The Greek philosopher Pythagoras of Samos (lived c. 570 – c. 495 BC) is most famous today for having allegedly discovered the Pythagorean theorem, but, historically speaking, he did not really discover this theorem and it is even questionable whether he ever engaged in any kind of mathematics at all. The historical Pythagoras of Samos seems to have been a sort of mystic sage and spiritual guru, who lived a far more bizarre and fascinating life than you ever would have guessed from what you learned in mathematics class.

The “Pythagorean” theorem?

First of all, Pythagoras could not have discovered the Pythagorean theorem because it was already known long before he was even born. Over a millennium, in fact. Plimpton 322, a Babylonian clay tablet written in around 1800 BC, demonstrates that the Babylonians knew a method for constructing Pythagorean triples roughly 1,230 some years before Pythagoras’s birth. The Egyptians also may have known about it because their method of constructing right angles shows they knew the Pythagorean triple of 3, 4, and 5.

ABOVE: Plimpton 322, the Babylonian clay tablet written around 1800 BC, which shows knowledge of the Pythagorean theorem over a millennium before Pythagoras

In any case, the legend about Pythagoras having discovered that the square of the hypotenuse of a right angel triangle is equal to the sum of the squares of the two legs is not attested until centuries after his death and is not mentioned by any of the earliest sources on his life (although it is worth mentioning that most of these sources are not especially forthcoming about him).

Some scholars have attempted to argue that, while he could not have discovered the theorem, Pythagoras could have been the first to construct a mathematical proof for it. This is hypothetically possible, but the argument is severely undermined by the fact that no one in antiquity ever credited Pythagoras with having constructed a mathematical proof for any theorem, let alone the one for which he is famous.

The concept of a mathematical proof as we know it does not seem to have existed in Pythagoras’s time and is more likely a later innovation originating with the later mathematician Eukleides, who lived in Alexandria, Egypt in the early third century BC. Finally, the idea of “proving” a theorem would have been antithetical to Pythagorean teachings, since the Pythagoreans believed that numbers were sacred in and of themselves and that their holiness was inherently self-evident to anyone who studied them. Consequently, their divine nature did not need to be proven.

ABOVE: A diagram illustrating the Pythagorean theorem

Along with discovering the Pythagorean theorem, Pythagoras is also credited with having (supposedly) been the first to discover the sphericity of the earth, first to divide the globe into five climate zones, first to identify the morning star and the evening star as the same celestial object, first to propose the idea of mathematical proportions, first to identify the five regular solids in geometry, first to discover the principles of Pythagorean tuning, and, while we are at it, the first person to ever call himself a φιλόσοφος (philósophos), or “philosopher,” meaning a “lover of wisdom.”

The three former of these accomplishments likely belong to Pythagoras’s younger contemporary Parmenides of Elea (lived c. 515 – c. 435 BC), who also lived in southern Italy, the same place where Pythagoras did most of his teaching. The next three are probably accomplishments made by some of his early followers. Only the last of these (the one about him having been the first to call himself a “philosopher”) has a high probability of being genuine.

So, now that we have established what Pythagoras did not discover, you are probably wondering: “What did he discover?” A better question might be what did he teach; the historical Pythagoras was probably more of a charismatic political and religious leader than a scientist or mathematician. All our earliest sources portray him as fundamentally a mystic sage.

Life of Pythagoras

Before we discuss Pythagoras’s teachings, we should first go over what we can accurately reconstruct about Pythagoras’s historical life. We know some information, but very little. Three biographies of Pythagoras have survived from antiquity, but these are all extremely late. The earliest of them, written in the third century AD by Diogenes Laërtios as part of his Lives and Opinions of Eminent Philosophers, is already full of legendary material. The later lives of Pythagoras, written by the Neoplatonist philosophers Porphyrios and Iamblichos respectively, are even longer and even more inundated with legends.

Nonetheless, we can reconstruct a few relatively certain details of Pythagoras’s life from the earliest references to him. The philosopher Herakleitos of Ephesos (lived c. 535 – c. 475 BC), who lived right across a narrow stretch of sea from Pythagoras’s birthplace on Samos, wickedly mocks Pythagoras in a preserved fragment from one of his now-lost works: “Pythagoras, son of Mnesarchos, practiced inquiry more than any other man, and selecting from these writings he manufactured a wisdom for himself—much learning, artful knavery.”

