But what branch of knowledge is there, my dear Glaucon, which is of the desired nature; since all the useful arts were reckoned mean by us?
Undoubtedly; and yet if music and gymnastic are excluded, and the arts are also excluded, what remains?
Well, I said, there may be nothing left of our special subjects; and then we shall have to take something which is not special, but of universal application.
What may that be?
A something which all arts and sciences and intelligences use in common, and which every one first has to learn among the elements of education.
What is that?
The little matter of distinguishing one, two, and three--in a word, number and calculation:--do not all arts and sciences necessarily partake of them?
Yes.
Then the art of war partakes of them?
To the sure.
Then Palamedes, whenever he appears in tragedy, proves Agamemnon ridiculously unfit to be a general.

Did you never remark how he declares that he had invented number, and had numbered the ships and set in array the ranks of the army at Troy; which implies that they had never been numbered before, and Agamemnon must be supposed literally to have been incapable of counting his own feet--how could he if he was ignorant of number?
And if that is true, what sort of general must he have been?
I should say a very strange one, if this was as you say.
Can we deny that a warrior should have a knowledge of arithmetic?
Certainly he should, if he is to have the smallest understanding of military tactics, or indeed, I should rather say, if he is to be a man at all.
I should like to know whether you have the same notion which I have of this study?
What is your notion?
It appears to me to be a study of the kind which we are seeking, and which leads naturally to reflection, but never to have been rightly used; for the true use of it is simply to draw the soul towards being.
Will you explain your meaning?
he said.
I will try, I said; and I wish you would share the enquiry with me, and say 'yes' or 'no' when I attempt to distinguish in my own mind what branches of knowledge have this attracting power, in order that we may have clearer proof that arithmetic is, as I suspect, one of them.
Explain, he said.
I mean to say that objects of sense are of two kinds; some of them do not invite thought because the sense is an adequate judge of them; while in the case of other objects sense is so untrustworthy that further enquiry is imperatively demanded.
You are clearly referring, he said, to the manner in which the senses are imposed upon by distance, and by painting in light and shade.
No, I said, that is not at all my meaning.
Then what is your meaning?
When speaking of uninviting objects, I mean those which do not pass from one sensation to the opposite; inviting objects are those which do; in this latter case the sense coming upon the object, whether at a distance or near, gives no more vivid idea of anything in particular than of its opposite.

An illustration will make my meaning clearer:--here are three fingers--a little finger, a second finger, and a middle finger.
Very good.
You may suppose that they are seen quite close: And here comes the point.
What is it?
Each of them equally appears a finger, whether seen in the middle or at the extremity, whether white or black, or thick or thin--it makes no difference; a finger is a finger all the same.

In these cases a man is not compelled to ask of thought the question, what is a finger?
for the sight never intimates to the mind that a finger is other than a finger.
True.
And therefore, I said, as we might expect, there is nothing here which invites or excites intelligence.
There is not, he said.
But is this equally true of the greatness and smallness of the fingers?
Can sight adequately perceive them?
and is no difference made by the circumstance that one of the fingers is in the middle and another at the extremity?
And in like manner does the touch adequately perceive the qualities of thickness or thinness, or softness or hardness?
And so of the other senses; do they give perfect intimations of such matters?
Is not their mode of operation on this wise--the sense which is concerned with the quality of hardness is necessarily concerned also with the quality of softness, and only intimates to the soul that the same thing is felt to be both hard and soft?
You are quite right, he said.
And must not the soul be perplexed at this intimation which the sense gives of a hard which is also soft?
What, again, is the meaning of light and heavy, if that which is light is also heavy, and that which is heavy, light?
Yes, he said, these intimations which the soul receives are very curious and require to be explained.
Yes, I said, and in these perplexities the soul naturally summons to her aid calculation and intelligence, that she may see whether the several objects announced to her are one or two.
True.
And if they turn out to be two, is not each of them one and different?
Certainly.
And if each is one, and both are two, she will conceive the two as in a state of division, for if there were undivided they could only be conceived of as one?
True.
The eye certainly did see both small and great, but only in a confused manner; they were not distinguished.
Yes.
Whereas the thinking mind, intending to light up the chaos, was compelled to reverse the process, and look at small and great as separate and not confused.
Very true.
Was not this the beginning of the enquiry 'What is great ?' and 'What is small ?' Exactly so.
And thus arose the distinction of the visible and the intelligible.
Most true.
This was what I meant when I spoke of impressions which invited the intellect, or the reverse--those which are simultaneous with opposite impressions, invite thought; those which are not simultaneous do not.
I understand, he said, and agree with you.
And to which class do unity and number belong?
I do not know, he replied.
Think a little and you will see that what has preceded will supply the answer; for if simple unity could be adequately perceived by the sight or by any other sense, then, as we were saying in the case of the finger, there would be nothing to attract towards being; but when there is some contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks 'What is absolute unity ?' This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of true being.
And surely, he said, this occurs notably in the case of one; for we see the same thing to be both one and infinite in multitude?
Yes, I said; and this being true of one must be equally true of all number?
Certainly.
And all arithmetic and calculation have to do with number?
Yes.
And they appear to lead the mind towards truth?
Yes, in a very remarkable manner.
Then this is knowledge of the kind for which we are seeking, having a double use, military and philosophical; for the man of war must learn the art of number or he will not know how to array his troops, and the philosopher also, because he has to rise out of the sea of change and lay hold of true being, and therefore he must be an arithmetician.
That is true.
And our guardian is both warrior and philosopher?
Certainly.
Then this is a kind of knowledge which legislation may fitly prescribe; and we must endeavour to persuade those who are prescribe to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being.
That is excellent, he said.
Yes, I said, and now having spoken of it, I must add how charming the science is! and in how many ways it conduces to our desired end, if pursued in the spirit of a philosopher, and not of a shopkeeper! How do you mean?
I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument.

You know how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply, taking care that one shall continue one and not become lost in fractions.
That is very true.
Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible,--what would they answer?
They would answer, as I should conceive, that they were speaking of those numbers which can only be realised in thought.
Then you see that this knowledge may be truly called necessary, necessitating as it clearly does the use of the pure intelligence in the attainment of pure truth?
Yes; that is a marked characteristic of it.
And have you further observed, that those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the dull if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would otherwise have been.
Very true, he said.
And indeed, you will not easily find a more difficult study, and not many as difficult.
You will not.
And, for all these reasons, arithmetic is a kind of knowledge in which the best natures should be trained, and which must not be given up.
I agree.
Let this then be made one of our subjects of education.