Summary by Wikipedia:
The Republic contains Plato's Allegory of the cave with which he explains his concept of The Forms as an answer to the problem of universals. The allegory of the cave is an attempt to justify the philosopher's place in society.
Plato imagines a group of people who have lived in a cave all of their lives, chained to a wall in the subterranean so they cannot see outside nor look behind them. Behind these prisoners is a constant flame that illuminates various statues that are moved by others, which cause shadows to flicker around the cave.
When the people of the cave see these shadows they realize how imitative they are of human life, and begin to ascribe forms to these shadows such as either "dog" or "cat". The shadows are as close as the prisoners get to seeing reality, according to Plato.
Plato then goes on to explain how the philosopher is a former prisoner who is freed from the cave and comes to understand that the shadows on the wall are not constitutive of reality at all. He sees that the fire and the statues which cause the shadows are indeed more real than the shadows themselves, and therefore apprehends how the prisoners are so easily deceived.
Plato then imagines that the freedman is taken outside of the cave and into the real world. The prisoner is initially blinded by the light. However when he adjusts to the brightness, he eventually understands that all of the real objects around him are illuminated by the sun (which represents the Form of the Good, the form which has caused the brightness). He also realizes it is the sun to which he is indebted for being able to see the beauty and goodness in the objects around him. The freedman is finally cognizant that the fire and statues in the cave were just copies of the real objects in the world.
The prisoner's stages of understanding correlate with the levels on the divided line that Plato imagines. The line is divided into what is the visible world, and what the intelligible world is, with the divider being the Sun. When the prisoner is in the cave, he is obviously in the visible realm that receives no sunlight, and outside he comes to be in the intelligible realm.
The shadows in the cave that the prisoners can see correspond to the lowest level on Plato's line, that of imagination and conjecture. Once the prisoner is freed and spots the fire's reflection onto the statues which causes the shadows in the cave, he reaches the second stage on the divided line, and that is the stage of belief, as the freedman comes to believe that the statues in the cave are real as can be.
On leaving the cave however, the prisoner comes to see objects more real than the statues inside of the cave, and this correlates with the third stage on Plato's line as being understanding. The prisoner is therefore able to ascribe Forms to objects as they exist outside of the cave. Lastly, the prisoner turns to the sun which he grasps as the source of truth, or the Form of the Good, and this last stage, named as dialectic, is the highest possible stage on the line. The prisoner, as a result of the Form of the Good, can begin to understand all other forms in reality.
Allegorically, Plato reasons that the freedman is the philosopher, who is the only person able to discern the Form of the Good, and thus absolute goodness and truth. At the end of this allegory, Plato asserts that it is the philosopher's burden to reenter the cave.
Those who have seen the ideal world, he says, have the duty to educate those in the material world, or spread the light to those in darkness. Since the philosopher is the only one able to recognize what is truly good, and only he can reach the last stage on the divided line, only he is fit to rule and educate society according to Plato.
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