For Plato, our lives are temporary and fleeting. Plato believed, however, in a realm beyond this life. He looked forward to a time when his soul would be liberated from his body and reach a heavenly realm of wisdom. There he could philosophize without bodily distractions and contemplate ultimate reality, or what he thought of as simply “the good.”
Plato tells us that the everyday world is less real than we think and illustrates this through an allegory. Imagine prisoners in a cave, sitting chained to a bench, facing a wall in front of them. Behind them, a fire is burning. Between the prisoners and the fire, there is a raised walkway. As people move across it, their shadows dance on the wall and their voices echo. The prisoners think the sounds come from the wall.
They have been sitting there for as long as they remember. Their world is a mutually experienced reality of shadows and sounds.
Plato suggests we are also prisoners. Our everyday world is a prison where we are trapped in bodies. In Plato’s cave, a fire projects a shadow world. All would be dark without it. In our world, the sun is our source of illumination. We see, thanks to it. Just as the fire enables the shadow world, so does the sun enable the visible world, and just as the prisoners in the cave were mistaken about the shadow world, so are we mistaken about the visible world. The visible world is a world of belief and imagination.
We can only attain understanding and knowledge in what Plato calls the intelligible world—a higher unchanging reality of forms. Many believe in a mathematical reality and that mathematics is reflected in nature—in the shapes of snowflakes, trees, or mountains. This is essentially the Platonic view that numbers and other mathematical objects have a reality reflected in the physical world. Plato thinks the mathematical square is more real than squares we draw on paper, make of wood, or otherwise construct. The mathematical object is more real than our replications.
Even if we could build squares with lines of single atoms, they would be imperfect. Atoms jiggle and have irregular shapes. Only mathematical objects are perfect. Mathematical objects are also such that we can reason about them and have true understanding of them. Plato, however, does not see mathematics as a source of the highest knowledge. Mathematics helps us to discipline our minds, but mathematics is not a source of wisdom. Mathematics cannot tell us what justice is or how to live the good life. These questions require philosophical—not mathematical—inquiry. Plato yearns to grasp the universal form of justice that exists apart from all concrete acts of justice.
Plato’s forms exist eternally and independently of our minds in a separate realm, where they are hierarchically arranged. The highest form is “the good,” comparable to the sun in the visible world. Without the sun, life would not be possible and there would be no visible world. In the intelligible world, the good brings another kind of illumination—the possibility of true knowledge. Without it, there could be a world neither of knowledge nor of understanding. Trying to grasp the intelligible world, we come closer to ultimate truth—the good. The quest for the good is the quest for reality. Ultimate reality is, for Plato, the good and the other forms. Whatever is not a form is, through reflection, what lies at the end of a chain beginning with a form. Thus, the entire visible world has a derived form of reality.
-  Plato introduces the allegory of the cave in Book VII of The Republic (Plato 2004).