ἢ ὧδε ποιεῖ ἑκάστη αὐτῶν· πρῶτον μὲν ἡ ἐπὶ τῷ σκληρῷ τεταγμένη αἴσθησις ἠνάγκασται καὶ ἐπὶ τῷ μαλακῷ τετάχθαι, καὶ παραγγέλλει τῇ ψυχῇ ὡς ταὐτὸν σκληρόν τε καὶ μαλακὸν αἰσθανομένη; οὕτως, ἔφη. οὐκοῦν, ἦν δʼ ἐγώ, ἀναγκαῖον ἔν γε τοῖς τοιούτοις αὖ τὴν ψυχὴν ἀπορεῖν τί ποτε σημαίνει αὕτη ἡ αἴσθησις τὸ σκληρόν, εἴπερ τὸ αὐτὸ καὶ μαλακὸν λέγει, καὶ ἡ τοῦ κούφου καὶ ἡ τοῦ βαρέος, τί τὸ κοῦφον καὶ βαρύ, εἰ τό τε βαρὺ κοῦφον καὶ τὸ κοῦφον βαρὺ σημαίνει; καὶ γάρ, ἔφη, αὗταί γε ἄτοποι τῇ ψυχῇ αἱ ἑρμηνεῖαι καὶ ἐπισκέψεως δεόμεναι. εἰκότως ἄρα, ἦν δʼ ἐγώ, ἐν τοῖς τοιούτοις πρῶτον μὲν πειρᾶται λογισμόν τε καὶ νόησιν ψυχὴ παρακαλοῦσα ἐπισκοπεῖν εἴτε ἓν εἴτε δύο ἐστὶν ἕκαστα τῶν εἰσαγγελλομένων. πῶς δʼ οὔ; οὐκοῦν ἐὰν δύο φαίνηται, ἕτερόν τε καὶ ἓν ἑκάτερον φαίνεται; ναί. εἰ ἄρα ἓν ἑκάτερον, ἀμφότερα δὲ δύο, τά γε δύο κεχωρισμένα νοήσει· οὐ γὰρ ἂν ἀχώριστά γε δύο ἐνόει, ἀλλʼ ἕν. ὀρθῶς. μέγα μὴν καὶ ὄψις καὶ σμικρὸν ἑώρα, φαμέν, ἀλλʼ οὐ κεχωρισμένον ἀλλὰ συγκεχυμένον τι. ἦ γάρ; ναί. διὰ δὲ τὴν τούτου σαφήνειαν μέγα αὖ καὶ σμικρὸν ἡ νόησις ἠναγκάσθη ἰδεῖν, οὐ συγκεχυμένα ἀλλὰ διωρισμένα, τοὐναντίον ἢ ʼκείνη. ἀληθῆ. οὐκοῦν ἐντεῦθέν ποθεν πρῶτον ἐπέρχεται ἐρέσθαι ἡμῖν τί οὖν ποτʼ ἐστὶ τὸ μέγα αὖ καὶ τὸ σμικρόν; παντάπασι μὲν οὖν. καὶ οὕτω δὴ τὸ μὲν νοητόν, τὸ δʼ ὁρατὸν ἐκαλέσαμεν. ὀρθότατʼ, ἔφη. ταῦτα τοίνυν καὶ ἄρτι ἐπεχείρουν λέγειν, ὡς τὰ μὲν παρακλητικὰ τῆς διανοίας ἐστί, τὰ δʼ οὔ, ἃ μὲν εἰς τὴν αἴσθησιν ἅμα τοῖς ἐναντίοις ἑαυτοῖς ἐμπίπτει, παρακλητικὰ ὁριζόμενος, ὅσα δὲ μή, οὐκ ἐγερτικὰ τῆς νοήσεως. μανθάνω τοίνυν ἤδη, ἔφη, καὶ δοκεῖ μοι οὕτω. τί οὖν; ἀριθμός τε καὶ τὸ ἓν ποτέρων δοκεῖ εἶναι; οὐ συννοῶ, ἔφη. ἀλλʼ ἐκ τῶν προειρημένων, ἔφην, ἀναλογίζου. εἰ μὲν γὰρ ἱκανῶς αὐτὸ καθʼ αὑτὸ ὁρᾶται ἢ ἄλλῃ τινὶ αἰσθήσει λαμβάνεται τὸ ἕν, οὐκ ἂν ὁλκὸν εἴη ἐπὶ τὴν οὐσίαν, ὥσπερ ἐπὶ τοῦ δακτύλου ἐλέγομεν·
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, 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 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 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 and tries to consider whether each of the things reported to it is one or two.” “Of course.” “And if it appears to be two, each of the two is a distinct unit.” “Yes.” “If, then, each is one and both two, the very meaning of “two” is that the soul will conceive them as distinct. 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. “Is not that so?” “Yes.” “And for the clarification of this, the intelligence is compelled to contemplate the great and small, 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?” “I cannot conceive,” he said. “Well, reason it out from what has already been said. For, if unity is adequately 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.
