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VIII. " This then is what I was just now trying to make you understand when I said that some sensations are suited to provoke thought, meaning those which make contrary impressions upon the senses at the same time, while other sensations do not awaken reflection, since they do not involve such contradictions."
" Now I understand," he said, " and agree with you."
" Well; to which of these two classes do you refer number and unity ? "
" I cannot decide," he answered.
" Well then, let what we have already said help you to reach a decision. If we gain a sufficient knowledge of simple unity by the sight or any other sense, as in the case of the finger we just now mentioned, there would be nothing in this experience to draw the mind toward being; but if unity always presents at the same time some contradiction, so that it shall appear to be no more unity than the opposite of unity, there will then be need of an arbiter to decide; the soul in such a case is necessarily perplexed, and inwardly arousing thought, is forced to pursue further inquiries and to ask herself this question, ' What of a truth is unity in itself ? ' And thus the study of the unit will be one of the agencies that lead and turn the soul to the contemplation of true being."
" And surely," he said, " this is precisely what happens in the visual perception of unity; for at the same moment we see the same object to be one and multiplied to infinity."
"If then this is true in the case of unity, is it not equally true of all numbers whatsoever they may be?"
" But surely calculation and arithmetic have to do exclusively with number."
" No doubt."
" And these, apparently, lead us towards truth."
" Yes, in a wonderful way."
" We may then, as it appears, include them among the studies -which we are seeking for. Indeed a knowledge of them is indispensable to the military man in marshalling his troops; to the philosopher as well, because he must rise above what is transitory and lay hold of true being; otherwise he can never become a real arithmetician."
" That is certain."
" But our guardian is at once a soldier and a philosopher."
" No doubt he is."
" It will be proper then, my dear Glaucon, to make the study of this subject a legal requirement, and to persuade those who are destined to occupy the highest official stations in the State to devote themselves to the science of reckoning, not in any superficial method, but to persevere until, by dint of sheer reflection, they come to look at the nature of numbers; to use them, not after the manner of merchants and shopkeepers, in buying and selling, but to apply them to the science of war, and to assist the soul herself in rising from transitory things toward truth and being."
" Admirably spoken."
" And verily," I said, " seeing we have entered upon the discussion of this science which treats of calculation, I have but just now discovered how elegant it is, and in how many ways it promotes the attainment of our end, provided one pursues it as a seeker after knowledge, and not as a shopkeeper."
" What do you mean ? " he asked.
" I mean, as we were just saying, that this science gives a powerful upward impulse, compelling the soul to reason about numbers themselves, absolutely declining the discussion when any one proposes to bring into the argument numbers which relate to visible or tangible objects. For you doubtless know that the expert arithmeticians ridicule and reject the attempt of any one, while computing, to divide unity itself; but as often as you divide it they multiply it for fear the unit should ever appear not one, but as a combination of many parts."
" Perfectly true."
" And suppose, Glaucon, a person were to say to them: ' Admirable men, about what kind of numbers are you speaking? Where are the numbers in which there is a unity such as you require, where each unit is perfectly equal to every other, showing not the slightest difference, and admitting no division into parts ?' What do you believe they would answer ? "
" They will reply, I believe, that they are speaking of numbers which are not objects of sense, and which cannot be apprehended otherwise than by the mind."
" Do you see then, my friend," I said, " that we cannot absolutely forego this science, since it manifestly compels the soul to employ the pure intelligence in the quest of pure truth ? "
" Yes," he replied; " it is certainly a powerful agent in bringing this about."
" Have you also observed this, that born calculators apply themselves with great facility to almost all kinds of learning; and even the sluggish minds, if they are trained and exercised in this science, though they derive no other advantage from it, yet they always make such progress as to gain keener insight than they had before?"
" That is certain."
" And further, as I think, you will not easily find many sciences which cost more toil in acquiring and practicing than this."
" No, indeed."
" Then, for all these reasons we cannot give up this study, but the best endowed minds must be trained in it."
" I agree with you."
IX. "Therefore, let this study," I continued, " be chosen as one part of our education. But in the next place let us consider whether the subject which is closely related to arithmetic is suited to our purpose or not."
" What is that ? Do you mean geometry ? "
" The very same," I replied.
" Manifestly," he said, " geometry does concern us, as far as it is related to the operations of war. For in pitching a camp, or taking possession of strongholds, in closing up or extending the lines of an army, and in executing all the evolutions which are employed in battle and in marching, it will make every difference to the general whether he is, or is not a geometrician."
" Yes," I answered, " but for all such purposes a slight knowledge of geometry and calculation will be sufficient. But the question to be considered is whether the principal and more advanced part of geometry has a tendency to promote our chief end, to enable the mind with greater ease to contemplate the idea of good. And thither, we may say, tend all the sciences which compel the soul to turn herself towards that region where dwells the most blessed part of being which the soul must, by all means, contemplate."
" Rightly spoken," he replied.
" If, therefore, geometry compels the soul to contemplate the essence of things, it concerns us, but if it limits her vision to what is transitory and perishable, it does not concern us."
" Yes, that is what we maintain."
" Very well," I said, " the man who has acquired even a smattering of geometry will not contradict us on one point at least, our assertion that this science has nothing in common with the language of those who practice it."
"How so?" he asked.
" They make use of a truly ridiculous and poverty-stricken language. For they speak of squaring, applying, adding and the like, as if they were engaged in some industrial pursuits, and all their demonstrations had a practical end in view; whereas, the whole aim and object of this science is knowledge."
" That is undoubtedly true."
" Must not this point also be conceded ? "
"What point?"