This tells us that Pythagoras’s father was named Mnesarchos and that Pythagoras himself had apparently studied extensively, but resorted to non-traditional teachings. The poets Ion of Chios (lived c. 480 – c. 421 BC) and Empedokles of Akragas (lived c. 494 – c. 434 BC) both laud Pythagoras as a wise sage and the Greek historian Herodotos of Halikarnassos (lived c. 484 – c. 425 BC) briefly mentions him in Book IV of his work The Histories, in which he calls him “one of the greatest Greek teachers.” From all these references, we can fairly certainly reconstruct that Pythagoras was already regarded as “one of the greats” within his own lifetime or shortly thereafter.

Our later sources are a bit thornier, so we need to be somewhat more skeptical. Isokrates, an Athenian orator who flourished during the fourth century BC, about a century after Pythagoras’s death, tells us that Pythagoras had visited Egypt as a young man. In the late fourth century BC, the Peripatetic philosophers Dikaiarchos, Aristoxenos, and Herakleides Pontikos all wrote accounts of the lives of Pythagoras, but none of these have survived. Thankfully, however, Diogenes Laërtios, Porphyrios, and Iamblichos incorporated some of the material from these accounts into their own Lives of Pythagoras, allowing us to piece together some of what they originally wrote.

It is possible, from what these authors preserve for us, to reconstruct a general outline of Pythagoras’s life: He was apparently born on the Ionian island of Samos to Mnesarchos, who was perhaps a gem-carver. Pythagoras may have travelled to other lands as a young man, but the extent of his travels is unclear. In around 530 BC or thereabouts, he relocated to the polis of Kroton in Magna Graecia in southern Italy. In Kroton, Pythagoras established a commune in which initiates swore oaths of solemn fealty and were bound by secrecy. They lived a rigorously ascetic lifestyle and devoted themselves fully to their master’s teachings.

In 510 BC, Kroton won a massive military victory over the neighboring city of Sybaris and several Pythagoreans, including Milon of Kroton, were apparently major generals in the battle. The victory instigated a proposal for a new, democratic constitution—a proposal which the Pythagoreans opposed. Two supporters of democracy named Kylon and Ninon rallied the people of Kroton against the Pythagoreans and, while the Pythagoreans were gathered in one of their meeting houses, Kylon, Ninon, and their mob of supporters, set fire to the house and murdered the Pythagoreans as they attempted to escape from the burning building.

Pythagoras himself may have been killed in this purge or he may have escaped to Metaponton, where he may have survived for several more years before eventually dying. (An amusing, but certainly apocryphal, legend retold by both Diogenes Laërtios and Iamblichos asserts that Pythagoras almost managed to escape, but that he came to a beanfield as he was being chased by his opponents and, since the Pythagoreans considered beans an abomination, he insisting on running around the field instead of cutting through it, allowing his pursuers to catch him and kill him.)

This bare outline is basically all we can really know about Pythagoras’s life for certain. The secrecy of the early Pythagoreans combined with their tragic early demise, which presumably resulted in the burning of any sacred texts they might have had, has left most of the details of their founder’s life obscure to us. Although Pythagoreanism in various forms did continue for several centuries after Pythagoras’s death, it is the earliest stage of the philosophy’s development that is the most obscure. Thankfully, Pythagoras’s teachings are much better attested than his biography.

Metempsychosis

The teaching we can most securely identify Pythagoras himself as having taught is the doctrine of metempsychosis (μετεμψύχωσις; metempsýchōsis), or the “transmigration of souls.” This teaching holds that all souls are immortal and that, after a person dies, his or her soul enters into a new body and is reincarnated. This applies both to humans and to animals and, in Pythagorean cosmology, it was possible for a human to be reincarnated as an animal or an animal to be reincarnated as a human.

We can be very certain that Pythagoras really taught metempsychosis because the earliest surviving mention of him, a satirical poem written in elegiac couplets by Pythagoras’s contemporary, the poet Xenophanes of Kolophon (lived c. 570 – c. 475 BC), clearly associates him with it. Here is a translation of the poem as quoted by Diogenes Laërtios in his Life of Pythagoras and translated by R. D. Hicks:

“They say that, passing a belaboured whelp,

He, full of pity, spake these words of dole :

‘Stay, smite not ! ‘Tis a friend, a human soul ;

I knew him straight whenas I heard him yelp !'”

