed to be new. Cf. e.g. Herbert Spencer, passim, and Dewey, How We Think, p. 12 we may recapitulate by saying that the origin of thinking is some perplexity, confusion, or doubt; also ibid, p. 62. Meyerson, Déduction relativiste p. 142, says Mais Platon . . . n’avait-il pas dit qu’il était impossible de raisonner si ce n’est en partant d’une perception? citing Rep. 523-524, and Rodier, Aristot. De anima, i. p. 191. But that is not Plato’s point here. Zeller, Aristot. i. p. 166 (Eng.), also misses the point when he says Even as to the passage from the former to the latter he had only the negative doctrine that the contradictions of opinion and fancy ought to lead us to go further and to pass to the pure treatment of ideas.) as to what significance for it the sensation of hardness has, if the sense reports the same thing as also soft? And, similarly, as to what the sensation of light and heavy means by light and heavy, if it reports the heavy as light, and the light as heavy?” “Yes, indeed,” he said, “these communications[*](For ἑρμηνεῖαι Cf. Theaet. 209 A.) to the soul are strange and invite reconsideration.” “Naturally, then,” said I, “it is in such cases as these that the soul first summons to its aid the calculating reason[*](Cf. Parmen. 130 A τοῖς λογισμῷ λαμβανομένοις.) and tries to consider whether each of the things reported to it is one or two.[*](Cf. Theaet. 185 B, Laws 963 C, Sophist 254 D, Hipp. Major 301 D-E, and, for the dialectic here, Parmen. 143 D.)” Of course. “And if it appears to be two, each of the two is a distinct unit.[*](Or, as the Greek puts it, both one and other. Cf. Vol. 1. p. 516, note f on 416 A. For ἕτερον Cf. What Plato Said, pp. 522, 580, 587-588.)” Yes. “If, then, each is one and both two, the very meaning[*](γεvi termini Cf. 379 B, 576 C, Parmen. 145 A, Protag. 358 C.) of two is that the soul will conceive them as distinct.[*](κεχωρισμένα and ἀχώριστα suggest the terminology of Aristotle in dealing with the problem of abstraction.) For if they were not separable, it would not have been thinking of two, but of one.” Right. “Sight too saw the great and the small, we say, not separated but confounded.[*](Plato’s aim is the opposite of that of the modern theorists who say that teaching should deal integrally with the total experience and not with the artificial division of abstraction.) “Is not that so?” Yes. “And for[*](The final use of διά became more frequent in later Greek. Cf. Aristot. Met. 982 b 20, Eth. Nic. 1110 a 4. Gen. an. 717 a 6, Poetics 1450 b 3, 1451 b 37. Cf. Lysis 218 B, Epin. 975 A, Olympiodorus, Life of Plato,Teubner vi. 191, ibid. p. 218, and schol.passim, Apsines, Spengel i. 361, line 18.) the clarification of this, the intelligence is compelled to contemplate the great and small,[*](Plato merely means that this is the psychological origin of our attempt to form abstract and general ideas. My suggestion that this passage is the probable source of the notion which still infests the history of philosophy, that the great-and-the-small was a metaphysical entity or principle in Plato’s later philosophy, to be identified with indeterminate dyad, has been disregarded. Cf. Unity of Plato’s Thought, 84. But it is the only plausible explanation that has ever been proposed of the attribution of that clotted nonsense to Plato himself. For it is fallacious to identify μᾶλλον καὶ ἦττον in Philebus 24 C, 25 C, 21 E, and elsewhere with the μέγα καὶ σμικρόν. But there is no limit to the misapprehension of texts by hasty or fanciful readers in any age.) not thus confounded but as distinct entities, in the opposite way from sensation.” True. “And is it not in some such experience as this that the question first occurs to us, what in the world, then, is the great and the small?” “By all means.” “And this is the origin of the designation intelligible for the one, and visible for the other.” “Just so,” he said. “This, then, is just what I was trying to explain a little while ago when I said that some things are provocative of thought and some are not, defining as provocative things that impinge upon the senses together with their opposites, while those that do not I said do not tend to awaken reflection.” “Well, now I understand,” he said, “and agree.” “To which class, then, do you think number and the one belong[*](To waive metaphysics, unity is, as modern mathematicians say, a concept of the mind which experience breaks up. The thought is familiar to Plato from the Meno to the Parmenides. But it is not true that Plato derived the very notion of the concept from the problem of the one and the many. Unity is a typical concept, but the consciousness of the concept was developed by the Socratic quest for the definition.)?” “I cannot conceive,” he said. “Well, reason it out from what has already been said. For, if unity is adequately[*](Cf. 523 B. The meaning must be gathered from the context.) seen by itself or apprehended by some other sensation, it would not tend to draw the mind to the apprehension of essence, as we were explaining in the case of the finger.

