In Book VII, Socrates presents the most beautiful and famous metaphor in Western philosophy: the allegory of the cave. This metaphor is meant to illustrate the effects of education on the human soul. Education moves the philosopher through the stages on the divided line, and ultimately brings him to the Form of the Good.

Socrates describes a dark scene. A group of people have lived in a deep cave since birth, never seeing the light of day. These people are bound so that they cannot look to either side or behind them, but only straight ahead. Behind them is a fire, and behind the fire is a partial wall. On top of the wall are various statues, which are manipulated by another group of people, lying out of sight behind the partial wall. Because of the fire, the statues cast shadows across the wall that the prisoners are facing. The prisoners watch the stories that these shadows play out, and because these shadows are all they ever get to see, they believe them to be the most real things in the world. When they talk to one another about “men,” “women,” “trees,” or “horses,” they are referring to these shadows. These prisoners represent the lowest stage on the line—imagination.

A prisoner is freed from his bonds, and is forced to look at the fire and at the statues themselves. After an initial period of pain and confusion because of direct exposure of his eyes to the light of the fire, the prisoner realizes that what he sees now are things more real than the shadows he has always taken to be reality. He grasps how the fire and the statues together cause the shadows, which are copies of these more real things. He accepts the statues and fire as the most real things in the world. This stage in the cave represents belief. He has made contact with real things—the statues—but he is not aware that there are things of greater reality—a world beyond his cave.

Next, this prisoner is dragged out of the cave into the world above. At first, he is so dazzled by the light up there that he can only look at shadows, then at reflections, then finally at the real objects—real trees, flowers, houses and so on. He sees that these are even more real than the statues were, and that those were only copies of these. He has now reached the cognitive stage of thought. He has caught his first glimpse of the most real things, the Forms.

When the prisoner’s eyes have fully adjusted to the brightness, he lifts his sight toward the heavens and looks at the sun. He understands that the sun is the cause of everything he sees around him—the light, his capacity for sight, the existence of flowers, trees, and other objects. The sun represents the Form of the Good, and the former prisoner has reached the stage of understanding.

The goal of education is to drag every man as far out of the cave as possible. Education should not aim at putting knowledge into the soul, but at turning the soul toward right desires. Continuing the analogy between mind and sight, Socrates explains that the vision of a clever, wicked man might be just as sharp as that of a philosopher. The problem lies in what he turns his sharp vision toward.

The overarching goal of the city is to educate those with the right natures, so that they can turn their minds sharply toward the Form of the Good. Once they have done this, they cannot remain contemplating the Form of the Good forever. They must return periodically into the cave and rule there. They need periodically to turn away from the Forms to return to the shadows to help other prisoners.

It is important to realize, when reading the allegory of the cave and of the line, that Plato means to depict not only four ways of thinking, but four ways of life. To use an example, imagine that a person in each of these stages were asked to say what courage is. The understanding of courage would differ widely from stage to stage.

Working with a possible interpretation of the imagination stage, an individual’s notion of courage in this stage would appeal to images from culture. Such an individual might try to explain courage by saying something like, “Luke Skywalker seems really courageous, so that’s courage.” An individual possessed of beliefs would also appeal to a particular example, but the example picked would be drawn from real life. There might be mention of the Marines or New York City firemen.

Someone at the stage of thought, in contrast, will try to give a definition of courage. Perhaps they will give the definition offered by Socrates in Book IV: courage as the knowledge of what is to be feared and what is not to be feared. What separates the person speaking from thought from the person possessed of understanding is that the person speaking from thought cannot inform his views with knowledge of the Form of the Good. They are working with unproven hypotheses rather than the true first principle. Even if their definition is correct, it is left open to attack and objection because their grasp of the relevant concepts stops at a certain point. Speaking from understanding, someone giving a definition comprehends all the terms in the definition and can defend each one of them based on the first principle, the Form of the Good.

Because the Form of the Good illuminates all understanding once it is grasped, knowledge is holistic. You need to understand everything to understand anything, and once you understanding anything you can proceed to an understanding of everything. All the forms are connected, and are comprehended together in the following way: you work your way up to the Form of the Good through thought until you grasp the Form of Good. Then, everything is illuminated.

Since the stages in the cave are stages of life, it seems fair to say that Plato thought that we must all proceed through the lower stages in order to reach the higher stages. Everyone begins at the cognitive level of imagination. We each begin our lives deep within the cave, with our head and legs bound, and education is the struggle to move as far out of the cave as possible. Not everyone can make it all the way out, which is why some people are producers, some warriors, and some philosopher-kings.

Given that the philosopher-kings have made it out of the cave, it might seem unfair that they are then forced back in. This is the worry that Socrates’ friends raise at the end of this section. Socrates has three lines of response to this concern. First, he reminds us again that our goal is not to make any one group especially happy, but rather to make the city as a whole as happy as possible. Second, he points out that the philosopher-kings are only able to enjoy the freedom above ground that they do because they were enabled by the education the city afforded them. They were molded to be -philosopher-kings so that they could return to the cave and rule. They owe the city this form of gratitude and service. Finally, he adds that the philosophers will actually want to rule—in a backhanded way—because they will know that the city would be less just if they refrained from rule. Since they love the Forms, they will want to imitate the Forms by producing order and harmony in the city. They would be loathe to do anything that would subject the city to disorder and disharmony. Socrates ends by remarking that the reluctance of the philosopher to rule is one of his best qualifications for ruling. The only good ruler rules out of a sense of duty and obligation, rather than out of a desire for power and personal gain. The philosopher is the only type of person who could ever be in this position, because only he has subordinated lower drives toward honor and wealth to reason and the desire for truth.

