Episode 11: The So-Called Pythagoreans

November 23, 2021

We turn to the enigmatic, charismatic philosopher Pythagoras and the following that he inspired. Though Pythagoras is today associated with the Pythagorean theorem, he developed a school whose secrets were jealously guarded. The Pythagoreans studied astronomy, mathematics, and music, but also developed a unique philosophy centering around numbers that heavily influenced Plato.


Good evening, and welcome to the Song of Urania, a podcast about the history of astronomy from antiquity to the present with new episodes every full moon. Or close to the full moon anyway, this episode is a few days late and I apologize to everyone who has been nervously glancing out their window every night wondering if their eyes were deceiving them. But it turns out that I had more to say about the Pythagoreans than I expected.

In the last episode we explored the philosophers of the Ionian School, a group of individuals who were active around Miletus on the western coast of Anatolia during the 6th century BC. To keep the episode at least nominally related to astronomy, I skipped over a few of these philosophers who did not have so much to say about astronomy or the nature of the world like Hecataeus of Miletus, who improved on the map of the world produced by Anaximander. Or philosophers who did have things to say about cosmology like Archelaus, who sort of combined the idea of Anaximander that the Infinite was the origin of all matter and the idea of Anaximenes that air was the origin of all matter into the idea that infinite air was the origin of all matter. But not much of his work survived.

But there was another philosopher from the region who was of tremendous influence: Pythagoras. But his philosophy was so radically different from that of the his compatriots in Ionia, that he in no way can be lumped together with the philosophers of the Ionian School. These philosophers, with Thales as the intellectual godfather of them all, were also, appropriately, called the natural philosophers. They were very much interested in explaining the physical world around them. Now, this was not their exclusive remit, and some of them, like Heraclitus, were more interested in explaining the follies of man, war, and ontology than the physical world. But especially relative to the later Greek philosophers, who were more interested in political philosophy, moral philosophy, epistemology, and other branches of what we traditionally think of as philosophy today, the philosophers of the Ionian School were far more focused on studying natural philosophy, or the nature of the physical world, how it came about, and the processes that went on within it like rainbows and meteors.

Pythagoras, however, cannot neatly be fit into the tendencies of this particular school, and his influence was so pervasive that we really have on choice but to treat him separately.

Now today, we think of Pythagoras as “the triangle guy”. His name is most closely associated with the Pythagorean Theorem, which states that the sum of the squares of the lengths of the two sides of a right triangle are equal to the square of the length of the hypotenuse. So from this, in modern times most people, to the extent they have heard of Pythagoras, tend to associate him with mathematics and think of him as an early mathematician. And this association is not entirely wrong because numbers played a central role in Pythagoras’s philosophy. But his life encompassed far more than mathematics alone. In the centuries after his death, Pythagoras acquired an almost mythical status. And the philosophical school he cultivated, a group of individuals called the Pythagoreans, was not really comparable to the other philosophical schools that sprang up around ancient Greece, but might be seen as something closer to a cult today. Pythagoras could plausibly be called a mathematician. But he could also plausibly be called a philosopher, a mystic, or a cult leader. The classicist Walter Burkert described Pythagoras’s position this way:

“The material seems to fall into the pattern each inquirer is looking for. The historian of science rediscovers Pythagoras the scientist; the religiously minded show us Pythagoras the mystic; … the anthropologist finds “shamanism”; and the philological scholar may play off against one another the contraction of the tradition, so that critical virtuosity may sparkle over a bog of uncertainty. Pythagoreanism is thus reduced to an impalpable will-o-the-wisp, which existed everywhere and nowhere.”

Now, as I have been saying somewhat monomaniacally is that one of the main difficulties in understanding the thought of the oldest Greek philosophers is that for the most part they wrote nothing down, and Pythagoras is no exception. There is some very tenuous circumstantial evidence that perhaps he may have written some treatise, but even if he did, it was lost and no fragments survive anywhere. So everything we know about Pythagoras comes to us secondhand or even third-hand or more through other sources. And here, the tremendous influence that Pythagoras had is both a curse and a blessing. It is, of course, a blessing in that because he influenced so many other thinkers his ideas were transmitted down the ages through a variety of sources. But it is simultaneously a curse in that through his influence on other thinkers it can be hard to understand where his ideas end and the ideas of others begin. Nowhere is this problem more acute than with Platonism. One of the most important sources we have for the ideas of Pythagoras and the Pythagoreans is Aristotle, who was writing maybe 150 to 200 years after Pythagoras. But of course, Aristotle was taught by Plato, and Plato was so heavily influenced by Pythagorean thought that many of the foundations of Platonism may have come originally from Pythagoras, but been so natural to Aristotle that he attributed them to Plato. Aristotle spends a great deal of time in his work Metaphysics trying to explain the ideas of Pythagoras and Plato and distinguish the two. But one of the central problems in the study of ancient philosophy is disentangling the contributions of Plato from the contributions of Pythagoras.

To make getting at Pythagoras the man even trickier, Aristotle himself never actually refers to Pythagoras. Instead he exclusively refers to the Pythagoreans. And in many instances he uses a phrasing that is usually translated as “the so-called Pythagoreans.” Now in modern English we interpret this phrasing as putting scare quotes around Pythagoreans, which may not be quite the same sense that Aristotle intended, but there is no doubt that he was attempting at the very least to put some rhetorical distance between the ideas he was attributing to the Pythagoreans and Pythagoras the man. If things had played out differently in history, we might know more about this from Aristotle. One of the rather depressing things about the classics is that in so many cases we know what we do not know. It would be one thing to be blissfully unaware that some ancient author had written a critical treatise. But in this case, we actually know that Aristotle wrote two texts specifically on Pythagoreanism, one called “On the Pythagoreans”, and the other called “A Reply to the Pythagoreans”. But with the exception of a few fragmentary quotations, these have been lost, so we can only dream about what might be contained in them.

Well, Aristotle is not the only source of writings about Pythagoreanism. One of the earliest references to Pythagoras is due to Heraclitus, the somewhat grouchy philosopher we learned about in the last episode. As you might expect, Heraclitus was not very impressed with Pythagoras. He writes:

Pythagoras, the son of Mnesarchus, practiced inquiry most of all men; he made a selection of these writings and created his own wisdom — much learning, artful knavery.

This is an interesting fragment because the formal introduction of “Pythagoras, the son of Mnesarchus”, tells us that this is the first time Heraclitus is writing about Pythagoras in the document, sort of like how today we might introduce a figure by their full name but then refer to their last name throughout the rest of the document. The statement that he “made a selection of these writings and created his own wisdom” implies that Heraclitus believed that Pythagoras didn’t have any original ideas but passed off other ideas as his own.

A very small number of Pythagoreans wrote their own philosophical treatises, but this should ipso facto make us somewhat suspicious because the Pythagoreans were notorious for their secrecy. What makes things even harder, too, is that it is easy to assume that the ideas that these later Pythagoreans wrote down, sometimes writing centuries later, were identical to the ideas of Pythagoras. But it is hard to believe that a philosophical school that had persisted for centuries would have had no new ideas in that time. So it seems very likely that later Pythagoreans developed new doctrines, but what did the later Pythagoreans say and what did Pythagoras himself say? It’s also entirely possible that Pythagoras was himself influenced by the ideas of some of his followers.