εἰ δʼ ἀεί τι αὐτῷ ἅμα ὁρᾶται ἐναντίωμα, ὥστε μηδὲν μᾶλλον ἓν ἢ καὶ τοὐναντίον φαίνεσθαι, τοῦ ἐπικρινοῦντος δὴ δέοι ἂν ἤδη καὶ ἀναγκάζοιτʼ ἂν ἐν αὐτῷ ψυχὴ ἀπορεῖν καὶ ζητεῖν, κινοῦσα ἐν ἑαυτῇ τὴν ἔννοιαν, καὶ ἀνερωτᾶν τί ποτέ ἐστιν αὐτὸ τὸ ἕν, καὶ οὕτω τῶν ἀγωγῶν ἂν εἴη καὶ μεταστρεπτικῶν ἐπὶ τὴν τοῦ ὄντος θέαν ἡ περὶ τὸ ἓν μάθησις. ἀλλὰ μέντοι, ἔφη, τοῦτό γʼ ἔχει οὐχ ἥκιστα ἡ περὶ αὐτὸ ὄψις· ἅμα γὰρ ταὐτὸν ὡς ἕν τε ὁρῶμεν καὶ ὡς ἄπειρα τὸ πλῆθος. οὐκοῦν εἴπερ τὸ ἕν, ἦν δʼ ἐγώ, καὶ σύμπας ἀριθμὸς ταὐτὸν πέπονθε τοῦτο; πῶς δʼ οὔ; ἀλλὰ μὴν λογιστική τε καὶ ἀριθμητικὴ περὶ ἀριθμὸν πᾶσα. καὶ μάλα. ταῦτα δέ γε φαίνεται ἀγωγὰ πρὸς ἀλήθειαν. ὑπερφυῶς μὲν οὖν. ὧν ζητοῦμεν ἄρα, ὡς ἔοικε, μαθημάτων ἂν εἴη· πολεμικῷ μὲν γὰρ διὰ τὰς τάξεις ἀναγκαῖον μαθεῖν ταῦτα, φιλοσόφῳ δὲ διὰ τὸ τῆς οὐσίας ἁπτέον εἶναι γενέσεως ἐξαναδύντι, ἢ μηδέποτε λογιστικῷ γενέσθαι. ἔστι ταῦτʼ, ἔφη. ὁ δέ γε ἡμέτερος φύλαξ πολεμικός τε καὶ φιλόσοφος τυγχάνει ὤν. τί μήν; προσῆκον δὴ τὸ μάθημα ἂν εἴη, ὦ Γλαύκων, νομοθετῆσαι καὶ πείθειν τοὺς μέλλοντας ἐν τῇ πόλει τῶν μεγίστων μεθέξειν ἐπὶ λογιστικὴν ἰέναι καὶ ἀνθάπτεσθαι αὐτῆς μὴ ἰδιωτικῶς, ἀλλʼ ἕως ἂν ἐπὶ θέαν τῆς τῶν ἀριθμῶν φύσεως ἀφίκωνται τῇ νοήσει αὐτῇ, οὐκ ὠνῆς οὐδὲ πράσεως χάριν ὡς ἐμπόρους ἢ καπήλους μελετῶντας, ἀλλʼ ἕνεκα πολέμου τε καὶ αὐτῆς τῆς ψυχῆς ῥᾳστώνης μεταστροφῆς ἀπὸ γενέσεως ἐπʼ ἀλήθειάν τε καὶ οὐσίαν. κάλλιστʼ, ἔφη, λέγεις. καὶ μήν, ἦν δʼ ἐγώ, νῦν καὶ ἐννοῶ, ῥηθέντος τοῦ περὶ τοὺς λογισμοὺς μαθήματος, ὡς κομψόν ἐστι καὶ πολλαχῇ χρήσιμον ἡμῖν πρὸς ὃ βουλόμεθα, ἐὰν τοῦ γνωρίζειν ἕνεκά τις αὐτὸ ἐπιτηδεύῃ ἀλλὰ μὴ τοῦ καπηλεύειν. πῇ δή; ἔφη. τοῦτό γε, ὃ νυνδὴ ἐλέγομεν, ὡς σφόδρα ἄνω ποι ἄγει τὴν ψυχὴν καὶ περὶ αὐτῶν τῶν ἀριθμῶν ἀναγκάζει διαλέγεσθαι, οὐδαμῇ ἀποδεχόμενον ἐάν τις αὐτῇ ὁρατὰ ἢ ἁπτὰ σώματα ἔχοντας ἀριθμοὺς προτεινόμενος διαλέγηται. οἶσθα γάρ που τοὺς περὶ ταῦτα δεινοὺς αὖ ὡς, ἐάν τις αὐτὸ τὸ ἓν ἐπιχειρῇ τῷ λόγῳ τέμνειν, καταγελῶσί τε καὶ οὐκ ἀποδέχονται, ἀλλʼ ἐὰν σὺ κερματίζῃς αὐτό, ἐκεῖνοι πολλαπλασιοῦσιν, εὐλαβούμενοι μή ποτε φανῇ τὸ ἓν μὴ ἓν ἀλλὰ πολλὰ μόρια. ἀληθέστατα, ἔφη, λέγεις.