Now we know what distinguishes the philosopher-king from everyone else: he knows the Form of the Good, and so he has an understanding of everything. But it is left for Socrates to tell us how to produce this sort of man. He must explain what sort of supplementary education is added to the general education we read about in Books II and III, in order to make the guardians turn their souls toward ultimate truth and seek out the Form of the Good. The answer, it turns out, is simple: t hey must study mathematics and philosophical dialectic. These are the two subjects that draw the soul from the realm of becoming—the visible realm—to the realm of what is—the intelligible realm.

Of these two, mathematics is the preparation and dialectic the ultimate form of study. Dialectic leaves behind sense perceptions and uses only pure abstract reasoning to reach the Good itself. Dialectic eventually does away with hy potheses and proceeds to the first principle, which illuminates all knowledge. Though he is enamored of dialectic, Socrates recognizes that there is a great danger in it. Dialectic should never be taught to the wrong sort of people, or even to th e right sort when they are too young. Someone who is not prepared for dialectic will “treat it as a game of contradiction.” They will simply argue for the sake of arguing, and lose all sense of truth instead of proceeding toward it.

After discussing mathematics and dialectic Socrates launches into a detailed description of how to choose and train the -philosopher-kings. The first step, of course, is to find the children with the right natures—those who are the most stable, courageous, and graceful, who are interested in the subjects and learn them easily, who have a good memory, love hard work, and generally display potential for virtue. From early childhood, th e chosen children must be taught calculation, geometry, and all other mathematical subjects which will prepare them for dialectic. This learning should not be made compulsory but turned into play. Turning the exercises into p lay will ensure that the children learn their lessons better, since one always applies oneself better to what is not compulsory. Second, it will allow those most suited to mathematical study to display their enjoyment, since only those who enjoy it will apply themselves when the work is presented as noncompulsory. Then, for two or three years, they must focus exclusively on compulsory physical training; they cannot do anything else during this time because they are so exhausted.

All along, whoever is performing best in these activities is inscribed on a list, and when physical training is over those on the list are chosen to proceed. The rest become auxiliaries. The children are now twenty, and those who have been chosen to go on with philosophical training now must integrate all of the knowledge of their early traing into a coherent whole. Those who manage this task successfully have good dialectical natures and the others have weak dialectical natures. Those who are best at this task, therefore, and also at warfare and various other activities, are then chosen out from among the rest at their thirtieth year and tested again, this time to see who among them can give up their reliance on the senses and proceed to truth on thought alone. Those who do well in this test will study dialectic for five years; the others will become auxiliaries.

After five years of dialectic, the young philosophers must “go back into the cave” and be in charge of war and other offices suitable to young people to gain experience in political rule. Here too, they are tested to see which of them remains steadfast in his loyalty and wisdom. After 15 years of this, at the age of 50, whoever performs well in these practical matters must lift up his soul and grasp the Form of the Good. Now philosopher-kings, they must model themselves, the other citizens, and the city on the Form of the Good that they have grasped. Though each of them will spend most of his time on philosophy, when his turn comes they must engage in politics and rule for the city’s sake. The other important task they are charged with is to educate the next generation of auxiliaries and guardians. When they die they are given the highest honors and worshipped as demi-gods in the city.

Now Socrates has finally completed describing the just city in every one of its aspects. He ends Book VII by explaining how we might actually go about instituting such a city. His shocking solution is go into an already existing city, banish everyone over ten years old and raise the children in the manner he has just -outlined.

Plato’s outline for the education of the philosopher-king may provide some insight into the education students received in the early days of the Academy. We know that mathematics was heavily emphasized at the Academy, and that, in fact, many of the subfields which Plato discusses here under the heading of “mathematics” could only have been learned at the Academy at that time. The mathematician Theaetetus and the mathematician and astronomer Eudoxus, both teachers at the Academy, were the only thinkers in the ancient world who understood these higher mathematical subjects well enough to transmit them to others. They were actually the only ones even working in some of these embryonic fields. In addition, there is some indication that Plato did not offer his students training in dialectic, since he believed that dialectic should not be taught to anyone under thirty.

Why did Plato put so much weight on mathemat ics? Mathematics draws us toward the intelligible realm because it is beyond the realm of sensible particulars. When we move beyond applied mathematics (e.g., beyond counting out particular objects, or trac ing the astronomical patterns of the planets we see) and begin to contemplate numbers in themselves, and to examine their relations to other numbers, then we begin to move from the sensible realm to the intelligible. Numbers, like Forms, are truly existing, non-sensible entities that we can only access through abstract thought. Contemplating numbers and numerical relations, then, shows us that there is some truth above the sensible, and that this truth i s higher than the sensible in that it explains and accounts for the sensible.

Mathematics, viewed in this way, was probably meant to play two roles in the education of the philosopher. First, it sets the students sights on truths above the sensible world. It indicates that there are such truths, and instills the desire to reach them. Second of all, by contemplating these truths the student cultivates his use of abstract reason and learns to stop relying on sensation to tell him about the world. Mathematics prepares the student to begin the final study of dialectic, in which he will eventually give up the images and unproven assumptions of mathematics and proceed entirely on the faculty of abstract thought which he has honed.

Plato puts little stock in human senses. The true philosopher must be trained to ignore his senses in his search for truth. He must rely on thought alone. The true philosopher probably makes no use of empirical investigation—that is, he does not observe the world in order to find truths. Plato is at odds with the typical scientific approach to knowledge, in which observation is the most important ingredient. Plato is also at odds with his most famous student, Aristotle, who himself was the first known proponent of the observational method of scientific investigation.