Plato's Allegory of the Cave and the Divided Line

Prisoners sit in an underground den, chained by the legs and neck so that they can look only at the wall in front of them. Behind them a fire burns, and between the fire and the prisoners people carry vessels and figures of animals along a low wall^[1].

The prisoners see only the shadows these things throw on the wall. Socrates says that for them the truth would be nothing but the shadows of the images^[1].

This is the image that opens Book VII of Plato’s Republic. It comes directly after two images at the end of Book VI, the sun and the divided line^[2].

The sun

In Book VI, Socrates calls the sun the child of the good. The sun is to sight and the things we see what the good is to the mind and the things it knows^[2].

The good gives truth to what is known and the power of knowing to the knower^[2,3]. It is also the source of the being of what is known, although, Socrates says, the good is not itself being but goes beyond it in dignity and power^[2].

The divided line

Socrates then takes a line cut into two unequal parts and divides each part again in the same proportion^[2]. The lower two sections stand for the visible world and the upper two for the intelligible world, and each section has its own state of mind^[3,4].

At the level of thought, the soul treats visible things as images and reasons from hypotheses to a conclusion, the way geometers do. At the level of understanding it rises from hypotheses to a first principle and works with Forms alone, using no images^[2].

What exactly the objects of thought are, and what the hypotheses are, is much disputed^[3].

The two middle sections

The passage does not say how unequal the first cut is. Whatever the ratio, the two middle sections have the same length.

Take the whole line as 1 and cut it in the ratio $a$ to $b$, the intelligible part taking $a$ and the visible part $b$. If each part is cut again so that its upper section takes $a$ and its lower section $b$, the four sections from the top are

$$\frac{a^2}{(a+b{)}^2},\frac{ab}{(a+b{)}^2},\frac{ab}{(a+b{)}^2},\frac{b^2}{(a+b{)}^2}$$

For a line of 9 units cut in the ratio 2 to 1, the parts are 6 and 3. Cutting them again gives 4 and 2, then 2 and 1, so belief and thought each get 2 units, one on each side of the boundary between the visible and the intelligible.

Out of the cave

When one prisoner is freed and made to stand up and turn toward the fire, the glare hurts, and he cannot make out the things whose shadows he used to see^[1]. He is then dragged up a steep, rough way into the sunlight, where his eyes adjust in stages^[1].

Last of all he can look at the sun in its own place rather than its reflection in water. He concludes that the sun brings the seasons and the years and is the guardian of everything in the visible world^[1].

Back in the cave

If he goes back to his old seat, his eyes are full of darkness. While they adjust he judges the shadows badly, and the others laugh and say that he went up and came back without his eyes^[1].

They conclude that it is better not even to think of climbing up. If anyone tried to free them and lead them up, they would kill him if they could catch him^[1].

What the image stands for

Socrates explains the image himself. The prison is the world of sight, the light of the fire is the sun, and the climb out is the soul’s ascent into the intelligible world^[1].

In that world the good is seen last and only with an effort, and once seen it is recognized as the cause of everything right and beautiful^[1]. The cave reinforces what the line says^[3].

Education as turning

From the image Socrates draws a conclusion about education^[1].

The eye cannot turn from darkness to light unless the whole body turns with it, and in the same way the soul has to turn as a whole^[1]. Education, on this view, is the turning of the soul from shadows and visible things to the Forms^[4].

Why the philosophers must rule

Those who have made the climb would rather stay above. In Socrates’ city they are made to go back down and share the work and honors of the prisoners^[1].

The law aims at the happiness of the whole city, not of one class in it, and Socrates holds that the city whose rulers are least eager to rule is governed best^[1]. The cave here is the city itself, and going back into it means ruling it^[4].

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