Imagine, Socrates says, humans living in a cave, whose entrance is above them and open to the light. They've been there since childhood, with their necks and legs chained, so that they can only see in front of them. There is a fire, above and behind them, which provides a little light. There is also a path behind them, a little higher than they are. Along the path there is a low wall, like a puppeteer's screen. People move along the wall, carrying models of objects and people. Some of those carrying the models are talking.
Socrates uses an allegory about prisoners in a cave to demonstrate the effects and importance of education for the philosopher.
The prisoners can see the shadows moving along the wall, and hear the people talking. From the prisoners' perspective, truth is the movement of the shadows along the wall, shadows they assume are real.
The shadows are "copies" or reflections of the models, which themselves are copies of the Ideal Forms of the objects.
One day a prisoner escapes. He looks towards the cave's mouth. Initially dazzled by the sun's light, he realizes that the objects he sees in the light are the real versions of the shadows he saw on the walls of the cave.
Socrates compares the visible realm of the world, the world of Belief, to the cave. The prisoner's upward journey to freedom and the things above is like the journey of the soul to the world of Ideas, the world of Forms, including the Form of Goodness.
The purpose of philosophy is to lead the soul to understanding the Form of the Good, because the form of the Good encompasses all other forms.
Education is the process of turning the soul around (much as the prisoner in the cave turned around to the light) and enticing people to look in the right place for knowledge. The philosopher must be taught to recognize the Form of Goodness. Then he must return to the cave, that is to the world of Belief, in order to teach others.
The philosopher-king, with all the other children, studies music and poetry. At eighteen the best students are chosen for advanced classes in physical training and warfare for several years. At twenty, the best are again selected for more education as future guardians. The rest will become the warriors. The future guardians will study mathematics, logic and dialectic, with the goal of reaching truth, not arguing for the sake of arguing. Only the right sort of people, at about thirty, should be taught dialectic. Then they must spend fifteen years in the real world of politics, from the age of 35 to 50.
Because mathematics moves from the real to the abstract, it is good preparation for dialectic, which emphasizes abstract reasoning and logic. Notice the emphasis is on abstract reasoning, not the empirical evidence of the senses that is favored by modern science. The real world education is the equivalent of "returning to the cave."