One of the more important sources here is from a Pythagorean named Philolaus. In the late fifth century BC, about 100 years after Pythagoras, Philolaus wrote extensively about astronomy among other things, so much of what we know about the astronomy of the Pythagoreans comes to us through him.

As we get more distanced from Pythagoras we also start seeing more works about the figure, and a particularly important one shows up in the late third century or early fourth century AD by a philosopher named Iamblichus who wrote a biography of Pythagoras that survives to today. But we see by this point the legends surrounding Pythagoras had fully flowered.

Some of these stories point to a sort of superhuman intellect, such as Pythagoras predicting an earthquake in the city of Metapontum three days in advance. Although we know today that this is not possible, or at least, we in the 21st century AD have no idea how something like this could be done, in the ancient world this probably would have been seen as an intellectual feat not too dissimilar from Thales predicting a solar eclipse. But other legends surrounding Pythagoras are more miraculous, or religious in nature for lack of a better term. Some of these tales indicate that Pythagoras had a special closeness with the gods. In ancient Greece natural features like rivers and springs were not simply physical entities, but had deities associated with them, and there is a tale of Pythagoras crossing a river and the river replying to him, “Good morning, Pythagoras!”

Many of the legends associated with Pythagoras have parallels to other religious figures. Like some Christian saints and mystics in India, some claimed that Pythagoras possessed the power of bilocation, being in two places at the same time. And like St. Francis of Assisi, Pythagoras was reputed to be able to communicate specially with animals. There is a tale of a bear that had been causing havoc in a town. Pythagoras approached the bear, patted it, gave it some barley and fruit, and then told it not to harm any living creature. The bear ambled off into the mountains and never harmed another soul again.

Another story tells of an ox that had been trampling on a farmer’s beans. Pythagoras advised the farmer to tell the ox not to step on the beans. The farmer mocked Pythagoras and said that he did not know how to speak oxish. So Pythagoras went and talked to the ox and the ox never again touched any beans.

Implausible as these stories may seem, they are not entirely arbitrary and in fact are consistent with certain aspects of Pythagoras’s philosophy. In particular, Pythagoras taught a doctrine of reincarnation which was later adopted by Plato as the idea of the “transmigration of souls.” People could be reincarnated as animals, so it was not implausible to suppose that to Pythagoras, who understood these things, the soul could still communicate with even if it was cloaked in another animal’s body. The story of the ox trampling on beans also points to the special relationship that the Pythagoreans had with beans, which we’ll talk about in more detail in a little bit.

One of the really fascinating things about Iamblichus’s biography is that certain elements parallel Christianity. Now, as I mentioned, Iamblichus was writing in the late 3rd century or early fourth century AD, long after the Gospels had been written, so it is more likely that he was influenced by Christianity rather than Christianity being influenced by Pythagoras. But it is clear that Iamblichus felt the need to cloak Pythagoras in the religious imagery available in his time. Christianity had been heavily Hellenized by this point, so it is not entirely surprising that Iamblichus would have reached for elements of the life of Jesus when telling his story of the life of Pythagoras. Pythagoras was not simply a philosopher of the natural world, but was such a well known mystic that philosophers were writing these hagiographies some seven centuries later.

One of these similarities is that Pythagoras meets some fishermen who have just pulled in a large catch of fish. They say that if Pythagoras can guess the number of fish caught they will do whatever he says. Pythagoras correctly states the number of fish they have caught and then tells them to return the fish to the sea. In his biography, Iamblichus tells this as one of the very first stories about Pythagoras which maps neatly onto the Gospel of Luke where one of the very first things Jesus does in his public ministry is cause Simon Peter, James, and John to catch a large number of fish after they had been fishing unsuccessfully all day. Moreover in the Gospel of John there is another story towards the end in which Jesus causes Simon Peter to pull in a large catch of fish and the exact number of fish caught is actually specified to be exactly 153. Now, it is almost always the case that anytime you see a number in a biblical text that there is numerological significance, and I think this is true independent of your interpretation of the Bible. In a secular reading it is no surprise that the authors of these texts would have constructed them in such a way as to imbue the numbers within them with special significance. But even if you take a highly literal reading, where the historical Jesus caused Simon Peter to pull exactly 153 fish from the sea, the number would still have numerological significance because if the Son of God did cause this miracle to occur, surely He would have chosen the number of fish to be caught to be of special cosmic significance, He would not have picked some number at random. After all, God does not play dice with the universe Now, there is no generally accepted numerological interpretation of this particular number, 153, in the Gospel of John. But intriguingly, it is a triangular number, the sum of the integers from one to 17, and the Pythagoreans, as we will see, imparted an almost mystical significance to at least some triangular numbers. This is probably a coincidence, but one might raise one’s eyebrows a little bit.

Well, back to Iamblichus, all this is not to say that we should dismiss Iamblichus’s biography entirely out of hand. Of course we cannot regard everything that appears here uncritically, but it is plausible that certain details of Pythagoras’s life like where he lived, when he lived there, may have had some factual basis, particularly if we can corroborate those details against other sources.

One last bit of evidence for the extent of Pythagoras’s influence comes to us through two coins produced between 425 and 430 BC, around 70 years after the death of Pythagoras. These coins both show on one side an image of bearded man around which is the name Pythagoras. In those days individual cities produced their own coins, and the master coiner was not permitted to put his own likeness on the coin, so he would usually choose some symbol: a statue of Apollo, an attacking warrior, a Pythian tripod. For whatever reason, the coiner of these particular coins chose Pythagoras as his symbol.

So, what do we know about Pythagoras the man? He was born around 572 BC on the island of Samos, which is right off the coast of modern-day Turkey, not very far from the city of Miletus. For its time, Samos was a very wealthy island. Sometime in Pythagoras’s youth the town managed to construct a tunnel through Mt. Kastro to provide it with water, today called the Eupalinian aqueduct. This tunnel is more than a kilometer in length. Even more impressive, the tunnel was dug from both ends and met in the middle. Ensuring that the two ends of the tunnel met in the same spot was an extraordinary engineering feat. And from modern surveying we can see that in the vertical plane, the two ends were only a few millimeters offset from each other. So Samos had to be quite well off to afford public works on this scale.

Now the sources say that at some point as a young man, Pythagoras traveled to the East and learned his wisdom there. We saw something similar with Thales. Later authors said that he went to Egypt and learned the secrets of astronomy and brought them back to Greece. So it was with Pythagoras. Most sources say he went to Egypt, where he learned mathematics. Some sources describe difficult trials that Pythagoras had to endure to learn the secrets of the Egyptian priests. The Pharaoh sends him to the temple in Heliopolis, but the priests pawn him off on the priests in Memphis by telling him that that the priests there are of an older tradition and that that is where the real secrets are known. And then the priests in Memphis tell him that if he really wants true knowledge he should go to the priests of Diospolis. So he passes from town to town until finally he comes to the oldest order of priests who try to discourage him by requiring him to follow a severe asceticism and strict set of rituals. But they are then astonished when he eagerly takes up all the trials they give him and finally teach him their secrets.

