What can be said about Pythagoras
Pythagoras of Samos lived in the sixth century BCE and, according to tradition, settled around 530 BCE in Croton, a Greek city in southern Italy, where a community of followers gathered around him. That is close to the limit of what can be stated with confidence. He left no surviving writings, and he may have written none. The earliest descriptions of his circle come from other Greek authors, and the long, detailed biographies (by Diogenes Laertius, Porphyry and Iamblichus) were written many centuries after his death.
This article surveys the school as a whole and tries to keep apart three layers: ideas attested fairly early, ideas reported by later admirers, and plain legend. The number six has a particular place in that story, because Greek arithmetic counted it among the perfect numbers, but the idea of the perfect number six is told separately. Here the subject is the wider school and why number mattered to it.
Number as the stuff of things
The best early witness is Aristotle, who was not a Pythagorean and wrote about two centuries after Pythagoras. In his surviving works he says that “the so-called Pythagoreans” gave mathematics a central place and came to believe that the principles of mathematics were the principles of all things. In some passages he describes things as being numbers, in others as imitating them. He is not always clear about which, and scholars still debate what exactly the claim meant.
Aristotle also names a later figure, Philolaus of Croton (fifth century BCE), who is the source of the few fragments commonly treated as genuinely Pythagorean. In them, reality is built from “limiters” and “unlimited” things, and number is what makes the world knowable by fitting them together. The attribution of particular ideas to Pythagoras himself, rather than to Philolaus or other early followers, is mostly unrecoverable.
Some of what Aristotle reports looks odd to a modern reader:
- Particular numbers were tied to abstractions, so that one number stood for justice, another for the soul, another for opportunity. Which number went with which is reported differently by different writers.
- A list of ten pairs of opposites, including limited and unlimited, odd and even, one and many, and right and left, was used as a framework for describing the world.
- Ten was regarded as a complete number, and Aristotle remarks that a hidden “counter-earth” was added to the Pythagorean astronomy partly so that the heavenly bodies would add up to ten. This is a good example of number shaping a theory, rather than the reverse.
Odd, even and the beginnings of number theory
Early Pythagorean work on number was probably less mystical than its reputation. The classification of numbers into odd and even, the study of figures made of pebbles or dots (the origin of triangular numbers and square numbers), and the first questions about divisors and factors all belong to the tradition that passed into Greek mathematics. How much of this was done by the early school, and how much by later mathematicians such as Archytas, is uncertain.
Later writers credited the school with studying prime numbers, perfect numbers and even amicable numbers, the pair 220 and 284 being the standard example. The evidence for such specific discoveries by Pythagoras himself is thin. It is safer to say that the school helped make number a subject of study in its own right.
Music and ratio
The most solid piece of Pythagorean science concerns musical intervals. Concordant intervals correspond to simple ratios of string lengths: 2:1 for the octave, 3:2 for the fifth and 4:3 for the fourth. Whether Pythagoras himself found this is unknown, but it was well established in the school by the fifth or fourth century BCE.
The famous story of Pythagoras hearing blacksmiths’ hammers and deriving the ratios from their weights is a later legend. It also does not work physically, since hammer weights do not give those intervals. The ratios themselves, however, are real, and they supported a lasting idea: that audible harmony and numerical proportion are the same thing seen from two sides. From there it was a short step to the “music of the spheres”, the notion that the heavenly bodies move in proportions that could be called harmonious. Ancient writers disagree on whether the sound was supposed to be literally audible.
The tetractys
The emblem most closely tied to the school is the tetractys, a triangle of ten points arranged in rows of one, two, three and four. It contains the numbers 1, 2, 3 and 4, which add up to ten, and it contains the ratios of the musical intervals just mentioned. Followers were reported to swear by it. Its ritual role and its later associations are the subject of the dedicated article on the Pythagorean tetractys, so it is only placed here as the school’s single most compact picture of its beliefs. Later writers read it as a diagram of the point, the line, the surface and the solid, and, still later, as a map of cosmic levels. These readings are interpretations, not evidence of early practice.
The theorem that carries the name
The right-triangle theorem is called Pythagorean, but the relationship was known earlier. Babylonian clay tablets from the early second millennium BCE show scribes working with right triangles in the Mesopotamian sexagesimal system, and Indian and Chinese sources contain related results. Greek tradition tied the theorem to Pythagoras, though no early source clearly says that he proved it. The usual story that a discovery about incommensurable lengths troubled the school, and that a member named Hippasus suffered for revealing it, is told in sources that are late and inconsistent, so the question of irrational numbers and Hippasus is best treated as legend with a mathematical core.
How later writers reshaped the legacy
The school faded as an organised community by the fourth century BCE, but its prestige did not. Plato absorbed Pythagorean themes, as the Timaeus shows. Around the first centuries CE, writers now called Neopythagoreans and Neoplatonists rebuilt the tradition. Nicomachus of Gerasa wrote an arithmetic that treated numbers as qualities with moral and cosmic meaning, and Iamblichus presented Pythagoras as a semi-divine sage. These authors are our main source for the colourful material, and they often projected later ideas back onto the founder.
The legacy continued through Christian thinkers, where Christian number symbolism drew on this vocabulary, and Augustine on number is a clear example of a theologian who treated number as a sign of divine order. Renaissance scholars revived the tradition again. Modern numerology claims Pythagoras as an ancestor, but its letter tables and personality readings are largely modern inventions, and the history of numerology shows little connection with the early school. Such readings are best understood as beliefs people held, not as demonstrated facts.
Documented, reported and legendary
| Layer | Examples |
|---|---|
| Reasonably documented | Number as a central principle in the school; musical ratios; odd and even; the counter-earth; Philolaus’ fragments |
| Reported by later writers | Specific numbers tied to abstractions; the tetractys oath; study of perfect and amicable numbers |
| Largely legend | The blacksmith story; Pythagoras’ supernatural powers; a strict “secret society” with fixed rules; punishment of Hippasus |
The picture that emerges is a real intellectual movement whose central insight, that structure and proportion can be expressed in number, outlived most of its particular claims.