Pythagoreanism: Philosophy, Numerology, Steam-Powered Pigeons & Farting 📐

Pythagoreanism and its philosophy are seldom taught in today’s classrooms, much unlike Pythagoras’s famous triangle theorem. It’s unlikely that any math class has the time to delve into the esoteric teachings of Pythagoras and his ‘cult’. What if we told you that Pythagoras may not have come up with his theorems but taken credit for the work of others?

Pythagoras’s contributions as a gifted mathematician are timeless but numeracy wasn’t the only gift he possessed; he was also a charismatic orator and used his gift of the gab to attract a devoted following. A following so devoted that they were willing to toil their lives away under him.

What is Pythagoreanism? ?

Pythagoreanism is a collection of beliefs that were held by Pythagoras and his followers. It’s difficult to describe every single belief held by the Pythagoreans but a core belief was that upon death the soul migrates to a new body, or metempsychosis in philosophical terms[efn_note][/efn_note]. The Pythagoreans also devoted time to researching new mathematical formulae, often in Pythagoras’s name.

Pythagoras himself, born on Samos in 570 BC, was an ancient Greek polymath (meaning one who excels in multiple fields). He is known for philosophy, numerological and musical theories and, of course, mathematics. We’ll leave it at that because this article is about his beliefs and following rather than a detailed biography of the man himself.

Image result for pythagoras bust
Pythagoras: philosopher, mathematician, musical theorist, spiritualist and cult leader.

Pythagoreanism trod a fine line between philosophy and religion. Some followers’ lifestyles were so austere they could be compared to monkhood, particularly the akousmatikoi and the Neopythagoreans. Mathematical breakthroughs were celebrated by ritual, one example being the slaughter of an ox in reverance of the discovery of the Pythagorean Theorem:

If we listen to those who wish to investigate ancient history, it is possible to find them referring this theorem back to Pythagoras and saying that he sacrificed an ox upon its discovery

Proclus, 5th century BC [efn_note][/efn_note].

We’ve included the diagram just as a reminder but we will skip the math to stay on topic.

Image result for pythagorean theorem
Pythagoras’s Theorem–in the unlikely event that you didn’t study it at school.

Perhaps ritual slaughter and borderline monkhood are what made Pythagoreanism blur the line between philosophy and religion, but it wasn’t the first ancient Greek cult to do so. Other notable examples include Thales of Miletus’s reverance of water and Socrates’ egotistical belief that he was guided by a divine voice. Yes, we are using the clichéd “but it was normal at the time” defence and, no, we won’t commit to an opinion on whether we think it was a religion or not.

Who Were the Pythagoreans? ?

Obviously they were Pythagoras’s followers and proponents of Pythagoreanism, but who were they exactly? What was their background and motivation? Where did they come from? Where did they live? We will answer these questions and more in this section.

Pythagoreans celebrate sunrise.
Fyodor Bronnikov, 1869.

Many scholars distinguish two separate Pythagorean sects: the original and the later (not to be confused with the Neo-Pythagoreans who emerged in the 1st century BC). The former existed between the 6th and 5th centuries BC, the latter existed from the 4th to the 3rd centuries BC. However, there was a gap between the two periods, hence the division. Thus the Pythagorean era lasted centuries after the death of its founder, but relatively short compared to other cults and religions.

Both sects inhabited Taranto, a coastal city nestled in the heel of Italy. Taranto, or Tarantum as the Romans called it, was a colony of Greater Greece or Magna Graecia. Various sects were scattered across southern Italy throughout both the original and later periods.

Image result for taranto italy
Taranto or Tarantum as it was known by the Romans.

Both sects were were similar in beliefs and customs but the later sects seemed to be more puritanical than the originals; Neopythagoreans practiced abstinence even though Pythagoras himself wasn’t abstinent[efn_note]Encyclopedia of Early Christianity: Second Edition [/efn_note]. Is it possible to be more Pythagorean than Pythagoras? Apparently they thought so.

Despite the question in the title, we aren’t able to give a straight answer to “Who were the Pythagoreans?” simply because of the lack of information. We know broadly where they came from, where they lived, what they ate and what they believed but we don’t know what segments of society they came from. We assume they were educated folk based on the fact they spent such large amounts of time studying math.