Some other sources expand Pythagoras’s travels beyond Egypt: he went to Babylonia to learn astronomy, he learned secrets from the Jews, the Phoenicians, even the Persians. The trope of the teacher gaining his wisdom from travels in the Orient is common enough that we cannot be sure that it in fact happened. But at the same time, in those days there was plenty of travel around the Mediterranean, so it is perfectly realistic that Pythagoras could have traveled to Egypt or elsewhere in the Near East for that matter.

But he then returned to Samos where he started a philosophical school, the semicircle of Pythagoras, but for one reason or another, left Samos at around age 40. Writers who are sympathetic to Pythagoras say that he prized freedom and had become fed up with the local tyrant. Authors who are less sympathetic say that he was a wannabe tyrant himself and was attempting to impose his rules on an unwilling city and was driven out.

Either way, he then travelled to southern Italy, a region that at the time was called Magna Graecia, or Greater Greece, because it had been extensively populated with Greek colonies. By around the year 300 BC these had became absorbed into the Roman Empire, but in Pythagoras’s day, these colonies were independent city-states. Pythagoras set up shop in the city of Croton and quickly won the trust of the city’s elite. The Senate entrusted the education of its children and women to Pythagoras and by all accounts he immediately became very popular and attracted a wide following. Pythagoras spent the next several decades in Croton, and it is here that Pythagoreanism as a sect or cult really took off. For this reason the Greeks associated Pythagorean philosophy with Italy rather than Pythagoras’s native Ionia.

Towards the end of the sixth century, political sentiment in Croton seems to have shifted and anti-Pythagorean riots broke out. The story as it is told to us is that a young nobleman named Cylon of Croton was extremely eager to join the Pythagoreans. However, Pythagoras observed that he did not have the temperament to be a Pythagorean and refused him. To exact his revenge Cylon whipped up a mob who then burned down the meeting house of the Pythagoreans while the Pythagoreans were discussing philosophy inside with Pythagoras. The famous wrestler Milon of Croton, whom Pythagoras had trained to be the best wrestler in the world, then sacrificed his own life to help Pythagoras escape from the burning building.

Regardless of how these riots began, Pythagoras escaped, probably along with a band of Pythagoreans, to the nearby city of Metapontum, also in Magna Graecia, where he remained for the last decade of his life. As you would expect for any self-respecting figure of his legendary status, there are a variety of tales of his death: he kills himself out of sorrow for his companions who died trying to save him; he is condemned to death by a tyrant, tries to escape, but is caught because he refuses to run through a bean field; he observes the luxury in which a king lives and starves himself to death, and so on.

The ancient sources are unanimous in regarding Pythagoras as being a uniquely charismatic individual, and given the devotion that Pythagoras inspired, founding a sect that persisted for centuries, it’s clear that there must have been something to this. Pythagoras is described as having a striking appearance and dressing unusually. He wore all white and trousers which was very rare in ancient Greece. It was said that he had a gold plate on his thigh, not usually visible, but once observed at one of the Olympic games when he was standing up. He wore a gold wreath in his hair. And he was an exceedingly gifted speaker. He tailored his speech to the group he was speaking to. Antisthenes, a pupil of Socrates says: “For being able to find the kind of wisdom digestible by each individual is a sign of wisdom. But it is a sign of stupidity to make use of one single kind of speech when addressing an audience composed of different sorts of people.” Sound advice, though difficult to do in podcast form.

The scholar Christoph Riedweg argues that Pythagoras is a canonical example of the charismatic leader in the taxonomy of authority introduced by Max Weber. Max Weber was a German sociologist of the early 20th century AD who is today probably best known for his treatise in which he conceived of the Protestant work ethic and argued that it was responsible for the differing economic development of northern vs. southern European countries, but he also developed a number of other ideas, some of which have become so completely accepted that we have generally forgotten his part in developing them, like the idea of the government as being the entity that has a monopoly on the legitimate use of force. But one of his ideas was that political authority can be divided into three classes.

The first of these is “traditional authority,” or figures who have the authority to lead simply because that is the way it has always been. A king has the authority to rule because he was the son of another king who had the authority to rule. A more modern kind of authority is what Weber called “legal authority.” These are individuals whose authority to rule is granted by the laws of a system. A politician’s party wins enough votes and they become prime minister. The most chaotic kind of authority, however, is what Weber called charismatic authority, whereby an individual, through the sheer force of their personality, attracts a following of people who will do their will. Of course these systems can overlap to an extent. You can have a charismatic politician who wins an election and can claim to rule by both charismatic authority and legal authority. But a charismatic politician who attracts enough people to vote for them so they win an election is a leader who is primarily operating in the domain of legal authority. A politician who attracts enough people to rig or overturn an election is primarily operating in the domain of charismatic authority. But the danger with charismatic authority is that it rests entirely on the personality of an individual. Once the individual is gone, the authority disappears and the system must either transform into one of traditional or legal authority, at least until another charismatic figure appears. Much of the superstructure of Frank Herbert’s science fiction series Dune centers around exploring what happens when a system controlled by a traditional authority figure collides with a new charismatic leader, and then how a new traditional or legal authority emerges after the charismatic leader has left.

In the case of the Pythagoreans, after about a century after Pythagoras died, there came to be two factions: the Acousmatics, and the Mathematicians, with the Acousmatics loosely operating in the domain of traditional authority, and the Mathematicians in the domain of legal authority. The Acousmatics took a more religious bent and occupied themselves with rules and rituals, studying the sayings of the master whereas the Mathematicians were more interested in developing natural philosophy and, unsurprisingly, mathematics.

So who was it, exactly, that Pythagoras ended up attracting and who called themselves Pythagoreans? We have strong evidence that the Pythagoreans were largely aristocratic. By far the most defining characteristic of the Pythagoreans was their secrecy. Unfortunately this secrecy means that, despite the relatively large number of ancient sources that talk about Pythagoras, we are almost certainly missing a great deal of his thought, whether that be mathematical theorems the Pythagoreans proved, or details about the mechanism of reincarnation. Iamblichus tells a story of a tyrant who wants to make a political alliance with the Pythagoreans but has been continually rebuffed. So he tells his soldiers to capture the Pythagoreans and bring them to him. As the Pythagoreans are moving from one town to another, the soldiers fall upon the Pythagoreans who then run until they come upon a bean field. Rather than escape through the bean field, they take up sticks and stones to fight the soldiers, but are all killed. Now, it happened that a Pythagorean named Myllias and his wife Timycha had been traveling somewhat behind the main group because Timycha was nine months pregnant. On their way back to the palace, the soldiers were relieved to stumble upon Myllias and Timycha because they had killed everyone they were supposed to bring back to the tyrant. So they easily captured Myllias and Timycha and brought them to the palace. There the tyrant apologized for the poor manners of his soldiers in killing all their fellow Pythagoreans and offers to compensate them and grant them a position in his court. Myllias and Timycha refuse both, and so the tyrant asks them to at the very least, tell him one thing: why was it that their friends didn’t escape through the bean field? Myllias replies, “my friends preferred death rather than to walk through a bean field. But I would rather walk through a bean field than tell you why.” Well, the tyrant is enraged and has a guard execute Myllias on the spot. He then calls for his wife Timycha to be tortured until she reveals the secret. Then Timycha bites off her own tongue and spits it out in front of the tyrant so that she will be unable to betray the group.

