Pythagoras: number and harmony

Pythagoras of Samos (c. 570–495 BCE) emigrated to Croton in southern Italy and founded not a school but a brotherhood (koinōnia) — a religious and philosophical community with shared rules of life, secrecy, and communal property. Its members are the Pythagoreans, and it is often impossible to separate the master’s own ideas from the movement’s.

Their central and startling thesis: the deep structure of reality is mathematical. Where the Milesians said everything is made of some stuff, and Heraclitus said everything flows by a law, the Pythagoreans said that the intelligible essence of things is number. As Aristotle reports (Metaphysics A.5), they held that the principles of mathematics are the principles of all things — and even that things are numbers.

The evidence that lit the fuse was musical harmony. Pythagoreans discovered that the concordant intervals correspond to simple whole-number ratios of string length: the octave is 2:1, the fifth is 3:2, and the fourth is 4:3. Beauty your ear recognizes turns out to be arithmetic your mind can state.

So they generalized. The universe is a kosmos (kosmos = ordered, beautiful arrangement) — indeed the Pythagoreans are credited with applying that word to the whole world — structured throughout by number and proportion. The heavenly bodies, moving in ratio-governed orbits, were said to produce a harmony of the spheres: a cosmic music we cannot hear only because we have never known its absence.

The brotherhood also held a religious doctrine: the soul is immortal and undergoes transmigration (metempsychōsis = the passing of the soul into successive bodies, human or animal). Life is therefore a project of purification (katharsis) — through ritual rules, and, crucially, through the study of mathematics and music, which turn the soul toward the eternal, orderly, and pure.

This is the Pythagoreans’ lasting gift: the idea that behind the shifting sensory world lies an intelligible, eternal, mathematical order — grasped by reason, not the senses. That template runs straight into Plato. His theory of Forms (eternal, non-sensible objects of knowledge), his conviction that geometry prepares the soul for philosophy, the mathematical cosmology of the Timaeus, and the immortal, reincarnated soul of the Phaedo and Republic all bear a deep Pythagorean stamp.