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 does especially involve this. For we see the same thing at once as one and as an indefinite plurality.” “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 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.” “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, by pure thought, not for the purpose of buying and selling, 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, 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 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, 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, they multiply, always on guard lest the one should appear to be not one but a multiplicity of parts.” “Most true,” he replied.
τί οὖν οἴει, ὦ Γλαύκων, εἴ τις ἔροιτο αὐτούς· «ὦ θαυμάσιοι, περὶ ποίων ἀριθμῶν διαλέγεσθε, ἐν οἷς τὸ ἓν οἷον ὑμεῖς ἀξιοῦτέ ἐστιν, ἴσον τε ἕκαστον πᾶν παντὶ καὶ οὐδὲ σμικρὸν διαφέρον, μόριόν τε ἔχον ἐν ἑαυτῷ οὐδέν;» τί ἂν οἴει αὐτοὺς ἀποκρίνασθαι; τοῦτο ἔγωγε, ὅτι περὶ τούτων λέγουσιν ὧν διανοηθῆναι μόνον ἐγχωρεῖ, ἄλλως δʼ οὐδαμῶς μεταχειρίζεσθαι δυνατόν. ὁρᾷς οὖν, ἦν δʼ ἐγώ, ὦ φίλε, ὅτι τῷ ὄντι ἀναγκαῖον ἡμῖν κινδυνεύει εἶναι τὸ μάθημα, ἐπειδὴ φαίνεταί γε προσαναγκάζον αὐτῇ τῇ νοήσει χρῆσθαι τὴν ψυχὴν ἐπʼ αὐτὴν τὴν ἀλήθειαν; καὶ μὲν δή, ἔφη, σφόδρα γε ποιεῖ αὐτό. τί δέ; τόδε ἤδη ἐπεσκέψω, ὡς οἵ τε φύσει λογιστικοὶ εἰς πάντα τὰ μαθήματα ὡς ἔπος εἰπεῖν ὀξεῖς φύονται, οἵ τε βραδεῖς, ἂν ἐν τούτῳ παιδευθῶσιν καὶ γυμνάσωνται, κἂν μηδὲν ἄλλο ὠφεληθῶσιν, ὅμως εἴς γε τὸ ὀξύτεροι αὐτοὶ αὑτῶν γίγνεσθαι πάντες