But if some contradiction is always seen coincidentally with it, so that it no more appears to be one than the opposite, there would forthwith be need of something to judge between them, and it would compel the soul to be at a loss and to inquire, by arousing thought in itself, and to ask, whatever then is the one as such, and thus the study of unity will be one of the studies that guide and convert the soul to the contemplation of true being.” “But surely,” he said, “the visual perception of it[*](See crit. note and Adam ad loc.) does especially involve this. For we see the same thing at once as one and as an indefinite plurality.[*](This is the problem of the one and the many with which Plato often plays, which he exhaustively and consciously illustrates in the Parmenides, and which the introduction to the Philebus treats as a metaphysical nuisance to be disregarded in practical logic. We have not yet got rid of it, but have merely transferred it to psychology.)” “Then if this is true of the one,” I said, “the same holds of all number, does it not?” Of course. “But, further, reckoning and the science of arithmetic[*](Cf. Gorg. 450 D, 451 B-C.) are wholly concerned with number.” “They are, indeed.” “And the qualities of number appear to lead to the apprehension of truth.” “Beyond anything,” he said. “Then, as it seems, these would be among the studies that we are seeking. For a soldier must learn them in order to marshal his troops, and a philosopher, because he must rise out of the region of generation and lay hold on essence or he can never become a true reckoner.[*](Cf. my review of Jowett, A.J.P. xiii. p. 365. My view there is adopted by Adam ad loc., and Apelt translates in the same way.)” “It is so,” he said. “And our guardian is soldier and philosopher in one.” Of course. “It is befitting, then, Glaucon, that this branch of learning should be prescribed by our law and that we should induce those who are to share the highest functions of state to enter upon that study of calculation and take hold of it, not as amateurs, but to follow it up until they attain to the contemplation of the nature of number,[*](It is not true as Adam says that the nature of numbers cannot be fully seen except in their connection with the Good. Plato never says that and never really meant it, though he might possibly have affirmed it on a challenge. Numbers are typical abstractions and educate the mind for the apprehension of abstractions if studied in their nature, in themselves, and not in the concrete form of five apples. There is no common sense nor natural connection between numbers and the good, except the point made in the Timaeus 53 B, and which is not relevant here, that God used numbers and forms to make a cosmos out of a chaos.) by pure thought, not for the purpose of buying and selling,[*](Instead of remarking on Plato’s scorn for the realities of experience we should note that he is marking the distinctive quality of the mind of the Greeks in contrast with the Egyptians and orientals from whom they learned and the Romans whom they taught. Cf. 525 D καπηλεύειν, and Horace, Ars Poetica 323-332, Cic. Tusc. i. 2. 5. Per contra Xen. Mem. iv. 7, and Libby, Introduction to History of Science, p. 49: In this the writer did not aim at the mental discipline of the students, but sought to confine himself to what is easiest and most useful in calculation, such as men constantly require in cases of inheritance, legacies, partition, law-suits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned.) as if they were preparing to be merchants or hucksters, but for the uses of war and for facilitating the conversion of the soul itself from the world of generation to essence and truth.” “Excellently said,” he replied. “And, further,” I said, “it occurs to me,[*](Cf. on 521 D, p. 147, note e.) now that the study of reckoning has been mentioned, that there is something fine in it, and that it is useful for our purpose in many ways, provided it is pursued for the sake of knowledge[*](Cf. Aristot. Met. 982 a 15 τοῦ εἰδέναι χάριν, and Laws 741 C. Montesquieu apud Arnold, Culture and Anarchy, p. 6: The first motive which ought to impel us to study is the desire to augment the excellence of our nature and to render an intelligent being more intelligent.) and not for huckstering.” “In what respect?” he said. “Why, in respect of the very point of which we were speaking, that it strongly directs the soul upward and compels it to discourse about pure numbers,[*](Lit. numbers (in) themselves, i.e. ideal numbers or the ideas of numbers. For this and the following as one of the sources of the silly notion that mathematical numbers are intermediate between ideal and concrete numbers, cf. my De Platonis Idearum Doctrina, p. 33, Unity of Plato’s Thought, pp. 83-84, Class. Phil. xxii. (1927) pp. 213-218.) never acquiescing if anyone proffers to it in the discussion numbers attached to visible and tangible bodies. For you are doubtless aware that experts in this study, if anyone attempts to cut up the one in argument, laugh at him and refuse to allow it; but if you mince it up,[*](Cf. Meno 79 C κατακερματίζῃς, Aristot. Met. 1041 a 19 ἀδιαίρετον πρὸς αὑτὸ ἕκαστον· τοῦτο δ’ ἦν τὸ ἑνὶ εἶναι, Met. 1052 b a ff., 15 ff. and 1053 a 1 τὴν γὰρ μονάδα τιθέασι πάντῃ ἀδιαίρετον. κερματίζειν is also the word used of breaking money into small change.) they multiply, always on guard lest the one should appear to be not one but a multiplicity of parts.