Although Plato did not produce a philosophical treatise as such, he did write about 30 dialogues in the form of conversations expressing his views. It is also through these dialogues that we have the only account of Socrates' teachings.
Plato's philosophy was based on his theory that the soul consists of three basic components: Reason, Emotion, and Appetite. Reason is given the greatest value, while Emotion and especially Appetite are regarded as the "lower passions".
According to Plato, a correctly functioning soul is governed by Reason, and therefore keeps the emotions and appetites under control.
He also believed that though the body dies and disintegrates, the soul continues to live forever. After death, the soul leaves the body and migrates to what Plato called the realm of the pure forms.
After a time, it is reincarnated in another body and returns to the world retaining only a dim recollection of the realm of forms, which it continues to yearn for. The body obstructs the soul's ability to recall the ideal forms.
In his "Theory of Forms" Plato states that the material world as we perceive it, is not the real world at all, but a shadow of the real world. He holds that the Realm of Forms is the true basis of reality and separate from our own world. What we see in our world is merely a representation of reality, not reality itself.
Plato regarded mathematics as being the finest training for the mind and what constituted a well-educated person. Over the door of his Academy in Athens, considered to be the first university, was written: "Let no one ignorant of mathematics enter here".
When it came to mathematics, Plato pushed for the idea of "proof" and insisted on accurate definitions and clear hypotheses. This laid the foundations for Euclid's systematic approach to mathematics.
All of the most important mathematical work of the fourth century was done by friends or pupils of Plato, including Eudoxus, Aristotle, and Archytas.
In mathematics the term Platonic solids comes from his dialogue called "Timaeus" in which he presents his scheme of the universe. He associates the elements earth, fire, air, and water with the cube, tetrahedron, octahedron, and icosahedron, repectively. The fifth Platonic solid, the dodecahedron, is Plato's model for the whole universe.
In Plato's major political work The Republic, he concerns himself with the matter of justice and poses the questions "what is a just state" and "who is a just individual?"
The ideal and just state, according to Plato, is composed of three classes; the merchant class (for economic structure), the military class (for security needs), and the philosopher-kings (for political leadership). A person's class is determined by his completed educational process, his interests and his abilities.
The just state characterizes society as a whole and one in which each class performs its own function well without infringing on the activities of the other classes. It is evident, however, that Plato's ideal educational system is primarily structured so as to produce philosopher-kings.
The last years of Plato's life were spent lecturing at the Academy and writing. When Plato was 60 years old, Aristotle, at age 18, arrived at his Academy as a student where he stayed until Plato's death in 347.
Plato's influence throughout history is monumental. The Academy remained open until 529 A.D., when the Eastern Roman Emperor, Justinian, ordered it closed because of what he considered to be pagan teachings.