Who Was Pythagoras?

Pythagoras of Samos (about 570 to 490 BCE) was a Greek philosopher and religious teacher who founded a close-knit community at Croton in southern Italy. He wrote nothing, and the detailed accounts of his life were written 700 years or more after his death. There is no reliable evidence that he discovered or proved the theorem named after him.

That last point surprises most people, so this page separates three things: what reasonably good evidence tells us, what later admirers added, and what modern historians conclude about Pythagoras and mathematics.

Pythagoras at a glance

How we know about Pythagoras, and why it is hard

For most early Greek thinkers, historians start from quotations of their own books. For Pythagoras there are none. What we have instead are a few brief remarks by people who lived at the same time, more careful reports from Aristotle's school about 150 years later, and long, admiring biographies from late antiquity.

• Contemporaries. The poet-philosopher Xenophanes mocked Pythagoras in a satirical poem for claiming to hear a dead friend's soul in the yelps of a beaten puppy. Heraclitus complained that "much learning" had not taught him understanding. These jibes are short, but they show that Pythagoras was real, famous and associated with reincarnation.
• The 4th century BCE. Aristotle wrote about "the so-called Pythagoreans" and a lost treatise on them. His students Aristoxenus and Dicaearchus wrote about Pythagoras himself. Their work survives only in fragments quoted by later authors, but historians treat it as the most reliable material we have.
• Late antiquity. Diogenes Laërtius (about 200–250 CE), Porphyry (234–about 305 CE) and Iamblichus (about 245–325 CE) wrote the full lives that survive. They preserve some early material, but they also present Pythagoras as a semi-divine wonder-worker, and their accounts grow more miraculous the further they are from his time.

There is not a single detail in the life of Pythagoras that stands uncontradicted. But it is possible, from a more or less critical selection of the data, to construct a plausible account.

— Walter Burkert, Lore and Science in Ancient Pythagoreanism (1972)

Life: from Samos to Croton

Pythagoras grew up on Samos when it was a busy trading port, a short sail from Miletus, where the early philosophers Anaximander and Anaximenes worked. Later biographers sent him on long study trips to Egypt, Babylon, Persia and even India; the Egyptian geometry he supposedly learned is examined on the Egyptian rope stretchers page. The list of countries grows with each retelling, and historians treat these journeys as embellishments, although Samos's trade links make some contact with Near Eastern learning plausible.

According to Aristoxenus, Pythagoras left Samos at about 40, during the rule of the tyrant Polycrates, which puts the move around 530 BCE. He settled in the Greek colony of Croton, on the coast of southern Italy. All the sources agree that he was charismatic and quickly gained influence there, both as a teacher and as an adviser to the city's leading families.

Around 510 BCE, after Croton defeated the neighboring city of Sybaris, opposition to the Pythagoreans' political power turned violent, and a meeting place of the group was attacked. The earliest and most reliable accounts suggest that Pythagoras moved to nearby Metapontum and died there around 490 BCE. More dramatic versions, such as the one in which he is caught because he refuses to run through a field of beans, are later legends.

The Pythagorean community

Plato remembered Pythagoras above all as the founder of a way of life. His followers at Croton shared meals and, according to later sources, their property ("friends have all things in common" was a Pythagorean saying). Members were bound to secrecy about the teachings, and late writers describe a five-year period of silence for newcomers.

• Reincarnation. The best-attested doctrine is metempsychosis: the soul is immortal and passes into new bodies, including animals. Xenophanes' puppy joke and Dicaearchus' summary of the teachings both point to it.
• Rules for living. Short oral sayings (akousmata) set out rules of ritual purity, such as putting on the right sandal before the left or not stirring a fire with a blade. Their original meanings were already obscure in antiquity.
• Diet. Pythagoreans are famous for avoiding beans and sometimes meat, but the ancient evidence contradicts itself. Aristoxenus, for example, said Pythagoras ate most kinds of meat except plough oxen and rams.
• Women members. Iamblichus' later catalogue of Pythagoreans includes women, and several sources suggest that women took an active part in the community.

