Book VII of Plato’s Republic contains the most famous metaphor of philosophy – the Allegory of the Cave. Socrates requests that his audience imagine a group of prisoners chained since birth to the bottom of a cave. The prisoners can only see the wall in front of them. They cannot turn their heads to either side. Behind the prisoners, puppeteers move statues in front of a fire. The statues cast shadows on the wall in front of the prisoners. The prisoners speak of these shadows as we speak of our world. They call the shadows – horses, dogs, men, etc.
Suppose a prisoner breaks free from his chains and turns around. His eyes, unaccustomed to the light of the fire, are unable to discern the statues. But after a period of time, his eyes adjust to the light, and he sees that the statues and the fire are more real than the shadows. The existence of the shadows depends upon the existence of the statues, not vice versa.
Now imagine that the prisoner ascends from the cave to the upper world. His eyes, unaccustomed to the light of the sun, are unable to look upon the objects of the world. But, in time, he gradually acquires the ability to see. He looks first upon shadows, then upon reflections in water, then upon the objects themselves, and finally upon the sun itself. He concludes that the sun is the reason why he can see the objects of the world, and also concludes that the sun is the source of all existence.
The Allegory of the Cave is a more elaborate and beautiful metaphor than the Simile of the Line, but the subject matter is the same. Plato uses both metaphors to illustrate his conception of reality and the different stages of knowledge. The cave represents the physical realm, the realm of becoming. The shadows symbolize images of physical objects, the statues symbolize the physical objects themselves, and the fire symbolizes the sun. The prisoners within the cave who are ignorant of the upper world are men who only possess knowledge of the physical realm.
The upper world represents the intelligible realm, the realm of being. The shadows symbolize the definitions of Beauty, Justice, Courage, etc.; the physical objects symbolize Beauty itself, Justice itself, Courage itself, etc.; and the sun symbolizes the Good. The prisoner who breaks free from his chains and ascends to the upper world is the philosopher who transcends the mundane world to contemplate the Forms of the intelligible realm and acquire true wisdom. Like the ascent from the bottom of the cave to the upper world, the journey of the philosopher from ignorance to knowledge is arduous and painful.
To acquire true wisdom, one must study mathematics. Mathematics directs the mind toward abstract truths because numbers exist in the intelligible realm, the realm of being. “The true use of math is simply to draw the soul toward being. We must endeavor to persuade those who are to be the principal men of our State to learn arithmetic because this will be the easiest way for her to pass from becoming to truth and being.” The more one practices mathematics, the better one will become at contemplating abstract truths.