We know members were presented to and selected by a council, much like in Freemasonry. New members were subjected to a grueling rite of passage: they were forbidden to speak for the first five years of their probation period, only being permitted to listen, and were only permitted to meet Pythagoras himself after this ‘probation’ period[efn_note]In Search of Pythagoreanism: Pythagoreanism as an Historiographical Category[/efn_note]. We don’t know how many had the patience for these unreasonably harsh expectations but it’s clear that the Pythagoreans took themselves very seriously. Filtering out the riff-raff was how they maintained their mystique and ensured they had only the best mathematicians.

Of a list of 235 members, 17 were women[efn_note]Pythagorean Women: Their History and Writings[/efn_note]. Perhaps we can then assume that roughly 7% of the members were female. This is surprising of a notoriously patriarchal and misogynistic society such as ancient Greece–it suggests that the Pythagoreans challenged more of their societal traditions than most think.

Cylon of Croton ⚔️

In fact, the Pythagoreans challenged society a little too much; Cylon of Croton led a bloody uprising against them circa 500 BC. Fifty or so years later, after the death of Pythagoras, more attacks followed. Settlements were sacked and razed, people were butchered and burned alive. Pythagoras’s death could be attributable to Cylon because he died shortly after an attack.[efn_note][/efn_note]

Image result for cylon of croton
Cylon of Croton’s deep contempt for the Pythagoreans was reflected in the merciless brutality of his assaults upon them.

Within the next fifty years there were barely any Pythagoreans left in Magna Graecia; most had fled to Greece due to the sheer viciousness of these unrelenting attacks. It’s unclear what upset Cylon of Croton and his barbarians so much but the savagery of their attacks makes it seem the Pythagoreans touched a nerve.

According to one source, Cylon was an aspiring Pythagorean but was rejected for membership by the council for some reason:

Cylon, a Crotoniate and leading citizen by birth, fame and riches, but otherwise a difficult, violent, disturbing and tyrannically disposed man, eagerly desired to participate in the Pythagorean way of life. He approached Pythagoras, then an old man, but was rejected because of the character defects just described. When this happened Cylon and his friends vowed to make a strong attack on Pythagoras and his followers. Thus a powerfully aggressive zeal activated Cylon and his followers to persecute the Pythagoreans to the very last man. Because of this Pythagoras left for Metapontium and there is said to have ended his days.

Iamblichus[efn_note]Iamblichus, Life of Pythagoras (translated into English by T Taylor) (London, 1818).[/efn_note]

Maybe he couldn’t keep his mouth shut for his 5 year silent probation period, no one knows.

Philolaus ♾️

Considered by some to have been the true successor to Pythagoras, Philolaus was a prominent Pythagorean who challenged some of his predecessor’s fundamental teachings. He’s notable for being the first recorded person to state that the Earth isn’t the centre of the universe.

Philolaus and Pythagoras tooting on some flutes.

He also believed that everything in existence has one of two properties: limited and unlimited. Unlimited materials were seen as abstract in shape, size and quantity such as the four elements plus space and time. Limited materials were just about everything else.[efn_note] [/efn_note]

Archytas (and His Steam-Powered Pigeon) ?️

Friend of Plato, Archytas was a student of Philolaus and is credited with founding mathematical mechanics. He also invented a steam-powered pigeon that allegedly flew 200 meters. This early ‘robot’ contained an airtight bladder and was powered by air pressure delivered by a steam boiler.[efn_note][/efn_note]

Archytas's steam-powered pigeon.
Archytas’s steam-powered pigeon.

The alleged flight of Archytas’s mechnical pigeon is poorly-documented and there is even the suggestion that it flew on a wire, which raises the question of whether it actually ‘flew’ at all. However, despite the skepticism over the flight of the pigeon, much to the inventor’s credit there is a crater on the moon named after him.

What Did the Pythagoreans Believe? ?

The original, later and neo-Pythagorean sects were further divided into different philosophical schools of thought: akousmatikoi and mathēmatikoi. Rivalry between the groups was bitter as neither would recognise the other as being true Pythagoreans–we won’t attempt to argue which school of thought we think was ‘true’ as we believe both were valid.

Beliefs of the Akousmatikoi ?

The most puritanical sect of Pythagoreans, the akousmatikoi (“listeners”) upheld an oral tradition of reciting and memorising Pythagoras’s teachings, revering those who could recite the most from memory. They frowned upon the continuous research and development of the mathēmatikoi and believed their dedication to mathematical innovation wasn’t true to Pythagoras’s intent (don’t worry, we can’t grasp how this makes sense either).

