In Republic, what does Plato mean when he says that "the realm of the visible should be compared to the prison dwelling, and the fire inside it the sun"?

In Republic, what Plato means by this is that the visible world, the world we see around us, is not the real world. Like the chained prisoners in the allegory of the cave, we mistake what we see around us as ultimately real. The fire inside the cave that casts the shadows represents our senses, which cannot give us philosophical truth, as represented in the allegory of the cave by the sun.

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In Plato's Republic, Socrates puts forward the famous allegory of the cave as a means of illustrating the difference between what is ultimately real and what isn't.

In the allegory, a number of men are chained up inside a cave. Because they are unable to move properly, all they can see are the shadows cast on the cave wall in front of them. As these are the only things they have ever known, they mistake the shadows for the objects that cast these shadows.

Those shadows are cast by a fire, which creates the illusion that the shadows on the wall are real. In that sense, the fire acts as a kind of artificial light, whereas the real light, the sun, is outside the cave. In allegorical terms, the fire represents our senses, which cannot give us a true picture of what is real.

For Plato, those of us who aren't philosophers are like the men in the cave. We mistakenly believe that the world around us, the world of other people, animals, rocks, plants, and trees, is ultimately real.

But this world is no such thing. It is, in fact, a mere copy of the real world, the world of Forms or Ideas, unchanging concepts like Truth, Beauty, and Justice. It is only in these Forms that we can discover ultimate reality.

In order for us to leave the cave, as it were, to achieve enlightenment, we need to be guided by philosophers such as Plato and Socrates. They can help us understand that ultimate reality does not reside in the everyday world given to us by the senses—as represented in the allegory by shadows and fire—but in the world of the intellect. This philosophical wisdom is represented in the allegory of the cave by the sun, which illuminates what is real.

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