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The Good, or Form of the Good
The highest object of knowledge. What makes justice and everything else useful and beneficial. Socrates says he cannot describe it directly, only its offspring and likeness.
Why the popular answers to "what is the Good" fail
Pleasure fails because there are bad pleasures. Knowledge fails because "knowledge of the good" is circular and explains nothing.
Simile of the Sun
As the sun gives visible things their visibility and also their coming to be and growth, the Good gives intelligible things their knowability and also their being.
The three-part condition of sight
Sight needs an eye, a visible object, and a third thing neither of them supplies - light. The sun is the source of that third thing, and the Good plays the same role for knowing.
"Beyond being"
The Good is not itself being. It exceeds being in dignity and power, and is the source of both being and knowledge, so it stands above both.
Divided line
A line cut into two unequal parts, visible and intelligible, each cut again in the same ratio. Four segments, each with its own kind of object and its own state of mind.
Eikasia, imagination or conjecture
Lowest segment of the line. Its objects are shadows and reflections in water and polished surfaces.
Pistis, belief or trust
Second segment. Its objects are ordinary visible things - animals, plants, manufactured items. Not knowledge, but not mere illusion.
Dianoia, thought
Third segment. Mathematics. It uses visible diagrams as images, starts from hypotheses it does not examine, and moves downward to conclusions.
Noesis, understanding or dialectic
Highest segment. It moves upward from hypotheses to an unhypothetical first principle, the Good, then back down using Forms alone, with no images at all.
The ratio that governs the line
Each segment stands to the next in clarity and truth as its objects stand in reality. The line maps degrees of being onto degrees of cognition.
Allegory of the Cave
Prisoners chained from birth watch shadows cast on a wall by puppets carried before a fire. One is freed, turned around, dragged up out of the cave into the sun, and then returns.
Cave, the shadows on the wall
The images the prisoners take for reality. They correspond to eikasia, the lowest segment of the line.
Cave, the puppets and the fire
The actual objects being carried and the firelight that casts their shadows. They correspond to pistis, the ordinary visible world.
Cave, the ascent
The painful, resisted climb out of the cave. It stands for the soul's journey from the visible world up to the intelligible.
Cave, the sun outside
The Form of the Good, seen last and with the greatest difficulty, and only after the eyes adjust by stages - first shadows, then reflections, then objects, then the stars, then the sun itself.
Cave, the return
The freed prisoner goes back down, stumbles in the dark, is mocked for having ruined his eyes, and Socrates says the prisoners would kill anyone who tried to free them. An unmistakable allusion to Socrates's own death.
Periagoge, the turning around
Education is not pouring sight into blind eyes. The capacity to know is already in every soul; education turns the whole soul away from becoming and toward being.
Education as the craft of turning
Since sight is already present, the real art is knowing how to turn the soul most easily and effectively, not how to implant knowledge.
Why philosophers must rule
Only those who have seen the Good can order a city or a soul correctly. They must be compelled to go back down into the cave, which is just because the city raised them for exactly that.
How sun, line, and cave fit together
All three model the same structure. The Good is to the intelligible what the sun is to the visible, and the cave's four stages map onto the line's four segments.
What the cave says about ordinary life
The condition of most people is not stupidity but being chained, facing one way, with their only evidence being shadows. Error here is structural, not personal.