Here Pythagoras intervenes to save a dog who is being beaten by his master, saying that the dog was a friend of his in a past life and that he recognized him by the sound of his voice. This is obviously intended satirically, but it clearly shows that Pythagoras was associated with metempsychosis from an early date. Xenophanes lived the later part of his life in Elea in southern Italy, not far from Kroton, where Pythagoras did most of his teaching, and may have either met Pythagoras himself in his old age or known people who had known him.

Pythagoras was not the first person to teach the doctrine of metempsychosis; certainly at least Pherekydes of Syros (lived c. 580 – c. 520 BC) had taught it before him and probably others. Nonetheless, we can be sure that Pythagoras taught this idea and, in later times, Pythagoras was the one most closely associated with it.

Pythagoras’s later biographers state that, in line with the doctrine of metempsychosis, he forbade his followers from eating meat. Eudoxos of Knidos takes this even further, stating that Pythagoras not only forbade eating meat, but also refused to even go anywhere in the presence of hunters or butchers. Earlier sources, however, contradict the idea that Pythagoras forbade meat, with many of them stating that he only forbade certain kinds of meat.

It is therefore likely that Pythagoras did not issue a wholesale prohibition against all meat, but rather specifically warned against particular varieties. One dietary prohibition that nearly all the ancient sources agree upon is that Pythagoras definitely forbade his followers from eating beans. They differ as to the reasons why he prohibited them, but the relatively strong consensus that he did forbid them strongly indicates that this tradition goes back to the historical Pythagoras himself.

The poet Empedokles of Akragas, one of our earliest sources of information about Pythagoras, at one point seems to imply that Pythagoras claimed that he could remember at least a few of his past lives. Diogenes Laërtios relates a report from the Peripatetic scholar Herakleides Pontikos which holds that Pythagoras claimed to be able to remember his past lives in detail. He told people he had once been a son of Hermes named Aithalides and his father had loved him so much that he promised to grant him one wish. Aithalides wished to be able to never forget anything that happened to him, even in death. After Aithalides died, he was reincarnated as Euphorbos, a minor hero in the Iliad who fought for Troy and was slain by Menelaos.

Then he was reincarnated as Hermotimos, a philosopher who proved his identity as the reincarnation of Euphorbos by recognizing his old shield hanging in the temple of Apollon, even though it was now so ancient and worn. Next, he was reincarnated as Pyrrhos, a fisherman on the island of Delos. Finally, after Pyrrhos died, he became Pythagoras. Because this report is consistent with what we know about Pythagoras from the earliest traditions and it can be traced back to at least the fourth century BC, so I am inclined to judge that it is probably along the lines of something Pythagoras himself really taught.

Mystical teachings

Pythagoras is credited with having devised the idea of the “music of the spheres,” a doctrine which holds that the planets moves according to mathematical formulae, which resonate and play specific musical frequencies, thus resulting in a tremendous, beautiful, cosmic symphony, which is inaudible to human ears. The fundamental basis of this teaching is that mathematics and music, although they are perceived as different, are actually the same thing. Whether this idea can really be traced back to Pythagoras is questionable, but it was definitely an important teaching for later Pythagoreans.

ABOVE: Detail of an Attic, white-ground polychrome kylix painting dated to circa 470-460 BC, showing a Muse tuning a kithara, a kind of ancient Greek lyre. The Pythagoreans believed that music was fundamentally mathematical in principle.

Numerology

We do not know if Pythagoras himself dealt with numerology, but we do know that numbers were of extreme mystical importance to later Pythagoreans. The Greek philosopher Aristotle (lived 384–322 BC) reports that the Pythagoreans of his own era believed that numbers were the fundamental basis of the entire universe and that “all things are made of numbers.” He also tells us that the Pythagoreans used numbers only for mystical purposes and that they utterly eschewed all practical applications of mathematics, regarding these as unholy.

The Pythagoreans believed that the number one, which they termed “the Monad,” was the source of all things. The number two, which they termed “the Dyad,” was matter itself. The number three was considered sacred because it represented the fewest number of coordinate points needed to draw a triangle, which the Pythagoreans revered as the symbol of the god Apollon.