The akousmatikoi were focused more on the metaphysical teachings of Pythagoras, particularly metempsychosis–the belief that the soul transfers to another body after death. Thus they were concerned more with the afterlife than the present one and committed to an austere lifestyle of silence, vegetarianism and anti-materialism. The closest parallel we can draw to the lifestyle of the akousmatikoi by today’s standards is Buddhist monkhood.[efn_note][/efn_note]

Beliefs of the Mathēmatikoi ?

The mathēmatikoi (“learners”) acknowledged the foundations of Pythagoras’s teachings and attempted to build upon them through mathematical and scientific innovation. The progressive mathēmatikoi were viewed as apostates by the fundamentalist akousmatikoi–a schism widened between the two groups though there is no evidence that it ever erupted into violence.

Contrary to Aristotle’s cosmological theories, the mathēmatikoi believed the Sun didn’t orbit the Earth, nor were planets of the solar system the centre of the universe. This sounds sensible by modern standards so far but they actually believed the solar system, including the Sun, orbited a greater cosmic entity–an eternal fire burning at the axis of the universe. We have no idea how they came to such conclusions given how primitive astronomical instruments were at the time, but they believed in much crazier stuff like metempsychosis so perhaps it’s not so far-fetched after all.[efn_note][/efn_note]

Numerology ?

The Pythagoreans believed that numbers are the essence of the universe and everything in existence can be reduced to single digit integers. They subscribed to the practice of numerology, the belief that the numbers in your life can reveal insights about yourself, others and your future.

Whilst we can’t credit the Pythagoreans for inventing numerology because the system we’re familiar with today is more sophisticated, we can credit them with building the foundations.

pythagorean numerology
Numerologists still use this today.

This proto-numerology was further developed upon by latter scholars who fused the findings of the Pythagoreans with Jewish Kabbalah to create Jewish Pythagorean numerology.

If you’re interested in learning more please read our beginner’s guide to numerology.

Vegetarianism (and Farting) ?

‘Pythagorean diet’ was used before the popularity of the word vegetarian in Medieval times. Pythagoreans believed that eating meat corrupted the soul because of the suffering involved in the slaughter of animals. They believed animals were intelligent and that no being deserved unnecessary suffering.[efn_note][/efn_note] However, the diet strictly prohibited the consumption of faba beans. Why? Because they make you fart.

This image (unintentionally) comically depicts Pythagoras’s fabled rejection of faba beans.

Well, at least according to Cicero that’s the reason Pythagoras famously reject faba beans:

Plato therefore encourages people to go to sleep with their bodies thus disposed that there be nothing which could introduce any wandering from or disturbance of sleep. From which it is thought that the Pythagoreans prohibited the consumption of beans, because that food causes a great flatulence which is contrary to the tranquility of a mind seeking the truth.

-Cicero, de Divinatione 1.30[efn_note][/efn_note]

Cicero’s conjecture was that excessive flatulence caused by eating the beans was ruining the sleep and zen-like concentration of the Pythagoreans. Whilst this is certainly the funniest explanation for why they rejected the humble faba bean, sadly the more sensible explanations are far less funny.

Pythagoras associated the beans with metempsychosis (explained below) because he believed they resembled human genitalia and foetuses. He even went so far as to bury some beans in soil and observed that they resembled human foetuses when he dug them up weeks later. Probably the most questionable contribution he has made to mankind’s wealth of knowledge.

Image result for faba beans
See anything particularly foetuslike about faba beans? Pythagoras did.

Faba beans also represented democracy; Ancient Greeks used them to cast ballots: white beans for yes, black beans for no. Pythagoras discouraged his followers from engaging in politics and democracy so perhaps this is another reason he said to “abstain from beans”; it wasn’t a dietary recommendation but an instruction to boycott democracy.[efn_note][/efn_note]

Pliny the Elder, Roman philosopher, wrote that faba beans were associated with death to their traditional consumption at funerals. It’s not clear why such a staple food should be associated with Hades but perhaps it’s due to one adverse effect occasionally caused by this bean.

Faba beans can kill you. Some people, particularly in the Mediterranean, are susceptible to allergic-like reactions from exposure to the beans, a genetic condition called favism. Even the smallest exposure to them can cause severe symptoms like heart failure. It’s a possibility that Pythagoras suffered from favism and attributed his severe reactions correctly to the beans but for superstitious reasons rather than scientific.

Legend has it that Pythagoras died after being chased to a field of faba beans, refusing to enter it and being caught and slain by his pursuer. We don’t know for certain who his killer was but some sources suggest it was Cylon of Croton.[efn_note] [/efn_note]

Metempsychosis ?

