Pythagoras

Pythagoras was born around 550 BCE on Samos, a prosperous island in the eastern Aegean just off the coast of Ionia, in what is now western Turkey ^[1]. Ancient sources name his father as Mnesarchus, often described as a gem engraver or merchant, and his mother as Pythais. Samos in the sixth century BCE was a hub of trade, engineering, and craft, ruled for much of Pythagoras's youth by the tyrant Polycrates, whose court attracted poets and artisans from across the Greek world ^[2].

Almost nothing about his education can be stated with certainty, because no writings by Pythagoras survive and the earliest detailed accounts of his life were composed centuries after his death ^[1]. Later biographers such as Diogenes Laertius, Porphyry, and Iamblichus claimed that he studied with the Milesian thinkers Thales and Anaximander and traveled to Egypt and Babylon to learn geometry, astronomy, and religious rites ^[3]. Modern scholars treat these travel stories with caution. They fit a common Greek pattern of crediting Eastern journeys to any sage, though contact between Samos and Egypt was real enough, since Polycrates maintained ties with the pharaoh Amasis ^[2].

What the evidence does support is that Pythagoras grew up in a region alive with new ways of thinking. Ionia in his lifetime produced the first Greek attempts to explain nature through observation and argument rather than myth alone, and that intellectual climate shaped the questions he would later pursue ^[4].

Around 530 BCE, when he was roughly forty, Pythagoras left Samos and sailed west to Croton, a Greek colony on the southern coast of Italy ^[1]. Ancient writers say he departed to escape the rule of Polycrates, though ambition may have mattered as much as politics. Croton was wealthy, famous for its athletes and physicians, and open to a charismatic newcomer with a reputation for wisdom ^[2].

In Croton he founded a community that was part school, part religious brotherhood, and part political club. Members shared property, followed strict rules of diet and conduct, and bound themselves to secrecy about the group's teachings ^[3]. Both men and women were admitted, which was unusual for the era; the tradition names Theano, sometimes described as his wife, among the community's members ^[5].

The society gained real political weight. Pythagoreans came to hold influence in Croton and neighboring cities, and ancient accounts credit their circle with a role in Croton's victory over the rival city of Sybaris around 510 BCE ^[2]. That entanglement of philosophy with power would eventually provoke a violent backlash.

Two doctrines can be traced to Pythagoras himself with reasonable confidence. The first is metempsychosis, the belief that the soul is immortal and passes after death into new bodies, human or animal ^[1]. His younger contemporary Xenophanes mocked this idea in verse, telling of Pythagoras stopping a man from beating a dog because he claimed to recognize the voice of a dead friend in its yelps. That joke, hostile as it is, counts as some of the earliest surviving testimony about him ^[4].

The second doctrine is the conviction that number underlies the order of the world. Pythagoreans held that musical harmony, the movements of the heavens, and the structure of things could all be expressed in numerical ratios ^[3]. Tradition credits Pythagoras with discovering that the basic musical intervals correspond to simple ratios of string lengths: the octave at 2:1, the fifth at 3:2, the fourth at 4:3 ^[6]. Whether he personally made the discovery or his followers did, the insight that quantity governs quality became the school's signature idea.

Daily life in the community followed detailed precepts known as akousmata, brief sayings that regulated everything from prayer to diet. The most famous rule forbade eating beans, a prohibition explained in wildly different ways by later writers ^[3]. Followers also abstained from certain meats, consistent with a belief that animals might house transmigrated souls. Because members swore silence, outsiders traded rumors, and the aura of secrecy fed both the school's prestige and the legends about its founder, including tales of a golden thigh and the ability to be in two places at once ^[1].

The question of what Pythagoras actually contributed to mathematics is among the hardest in the history of ancient science. The Pythagorean theorem, which states that in a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides, carries his name, yet Babylonian scribes knew the numerical relationship more than a thousand years before he was born ^[6]. A clay tablet catalogued as Plimpton 322, dated to roughly 1800 BCE, lists numbers connected to right triangles ^[6].