[*](Numbers are the aptest illustration of the principle of the Philebus and the Parmenides that thought has to postulate unities which sensation (sense perception) and also dialectics are constantly disintegrating into pluralities. Cf. my Ideas of Good in Plato’s Republic, p. 222. Stenzel, Dialektik, p. 32, says this dismisses the problem of the one and the many das ihn (Plato) später so lebhaft beschäftigen sollte. But that is refuted by Parmen. 159 C οὐδὲ μὴν μόριά γε ἔχειν φαμὲν τὸ ὡς ἀληθῶς ἕν. The problem was always in Plato’s mind. He played with it when it suited his purpose and dismissed it when he wished to go on to something else. Cf. on 525 A, Phaedr. 266 B, Meno 12 C, Laws 964 A, Soph. 251.)” “Most true,” he replied.

“Suppose now, Glaucon, someone were to ask them, My good friends, what numbers[*](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.) 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 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 were[*](Cf. Isoc. Antid. 267 αὐτοὶ 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 7 ἀλλὰ 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.)?” “It is so,” he said. “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.[*](The translation is, I think, right. Cf. A.J.P. xiii. p. 365, and Adam ad loc.)” “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 next[*](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.) is suited to our purpose.” “What is that? Do you mean geometry,” he said. “Precisely that,” said I. “So much of it,” he said, “as applies to the conduct of war[*](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.) 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 modicum[*](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.) of geometry and calculation would suffice. What we have to consider is whether the greater and more advanced part of it tends to facilitate the apprehension of the idea of good.[*](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.) 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,[*](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.) 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,[*](Or becoming. Cf. 485 B, 525 B.) it is not.” “So we affirm.[*](γε δή is frequent in confirming answers. Cf. 557 B, 517 C, Symp. 172 C, 173 E, Gorg. 449 B, etc.)”

“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.[*](Geometry (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 triangle is simply a mental picture of the little equilateral triangle that stands for the word in text-books of geometry. There have been some attempts to prove (that of Mr. F. M. Cornford in Mind, April 1932, 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,[*](The very etymology of geometry implies the absurd practical conception of the science. Cf. Epin. 990 C γελοῖον ὄνομα.) though they cannot help it,[*](Cf. 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[*](Cf. 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[*](For φθεγγόμενοι cf. on 505 C, p. 89, note g.) is of squaring and applying[*](Cf. Thompson on Meno 87 A.) and adding and the like,[*](E. 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.[*](Cf. 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,[*](i.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[*](καλλιπόλει: 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[*](Plato 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[*](For 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,[*](Xen. 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,[*](For 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[*](Cf. on 499 D-E, p. 66, note a.) may suppose you to be recommending useless studies.[*](Again 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[*](Cf. 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[*](Another 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[*](For 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.[*](This 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. (1914) p. 352.) For they can see no other benefit from such pursuits worth mentioning.

