Complete Works (242 page)

Of course.

If it’s evidently two, won’t each be evidently distinct and one?

Yes.

Then, if each is one, and both two, the soul will understand that the two are separate, for it wouldn’t understand the inseparable to be two, but rather one. [c]

That’s right.

Sight, however, saw the big and small, not as separate, but as mixed up together. Isn’t that so?

Yes.

And in order to get clear about all this, understanding was compelled to see the big and the small, not as mixed up together, but as separate—the opposite way from sight.

True.

And isn’t it from these cases that it first occurs to us to ask what the big is and what the small is?

Absolutely.

And, because of this, we called the one the intelligible and the other the visible.

That’s right. [d]

This, then, is what I was trying to express before, when I said that some things summon thought, while others don’t. Those that strike the relevant sense at the same time as their opposites I call summoners, those that don’t do this do not awaken understanding.

Now I understand, and I think you’re right.

Well, then, to which of them do number and the one belong?

I don’t know.

Reason it out from what was said before. If the one is adequately seen itself by itself or is so perceived by any of the other senses, then, as we were saying in the case of fingers, it wouldn’t draw the soul towards being. But if something opposite to it is always seen at the same time, so that [e] nothing is apparently any more one than the opposite of one, then something would be needed to judge the matter. The soul would then be puzzled, would look for an answer, would stir up its understanding, and would ask what the one itself is. And so this would be among the subjects that lead the soul and turn it around towards the study of that which is.
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But surely the sight of the one does possess this characteristic to a remarkable degree, for we see the same thing to be both one and an unlimited number at the same time.

Then, if this is true of the one, won’t it also be true of all numbers?

Of course.

Now, calculation and arithmetic are wholly concerned with numbers.

That’s right.

[b] Then evidently they lead us towards truth.

Supernaturally so.

Then they belong, it seems, to the subjects we’re seeking. They are compulsory for warriors because of their orderly ranks and for philosophers because they have to learn to rise up out of becoming and grasp being, if they are ever to become rational.

That’s right.

And our guardian must be both a warrior and a philosopher.

Certainly.

Then it would be appropriate, Glaucon, to legislate this subject for those who are going to share in the highest offices in the city and to persuade them to turn to calculation and take it up, not as laymen do, but staying [c] with it until they reach the study of the natures of the numbers by means of understanding itself, nor like tradesmen and retailers, for the sake of buying and selling, but for the sake of war and for ease in turning the soul around, away from becoming and towards truth and being.

Well put.

Moreover, it strikes me, now that it has been mentioned, how sophisticated the subject of calculation is and in how many ways it is useful for [d] our purposes, provided that one practices it for the sake of knowing rather than trading.

How is it useful?

In the very way we were talking about. It leads the soul forcibly upward and compels it to discuss the numbers themselves, never permitting anyone to propose for discussion numbers attached to visible or tangible bodies. You know what those who are clever in these matters are like: If, in the course of the argument, someone tries to divide the one itself, they laugh [e] and won’t permit it. If you divide it, they multiply it, taking care that one thing never be found to be many parts rather than one.

That’s very true.

Then what do you think would happen, Glaucon, if someone were to
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ask them: “What kind of numbers are you talking about, in which the one is as you assume it to be, each one equal to every other, without the least difference and containing no internal parts?”

I think they’d answer that they are talking about those numbers that can be grasped only in thought and can’t be dealt with in any other way.

[b] Then do you see that it’s likely that this subject really is compulsory for us, since it apparently compels the soul to use understanding itself on the truth itself?

Indeed, it most certainly does do that.

And what about those who are naturally good at calculation or reasoning? Have you already noticed that they’re naturally sharp, so to speak, in all subjects, and that those who are slow at it, if they’re educated and exercised in it, even if they’re benefited in no other way, nonetheless improve and become generally sharper than they were?

That’s true.

Moreover, I don’t think you’ll easily find subjects that are harder to learn or practice than this. [c]

No, indeed.

Then, for all these reasons, this subject isn’t to be neglected, and the best natures must be educated in it.

I agree.

Let that, then, be one of our subjects. Second, let’s consider whether the subject that comes next is also appropriate for our purposes.

What subject is that? Do you mean geometry?

That’s the very one I had in mind.

Insofar as it pertains to war, it’s obviously appropriate, for when it [d] comes to setting up camp, occupying a region, concentrating troops, deploying them, or with regard to any of the other formations an army adopts in battle or on the march, it makes all the difference whether someone is a geometer or not.

But, for things like that, even a little geometry—or calculation for that matter—would suffice. What we need to consider is whether the greater and more advanced part of it tends to make it easier to see the form of the good. And we say that anything has that tendency if it compels the [e] soul to turn itself around towards the region in which lies the happiest of the things that are, the one the soul must see at any cost.

You’re right.

Therefore, if geometry compels the soul to study being, it’s appropriate, but if it compels it to study becoming, it’s inappropriate.

So we’ve said, at any rate.

Now, no one with even a little experience of geometry will dispute that
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this science is entirely the opposite of what is said about it in the accounts of its practitioners.

How do you mean?

They give ridiculous accounts of it, though they can’t help it, for they speak like practical men, and all their accounts refer to doing things. They talk of “squaring,” “applying,” “adding,” and the like, whereas the entire subject is pursued for the sake of knowledge. [b]

Absolutely.

And mustn’t we also agree on a further point?

What is that?

That their accounts are for the sake of knowing what always is, not what comes into being and passes away.

That’s easy to agree to, for geometry
is
knowledge of what always is.