Later sources describe a split among the followers into "listeners" (akousmatikoi), who kept the sayings and rituals, and "learners" (mathematikoi), who pursued mathematical study. The word "mathematics" comes from the same Greek root, meaning "what is learned".

Numbers, music and "all is number"

The discovery most often credited to Pythagoras is that the main musical intervals match simple whole-number ratios. If a stretched string sounds a note, half the string sounds the octave above it (ratio 2 : 1), two-thirds sounds a fifth (3 : 2) and three-quarters sounds a fourth (4 : 3).

The slogan "all is number" is not a quotation from Pythagoras. It is a modern summary of Aristotle's report that the "so-called Pythagoreans" came to believe that the principles of mathematics were the principles of everything that exists. Aristotle never attributes this view to Pythagoras himself, and many scholars think the developed philosophy of number was the work of Philolaus and his generation.

Pythagoreans also studied "figurate" numbers, arranging pebbles or dots into triangles and squares. The tetractys is one example. Square numbers are another, and they lead straight to Pythagorean triples, as the next section shows.

Did Pythagoras prove the Pythagorean theorem?

Probably not, and certainly not first. The relationship a² + b² = c² was in routine use in Babylon more than 1,000 years before he was born; tablets such as Plimpton 322 list triples far larger than 3, 4, 5. The real questions are whether Pythagoras knew the theorem, and whether he or his early followers proved it. The evidence on both is weak:

• No early source, including Plato, Aristotle and Aristotle's students, connects Pythagoras with the theorem.
• Eudemus, a student of Aristotle, wrote a history of geometry. As far as it can be recovered from later quotations, it did not credit Pythagoras with the theorem.
• The earliest link is a two-line verse about an ox sacrifice, of uncertain date (see the legend below). Even that celebrates finding a diagram, not proving it.
• The oldest proof that survives is Euclid's proof in the Elements (about 300 BCE), some two centuries after Pythagoras died. Proclus, writing on Euclid, praised the proof as Euclid's own achievement.

Historians still disagree about how much mathematics Pythagoras did. Walter Burkert argued that he was a religious and political teacher, and that real Pythagorean mathematics began with later figures such as Philolaus and Archytas. Leonid Zhmud has argued the opposite, that Pythagoras and the early Pythagoreans did genuine mathematics and deserve some of the credit tradition gives them. More traditional histories, such as the MacTutor biography, allow that he may have been the first to prove the theorem. What nobody can show is a proof by Pythagoras, because none survives; the hundreds of proofs of the theorem we have all come later.

The Pythagorean rule for triples

One piece of mathematics that ancient writers did attach to Pythagoras is a recipe for Pythagorean triples. Proclus credits him with this rule: start with any odd number n, and the three sides are

With n = 3 you get the 3-4-5 triangle; with n = 5 you get the 5-12-13 triangle; then 7, 24, 25 and 9, 40, 41. Dot pictures of square numbers show why it works.

Going from k² dots to (k + 1)² dots always adds 2k + 1 dots, an odd number. If 2k + 1 = n², then k² + n² = (k + 1)², and solving 2k + 1 = n² gives k = (n² − 1)/2, exactly the rule above. Whether Pythagoras found the rule this way is unknown, but figurate numbers were a genuine Pythagorean interest. The rule only produces triples whose hypotenuse is one more than a leg; Euclid's formula produces them all.

Pythagoreans are also linked, through a story told centuries later, to the shocking discovery that √2 is irrational, supposedly made by a member named Hippasus. That story is a legend too, though the discovery itself was real and Greek.

Why the theorem still carries his name

By the 1st century BCE Cicero could allude to the ox story without explaining it, and around 100 CE Plutarch mentions the attribution in passing, as if everyone knew it. Proclus' commentary on Euclid then carried it into the medieval and modern traditions, and textbooks have used the name ever since. The full story of how other cultures found the same result, and why the label stuck anyway, is on the history of the Pythagorean theorem page. If you want to see what the theorem itself says, start with the Pythagorean theorem formula.