So, the general lesson here is that initiates were chosen and trained to maintain this secrecy and it was said that mastery of the tongue was the most difficult skill to acquire. A new initiate had to spend the first five years in silence. During meetings they could listen, but could not speak any ideas of their own. During this period they were called an Acousmatic, which is where the later sect within the Pythagoreans got its name. After this five year probationary period, a new Pythagorean would give all their property to the group.

If at any time a member had to be ejected, they would be given back twice the amount of their original property and a tombstone would be erected with their name on it. From that point forward, the Pythagoreans would treat the individual as a new person that they didn’t previously know.

One of the features of Pythagoreanism was that it was totalizing. You could not be a Pythagorean every now and again or for a little bit at a time, or even just on your own. It was for the entire social group: men, women and children. The rules and rituals that governed Pythagoreanism were so elaborate that a man could not be a Pythagorean alone — it was an endeavor for the whole family. Some of the rules Pythagoras gave to everyone, Pythagorean or not. In his speech to the Senate at Croton, he implored them to follow a number of principles and rules. A few of these, which I will loosely quote from Riedweg’s book on Pythagoras are:

To regard the state as a common pledge, which the leaders have received from the mass of the citizens. To be like one’s fellow citizens in every way, and to be superior to them only in justice. Not to abuse any of the gods by swearing oaths by them, but rather to speak in such a way that what is said is credible without oaths. To strive to make children love their parents not because of the blood-bond (which they had no choice in) but rather by free choice. Not to carry on any extramarital affairs. To be a model of discipline and temperance for all, and to avoid sluggishness in action (there is nothing more important than the right moment for each thing); Not to tear asunder children and parents (which is the worst injustice).

Many of these read like the ten commandments. The prohibition on extramarital affairs, while it might seem somewhat pedestrian to us today, was somewhat unusual for a culture in the ancient Mediterranean, at least insofar as it applied to men. But in addition to these rules, which Pythagoras recommended to all, there were a huge number of rules and rituals that the Pythagoreans were called to observe. Reading the examples we have of these reminds one of reading the Book of Leviticus with its pages of arcane dietary laws and specifications for rituals. Loosely quoting Riedweg again, a few of them are:

Refrain from blood sacrifices in prophecy; forego animal skins as clothing; wear white, clean garments when going into the shrine; when blood is involuntarily shed in the shrine, it must be cleaned up with gold or with sea water; it is forbidden to cut one’s hair or nails during a festival; the gods are to be worshipped with cedar, laurel, cypress, oak, and myrtle; coffins made of cypress wood are not to be used; it is forbidden to burn the dead; it is forbidden to offer up libations with closed eyes; when it thunders, touch the earth; one should enter shrines on the right side and leave them on the left; do not touch fish that are sacred; pour libations to the gods from a drinking cup’s handle.

And so on. For some of these rituals we know of a philosophical justification. The requirement that libations be poured from a drinking cup’s handle was so that one did not accidentally touch the part of the cup that the libation flowed out of or drank from that part, thereby mixing the human and the divine. But other rules, like “do not poke the fire with a long knife” or “do not take swallows into the house” don’t have any obvious rationale that anyone has been able to find. And given the secrecy of the Pythagoreans this is really no surprise. In fact, we should probably expect that there were many more rules that the Pythagoreans observed than what has survived down to us today.

Now, it’s probably the case the level of detail in these rules and regulations was not exactly unique to the Pythagoreans. The temple priests and priestesses of the various mystery cults in ancient Greece also followed a highly regimented suite of regulations, although due to the secrecy of their rites, many of the rules they had to follow did not survive down to today. In fact, Pythagoreanism bears many similarities to a variant of Greek paganism called Orphism which probably developed somewhat before Pythagoras’s time. But what made the Pythagoreans unique was that these rules were extended to the entire society, rather than being limited to the priests and priestesses. It is a little like the later Christian notion of the priesthood of all believers in which everyone in a Christian society is expected to perform a priestly role.

The Pythagoreans followed a common schedule. Pythagoras believed that memory was the most important intellectual skill, so upon waking up, before getting out of bed, a Pythagorean would exercise their memory by recalling everything they had done the previous day in as much detail as possible. Then in the morning they would take a walk in a quiet location like a grove with one or two companions. After their morning walk they would go to the temple or meeting place and would discuss or be instructed in philosophy. And in the afternoon they would exercise by wrestling, running, or other athletic activities and generally care for their bodies with oil rubs. Finally in the evening they would take some time for quiet reflection. Now you’ll notice that between the peaceful morning walks and philosophical discussions, this does not leave a lot of room in the schedule for work. This is clearly the schedule of an aristocrat, and as you’ll recall, the Pythagoreans were drawn from the aristocracy.

As you might imagine, being a Pythagorean was a full time commitment. And more than that, it was certainly not something you could do alone. A man or woman could not decide to become a Pythagorean and just do that on their own. To live this way, the whole family had to be on board. Pythagoras therefore had something to say to everyone: children and women along with men. We heard some of Pythagoras’s opinions on children already: that the goal should be to get them to love you of their own free will and that tearing children away from parents is the worst injustice. Pythagoras had a high opinion of children. He believed them to be especially close to the gods. But at the same time he instructed children not to contradict their parents because they were just starting their journey of life whereas their parents had already traveled a considerable distance and had seen things that they hadn’t.

Pythagoras was especially notable in that he taught women. In fact he was the first western philosopher who taught women and nearly all of the female philosophers of ancient Greece were Pythagoreans. Pythagoras was relatively egalitarian and expected women to be held to the same moral standard as men and more or less follow the same ritual practices. In some ways he even believed women to be morally superior to men. He noted that women form tighter bonds with each other than men do and that they will lend things to each other without witnesses. But nevertheless he had things to say to them. He implored them to have the humility to bring their sacrifices to the temple themselves rather than having their servants do it for them, and to dress simply in clean white garments rather than ostentatiously with expensive cloth. Pythagoras was also opposed to makeup, which was in retrospect a salubrious recommendation since the makeup of his day contained lots of lead.

It’s no surprise that Pythagoras had things to say on the relations between the sexes, too. I already mentioned Pythagoras’s prohibition on extramarital affairs. For its culture, this was quite a radical demand, at least for men. Women, of course, were expected to be chaste. But an aristocratic man generally did not limit himself to his wife. His wife produced legitimate children, but it was common practice to also employ a woman called a hetaira, which means “companion,” who was a high class prostitute. A hetaira would only have a few clients at any time and would enter into a long-term relationship with each of her clients, being paid for a long period of time rather than individual sex acts as the lower class prostitutes called pornai did. A hetaira was expected to entertain as well, perhaps to sing or dance, and to be enjoyable to converse with. But a hetaira was expensive, so an aristocratic man generally also had one or more slaves in his household that he would use as concubines.

According to some of the sources one of the first things the women of Croton ask of Pythagoras is that their husbands get rid of their concubines. And whereas Greek city-states, which were in a perpetual shortage of men because they all got killed in wars, generally conferred a limited form of citizenship to illegitimate children a man had with a concubine, Pythagoras restricted citizenship to legitimate children, further discouraging the practice of keeping concubines.

