Pythagoreanism

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Pythagoreanism is the body of teaching ascribed to Pythagoras of Samos, who in the sixth century BCE settled at Croton in southern Italy and gathered around himself a community bound by shared conviction and a common way of life. It is at once a school, a brotherhood, and a rule of conduct, and the difficulty stated at the outset is that none of the three can be heard in the master’s own voice. Pythagoras left no writings, and almost nothing reported of him can be traced to his own mouth. What survives is what later authors said the Pythagoreans held — Aristotle chief among them, writing nearly two centuries on, already laboring to sort the master from the school, and behind Aristotle a long file of biographers who set down their portraits seven and eight centuries after the man had died.

Number as principle

At the heart of the reports stands a single claim: that number is the principle of all things. The Pythagoreans, Aristotle records, held that the elements of number were the elements of everything that exists, and that the whole heaven is a harmony and a number. The intuition seems to have grown from a real discovery — that musical intervals correspond to simple ratios of string length, so that what the ear hears as concord is, underneath, plain arithmetic. The octave answers to a doubling, two to one; the fifth to three to two; the fourth to four to three. From this the Pythagoreans reasoned outward to a cosmos ordered by ratio throughout, a world in which the audible proof of proportion was only the nearest instance of a law that governed the stars no less than the strings.

The doxographer Diogenes Laertius, drawing on a Hellenistic compendium, preserves the bare scaffold of that cosmology: the principle of all things is the monad or unit, and from the monad arises the undefined dyad as a material substratum; from the monad and the dyad spring numbers, from numbers points, from points lines, from lines plane figures, from planes solids, and from solids the sensible bodies whose four elements are fire, water, earth, and air. The world that results is “animate, intelligent, spherical, with the earth at its centre” — a single living thing whose order is, at every level, the unfolding of unity into multiplicity and back again. The qualitative side of this scheme — the distinct character assigned to each of the first ten numbers — became its own developed art, treated under Pythagorean arithmology; what concerns the school as a whole is the prior and broader thesis that number is not counted onto the world from outside but found already in it, holding it together.

Caution is owed here that the ancient sources rarely supply. When Aristotle writes that the Pythagoreans made things out of numbers, he is interpreting, and modern scholarship has shown how much his formula flattens. The one early Pythagorean whose words survive in quotable fragments, Philolaus of Croton, does not say that things are numbers. He says that everything that is known has number, and that without number nothing could be thought or known — a claim about the conditions of knowledge, not a claim that bodies are made of arithmetic. The cosmos, for Philolaus, is composed of limiters and unlimiteds fitted together by harmony, with number as the bond that makes the fitting knowable. The slide from Philolaus’s careful epistemology to the ringing metaphysical slogan — all is number — is itself a piece of the tradition’s history, and one of the places where the master’s school and the master’s later legend pull apart.

The harmony of the spheres

The boldest extension of the number-cosmology is the doctrine later called the harmony of the spheres: that the heavenly bodies, vast and swift and set at intervals that answer to the ratios of the musical scale, sound as they move, producing a music too constant and too total ever to be heard, since it has filled the ears from birth and admits no contrasting silence. The fully worked version belongs to Philolaus and the fifth-century Pythagoreans rather than to Pythagoras himself, and it is bound up with Philolaus’s striking astronomy, in which the earth is not the center of the universe but one body among ten circling a central fire — the first cosmology in the Greek tradition to set the earth in motion. The detail of celestial harmonics is developed in its own right under Pythagorean-Platonic harmonics; for the school it stands as the grandest single image of its governing conviction, that the order of the cosmos is, at bottom, a proportion the mind can grasp because the mind shares in it.

The soul reborn, and the Pythagorean life

Alongside the mathematics ran a doctrine of the soul, and it is this, not the number-theory, that the earliest evidence attaches most firmly to Pythagoras. The Pythagoreans taught metempsychosis — the transmigration of souls — the teaching that the soul outlives the body and is reborn into others, human and animal alike. The doctrine is attested in Pythagoras’s own lifetime: his near-contemporary Xenophanes mocked him for it, telling how Pythagoras, passing a puppy being beaten, called out for the beating to stop because he recognized in its cries the voice of a dead friend. Herodotus, a little later, links the belief to Egyptian teaching about the soul’s passage through the animals. That the same soul might wear many bodies made the boundary between human and beast permeable, and from this followed the community’s most visible mark, its abstinence from the flesh of animals that might house a kindred soul.

