A Cult on the Italian Coast
Pythagoras was born on Samos about 570 BCE and left, according to the tradition, to escape the tyrant Polycrates, settling around 530 at Croton in southern Italy. There he founded a community bound by rules that were as much religious as intellectual: shared property, periods of silence for initiates, dietary restrictions including the notorious ban on beans, and a doctrine of the soul's transmigration through successive bodies, human and animal. Members were divided into an inner circle and an outer one, and the group acquired real political power in the city.
That power ended badly. Sometime around 500 BCE, resentment against the society boiled over, meeting houses were burned and members killed, and Pythagoras is said to have fled to Metapontum, where he died. The scattering pushed his followers across the Greek world. Two rival branches emerged, one devoted to religious observance and one to mathematical and cosmological enquiry, and they argued about which was faithful to the founder. Nearly everything later attributed to Pythagoras reaches us through these competing partisans, writing generations afterwards and each claiming the founder for their own side.
All Things Are Number
Pythagoras himself wrote nothing that survives, and the earliest sources say strikingly little about his mathematics. Heraclitus, nearly a contemporary, mocked him as a practitioner of wide learning and fraud. What later Pythagoreans developed was the claim that number is not a description of things but their principle: that the cosmos is ordered by ratio, and that understanding proportion is understanding reality. From this came the study of numerical relations for their own sake, the beginning of mathematics as a subject rather than a set of surveying tricks.
The most concrete piece of evidence is musical. Sounding strings whose lengths stand in simple whole-number ratios of two to one, three to two and four to three produce the octave, fifth and fourth. Here was an audible, testable case of the world obeying arithmetic. From it the school extrapolated a harmony of the spheres, in which the heavenly bodies sound intervals as they move. The extrapolation was wrong, but the instinct that physical phenomena might be governed by mathematical law runs straight through to Kepler and Newton.
Credit, Legend and Consequence
The relationship between the right triangle and the squares on its sides was known and used in Babylonia and in Egypt more than a thousand years before Pythagoras was born, and the Indian Sulba Sutras record it independently. What Greek tradition credits to the school is proof, the demonstration that the relation must hold for every right triangle rather than for the cases anyone has checked. Even that attribution is late, and no source close to his lifetime connects Pythagoras personally to the theorem that carries his name.
The discovery that the diagonal of a square is incommensurable with its side, that no ratio of whole numbers can express it, is credited to the Pythagorean Hippasus, who legend says was drowned at sea for revealing it. The story is probably fiction. The mathematical crisis was real, and it forced Greek geometry toward the rigorous theory of proportion that Eudoxus supplied and Euclid recorded. Pythagoras survives less as a discoverer than as the origin point of a conviction that the universe is legible in number.