Now, in some ways, Pythagoras was careful about sex. He advised couples to only have sex during the winter, and even then only infrequently. Sex should only be done sober, and deliberately. This would produce sober, deliberate children. By contrast wild, uncontrolled sex would produce wild, uncontrolled children. When someone asked him how often to have sex, Pythagoras replied, “whenever you want to be weaker than yourself.” But at the same time, Pythagoras encouraged couples to have children so as to produce more worshippers of the gods. And whereas priests and priestesses of the mystery cults generally had to observe purity rituals around sex — having sex would render the priest or priestess impure and they would then need to go through a purity ritual before they could offer sacrifices again — the Pythagoreans did not do this. The Pythagorean philosopher Theano, who was probably Pythagoras’s wife, was asked when intercourse conferred ritual impurity. She replied that a woman was pure immediately after sleeping with her husband, and was never pure after sleeping with anyone else. She also recommended to wives that when coming to their husbands, they remove their modesty along with their clothes.

Pythagoras also insisted that women in particular delay marriage until they were older. Men, according to Pythagoras, should not marry until twenty, but women should not marry until the age of 18. In Athens girls would be married around the age of 12 or 13, shortly after puberty, so waiting until 18 was quite late by Greek standards.

The final thing I’ll mention on the lifestyle of the Pythagoreans was their diet. By this point it will come as no surprise to you that a philosophy that regulated every aspect of its members lives had quite a great deal to say about what they ate. After their secrecy, the second most notable characteristic of the Pythagoreans that the ancient sources discuss is their vegetarianism, which was a highly unusual practice for the ancient world. On this point the sources are, in fact, somewhat mixed. In fact, if you remember Milon of Croton, the famous wrestler who saved Pythagoras from a burning building during the anti-Pythagorean riots, he was trained by Pythagoras and became a champion despite his smaller than average stature because Pythagoras recommended he eat a diet of meat rather the diet of figs and cheese that most wrestlers of the day ate. We also see references to Pythagoreans eating meat that was sacrificed. And, in fact, it is hard to see how they could have . During festivals, a number of animals would be sacrificed and the entire city-state would share the meat together. Refusing this meat would have been seen as an insult, sort of like coming over to a Thanksgiving dinner and refusing to eat any of the host’s food, so to maintain their social ties most Pythagoreans would have had to eat meat at least occasionally. So it is probably the case that the Pythagoreans were not strict vegetarians.

But there were many prohibitions around the consumption of meat. The meat of many animals was ruled out of hand: white roosters, various kinds of fish. And if we consider the Pythagorean philosophy, this makes sense. After all, the Pythagoreans believed in the transmigration of souls. These animals may have once been human, and might be human once more in the future. It was the same soul and it would be abhorrent to eat it simply because it happened to be in a different body at this moment. It seems to be the case that the Pythagoreans divided animals into rational and irrational. Souls could enter into the rational animals, but not the irrational animals, so it was permitted to eat irrational animals, but not rational animals.

But even of the irrational animals, the Pythagoreans heavily restricted what parts of the animal they could eat. They could not eat the heart, feet, womb, brain, heads, testicles, or bone marrow. The common thread here was that they were not allowed to eat organs that are, as the author Porphyry says, associated with “origin, growth, beginning, end, and also that which from which the first foundation of all things comes into being.” So gonads, being associated with origin and growth are out. But also feet, being at the foundation of the animals, along with the head and brain which are at the end.

This respect for organs which are associated with origin and growth also helps to explain the Pythagorean’s most famous dietary prohibition: that against beans. We already saw in the story of Myllias and his wife Timycha that the Pythagoreans preferred to die rather than walk through a bean field. One thing to get out of the way first, though, is that the beans the Pythagoreans were referring to are not generally the beans we eat today: black beans, pinto beans, green beans, kidney beans, these all originated in the New World and were not known in Ancient Greece. The beans they referred to were fava beans. Aristotle provides us with a hodgepodge of explanations as to what the deal was with the Pythagoreans — sorry, the so-called Pythagoreans — and beans. He says:

They resembled genitals, or the gates of Hades, or because it is harmful or because it resembles the nature of the universe, or because it is not oligarchical

Given the Pythagorean prohibition on eating anything associated with origin or growth, it is perhaps not surprising that they would be wary of eating a food that looked like genitals. But it may be that their rationale was even deeper than a superficial resemblance. It seems in the creation of the world they believed that both humans and beans sprouted from the same mud. As proof for the relationship between beans and new life, the author Porphyry says that Pythagoras proposed the following experiment. Partially chew some fava beans and then spit them out into the sun. Leave them there for some time. When you come back, you will find that they smell like semen. A researcher in the 1960s replicated this experiment and verified that, indeed, the result does smell, “something like sperm.” Furthermore, Pythagoras says that if some of those beans were in bloom and had turned black, and then you put it in a pot, seal it, and bury it underground, after ninety days you will find that there will be a child’s head there or the sexual organs of a woman. As far as I know, however, no modern researcher has been able to replicate this second experiment.

Reading between the lines of the available sources, one plausible interpretation of the Pythagorean relationship with beans was that they were a sort of portal from Hades to the Earth. After death, the soul descended into Hades, but then reentered the world through bean plants, and would be transferred into a new animal. Eating beans, or trampling on a bean field, would therefore disrupt the process of reincarnation.

This picture fits very nicely, too, with the famous association between beans and flatulence. Given that beans had souls, it is entirely reasonable that they should cause flatulence since souls were associated with wind and air. Indeed the word for soul was breath. This avoidance of beans was actually not limited to the Pythagoreans. Other Greek priests and priestesses avoided beans, at least on certain occasions, because it left them ritually impure. But the Pythagoreans, who generally extended the priestly rules and regulations to the entire community, avoided beans as a general principle. That said, the Pythagoreans certainly took it to another level. Heraclides Ponticus says that to Pythagoras, “eating beans and eating the heads of one’s parents amounts to the same thing.”

Okay, well we have been talking about the ritual life and some aspects of the moral philosophy of the Pythagoreans. But I haven’t said too much about the details of the philosophy of Pythagoras. So let’s turn to that now.

Now, when we talked about the philosophers of the Ionian School, one of the things they were very interested in doing was finding what they called the arche, or the fundamental nature of all things. For Thales this was water, for Anaximenes it was air, for Heraclitus it was fire, and for Anaximander it was something he called the apeiron or the Infinite. Now, Pythagoras never put it in quite these terms, but if he did, we could say that the arche of Pythagoras was number. To Pythagoras and the Pythagoreans, all things were number. To them, numbers held mystical significance, both in the creation of the world, and in its operation.

It’s at this point where things really start to become inextricably mixed in with Platonism, so we have to proceed with a bit of caution in saying that for sure all of these ideas can truly be attributed to Pythagoras. But it seems that the sort of primal forces in his worldview were the Limited and the Unlimited, associated with the odd and even numbers, respectively. These two opposing forces, however, nevertheless came together to form the One, which is in their view, both even and odd. The even numbers are associated with the feminine and the odd numbers with the masculine. This association comes about because if you take an odd number of pebbles and arrange them symmetrically, there will be an extra pebble in the middle, and if you take an even number of pebbles there will be a void in the center. Since the One is both even and odd, later sources describe it as bisexual.

Aristotle then writes:

When the One had been constructed — whether of planes or surfaces or seed or something they cannot express, then immediately the nearest part of the Unlimited began to be grown and limited by the Limit.

