The Pythagoreans were a secretive philosophical community founded by the Greek thinker Pythagoras around 530 B.C. in Croton, a Greek colony in southern Italy. Part religious brotherhood, part intellectual school, and part political force, they lived by strict rules, pursued mathematics as a path to understanding reality, and believed that numbers formed the hidden structure of the universe.
How the School Began
Pythagoras arrived in Croton when he was between fifty-five and sixty years old, and the school he opened there quickly became one of the most influential institutions in the ancient world. People from every economic class came to hear him speak. Business and professional men attended his night lectures. Women broke local law to attend, something almost unheard of in the Greek world at the time.
The community functioned as what ancient sources describe as a semi-religious cult. New members went through a secret initiation, and every member swore an oath never to reveal the school’s teachings. Breaking that oath, at least according to tradition, meant death. Daily life was austere: simple food, severe discipline, and a constant emphasis on temperance, purity, and obedience. The Pythagoreans were a closed corporation, and for a time they held real political power in Croton and surrounding cities.
Two Types of Pythagoreans
Not all members followed the same path. The school eventually split into two distinct groups, each claiming to be the true Pythagoreans. The first group, called the acusmatici (from the Greek for “things heard”), focused on rules for daily living. They memorized Pythagoras’ oral instructions on how to act, what to eat, and how to conduct themselves, without necessarily understanding the reasoning behind those rules. Their Pythagoreanism was a way of life more than an intellectual pursuit.
The second group, the mathematici, devoted themselves to mathematics, scientific inquiry, and logical proof. They argued they were Pythagoreans “in a truer sense” because they grasped the reasons behind the teachings. According to their own account, the split originated because the leading men in Italian cities who first came to Pythagoras simply didn’t have time to learn the sciences and proofs, so Pythagoras gave them instructions without explanation. Those followers became the first acusmatici. The ones who stayed long enough to learn the deeper reasoning became the mathematici.
A figure named Hippasus sat awkwardly between the two camps. He was closely connected to the mathematici but ended up being disavowed by both sides. In practice, many Pythagoreans blended elements of both paths. The philosopher Archytas, for instance, did serious original research in mathematics while also living the strict ascetic lifestyle. Others, like Cleinias and Xenophilus, lived the Pythagorean life without doing any scientific work at all.
Numbers as the Key to Reality
The Pythagoreans’ most lasting idea was that numerical relationships underlie everything in the natural world. Their clearest demonstration of this came from music. By experimenting with strings of different lengths, they discovered that the intervals humans hear as most pleasing correspond to simple whole-number ratios. A string half the length of another produces a note one octave higher (a 2:1 ratio). A ratio of 3:2 produces the interval called a perfect fifth. A ratio of 4:3 produces a perfect fourth.
This was a genuinely revolutionary insight. It showed that something as subjective as musical beauty could be described with precise mathematics. The Pythagoreans took this as evidence that the entire universe operated on the same principle, and they used the Greek word “harmonia” to describe the hidden mathematical order they believed governed all things. They gave almost mystical importance to the numbers 1 through 4 and to their sum (10), and they searched for arithmetic harmony in geometry, astronomy, philosophy, and even ethics.
This idea eventually expanded into the concept of the “music of the spheres,” the notion that the planets and stars moved according to mathematical ratios and produced a kind of cosmic harmony. Plato later described the universe as comprising rotating wheels bearing the planets, each associated with a pitch, together producing this celestial music. Whether Plato meant this literally or allegorically has been debated for centuries, but the underlying Pythagorean conviction that mathematics reveals the structure of nature became one of the foundational assumptions of Western science.
The Famous Theorem
The relationship most people associate with Pythagoras, that in a right triangle the square of the longest side equals the sum of the squares of the other two sides, was not actually discovered by the Pythagoreans. Babylonian tablets dating to around 1000 B.C. contain exercises that depend on knowing specific cases of this relationship, and some tablets include diagrams that amount to geometric proofs for right triangles with equal sides. There is also evidence that Chinese and Indian mathematicians knew the principle early on, though exactly how early is uncertain.
What the Pythagorean school likely contributed was a formal, general proof that the relationship holds for all right triangles, not just specific examples. This distinction between knowing particular cases and proving a universal rule captures something essential about Pythagorean thinking. They were driven to show that mathematical truths were absolute and universal, not merely useful tricks for builders and surveyors.
Women in the School
Pythagoreanism gave an unusually large role to women for an ancient philosophical school. A catalogue compiled by the later writer Iamblichus lists 218 male Pythagoreans followed by 17 Pythagorean women by name. The most famous was Theano, who appears in different sources as either Pythagoras’ wife, his daughter, or his pupil. The philosopher Dicaearchus, writing in the fourth century B.C., confirmed that Pythagoras had women among his followers and that Theano in particular became famous. Later writers attributed several works to her, including a treatise called “On Piety,” along with aphorisms and letters, though scholars believe most or all of these were forged under her name in later centuries.
The second most famous woman on the list was Timycha, whose story captures the intensity of Pythagorean loyalty. When the tyrant Dionysius II of Syracuse persecuted her and her husband Myllias, Timycha, reportedly ten months pregnant at the time, bit off her own tongue and spat it in the tyrant’s face rather than risk revealing Pythagorean secrets under torture. Works were also forged in the names of other Pythagorean women, including Periktione (credited with treatises “On the Harmony of a Woman” and “On Wisdom”), Melissa, Myia, and Phintys.
Rules, Rituals, and the Bean Taboo
The acusmatici side of Pythagoreanism produced an elaborate set of rules governing daily behavior, some practical and some deeply strange to modern eyes. The most famous prohibition was against eating, and possibly even touching, fava beans. Ancient explanations for this taboo varied widely. Some sources said Pythagoras believed beans contained the souls of the dead, making eating them a form of cannibalism. Others interpreted it symbolically, connecting the bean’s role in planting and decomposition to cycles of life and death. Still others suggested that beans were considered spiritually impure because of their association with flatulence and bodily disorder.
One of the most persistent legends about Pythagoras’ death ties directly to this rule. According to the story, while fleeing from enemies, he reached the edge of a bean field and chose to stop rather than trample through it. His hesitation allowed his pursuers to catch and kill him. Whether the story is true is impossible to verify, but it illustrates how seriously the Pythagoreans took their prohibitions. For the acusmatici especially, following the rules was itself the point. The rules were not guidelines to be weighed against circumstances. They were sacred, absolute, and worth dying for.
Political Power and Collapse
The Pythagorean school was never just an academic institution. In Croton, the community achieved what ancient sources describe as temporary supremacy in the state. Pythagoreans held political offices and shaped civic life according to their principles. This concentration of power, combined with the group’s secrecy and exclusivity, eventually provoked a violent backlash. Anti-Pythagorean revolts broke out in several southern Italian cities, and meeting houses were burned. Many Pythagoreans were killed or driven into exile.
The school as a physical institution did not survive these attacks, but Pythagorean ideas proved far more durable than the community that generated them. The conviction that mathematical relationships explain the physical world passed directly into the work of Plato and, through him, into the entire tradition of Western philosophy and science. Every physicist who writes an equation describing the behavior of nature is, in a sense, working within the framework the Pythagoreans established twenty-five centuries ago in a small Greek colony on the coast of Italy.