Arithmetic has a practical and also a philosophical use, the latter the higher. 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 [C]who are 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 [D]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?
The higher arithmetic is concerned, not with visible or tangible objects, but with abstract numbers. 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 [E]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[4], taking care that one shall continue one and not become lost in fractions. 228
[4] Meaning either (1) that they integrate the number because they deny the possibility of fractions; or (2) that division is regarded by them as a process of multiplication, for the fractions of one continue to be units.