The Pythagoreans thought that all even numbers were female and all odd numbers were male. Odd numbers were superior to even numbers because they had a middle. The number five represented marriage, because it was the sum of two, the first even number, and three, the first odd number. The number seven was seen as important because there were seven planets, seven strings on a kithara (a kind of ancient Greek lyre), and because Apollon’s birthday was celebrated on the seventh day of every month.

ABOVE: Photograph of the ceiling of the Porta Maggiore Basilica, a meeting house in Rome that was used in the first century BC by Neopythagorean philosophers and was built according to Pythagorean numerology

Above all other numbers, the Pythagoreans revered the number ten, which they considered to be the “perfect number” because it was the sum of the first four numbers: one, two, three, and four. In fact, the number ten was regarded as so sacred and holy that the Pythagoreans never gathered in numbers more than ten because they believed that to do so would disrespect the number ten’s holiness.

The most important symbol to the Pythagoreans was the sacred tetraktys, which consisted of an equilateral triangle drawn with four tiers of dots, with four dots in the bottom tier, three in the second tier, two in the third, and only one in the very top. The total number of dots add up to exactly the number ten. Each of the three sides of the triangle is composed of exactly four dots.

The Roman orator Cicero (lived 106 – 43 BC) recounts a story claiming that the tetraktys was discovered by Pythagoras himself and that, after he discovered it, he was so grateful to the gods that he sacrificed an entire ox, or possibly even a whole hekatomb (one hundred oxen).

Cicero rejects this story as clearly spurious, since everyone knew that Pythagoras was a vegetarian, so he obviously would have never sacrificed a real ox. The later philosopher Porphyrios insisted that the story really happened, but that the ox was actually a decoy made of dough. Iamblichos in his treatise On the Pythagorean Way of Life declares that Pythagoras’s students revered the tetraktys as so utterly holy and sacred that they would swear oaths by it.

ABOVE: The tetraktys, a mathematical symbol which the Pythagoreans revered as the most sacred and holy object in existence, an emblem so utterly divine that they would swear oaths by it as though it were a god

So do all these bizarre ideas about numbers really go back to Pythagoras himself, or were they instead invented by his later followers? The German scholar Walter Burkert, in his landmark study Lore and Science in Ancient Pythagoreanism, possibly the most important work on the subject written in the twentieth century, concludes that Pythagoras probably taught nothing at all about numbers and that the number philosophy of the Pythagoreans is entirely the invention of later Pythagoreans, mostly the Pythagorean philosopher Philolaos of Kroton (lived c. 470 – c. 385 BC). Later scholars, however, have argued against this, instead asserting that the basis of the philosophy can indeed be traced back to Pythagoras himself.

As for the story of the discovery of the tetraktys, it is precisely the detail about the ox that Cicero and Porphyrios found so troubling that indicates most strongly in favor of the story’s origins in the earliest days of Pythagoreanism because, while later Pythagoreans were vegetarians, it is unlikely that the earliest ones were. Therefore, we can probably safely conclude that this story is either genuine or at least a story that originates from the earliest followers of Pythagoras.

ABOVE: Detail of Pythagoras from Raphael’s famous fresco The School of Athens (painted 1509–1511). The slate in front of him shows a lyre and a tetraktys. For more information about this painting, see this article I wrote in which I explore it and some of the symbolism used in it.

Rationality of the universe

Perhaps the greatest legacy of Pythagoreanism today is the idea that the universe is inherently rational and that it can be understood through study and contemplation. This assumption is, of course, the fundamental basis of modern science. It is an idea that goes hand-in-hand with the Pythagorean number philosophy and the idea that “all things are made of numbers.” Earlier Greek philosophers such as Thales of Miletos and his student Anaximandros had argued for rational explanations for natural phenomena, but we have no direct record of them arguing that the universe was inherently rational; whereas, this seems to have been an important idea for at least some Pythagoreans. Whether Pythagoras himself actually taught this particular idea, however, is debatable. There is no reason why he could not have taught it, though, and, if he did, it would represent his most significant contribution to modern science—one that would excel by far the Pythagorean theorem that he did not actually discover.