Pythagoras

Wrote nothing, is known only through sources centuries later, probably did not prove the theorem — and his best-attested doctrine is reincarnation, not mathematics.

Who they were

Pythagoras of Samos (c. 570 – c. 495 BCE) is one of the most influential figures in Western thought and one of the least recoverable.

He is credited with the theorem, with the discovery of numerical ratios in music, with the doctrine that reality is fundamentally numerical, and — by modern numerologists — with a system of letter-values he could not possibly have devised. Very little of this is secure.

Where they came from

He was born on Samos and emigrated around 530 BCE to Croton in southern Italy, where he founded a community that was religious, philosophical and political at once, and influential enough that it was eventually attacked and dispersed.

The evidence problem is severe and should be stated first. He wrote nothing that survives, and may have written nothing at all. The earliest references are brief and often hostile — Heraclitus and Xenophanes, within living memory, mention him dismissively. The full biographies come from Porphyry, Iamblichus and Diogenes Laertius in the third and fourth centuries CE: eight hundred years later, in a tradition that had every reason to make him a wonder-worker, and which duly reports him having a golden thigh and being seen in two cities at once.

Walter Burkert's Lore and Science in Ancient Pythagoreanism (1962) is the study that reorganised the field. Burkert argued that the early evidence presents Pythagoras primarily as a religious teacher concerned with the soul, and that the mathematical and scientific achievements were attributed to him later by his followers, who ascribed the school's collective work to its founder.

What they taught

Metempsychosis — the transmigration of souls. This is the most securely attested doctrine, because it is what the earliest and least friendly sources mention. Xenophanes tells a story of Pythagoras stopping a man beating a puppy because he recognised the voice of a dead friend in its cry. Whatever else he was, he taught that souls pass between bodies, human and animal.

A way of life. The community observed dietary and behavioural rules — vegetarianism, and a famous prohibition on beans whose rationale nobody has convincingly explained. Members were divided into akousmatikoi, who received sayings, and mathematikoi, who received reasoning.

"All is number." The doctrine that number is the principle of things. This is genuinely Pythagorean in the sense that the school held it; whether the man did is less certain.

Musical ratios. The Pythagoreans identified that consonant intervals correspond to simple whole-number ratios of string length — 2:1 for the octave, 3:2 for the fifth, 4:3 for the fourth. This part is true and is checkable on any stringed instrument. The famous story of Pythagoras discovering it by hearing hammers of different weights at a forge is physically false — pitch does not scale that way with hammer mass — but the monochord relationships are real, and the discovery that something as subjective as musical beauty sits on simple arithmetic is one of the genuinely astonishing findings of antiquity. It is the origin of the idea that number underlies quality.

The theorem. The relationship was known to Babylonian mathematicians roughly a thousand years earlier — the tablet Plimpton 322 lists Pythagorean triples — and to Indian and Chinese mathematicians independently. Whether the Pythagoreans supplied the first proof is unknown and disputed.

What we can and cannot say

We can say his influence is enormous and largely indirect. It runs through Plato, for whom mathematics is the route to reality, and through Plato into the entire Western tradition — including the sacred geometry that follows.

We can say the man is essentially unrecoverable, and that confident statements about what Pythagoras believed should be treated with suspicion regardless of who makes them.

We can say that "Pythagorean numerology" has no connection to him whatever. The system described under numerology assigns numbers to letters of the Latin alphabet — a script that did not exist in a usable form in his lifetime, for a language he did not speak, in a scheme devised in New Jersey in 1908. Attaching his name to it was a marketing decision.

We can say the musical discovery is real and is the strongest thing in his tradition. It is also the seed of everything else: once you have found that harmony is arithmetic, the leap to supposing that everything is arithmetic is short, and the Pythagoreans took it.

We cannot say whether he was a mathematician, a mystic, a political leader or a charismatic teacher whose followers built the rest. Probably some combination, and the ancient sources disagreed about it too.

Who they were

Where they came from

What they taught

What we can and cannot say

Further reading