Plato's philosophies examine issues such as immortality, man, mind, logic and legal philosophy (including rhetoric). Platonism is a term that is used to describe Socrates' denial of material reality. In "The Republic," Socrates discusses the assumption that most people have that what they can touch or sense must be proof of reality. Socrates uses the Allegory of the Cave to explain our disconnection between the senses and reality.
Perhaps Plato's most important discovery in mathematics was his belief that the study of math greatly contributed to a well-rounded statesman. In founding an education institution known as the Academy, Plato placed great emphasis on providing proof in mathematics, and in the development of clear hypothesis. This helped to formulated Euclid's systemic approach to math. Plato is also associated to the Platonic Solids, a mathematical construction of the four elements: earth, fire, air and water, which are represented by the cube, tetrahedron, octahedron and icosahedrons. The fifth Platonic solid is known as the dodecahedron, which is a model Plato devised for the entire universe.
Much of "The Republic" comprises Plato's views on the importance of education in creating intelligent, forward-thinking leaders in society. He posited the idea that everyone should be educated to the best of their abilities, which would result in three classes of people: the military class, who are ruled by virtue of courage; the minority ruling class with the highest education, who are ruled by virtue of wisdom; and the merchant class, who are ruled by virtue of temperance. The fourth virtue of justice would characterize society as a whole.