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The Importance of Mathematics in the Education of the Philosopher

390-360 b.C.

The Republic, Book VII, 521 C - 527 D

The passage presented follows the celebre allegory of the Myth of the Cave. The text helps to understand Plato's view on mathematics and its function as mean to get to the contemplation of the Goodness. In the Platonic thought, the mathematical entities are intermediate between the multiple becoming of the sensible world and the immutable identity of the ideas. In the following excerpt, Plato focuses on the essential subjects for the education of the philosophers. The first proposal is arithmetic, described as “indispensable for us, since it plainly compels the soul to employ pure thought with a view to truth itself”. It helps pjilosophers to understand that the unity of which we talk about is not a single sensible and divisible thing. Furthermore, those who have ability in calculation acquires readiness also in all the other sciences. In addition to arithmetic, geometry is defined as “the knowledge of the eternally existent”. That is why it is the second essential discipline for the education of the philosophers.

“What, then, Glaucon, would be the study that would draw the soul away from the world of becoming to the world of being? A thought strikes me while I speak: Did we not say that these men in youth must be athletes of war

“We did.”

“Then the study for which we are seeking must have this additional qualification.”

“What one?”

“That it be not useless to soldiers.

“Why, yes, it must,” he said, “if that is possible.” [521e]

“But in our previous account they were educated in gymnastics and music.

“They were, he said.

“And gymnastics, I take it, is devoted to that which grows and perishes; for it presides over the growth and decay of the body.


“Then this cannot be the study [522a] that we seek.”


“Is it, then, music, so far as we have already described it?

“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, 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.”

“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.”


“Why, for example, this common thing that all arts and forms of thought and all sciences employ, and which is among the first things that everybody must learn.”

“What?” he said.

“This trifling matter,” 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, “Palamedes in the play is always making Agamemnon appear a most ridiculous 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.” [...]

“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, 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, [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 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 [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, they multiply, always on guard lest the one should appear to be not one but a multiplicity of parts.

“Most true,” he replied. 

[526a] “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 [526b] 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. [526c]

“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, [526d] “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 [526e] 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.” [527a]

“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.

“How so?” he said.

“Their language is most ludicrous, though they cannot help it, for they speak as if they were doing something and as if all their words were directed towards action. For all their talk is of squaring and applying and adding and the like, whereas in fact [527b] the real object of the entire study is pure knowledge.

“That is absolutely true,” he said.

“And must we not agree on a further point?”


“That it is the knowledge of that which always is, 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. [527c]

“Then nothing is surer,” said I, “than that we must require that the men of your Fair Cityshall 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 there will be an immeasurable 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. [527c]

Source of the English digital text: Perseus Digital Library at the Tufts University