Decide, then, on the spot, to which party you address yourself. Or are you speaking to neither, but chiefly carrying on the discussion for your own sake,[*](Cf. Charm. 166 D, Phaedo 64 C, Soph. 265 A, Apol. 33 A.) without however judging any other who may be able to profit by it?” “This is the alternative I choose,” he said, “that it is for my own sake chiefly that I speak and ask questions and reply.” “Fall back[*](ἄναγε is a military term. Cf. Aristoph. Birds 383, Xen. Cyr. vii. 1.45, iii. 3. 69.) a little, then,” said I; “for we just now did not rightly select the study that comes next[*](ἑξῆς Cf. Laches 182 B.) after geometry.” “What was our mistake?” he said. “After plane surfaces,” said I, “we went on to solids in revolution before studying them in themselves. The right way is next in order after the second dimension[*](Lit. increase Cf. Pearson, The Grammar of Science, p. 411: He proceeds from curves of frequency to surfaces of frequency, and then requiring to go beyond these he finds his problem lands him in space of many dimensions.) to take the third. This, I suppose, is the dimension of cubes and of everything that has depth.” “Why, yes, it is,” he said; “but this subject, Socrates, does not appear to have been investigated yet.[*](This is not to be pressed. Plato means only that the progress of solid geometry is unsatisfactory. Cf. 528 D. There may or may not be a reference here to the Delian problem of the duplication of the cube (cf. Wilamowitz, Platon, i. p. 503 for the story) and other specific problems which the historians of mathematics discuss in connection with this passage. Cf. Adam ad loc. To understand Plato we need only remember that the extension of geometry to solids was being worked out in his day, perhaps partly at his suggestion, e.g. by Theaetetus for whom a Platonic dialogue is named, and that Plato makes use of the discovery of the five regular solids in his theory of the elements in the Timaeus. Cf. also Laws 819 E ff. for those who wish to know more of the ancient traditions and modern conjectures I add references: Eva Sachs, De Theaeteto Ath. Mathematico, Diss. Berlin, 1914, and Die fünf platonischen Körper (Philolog. Untersuch. Heft 24), Berlin, 1917; E. Hoppe, Mathematik und Astronomie im klass. Altertum, pp. 133 ff.; Rudolf Eberling, Mathematik und Philosophie bei Plato, Münden, 1909, with my review in Class. Phil. v. (1910) p. 114; Seth Demel, Platons Verhältnis zur Mathematik, Leipzig, with my review, Class. Phil. xxiv. (1929) pp. 312-313; and, for further bibliography on Plato and mathematics, Budé, Rep. Introd. pp. lxx-lxxi.)” “There are two causes of that,” said I: “first, inasmuch as no city holds them in honor, these inquiries are languidly pursued owing to their difficulty. And secondly, the investigators need a director,[*](Plato is perhaps speaking from personal experience as director of the Academy. Cf. the hint in Euthydem. 290 C.) who is indispensable for success and who, to begin with, is not easy to find, and then, if he could be found, as things are now, seekers in this field would be too arrogant[*](i.e. the mathematicians already feel themselves to be independent specialists.) to submit to his guidance. But if the state as a whole should join in superintending these studies and honor them, these specialists would accept advice, and continuous and strenuous investigation would bring out the truth. Since even now, lightly esteemed as they are by the multitude and hampered by the ignorance of their students[*](This interpretation is, I think, correct. For the construction of this sentence cf. Isoc. xv. 84. The text is disputed; see crit. note.) as to the true reasons for pursuing them,[*](Lit. in what respect they are useful. Plato is fond of the half legal καθ’ ὅ τι. Cf. Lysis 210 C, Polit. 298 C.) they nevertheless in the face of all these obstacles force their way by their inherent charm[*](An eminent modern psychologist innocently writes: The problem of why geometry gives pleasure is therefore a deeper problem than the mere assertion of the fact. Furthermore, there are many known cases where the study of geometry does not give pleasure to the student. Adam seems to think it may refer to the personality of Eudoxus.) and it would not surprise us if the truth about them were made apparent.” “It is true,” he said, “that they do possess an extraordinary attractiveness and charm. But explain more clearly what you were just speaking of. The investigation[*](πραγματείαν: interesting is the development of this word from its use in Phaedo 63 A (interest, zeal, inquiring spirit. Cf. Aristot. Top. 100 a 18, Eth. Nic. 1103 b 26, Polyb. i. 1. 