ἐπιδιδόασιν; ἔστιν, ἔφη, οὕτω. καὶ μήν, ὡς ἐγᾦμαι, ἅ γε μείζω πόνον παρέχει μανθάνοντι καὶ μελετῶντι, οὐκ ἂν ῥᾳδίως οὐδὲ πολλὰ ἂν εὕροις ὡς τοῦτο. οὐ γὰρ οὖν. πάντων δὴ ἕνεκα τούτων οὐκ ἀφετέον τὸ μάθημα, ἀλλʼ οἱ ἄριστοι τὰς φύσεις παιδευτέοι ἐν αὐτῷ. σύμφημι, ἦ δʼ ὅς. τοῦτο μὲν τοίνυν, εἶπον, ἓν ἡμῖν κείσθω· δεύτερον δὲ τὸ ἐχόμενον τούτου σκεψώμεθα ἆρά τι προσήκει ἡμῖν. τὸ ποῖον; ἢ γεωμετρίαν, ἔφη, λέγεις; αὐτὸ τοῦτο, ἦν δʼ ἐγώ. ὅσον μέν, ἔφη, πρὸς τὰ πολεμικὰ αὐτοῦ τείνει, δῆλον ὅτι προσήκει· πρὸς γὰρ τὰς στρατοπεδεύσεις καὶ καταλήψεις χωρίων καὶ συναγωγὰς καὶ ἐκτάσεις στρατιᾶς καὶ ὅσα δὴ ἄλλα σχηματίζουσι τὰ στρατόπεδα ἐν αὐταῖς τε ταῖς μάχαις καὶ πορείαις διαφέροι ἂν αὐτὸς αὑτοῦ γεωμετρικός τε καὶ μὴ ὤν. ἀλλʼ οὖν δή, εἶπον, πρὸς μὲν τὰ τοιαῦτα καὶ βραχύ τι ἂν ἐξαρκοῖ γεωμετρίας τε καὶ λογισμῶν μόριον· τὸ δὲ πολὺ αὐτῆς καὶ πορρωτέρω προϊὸν σκοπεῖσθαι δεῖ εἴ τι πρὸς ἐκεῖνο τείνει, πρὸς τὸ ποιεῖν κατιδεῖν ῥᾷον τὴν τοῦ ἀγαθοῦ ἰδέαν. τείνει δέ, φαμέν, πάντα αὐτόσε, ὅσα ἀναγκάζει ψυχὴν εἰς ἐκεῖνον τὸν τόπον μεταστρέφεσθαι ἐν ᾧ ἐστι τὸ εὐδαιμονέστατον τοῦ ὄντος, ὃ δεῖ αὐτὴν παντὶ τρόπῳ ἰδεῖν. ὀρθῶς, ἔφη, λέγεις. οὐκοῦν εἰ μὲν οὐσίαν ἀναγκάζει θεάσασθαι, προσήκει, εἰ δὲ γένεσιν, οὐ προσήκει. φαμέν γε δή.
“Suppose now, Glaucon, someone were to ask them, “My good friends, what numbers 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?” “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.” “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 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 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 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. 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, 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, it is not.” “So we affirm.”
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