So the masculine One, which is limited, becomes penetrated by the feminine unlimited. And this process produces the number two. Now, we’ll talk about Plato in a later episode, but many ideas here are very similar to the ideas of Plato. But a crucial difference seems to be that Plato envisioned these concepts, the Unlimited and the Limited, the One and the Indefinite Dyad, as being ontological and logical concepts. Pythagoras, however, seems to view them as cosmological concepts. In other words, he understands them to be very literally involved in the creation of the universe, not just in some abstract way, but physically.

In a similar way, the Pythagoreans then viewed the rest of the numbers as being composed of the earlier numbers, and they associated these numbers with particular characteristics. So the number three has the essence of the “whole”, since it has a beginning, middle, and an end. The numbers four and nine represent justice, since justice is receiving that which you have given and the number four was composed of two by two and the number nine was three by three. The number five was associated with weddings, since it was the coming together of two, the first even and therefore feminine number, and three, the first odd, and therefore masculine number. The number seven occupied a special place in the Pythagorean numerology since it was the only number below ten that was not composed of any other number nor composed any other number below ten. In some sense it stood apart from the other numbers and therefore was called the virgin number and was associated with the goddess Athena, who sprung from the head of Zeus fully formed. But by far the most important number in Pythagorean numerology was ten. It’s not entirely clear why the number ten was so significant, but it seems to have occupied its special place because it was the combination of the other most significant numbers, namely it is the sum of the numbers one through four. In the Pythagorean system these numbers are foundational not just mathematically because they are the smallest integers, but also cosmologically. The number one is characterized by a single point. The number two consists of two points, and so from that a point becomes a line. From the number three we now extend the line to a plane. And with four points, we can now create a tetrahedron, the simplest solid. Add all these together and we get the synthesis of all things in the universe.

Now, in addition to this numerology, the Pythagoreans were also very interested in what we would today regard as more traditional mathematics. Of course, Pythagoras’s name is forever associated with the theorem about right triangles that bears his name, but he was almost certainly not the first to discover it. The Babylonians and Egyptians had been aware of the theorem for centuries, or at least had known of Pythagorean triples. Pythagoras may have learned of the relationship during his travels in the East and simply brought it to the Greeks. Regardless the Pythagoreans apparently did quite a lot of other math, although, thanks to their strict secrecy, much of it is unknown. Nevertheless, some sources say that there was a custom whereby a Pythagorean who had fallen on hard times and needed to make some money could be exempted from the rules of secrecy for a time and be permitted to teach geometry to the broader public for money. One story tells of a Pythagorean named Hippasus of Metapontum who had lost all his money and was reduced to the lowly state of having to peddle his discovery of the construction of a pentagonal dodecahedron. Another, darker story, says that he betrayed the group and told of his discovery to a non-Pythagorean, and consequently was drowned in the ocean.

Hippasus meets another grisly end as the reputed discoverer of irrational numbers, numbers which cannot be represented as a ratio between two integers. Hippasus found that if he took a pentagon, the distance between two opposite corners could not be represented by any number that evenly divided the length of the side of the pentagon. It was not simply that such a representation was impractical or took too many steps, but that it produced a logical contradiction. It had always been assumed that it was possible to represent the length of any geometric construction as the ratio between two integers, and, in fact, the entire Pythagorean worldview depended on it. Remember that in the beginning there were the two primordial forces of even and odd, the limited and unlimited. The limited produced the number one and was then penetrated by the unlimited and became two. And from these sprang forth all the integers, and with them the entire universe, because, after all, remember that all things are numbers. Not just in some vague abstract sense, but literally. There was a number associated with you. But the discovery of irrational numbers shattered all this. Because here was a part of the universe which could not in principle be represented with the numbers of their system. So, according to the story, Hippasus had made his discovery while at sea and was consequently thrown from the boat and drowned. Incidentally, the history of astronomy has its own famous discovery at sea, namely Subrahmanyan Chandrasekhar, who in the year 1930 derived the maximum mass of a white dwarf on his voyage from India to England. Although Chandrasekhar’s result was also initially met with a mixed reception — the great astrophysicist of the time, Arthur Eddington, publicly disparaged it — Chandrasekhar at least was not drowned at sea for his work. So there was some progress.

Well, I’ve talked now for some time about a lot of different things that don’t really have much connection with astronomy: the transmigration of souls, how the Pythagoreans structured their day, what their deal with beans was, their numerology. But I went through all of this because the Pythagoreans didn’t see astronomy as just one thing they did, independent of everything else. They didn’t sometimes do astronomy, and then sometimes do some math, and then sometimes observe some ritual about how to put on their shoes. Pythagoreanism was a complete way of life, and many of the rituals they observed had motivations from their philosophy of the cosmos. To them this philosophy informed their understanding of the universe, and they connected to this understanding of the universe through even trifling daily activities. But, now that we have laid the foundations of the Pythagorean philosophy, it is finally time, at long last, to turn to their astronomy, because this is, after all, a podcast about the history of astronomy.

Now, as I mentioned towards the beginning of this episode, most of our understanding of Pythagorean astronomy comes to us through a Pythagorean named Philolaus, who lived about a century after Pythagoras. Now, why he dared to write these ideas down, given the secrecy of the Pythagoreans, we cannot say. Perhaps he had fallen on hard times. But because our sources are so limited, we really have a hard time saying to what extent the cosmology that Philolaus set down was due to Pythagoras himself, held by the Pythagoreans generally, or were his own idiosyncratic ideas. Based on some remarks in Aristotle it seems as though Pythagoras’s cosmology was originally geocentric, but by Philolaus’s time it seems that the Pythagorean cosmology had moved beyond geocentrism.

Now, most cosmologies really throughout history were either geocentric and placed the Earth at the center of the universe, or heliocentric and placed the Sun at the center of the universe. But the Pythagorean cosmology was neither. Instead, they placed at the center of the universe an object which we will call the Central Fire, but they had many names for it, “The Tower of Zeus”, the “Throne of Zeus”, the “Hearth of the Universe”, the “Mother of the Gods”, the “Altar, Bond, and Measure of Nature”.

Now, this Central Fire is not the Sun. In fact, the order of the celestial bodies goes like this: you have the Central Fire at the center of the universe, around it orbits an object called the Counter-Earth, beyond that is the Earth, then you have the Moon, then the Sun, and then the planets Mercury, Venus, Mars, Jupiter, and Saturn, and finally you have the fixed stars beyond. Now, the way this is structured, the Earth rotates around the Central Fire every 24 hours, however just as one face of the Moon is always turned towards the Earth, so one face of the Earth is always turned towards the Central Fire. We are on the side of the Earth facing away from the Central Fire so we never see it, nor do we see this mysterious other body, the Counter-Earth. Now, the Central Fire exists in order to provide the creative force of the universe. It animates all life in the universe. Well, fair enough. But what about the Counter-Earth? Why did the Pythagoreans add that? It seems the Pythagoreans had three possible motivations. The first was to explain an empirical phenomenon, namely that lunar eclipses are more frequent than solar eclipses. A solar eclipse is caused by the Moon coming between the Earth and the Sun. A lunar eclipse could be caused by the Earth coming between the Moon and the Sun. But if that were the only way to cause lunar eclipses, the Pythagoreans reasoned that lunar eclipses should be as rare as solar eclipses, when in fact we see many more lunar eclipses than solar eclipses. However, if there was another object closer to the center of the universe than the Earth, then it could also come between the Moon and the Sun and cause lunar eclipses as well. The second motivation for the Counter-Earth only applies to some versions of the Pythagorean cosmology. In some versions, the Counter-Earth is at the same distance from the Central Fire as the Earth, but always on the opposite side of it. Here the idea was that the celestial bodies were in general insubstantial. They were lights in the sky, some small and pointlike, others like the Moon and Sun large and bright. But they were not made of solid matter as the Earth was. But if the Earth was alone in being made of matter, it would throw the universe off balance. So there had to be a second, Counter-Earth of equal size and composition on the opposite side of the Central Fire as a sort of counterweight.