4, etc.) of plane surfaces, I presume, you took to be geometry?” Yes, said I. “And then,” he said, “at first you took astronomy next and then you drew back.” Yes, I said, “for in my haste to be done I was making less speed.[*](An obvious allusion to the proverb found in many forms in many languages. Cf. also Polit. 277 A-B, 264 B, Soph. Antig. 231 σχολῇ ταχύς, Theognis 335, 401 μηδὲν ἄγαν σπεύδειν, Suetonius, Augustus 25, Aulus Gellius x. 11. 4, Macrob. Sat. vi. 8. 9, festina lente, hâtez-vous lentement (Boileau, Art poétique, i. 171), Chi va piano va sano e va lontano (Goldoni, I volponi,I. ii.), Eile mit Weile and similar expressions; Franklin’s Great haste makes great waste, etc.) For, while the next thing in order is the study[*](μέθοδον: this word, like πραγματεία came to mean treatise.) of the third dimension or solids, I passed it over because of our absurd neglect[*](This is the meaning. Neither Stallbaum’s explanation, quia ita est comparata, ut de ea quaerere ridiculum sit,” nor that accepted by Adam, quia ridicule tractatur, is correct, and 529 E and 521 A are not in point. Cf. 528 B p. 176, note a.) to investigate it, and mentioned next after geometry astronomy,[*](Cf. Laws 822 A ff.) which deals with the movements of solids.” “That is right,” he said. “Then, as our fourth study,” said I, “let us set down astronomy, assuming that this science, the discussion of which has been passed over, is available,[*](i.e. assuming this to exist, vorhanden sein, which is the usual meaning of ὑπάρχειν in classical Greek. The science, of course, is solid geometry, which is still undeveloped, but in Plato’s state will be constituted as a regular science through endowed research.) provided, that is, that the state pursues it.”

“That is likely,” said he; “and instead of the vulgar utilitarian[*](Cf. Vol. I. p. 410, note c, on 442 E, Gorg. 482 E, Rep. 581 D, Cratyl. 400 A, Apol. 32 A, Aristot. Pol. 1333 b 9.) commendation of astronomy, for which you just now rebuked me, Socrates, I now will praise it on your principles. For it is obvious to everybody, I think, that this study certainly compels the soul to look upward[*](Cf. my review if Warburg, Class. Phil. xxiv. (1929) p. 319. The dramatic misunderstanding forestalls a possible understanding by the reader. Cf. on 523 B. The misapprehension is typical of modern misunderstandings. Glaucon is here the prototype of all sentimental Platonists or anti-Platonists. The meaning of higher things in Plato’s allegory is obvious. But Glaucon takes it literally. Similarly, modern critics, taking Plato’s imagery literally and pressing single expressions apart from the total context, have inferred that Plato would be hostile to all the applications of modern science to experience. They refuse to make allowance for his special and avowed educational purpose, and overlook the fact that he is prophesying the mathematical astronomy and science of the future. The half-serious exaggeration of his rhetoric can easily be matched by similar utterances of modern thinkers of the most various schools, from Rousseau’s écarter tous les faits to Judd’s Once we acquire the power to neglect all the concrete facts . . . we are free from the incumbrances that come through attention to the concrete facts. Cf. also on 529 B, 530 B and 534 A.) and leads it away from things here to those higher things.” “It may be obvious to everybody except me,” said I, “for I do not think so.” “What do you think?” he said. “As it is now handled by those who are trying to lead us up to philosophy,[*](ἀνάγοντες is tinged with the suggestions of 517 A, but the meaning here is those who use astronomy as a part of the higher education. φιλοσοφία is used in the looser sense of Isocrates. Cf. A.J.P. xvi. p. 237.) I think that it turns the soul’s gaze very much downward.” “What do you mean?” he said. “You seem to me in your thought to put a most liberal[*](For οὐκ ἀγεννῶς Gorg. 462 D, where it is ironical, as here, Phaedr. 