Socrates “No.” “Is it, then, music, so far as we have already described it?¹” “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,² 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 522b 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.³” “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 522c something that applies to all alike.⁴” “What?” “Why, for example, this common thing that all arts and forms of thought⁵ and all sciences employ, and which is among the first things that everybody must learn.” “What?” he said. “This trifling matter,⁶” 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. 522d “Certainly, then,” said I, “Palamedes⁷ in the play is always making Agamemnon appear a most ridiculous⁸ 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.” 522e

“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.⁹” “Do you observe then,” said I, “in this study what I do?” “What?”

“It seems likely 523a 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 use¹⁰ 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 clearly¹¹ 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 523b do not provoke thought to reconsideration because the judgement¹² of them by sensation seems adequate,¹³ while others always invite the intellect to reflection because the sensation yields nothing that can be trusted.¹⁴” “You obviously mean distant¹⁵ appearances,” he said, “and shadow-painting.¹⁶” “You have quite missed my meaning,¹⁷” said I. “What do you mean?” he said. “The experiences that do not provoke thought are those that do not 523c at the same time issue in a contradictory perception.¹⁸ 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 impact¹⁹ 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 523d equally a finger,²⁰ 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 awaken²¹ 523e 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 situated²² 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? 524a In the first place, the sensation that is set over the hard is of necessity related also to the soft,²³ 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 loss²⁴ 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?” 524b “Yes, indeed,” he said, “these communications²⁵ 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 reason²⁶ and tries to consider whether each of the things reported to it is one or two.²⁷” “Of course.” “And if it appears to be two, each of the two is a distinct unit.²⁸” “Yes.” “If, then, each is one and both two, the very meaning²⁹ of “two” is that the soul will conceive them as distinct.³⁰ For if they were not separable, 524c 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.³¹ “Is not that so?” “Yes.” “And for³² the clarification of this, the intelligence is compelled to contemplate the great and small,³³ 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.” 524d “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 belong³⁴?” “I cannot conceive,” he said. “Well, reason it out from what has already been said. For, if unity is adequately³⁵ seen by itself 524e 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, 525a 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 it³⁶ does especially involve this. For we see the same thing at once as one and as an indefinite plurality.³⁷” “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 arithmetic³⁸ are wholly concerned with number.” 525b “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.³⁹” “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 525c 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,⁴⁰ by pure thought, not for the purpose of buying and selling,⁴¹ 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,⁴² now that the study of reckoning has been mentioned, 525d 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 knowledge⁴³ 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,⁴⁴ never acquiescing if anyone proffers to it in the discussion numbers attached to visible and tangible bodies. For you are doubtless aware 525e 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,⁴⁵ they multiply, always on guard lest the one should appear to be not one but a multiplicity of parts.⁴⁶” “Most true,” he replied.