Plato’s works, perhaps the most consistently popular and influential philosophic writings ever published, consist of a series of dialogues in which the discussions between Socrates and others are presented with infinite charm. Most of our knowledge of Socrates is from these dialogues, and which views are Socrates’ and which are Plato’s is anybody’s guess. (Plato cautiously never introduced himself into any of the dialogues.)
Like Socrates, Plato was chiefly interested in moral philosophy and despised natural philosophy (that is, science) as an inferior and unworthy sort of knowledge. There is a famous story (probably apocryphal and told also of Euclid of a student asking Plato the application of the knowledge he was being taught. Plato at once ordered a slave to give the student a small coin that he might not think he had gained knowledge for nothing, then had him dismissed from school. To Plato, knowledge had no practical use, it existed for the abstract good of the soul.
Plato was fond of mathematics because of its idealized abstractions and its separation from the merely material. Nowadays, of course, the purest mathematics manages to be applied, sooner or later, to practical matters of science. In Plato’s day this was not so, and the mathematician could well consider himself as dealing only with the loftiest form of pure thought and as having nothing to do with the gross and imperfect everyday world. And so above the doorway to the Academy was written, “Let no one ignorant of mathematics enter here.”
Plato did, however, believe that mathematics in its ideal form could still be applied to the heavens. The heavenly bodies, he believed, exhibited perfect geometric form. This he expresses most clearly in a dialogue called Timaeus in which he presents his scheme of the universe. He describes the five (and only five) possible regular solids — that is, those with equivalent faces and with all lines and angles, formed by those faces, equal. These are the four-sided tetrahedron, the six-sided hexahed ron (or cube), the eight-sided octahedron, the twelve-sided dodecahedron, and the twenty-sided icosahedron. Four of the five regular solids, according to Plato, represented the four elements, while the dodecahedron represented the universe as a whole. These solids were first discovered by the Pythagoreans, but the fame of this dialogue has led to their being called the Platonic solids ever since.