The final motivation was probably the one that was the most persuasive to the Pythagoreans and is probably the least persuasive to us today, namely that the Counter-Earth was needed to bring the number of celestial bodies to ten. With the Central Fire, Sun, Earth, and Moon, along with with the five classical planets and the sphere of the fixed stars, we have nine bodies in the universe. But in the Pythagorean worldview, ten was a number of prime importance, representing the totality of all things. So how could the entire universe be composed of anything except ten objects? So an extra object needed to be added. But of course, like the Central Fire, we don’t see it in the sky, so it has to be closer to the center of the universe than we are, or at least on the opposite side of the Central Fire from us. This construction also fit with their cosmogony, their understanding of the creation of the world, where we see the number one first being created, being manifested by the Central Fire, then the number two, being represented by the Counter-Earth, and then the subsequent numbers, each one becoming manifested in a celestial body. The Earth becomes associated with wholeness as the number three, having a beginning, middle, and end, and by the time you get to the most distant object, the fixed stars, you arrive at the number ten, which encompasses the totality of all things, just as the stars represent the furthest limit of the universe and encompass all things. And again, here, it’s very easy to apply a Platonic interpretation of this worldview and suppose that there is an analogy between the numbers, the concepts associated with the numbers, and the celestial bodies. But Aristotle is very clear that one of the key differences between Platonic and Pythagorean philosophy was that the Pythagoreans interpreted this quite literally. Aristotle personally believes the Pythagorean worldview to be somewhat confused in that it mixes up very different concepts together in the word “number”: a number can refer to an ordering, an abstract notion of amount, a spatial extent, a ratio, and so forth. We, along with Aristotle, understand these meanings to all be different concepts even if we as shorthand use the same word, “number” to refer to all of them, but nevertheless, the Pythagoreans did not seem to hold these distinctions as being meaningful.

Now, there are a few interesting things to note about this cosmology. The first is that it is not geocentric. Now, from our perspective, the Pythagoreans did not have very compelling reasons to reject a geocentric model. But the fact that they seem to have had no reservations about displacing the Earth from the center of the universe should tell us something about their freethinking ways. The other thing to note about this cosmology is that it is fairly implausible, at least in any model that attaches distances to the heavenly bodies. Some of these models have the distances between the objects growing exponentially by factors of three. So the distance from the Earth to the Counter-Earth is three times the distance between the Counter-Earth and the Central Fire, and the distance between the Moon and the Earth is nine times larger, and the Sun and Moon is 27 times larger and so on. The problem with this is that the parallax involved is tremendous. As the Earth revolves around the Central Fire every day, the apparent position of the Moon would vary across half the sky. So the only way this can work is if the Earth and Counter-Earth are much closer to the Central Fire than the rest of the celestial bodies. But it is more likely the case that the Pythagoreans simply weren’t interested in pursuing the empirical consequences of their theories. It just may not have been something that occurred to them. After all, they were the forerunners of mathematicians, and even mathematicians today are generally not very interested in the empirical consequences of their ideas. The intellectual edifice that they create is interesting enough to warrant their attention, regardless of its application to the real world.

The Pythagoreans had a few things to say about nature of the celestial bodies beyond just their overall arrangement. For the Sun, they believed it to be spherical, but not to produce its own light. Instead they thought that it was composed of a glassy substance, which either reflected light from the Central Fire like a mirror, or refracted light from the Milky Way or both, and either way, intensified it.

Now, one thing to note here is that it seems as though the light from the Central Fire was actually rather feeble. After all, the Moon is lit up by the Sun and goes through phases depending on where it is relative to the Sun, but it isn’t really lit up by the Central Fire even though the side of the Moon that’s facing us should be constantly illuminated by the Central Fire. So the light from the Central Fire cannot be all that strong, which means that the Sun must somehow be concentrating this light.

As for the Moon, it seems that Pythagorean opinions changed over time. I mentioned that one reason that the Pythagoreans added the Counter-Earth to their models was in order to balance the weight of the Earth, and this seems to imply that the Moon must be insubstantial since it does not need its own counterweight. But later Pythagoreans came to be of the opinion that the Moon was solid and moreover plants and animals lived there. But these were no ordinary plants and animals, but were fifteen times stronger than the plants and animals we have here on Earth. Why fifteen times stronger? That number probably came about because a lunar day is about fifteen 24-hour periods. It’s a bit of an apples to oranges comparison since on the one hand we’re looking at the time of daylight on the Moon and comparing it to a period of both day and night on the Earth. If you did a fair comparison, those plants and animals would be more like thirty times stronger, but there you go. Given the paucity of sources we have, it can oftentimes be hard to understand discrepancies like this. Maybe the Pythagoreans just didn’t think about this particular detail very hard or didn’t think this was a big issue. Or maybe had a perfectly reasonable explanation for this and we’re missing something in the sources. Or maybe the sources made a mistake in relaying the Pythagorean ideas. It’s hard to say.

We do seem to see what we might consider a regression in Pythagorean thought over the centuries. Whereas earlier Pythagoreans believed that the phases of the Moon were due to light reflected from the Sun, which is the modern opinion, later Pythagoreans claimed that a great fire periodically swept across the face of the Moon. Though, maybe this explanation was not inconsistent with the earlier explanation and the fire was caused by the Sun.

The last strictly astronomical aspect of Pythagorean thought that I’ll mention is their thoughts on the Milky Way, though here they did not have any particularly unique ideas. Here they generally held that the Milky Way was related to the myth of Phaethon. If this were Episode 7 I would go into the entire backstory of this myth, but in brief, Phaethon was the son of the nymph Clymene and the Sun god Helios, but was raised by Clymene alone since Helios had to drive the chariot of the Sun across the sky every day. Phaethon knew that his father was divine, but his friends teased him since he could not prove it. So one day he went to his mother for advice and she told him to travel east to India where he would find Helios’s palace. And there he could ask Helios himself and request some token of his heritage. So Phaethon traveled east and presented himself before Helios and asked if the god would recognize him as his son. Helios replied that he could not deny his own son anything and told Phaethon that he would grant him whatever he desired. Phaethon, being a brash young man, demanded to drive the chariot of the sun across the sky for a day. Helios’s countenance immediately turned from gaiety to grimness. He told the young man you don’t know what it is that you’re asking for. Driving the chariot of the Sun is no easy thing. The horses are strong, in the morning the way is steep, at noon the heights are terrifying, and at evening the descent is very fast. Even I as a God can hardly keep control of the chariot. And the journey is not one of forests and towns, but is filled with monsters. You must avoid the horns of the bull, the sting of the scorpion, the jaws of the Great Bear. Ask anything but this.

