Founder of semi-mystical brotherhood contemplating the mathematical nature of the divine cosmos, Pythagoras wrote none of his ideas down. Yet more is recorded of his life and beliefs than of any other pre-Socratic philosopher. It is difficult to separate the legend from the history. Later Pythagoreans often attributed teachings borrowed from other schools to their founder. He left his birthplace, Samos, when Polycrates was tyrant and founded a school at Croton in Southern Italy. This was a sort of secret brotherhood which apparently followed special dietary rules which forbade the eating of beans. It is said that disciples were required to undertake five years of silence and pass a test before being admitted.

Pythagoras is said to have discovered the numerical basis of musical harmony. This became the paradigm for his understanding of the entire cosmos which was seen to be intrinsically mathematical. For the Pythagoreans, philosophy seems to have been a means of purification, a way of salvation. By contemplating the intelligent mathematical reality of the divine cosmos, their souls would partake more of divinity and eventually be released from an endless series of reincarnations. Diogenes Laertius tells us that Pythagoras could list his previous incarnations. Pythagoras probably knew some of the theorems later proved by Euclid including the theorem of the right-angled triangle that bears his name. Pythagorean physics recognised four elements: earth, air, fire and water, which could all interchange. Citizens, angry at the political power of his school, forced him to flee Croton; he died in Metapontum.