Pythagoras and Plato - Soul Math and 5 Elements

Although Socrates influenced Plato directly as related in the dialogues, the influence of Pythagoras upon Plato, or in a broader sense, the Pythagoreans, such as Archytas also appears to have been significant. Aristotle claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans, and Cicero repeats this claim: "They say Plato learned all things Pythagorean." It is probable that both were influenced by Orphism, and both believed in metempsychosis, transmigration of the soul. Pythagoras held that all things are number, and the cosmos comes from numerical principles. He introduced the concept of form as distinct from matter, and that the physical world is an imitation of an eternal mathematical world. These ideas were very influential on Heraclitus, Parmenides and Plato. For Numenius it is just that Plato wrote so many philosophical works, whereas Pythagoras' views were originally passed on only orally. The platonic Republic might be related to the idea of "a tightly organized community of like-minded thinkers", like the one established by Pythagoras in Croton. The idea that mathematics and, generally speaking, abstract thinking is a secure basis for philosophical thinking as well as "for substantial theses in science and morals". They shared a "mystical approach to the soul and its place in the material world". Plato may have studied under the mathematician Theodorus of Cyrene, and has a dialogue named for and whose central character is the mathematician Theaetetus. While not a mathematician, Plato was considered an accomplished teacher of mathematics. Eudoxus of Cnidus, the greatest mathematician in Classical Greece, who contributed much of what is found in Euclid's Elements, was taught by Archytas and Plato. Plato helped to distinguish between pure and applied mathematics by widening the gap between "arithmetic", now called number theory and "logistic", now called arithmetic. Assignment to the elements in Kepler's Mysterium Cosmographicum. In the dialogue Timaeus Plato associated each of the four classical elements (earth, air, water, and fire) with a regular solid (cube, octahedron, icosahedron, and tetrahedron respectively) due to their shape, the so-called Platonic solids. The fifth regular solid, the dodecahedron, was supposed to be the element which made up the heavens.