What we know of Pythagoras of Samos (c. 570–495 BCE) is interwoven with legend; no writing of his survives, and his teachings were transmitted through the school — both philosophical and religious — that he founded at Croton in southern Italy. Yet an idea attached to him shaped all of Western thought: what underlies reality is not matter but number.

At the core of this idea lies a concrete discovery. The Pythagoreans noticed a relationship between the length of a taut string and the note it sounds: press a string at exactly its half and you hear an octave, at two-thirds a fifth, at three-quarters a fourth. That is, the musical harmony pleasing to the ear corresponded to simple whole-number ratios (2:1, 3:2, 4:3). Audible beauty was the expression of an invisible mathematics. If music is number, perhaps everything is number.

The Pythagoreans carried this intuition to the cosmos: in the motions of the heavenly bodies, the cycle of the seasons, geometric forms, they sought the same numerical order — it was they who named the universe "kosmos," meaning "order." The theorem of the right triangle that bears his name (a² + b² = c²) is far older, but it was this school that bound it to a tradition of proof.

The philosophical legacy of this thought is vast. The idea that numbers are real in themselves, beyond sensible objects, directly fed Plato's theory of "Forms"; the belief that behind the visible world lies an unchanging reality grasped by reason was born here. Two thousand years later, when Galileo said "the book of nature is written in the language of mathematics," he was repeating Pythagoras's intuition. The root of the belief that science can speak in mathematics was set down in this half-mystical community.