Plato
The Divided Line

from The Republic

Conceive, then, that there are these two powers I speak of, the Good reigning over the domain of all that is intelligible, the Sun over the visible world—or the heaven as I might call it; only you would think I was showing off my skill in etymology. At any rate you have these two orders of things clearly before your mind: the visible and the intelligible?

I have.

Now take a line divided into two unequal parts, one to represent the visible order, the other the intelligible; and divide each part again in the same proportion, symbolizing degrees of comparative clearness or obscurity. Then (A) one of the two sections in the visible world will stand for images. By images I mean first shadows, and then reflections in water or in close-grained, polished surfaces, and everything of that kind, if you understand.

Yes, I understand.

Let the second section (B) stand for the actual things of which the first are likenesses, the living creatures about us and all the works of nature or of human hands.

So be it.

Will you also take the proportion in which the visible world has been divided as corresponding to degrees of reality and truth, so that the likeness shall stand to the original in the same ratio as the sphere of appearances and belief to the sphere of knowledge?

Certainly.

Now consider how we are to divide the part which stands for the intelligible world. There are two sections. In the first (C) the mind uses as images those actual things which themselves had images in the visible world; and it is compelled to pursue its inquiry by starting from assumptions and traveling, not up to a principle, but down to a conclusion. In the second (D) the mind moves in the other direction, from an assumption up towards a principle which is not hypothetical; and it makes no use of the images employed in the other section, but only of Forms, and conducts its inquiry solely by their means.

I don’t quite understand what you mean.

Then we will try again; what I have just said will help you to understand. (C) You know, of course, how students of subjects like geometry and arithmetic begin by postulating odd and even numbers, or the various figures and the three kinds of angle, and other such data in each subject. These data they take as known; and, having adopted them as assumptions, they do not feel called upon to give any account of them to themselves or to anyone else, but treat them as self-evident. Then, starting from these assumptions, they go on until they arrive, by a series of consistent steps, at all the conclusions they set out to investigate.

Yes, I know that.

You also know how they make use of visible figures and discourse about them, though what they really have in mind is the originals of which these figures are images: they are not reasoning, for instance, about this particular square and diagonal which they have drawn, but about the Square and the Diagonal; and so in all cases. The diagrams they draw and the models they make are actual things, which may have their shadows or images in water; but now they serve in their turn as images, while the student is seeking to behold those realities which only thought can apprehend.

True.

This, then, is the class of things that I spoke of as intelligible, but with two qualifications: first, that the mind, in studying them, is compelled to employ assumptions, and, because it cannot rise above these, does not travel upwards to a first principle; and second, that it uses as images those actual things which have images of their own in the section below them and which, in comparison with those shadows and reflections, are reputed to be more palpable and valued accordingly.

I understand: you mean the subject-matter of geometry and of the kindred arts.

(D) Then by the second section of the intelligible world you may understand me to mean all that unaided reasoning apprehends by the power of dialectic, when it treats its assumptions, not as first principles, but as hypotheses in the literal sense, things ‘laid down’ like a flight of steps up which it may mount all the way to something that is not hypothetical, the first principle of all; and having grasped this, may turn back and, holding on to the consequences which depend upon it, descend at last to a conclusion, never making use of any sensible object, but only of Forms, moving through Forms from one to another, and ending with Forms.

I understand, he said, though not perfectly; for the procedure you describe sounds like an enormous undertaking. But I see that you mean to distinguish the field of intelligible reality studied by dialectic as having a greater certainty and truth than the subject matter of the ‘arts,’ as they are called, which treat their assumptions as first principles. The students of these arts are, it is true, compelled to exercise thought in contemplating objects which the senses cannot perceive; but because they start from assumptions without going back to a first principle, you do not regard them as gaining true understanding about those objects, although the objects themselves, when connected with a first principle, are intelligible. And I think you would call the state of mind of the students of geometry and other such arts, not intelligence, but thinking, as being something between intelligence and mere acceptance of appearances.

You have understood me quite well enough, I replied. And now you may take, as corresponding to the four sections, these four states of mind: intelligence for the highest, thinking for the second, belief for the third, and for the last imagining. These you may arrange as the terms in a proportion, assigning to each a degree of clearness and certainty corresponding to the measure in which their objects possess truth and reality.

I understand and agree with you. I will arrange them as you say.