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The dialogue between Socrates and Glaucon is probably fictitious and composed by Plato; whether or not the allegory originated with Socrates, or if Plato is using his mentor as a stand-in for his own idea, is unclear.
In the dialogue, Socrates asks Glaucon to imagine a cave, in which prisoners are kept. These prisoners have been in the cave since their childhood, and each of them is held there in a peculiar manner - they are all chained so that their legs and necks are immobile, forced to look at a wall in front of them. Behind the prisoners is a fire and between the fire and the prisoners is a raised walkway, on which people can walk.
These people are puppeteers, and they are carrying objects, in the shape of human and animal figures, as well as everyday items. The prisoners could only see these flickering images on the wall, since they could not move their heads; and so, naturally enough, they presumed the images to be real, rather than just shadowy representations of what is actually real. In fact, Socrates claimed, the images on the wall would be so real that the prisoners would assign prestige among each other to the one who could recall the most detail about the shapes, the order in which they appeared and which might typically be found together or in tandem. Of course, Socrates would point out, this was hollow praise, since in fact the images were not real.
Then Socrates offered a twist in the plot - what if one of the prisoners were to be freed and made to turn and look at the fire? The bright light would hurt his eyes, as accustomed as he was to the shadows, and even in turning back to the wall and its flickering images (which would be only natural), the prisoner couldn't help but notice that they weren't real at all, but only shadows of the real items on the walkway behind him.
If the prisoner was then taken from the cave and brought into the open, the disorientation would be even more severe; the light of the sun would be much more brilliant than the fire. But as his eyes adjusted, the newly freed prisoner would be able to see beyond only shadows; he would see dimensions and reflections in the water (even of himself). After learning of the reality of the world, the prisoner now sees how 'pitiable' his former colleagues in the cave really are. If he returned to the cave and rejoined them, he would take no pleasure in their accolades or praise for knowledge of the shadow-figures; for their own part, the prisoners would see him as deranged, not really knowing what reality is and would say of him that he left the cave and returned with corrupted eyes.
Socrates's (and Plato's) point is that, once we understand what reality is (the forms), it is the job of the informed to lead the ignorant 'out of the cave' and into true knowledge. This means, of course, that those who still are uninformed will resist since, after all, the cave is all they've ever known. But, this doesn't change the obligation of the enlightened philosopher to try (and keep trying) to help his fellow citizens.
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