Metempsychosis is the supposed act of the soul migrating to a new body after death. It’s analogous to reincarnation in Buddhism and the ancient Egyptian religion. Pythagoras isn’t credited with theorising it but he played an important role in popularising it in local philosophical circles of the era. Its actual origins are attributed to Orphism, a religion that arose from the texts of Orpheus.[efn_note][/efn_note]

metempsychosis definition
Google’s definition.

Relationship with Numbers ?

Pythagoreanism blurred the lines between mathematics and philosophy, with a particular focus on exploring geometry solely through the use of integers (whole numbers). The ancient Greeks at the time were well-versed in the use of fractions and decimals but the Pythagoreans seldom or never used them over integers.

pythagorean triangular numbers
The Pythagoreans explored mathematics visually, favouring triangular diagrams over traditional numbers, unlike the mainstream ancient Greek populace.

As depicted in the diagram above, the Pythagoreans preferred the use of graphical representations of numbers, mainly through dots, over the traditional numerical symbols of ancient Greece. They called these dots psiphi (“pebbles”) and typically arranged them in geometric patterns like the pyramids above.

The Pythagoreans had a fixation on odd and even numbers, which are more visually distinguishable by the alternating tiers of the pebble pyramids above. They also fixated upon what they called ‘perfect numbers’, numbers that equal the sum of their divisors.

pythagorean perfect number
The Pythagoreans believed that 6 is a perfect number.

Numbers held special abstract meanings, particularly 1, 2, 3 and 4. Whilst not strictly the same as modern numerology the Pythagoreans’ assigned abstract properties to certain numbers:

1: Intelligence and existence
2: Sentience
4: Justice

We left 3 until last because it held the most special significance; they believed it represented the sum of the beginning, middle and end of everything. Also let’s not forget how important certain three-sided shapes were to the Pythagoreans . . .

They also take credit for discovering three of the regular polyhedra (three-dimensional shapes with flat sides and corners): the tetrahedron (pyramid), the cube and the dodecahedron.

pythagorean regular polyhedra
The Pythagoreans take credit for discovering 1, 2 and 4.

The dodecahedron (not the tetrahedron, surprisingly) held special significant because it consists of 20 pentagons, a shape which represented health (we’re not sure why either).[efn_note]The Founders of Western Thought – The Presocratics: A Diachronic Parallelism Between Presocratic Thought and Philosophy and the Natural Sciences[/efn_note]

Music ?

The Pythagorean exploration of music was mathematical and spiritual; they distilled music down to the essence of its parts and attempted to explain why it changes the way we think and feel. Sadly we have no records of what music the Pythagoreans composed and played for their own enjoyment outside the practice of experimentation.

There is geometry in the humming of the strings, there is music in the spacing of the spheres.


Pythagoras applied his knowledge of ratios to examine the relationship between the length and tone of a plucked string. He observed that the most musically satisfying dyads (two note chord) were the eighth, the fifth and the fourth.

piano octave chord
8th dyad, more commonly known as an octave.
piano 5th chord
5th dyad.
4th dyad.

Although we have represented each dyad with piano keys, Pythagoras used a stringed instrument for his investigations. To play a note exactly one octave or eight notes higher on a string you simply half the length of it; no matter the length of the string it will always play an octave higher. Likewise for a fifth you must pluck 2/3 of the string and for a fourth you must pluck 3/4 of it. If we add each of these numbers, including 1 (original string length) we get 10:

1 + 2 + 3 + 4 = 10

The number 10 held a special significance for Pythagoras because he believed it to be the perfect number, containing “the whole essential nature of numbers”. Amazingly today we rely on these principles for the measurement of sound travel in space.[efn_note]The Founders of Western Thought – The Presocratics: A Diachronic Parallelism Between Presocratic Thought and Philosophy and the Natural Sciences[/efn_note]

What is Neopythagoreanism? ?

A revival of Pythagoreanism that lasted from 100 BC to 200 AD, Neopythagoreanism sought to expand upon the original movement’s doctrines and combine them with monotheism (worship of a single god; opposite of polytheism).

They seemed less concerned about mathematics than their predecessors, fixating on metempsychosis, thus they had more in common with the akousmatikoi sect than the Mathēmatikoi.

To complicate things further, Neopythagoreans were also influenced by Plato and were far more religious than their earlier counterparts; they lived simple, ascetic lifestyles and engaged in rigorous prayer and abstinence. Pythagoras himself didn’t abstain from sex so it’s questionable how closely they represented his beliefs.[efn_note]The World that Shaped the New Testament[/efn_note]

Another great post on Pythagoreanism.