264 B, Euthyph. 2 C, Theaet. 184 C. In Charm. 158 C it is not ironical.) interpretation on the study of higher things,” I said, “for apparently if anyone with back-thrown head should learn something by staring at decorations on a ceiling, you would regard him as contemplating them with the higher reason and not with the eyes.[*](The humorous exaggeration of the language reflects Plato’s exasperation at the sentimentalists who prefer star-gazing to mathematical science. Cf. Tim. 91 D on the evolution of birds from innocents who supposed that sight furnished the surest proof in such matters. Yet such is the irony of misinterpretation that this and the following pages are the chief support of the charge that Plato is hostile to science. Cf. on 530 B, p. 187, note c.) Perhaps you are right and I am a simpleton. For I, for my part, am unable to suppose that any other study turns the soul’s gaze upward[*](Cf. Theaet. 174 A ἄνω βλέποντα.) than that which deals with being and the invisible. But if anyone tries to learn about the things of sense, whether gaping up[*](Cf. Aristoph. Clouds 172.) or blinking down,[*](συμμύω probably refers to the eyes. But cf. Adam ad loc.) I would never say that he really learns—for nothing of the kind admits of true knowledge—nor would I say that his soul looks up, but down, even though he study floating on his back[*](Cf. Phaedr. 264 A, and Adam in Class. Rev. xiii. p. 11.) on sea or land.” “A fair retort,[*](Or rather, serves me right, or, in the American language, I’ve got what’s coming to me. The expression is colloquial. Cf. Epist. iii. 319 E, Antiphon cxxiv. 45. But δίκην ἔχει in 520 B = it is just.)” he said; “your rebuke is deserved. But how, then, did you mean that astronomy ought to be taught contrary to the present fashion if it is to be learned in a way to conduce to our purpose?” Thus, said I, “these sparks that paint the sky,[*](Cf. Tim. 40 A κόσμον ἀληθινὸν αὐτῷ πεποικιλμένον, Eurip. Hel. 1096 ἀστέρων ποικίλματα, Critias, Sisyphus, Diels ii.3 p. 321, lines 33-34 τό τ’ ἀστερωπὸν οὐρανοῦ δέμας χρόνου καλὸν ποίκιλμα τέκτονος σοφοῦ. Cf. also Gorg. 508 A, Lucretius v. 1205 stellis micantibus aethera fixum, ii. 1031 ff., Aeneid iv. 482 stellis ardentibus aptum, vi. 797, xi. 202, Ennius, Ann. 372. The word ποικίλματα may further suggest here the complication of the movements in the heavens) since they are decorations on a visible surface, we must regard, to be sure, as the fairest and most exact of material things but we must recognize that they fall far short of the truth,[*](The meaning of this sentence is certain, but the expression will no more bear a matter-of-fact logical analysis than that of Phaedo 69 A-B, or Rep. 365 C, or many other subtle passages in Plato. No material object perfectly embodies the ideal and abstract mathematical relation. These mathematical ideas are designated as the true,ἀληθινῶν, and the real,ὄν. As in the Timaeus (38 C, 40 A-B, 36 D-E) the abstract and ideal has the primacy and by a reversal of the ordinary point of view is said to contain or convey the concrete. The visible stars are in and are carried by their invisible mathematical orbits. By this way of speaking Plato, it is true, disregards the apparent difficulty that the movement of the visible stars then ought to be mathematically perfect. But this interpretation is, I think, more probable for Plato than Adam’s attempt to secure rigid consistency by taking τὸ ὂν τάχος etc., to represent invisible and ideal planets, and τὰ ἐνόντα to be the perfect mathematical realities, which are in them. ἐνόντα would hardly retain the metaphysical meaning of ὄντα. For the interpretation of 529 D cf. also my Platonism and the History of Science, Am. Philos. Soc, Proc. lxvi. p. 172.) the movements, namely, of real speed and real slowness in true number and in all true figures both in relation to one another and as vehicles of the things they carry and contain. These can be apprehended only by reason and thought, but not by sight; or do you think otherwise?” “By no means,” he said. Then, said I, “we must use the blazonry of the heavens as patterns to aid in the study of those realities, just as one would do who chanced upon diagrams drawn with special care and elaboration by Daedalus or some other craftsman or painter.