The bios — the Pythagorean way of life — wove this conviction into a daily discipline. Property was held in common; the maxim ascribed to the founder, that friends have all things in common, was lived as a rule. Newcomers passed through a long probation of silence, the echemythia, hearing the master’s discourses from behind a screen, for as much as five years, before they were admitted to see him and to speak. Dress was white and plain; the day was framed by self-examination and by hymns; divination by sacrifice was rejected in favor of bloodless offering. The soul was understood as immortal and as the true self, and the regimen of the body — diet, purity, restraint — was the care of that self across its long migration. The discipline was austere enough that ancient critics, the musicologist Aristoxenus among them, pushed back, denying that Pythagoras had been a vegetarian at all and claiming he ate the flesh of young animals; the dispute is itself evidence that the dietary rule was, by the fourth century, the school’s signature.

The akousmata, the symbola, and the bean

Bound into the life was a body of oral precepts, terse and riddling, known as akousmata — “things heard” — or symbola, tokens whose surface masked a hidden sense. Diogenes Laertius lists a string of them as the master’s “watchwords or precepts”: don’t stir the fire with a knife, don’t step over the beam of a balance, don’t sit on your grain-measure, don’t eat your heart, don’t help a man set down a burden but help him take it up, don’t leave the pan’s imprint in the ashes, turn the sharp blade away, when you go abroad don’t turn round at the frontier. To the literal-minded these are taboos; to the school they were ciphers. “Don’t stir the fire with a knife,” the same source explains, means do not provoke the anger of the powerful; “don’t eat your heart” means not to waste one’s life in grief; the rule against turning round at the frontier counsels the dying not to cling, at the threshold, to the pleasures they are leaving. The same enigmatic register governed the school’s catechism of definitions, where, by Iamblichus’s report, “What is the oracle at Delphi?” is answered “The tetractys,” and the islands of the blessed turn out to be the sun and moon.

The most notorious of the prohibitions is the rule against beans. It is the oldest attested of all the Pythagorean taboos — Aristotle already assigns it to Pythagoras himself — and the ancients could not agree on its reason. In a single sentence Diogenes preserves a whole cabinet of explanations drawn from Aristotle’s lost treatise: that the bean was shunned because it resembles the genitals, or the gates of Hades, or because it is alone among plants unjointed, or because it is injurious, or because it figured the shape of the universe, or because it belonged to oligarchy, the lot being cast with beans. The image that most caught later imaginations — the hollow, seemingly unbroken stem as a channel between the living world and the dead — belongs to other strands of the ancient record rather than to Diogenes’ list. The prohibition passed into legend at the hour of the master’s death: in the story Diogenes tells, Pythagoras in flight reached the edge of a bean-field and stopped there, declaring he would sooner be caught than trample it, and so was overtaken and killed — choosing the taboo over his life.

The hearers and the learners

The community did not hold its teaching all on one level. Ancient sources already divided the followers into two strands, the akousmatikoi, who kept the precepts as oral rule and lived by the akousmata without demanding the reasoning behind them, and the mathematikoi, who pursued the demonstrations and the sciences — arithmetic, geometry, harmonics, astronomy, the disciplines that still carry the Pythagorean name of mathēmata, “the things learned.” The two wings later quarreled over their own legitimacy. On the reconstruction modern scholarship favors — Burkert’s, following Iamblichus’s earlier report in his treatise on general mathematical science — the akousmatikoi refused to count the mathematikoi as true Pythagoreans, claiming the learned wing’s doctrines derived not from Pythagoras but from Hippasus, the same Hippasus whom another strand of legend drowns for divulging a secret of the school; Iamblichus’s own Pythagorean Life, oddly, states the dispute the other way about. Scholars read the division either as two contemporary grades of an early initiatory order or as a later schism projected back onto the founder; what is clear is that the tension between a rule to be obeyed and a reason to be understood ran inside Pythagoreanism from early on.

Croton and the burning of the houses

For a generation the brotherhood was a power in the Greek cities of southern Italy. Its members governed or steadied the governments of Croton, Metapontum, Tarentum, and their neighbors; the lawgivers Zaleucus and Charondas were counted among those Pythagoras had formed. Political eminence drew political hatred. The sources tell of a man named Cylon, rejected for membership, who turned the resentment of the excluded against the order. The catastrophe came when the Pythagoreans were gathered in the house of the athlete Milo at Croton and the house was set ablaze; most inside were burned or cut down as they fled, and the survivors scattered. Diogenes records the toll at some forty dead, with only a few escaping, Archippus and Lysis among them. Later writers multiplied the conflagration into a wave of risings across the Italian cities in the middle of the fifth century BCE, in which the meeting-houses were burned and the brotherhood broken as an organized force. The way of life lingered in isolated holdouts — Aristoxenus names a last generation at Phlius around the middle of the fourth century BCE — and then the original communities died out, leaving the doctrines without the discipline that had carried them.