Then it draws the soul towards truth and produces philosophic thought by directing upwards what we now wrongly direct downwards.

As far as anything possibly can.

[c] Then as far as
we
possibly can, we must require those in your fine city not to neglect geometry in any way, for even its by-products are not insignificant.

What are they?

The ones concerned with war that you mentioned. But we also surely know that, when it comes to better understanding any subject, there is a world of difference between someone who has grasped geometry and someone who hasn’t.

Yes, by god, a world of difference.

Then shall we set this down as a second subject for the young?

Let’s do so, he said.

And what about astronomy? Shall we make it the third? Or do you disagree? [d]

That’s fine with me, for a better awareness of the seasons, months, and years is no less appropriate for a general than for a farmer or navigator.

You amuse me: You’re like someone who’s afraid that the majority will think he is prescribing useless subjects. It’s no easy task—indeed it’s very difficult—to realize that in every soul there is an instrument that is purified and rekindled by such subjects when it has been blinded and destroyed [e] by other ways of life, an instrument that it is more important to preserve than ten thousand eyes, since only with it can the truth be seen. Those who share your belief that this is so will think you’re speaking incredibly well, while those who’ve never been aware of it will probably think you’re talking nonsense, since they see no benefit worth mentioning in these subjects. So decide right now which group you’re addressing. Or are your
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arguments for neither of them but mostly for your own sake—though you won’t begrudge anyone else whatever benefit he’s able to get from them?

The latter: I want to speak, question, and answer mostly for my own sake.

Then let’s fall back to our earlier position, for we were wrong just now about the subject that comes after geometry.

What was our error?

After plane surfaces, we went on to revolving solids before dealing with solids by themselves. But the right thing to do is to take up the third [b] dimension right after the second. And this, I suppose, consists of cubes and of whatever shares in depth.

You’re right, Socrates, but this subject hasn’t been developed yet.

There are two reasons for that: First, because no city values it, this difficult subject is little researched. Second, the researchers need a director, for, without one, they won’t discover anything. To begin with, such a director is hard to find, and, then, even if he could be found, those who [c] currently do research in this field would be too arrogant to follow him. If an entire city helped him to supervise it, however, and took the lead in valuing it, then he would be followed. And, if the subject was consistently and vigorously pursued, it would soon be developed. Even now, when it isn’t valued and is held in contempt by the majority and is pursued by researchers who are unable to give an account of its usefulness, nevertheless, in spite of all these handicaps, the force of its charm has caused it to develop somewhat, so that it wouldn’t be surprising if it were further developed even as things stand.

The subject
has
outstanding charm. But explain more clearly what you [d] were saying just now. The subject that deals with plane surfaces you took to be geometry.

Yes.

And at first you put astronomy after it, but later you went back on that.

In my haste to go through them all, I’ve only progressed more slowly. The subject dealing with the dimension of depth was next. But because it is in a ridiculous state, I passed it by and spoke of astronomy (which deals with the motion of things having depth) after geometry. [e]

That’s right.

Let’s then put astronomy as the fourth subject, on the assumption that solid geometry will be available if a city takes it up.

That seems reasonable. And since you reproached me before for praising astronomy in a vulgar manner, I’ll now praise it your way, for I think it’s clear to everyone that astronomy compels the soul to look upward and
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leads it from things here to things there.

It may be obvious to everyone except me, but that’s not my view about it.

Then what
is
your view?

As it’s practiced today by those who teach philosophy, it makes the soul look very much downward.

How do you mean?

In my opinion, your conception of “higher studies” is a good deal too generous, for if someone were to study something by leaning his head back and studying ornaments on a ceiling, it looks as though you’d say he’s studying not with his eyes but with his understanding. Perhaps you’re [b] right, and I’m foolish, but I can’t conceive of any subject making the soul look upward except one concerned with that which is, and that which is is invisible. If anyone attempts to learn something about sensible things, whether by gaping upward or squinting downward, I’d claim—since there’s no knowledge of such things—that he never learns anything and that, even if he studies lying on his back on the ground or floating on it [c] in the sea, his soul is looking not up but down.

You’re right to reproach me, and I’ve been justly punished, but what did you mean when you said that astronomy must be learned in a different way from the way in which it is learned at present if it is to be a useful subject for our purposes?

It’s like this: We should consider the decorations in the sky to be the most beautiful and most exact of visible things, seeing that they’re embroidered on a visible surface. But we should consider their motions to fall far short of the true ones—motions that are really fast or slow as measured [d] in true numbers, that trace out true geometrical figures, that are all in relation to one another, and that are the true motions of the things carried along in them. And these, of course, must be grasped by reason and thought, not by sight. Or do you think otherwise?

Not at all.

Therefore, we should use the embroidery in the sky as a model in the study of these other things. If someone experienced in geometry were to come upon plans very carefully drawn and worked out by Daedalus or [e] some other craftsman or artist, he’d consider them to be very finely executed, but he’d think it ridiculous to examine them seriously in order to
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find the truth in them about the equal, the double, or any other ratio.

How could it be anything other than ridiculous?

Then don’t you think that a real astronomer will feel the same when he looks at the motions of the stars? He’ll believe that the craftsman of the heavens arranged them and all that’s in them in the finest way possible for such things. But as for the ratio of night to day, of days to a month, of a month to a year, or of the motions of the stars to any of them or to each other, don’t you think he’ll consider it strange to believe that they’re [b] always the same and never deviate anywhere at all or to try in any sort of way to grasp the truth about them, since they’re connected to body and visible?