No source from Pythagoras's own lifetime attributes the theorem to him, and the first surviving proof appears in Euclid's Elements, written around 300 BCE ^[4]. Some historians think Pythagoras or his early followers may have supplied a general proof of a relationship the Babylonians used only in particular cases; others doubt he engaged in deductive geometry at all and see him chiefly as a religious teacher whose school later turned mathematical ^[1]. The truth is probably layered: the community he founded made number theory central, and achievements of later Pythagoreans such as Philolaus and Archytas were folded back onto the founder's name ^[3].

One discovery associated with the school had unsettling consequences for its own worldview. The diagonal of a square cannot be expressed as a ratio of whole numbers, which means some magnitudes are incommensurable, or in modern terms irrational ^[4]. Late legends say the Pythagorean Hippasus was drowned at sea for revealing this secret. The drowning story is almost certainly embroidery, but the mathematics is genuine, and it showed that the claim that all things are number needed refinement ^[6].

Among Pythagoras facts that rest on firmer ground is the school's fascination with figurate numbers, especially the tetractys, a triangular arrangement of ten points in rows of one, two, three, and four. Pythagoreans treated it as a sacred symbol summing up the ratios of musical harmony, and reportedly swore oaths by it ^[3].

The political influence of the Pythagoreans in Croton eventually bred resentment. Ancient accounts describe a revolt led by a prominent Crotoniate named Cylon, said to have been rejected for membership in the society, who turned the citizenry against the order ^[2]. Meeting houses of the Pythagoreans were attacked and burned, and members were killed or driven out.

The chronology of these events is tangled, and historians distinguish an early disturbance during Pythagoras's lifetime from a larger wave of anti-Pythagorean violence across southern Italy in the mid fifth century BCE ^[3]. During the first upheaval Pythagoras withdrew from Croton to Metapontum, another Greek city on the Italian coast, where he died around 490 BCE ^[1]. Stories about his end vary: one has him starving himself in grief, another has him refusing to escape pursuers by crossing a bean field. None can be verified, and the bean field tale in particular reads like a moral fable spun from his own dietary rule ^[3].

In Metapontum his memory was honored long after his death. When the Roman writer Cicero visited the city in the first century BCE, locals still pointed out the house where Pythagoras was said to have died ^[5].

Anyone asking who was Pythagoras confronts a double figure: the historical teacher, dimly visible behind late and contradictory sources, and the legendary sage constructed over centuries. Both mattered enormously. The scattered Pythagorean communities survived the persecutions, and thinkers in the tradition, above all Philolaus of Croton and Archytas of Tarentum, developed serious work in harmonics, astronomy, and number theory during the fifth and fourth centuries BCE ^[3].

Through Archytas, a friend of Plato, Pythagorean ideas flowed into the mainstream of Greek philosophy. Plato's Timaeus, with its mathematically structured cosmos, is saturated with Pythagorean themes, and Aristotle wrote a study of the school, now lost, that remains one of our best indirect witnesses ^[4]. Centuries later, Neopythagoreans and Neoplatonists of the Roman era rebuilt Pythagoras into a near divine authority, and the biographies by Porphyry and Iamblichus from the third century CE preserve, amid much invention, fragments of older tradition ^[1].

The long afterlife of Pythagoras achievements extends well beyond antiquity. The idea of the harmony of the spheres, the notion that the planets produce a music governed by ratio, echoed through medieval and Renaissance thought and inspired Johannes Kepler's search for mathematical law in planetary motion ^[6]. Copernicus cited Pythagorean precedents for a moving Earth. And every student who learns that the square of the hypotenuse equals the sum of the squares of the other two sides repeats his name, whether or not he ever proved it ^[4].

His deepest legacy may be a habit of mind. Pythagoras and his followers wagered that the world is intelligible through number, and that wager, refined and corrected over two and a half millennia, became the working assumption of mathematical science ^[6].