The afterlife: Plato, the revival, and after

The school’s afterlife outweighs its history. Plato absorbed the number-cosmology and the soul-doctrine so thoroughly that the two can scarcely be pried apart: the Timaeus builds the soul of the world out of musical ratios, and the doctrine of the soul’s recurrence and recollection in the Phaedo and elsewhere carries an unmistakably Pythagorean charge. Plato’s own immediate circle leaned harder still on number — Speusippus and Xenocrates in the early Academy wrote on the Pythagorean decad — so that within a generation of his death the line between a Pythagorizing Platonist and a Platonizing Pythagorean had already begun to blur. Through Plato the conviction that mathematical form underlies the sensible world entered the Western mainstream and never left it.

In the Roman era a self-conscious revival, Neopythagoreanism, recovered the master as sage and miracle-worker and rebuilt a metaphysics in his name, producing under his and his disciples’ names a literature of treatises now recognized as centuries too late to be theirs. The mathematician Nicomachus of Gerasa, in the first and second centuries CE, gave the number-doctrine its durable handbook form, and the philosopher Numenius of Apamea read Plato through Pythagoras toward a graded hierarchy of divine principles. This current fed directly into Neoplatonism, where the supreme One, the procession of being from intellect, and the descent of emanation carry a Pythagorean stamp, and where Iamblichus and Porphyry composed the lives of Pythagoras that fixed his image for posterity. Much that later ages would call Pythagorean — the systematic theology of number, the reincarnation of the soul read as a ladder of ascent — reached them through this late-antique synthesis rather than from the sixth-century master himself. Later esotericism kept the thread alive in its own key, treating numbers as bearers of hidden virtue and Pythagoras as the fountainhead of a perennial wisdom.

The resemblance between that ancient arithmetic and the number-mysticism of later traditions is exact in one respect and overdrawn in many. What the early Pythagoreans appear to have meant was something disciplined: that the order of things is, at bottom, ratio and proportion, a structure the mind can find because the mind is part of it. That is a claim the sciences would later make in their own language, in Kepler’s hunt for the harmonies of the planets and in the mathematical physics that followed — having, in a sense, forgotten where they first heard that the book of the world is written in number.

Sources and scholarship

The whole of what is reportable about Pythagoreanism comes through later hands, and the modern study of the school is in large part a study of those hands. The three ancient lives are foundational and, being centuries old, freely readable. Diogenes Laertius, Lives of Eminent Philosophers, Book VIII (third century CE), gives the doxographical core — the cosmology of monad and dyad, the catalog of akousmata, the bean prohibition with Aristotle’s half-dozen reasons, and the burning of Milo’s house; the standard accessible translation is by R. D. Hicks, openly available at Wikisource. Iamblichus, On the Pythagorean Life (De vita Pythagorica, early fourth century CE), is the fullest ancient portrait and the source for the akousmatikoi/mathematikoi division and the quinquennial silence; Thomas Taylor’s 1818 English version is on Project Gutenberg. Porphyry’s shorter Life of Pythagoras (late third century CE) preserves the early testimony of Dicaearchus that the soul’s immortality and its transmigration were the master’s best-known teachings. The Golden Verses, a late didactic poem gathering the moral precepts, contains the famous oath by the tetractys; Florence Firth’s 1904 compilation, with the commentary of Hierocles, is at sacred-texts.com.

The decisive modern intervention is Walter Burkert, Lore and Science in Ancient Pythagoreanism (English translation 1972), which sifted the evidence to argue that the historical Pythagoras was a religious teacher and wonder-worker rather than a mathematician, and that the scientific Pythagoreanism belongs to Philolaus and the fifth century. The accessible overviews are Charles H. Kahn, Pythagoras and the Pythagoreans: A Brief History (Hackett, 2001), and Christoph Riedweg, Pythagoras: His Life, Teaching, and Influence (Cornell, 2005). The reconstruction of the one early Pythagorean we can read directly is Carl A. Huffman, Philolaus of Croton: Pythagorean and Presocratic (Cambridge, 1993); Huffman also edited the state-of-the-art collection A History of Pythagoreanism (Cambridge, 2014). Huffman’s open-access entries for the Stanford Encyclopedia of Philosophy gather the current consensus and the evidence behind it: Pythagoreanism, Pythagoras, and Philolaus.

→ In the library: Plato — Timaeus (Jowett) · Plotinus — The Enneads (MacKenna): On Numbers · Westcott — Numbers: Their Occult Power and Mystic Virtues (1911) · Mead — Apollonius of Tyana (1901)

→ Related: Pythagoras · Neopythagoreanism · Pythagorean Arithmology · Pythagorean Platonic Harmonics · Nicomachus Of Gerasa · Numenius Of Apamea · Plato · Platonism · Neoplatonism · The One · Nous · Emanation · Reincarnation · Alcmaeon