But Phaethon was undeterred and demanded to drive the chariots. So as dawn approached, Helios instructed him on how to keep a firm rein on the horses to keep control of them and avoid the monsters in the sky. But as Phaethon took off, he found that the horses were too strong for him, and the monsters in the sky more terrifying than he had imagined. As he narrowly missed the stinger of the Scorpion, he let loose of the reins entirely and the horses began careering around the sky. The Sun came close to Libya, drying up the land and burning the skin of the Æthiopians black, many cities burst into flames and many people perished. Finally, Zeus, who had been watching the chaos from Olympus could permit this no more and sent down a lightning bolt to strike Phaethon dead and end the sorry affair.

According to the Pythagoreans then, the Milky Way came about during the incident of Phaethon’s journey across the sky. There were a variety of particular explanations, that it was a trail of stars that the had been dragged from their original place by the Sun, or that it was bits of the Sun that had broken off during its chaotic journey, or that this was the original path of the Sun before Phaethon had modified it. Some more naturalistic explanations held that the Milky Way was nothing more than refracted light from the Sun, similar to a rainbow.

Okay, well there is one other major component to the astronomy of the Pythagoreans, but this ties into another important area of investigation of the Pythagoreans, namely music theory. This is the idea of the music of the spheres, which came to be extremely influential in Western civilization and some two thousand years later motivated the astronomer Johannes Kepler in his studies of the heavens.

The Pythagoreans were the first thinkers we are aware of to systematically study the theory of musical tones. Music was an important part of daily life among Pythagoreans. They likely heard and played music daily and had music to help the ill recover. It is very likely, too, that Pythagoras himself was responsible for the early investigations into music theory since the sources all attribute these discoveries to him directly rather than the Pythagoreans more generally. One of the later sources tells a story of Pythagoras walking past a forge and hearing the hammers hitting upon the anvils. However, he notices that these hammers produced sounds in harmony. He recognized that some of them produced an interval of an octave, others an interval of a fifth, and still others an interval of a fourth. He then runs into the forge and demands to see the hammers. He tries hitting the anvil with different amounts of force and at different locations, but finds that the tone is solely determined by the weight of the hammer. So he weighs the hammers and goes home.

Then at home he attaches four strings of equal length to a horizontal peg and hangs the same weights that he measured at the forge from each of the strings. Plucking the strings, he determines that he hears the same intervals that he did at the forge, an octave, a fifth, and a fourth. He notices that the string with the most weights on it produces the highest tone, and the string with the least weight on it the lowest tone. Moreover, the highest tone, one octave above the lowest, had twelve weights on it and the lowest tone had six on it, so there was twice as much weight to produce an octave. Then he found that the string that produced a tone a fifth above the lowest tone had nine weights on it compared to the six of the lowest tone. So there was a 3:2 ratio to produce a fifth. And the string which produced the interval of a fourth had eight weights on it, which made a 4:3 ratio. And the dissonant pair had a ratio of 9:8.

This is a wonderful story of early scientific experimentation, but unfortunately we know that it is fabricated because it is physically impossible. First of all, Pythagoras’s original observation that the tone produced by a hammer striking an anvil is determined by the weight of the hammer is not true. But a more serious issue is that the tone that a string of fixed length produces is proportional to the square root of the tension of the string, so Pythagoras would have had to square all the weights he attached to the string. In other words, if he had six weights on the string with the lowest tone, he would have needed not 12, but 144 weights to produce an octave. And similarly 81 weights rather than 9 to produce a fifth. Other philosophers of antiquity seem to have realized that this was a mistake, but certainly by the 1600s AD natural philosophers were well aware that this reported experiment by Pythagoras did not work.

Nevertheless, the ultimate conclusion, that an octave is associated with the ratio 2:1, the fifth with the ratio 3:2, and the fourth with the ratio 4:3, is correct. The more modern understanding of these ratios is that they have to do with the frequency of sound produced. One tone an octave above another will have twice the frequency of the fundamental. Of course, the Pythagoreans, having no theory of sound, could not have had this interpretation, but another more direct way these ratios show up is in the lengths of strings needed to produce the tones, assuming that the strings are held at fixed tension. One string half as long as another will produce a sound an octave above the fundamental when plucked.

At this point I will pause and give a brief warning that to illustrate some of these musical ideas I am going to play some tones. If you happen to listen to your podcasts at faster than real-time speeds, or slower for that matter, I recommend you reset to playing at real-time for the time being because the speedup can distort musical tones.

Okay, so the interval of an octave has a ratio of 2:1 and sounds like this:

And the interval of a fifth has a ratio of 3:2 and sounds like this:

And then the interval of a fourth has a ratio of 4:3 and sounds like this:

However, if you play the upper notes of the fourth and fifth, you get a whole tone, which when played together, is dissonant:

Okay, for the time being you can return your podcast to its original speed. There will be a few more tones later on, but don’t worry I’ll warn you again.

Now you’ll notice the ratios I mentioned made use of the numbers one, two, three, four. If you add them together you get the number ten, the most important number in Pythagorean numerology, and if you play them, you can build up all the harmonies of music. So we can see how in the Pythagorean worldview the idea of number permeated everything: ontology, cosmology, and musicology.

Now Greek music was structured around the tetrachord. This consisted of a fourth [play fourth] with two notes in between. What exactly these two notes were depended on the system, the region, and the musician. But in the Pythagorean system, they were a half-tone followed by a whole tone. Pythagoras’s innovation was to essentially stack two tetrachords on top of each other separated by a whole tone, this dissonant 9:8 ratio he had earlier discovered. This then builds out a full scale, in what we today call the Phrygian mode. The seven unique notes it produced then make up the heptachord, although with the octave on top it is sometimes called the octachord.

It was clear to the Pythagoreans that the tones of the heptachord must apply to the heavens. After all, the various celestial objects were so large and moved so rapidly that it was inconceivable that they would not produce any sound at all. So they should be continually emanating a music of the spheres. Why then do we not hear this music of the spheres? Here the Pythagoreans had a very clever explanation which Aristotle relates to us. Sounds are only audible in terms of their contrast. We hear speech as a change from silence and then a change back to silence. But the music of the spheres presents no contrasts. It is simply a constant, never-changing sound. Since it is present from the moment of our birth to the time that we die, we can never perceive it. It is like water to a fish. In fact the Pythagoreans claimed that only Pythagoras had attuned himself to the cosmos so acutely that he was able to perceive it. As you might imagine, Aristotle rejects this argument. Maybe we don’t perceive it with our ears directly, but there are other ways to notice sounds. If there is a very loud sound, objects start vibrating and even shatter. But we don’t see trees and rocks shattering from the music of the heavens even though it must be a tremendous, pervasive roar.

Okay, well there Aristotle goes being a party-pooper again. But what did the music of the spheres sound like? According to the Pythagoreans, the tone produced by each of the spheres was proportional to the speed at which it rotated. More slowly moving spheres produced lower tones, and faster moving spheres produced higher tones. Now here we have some conflicting sources, because it’s not exactly clear how rapidly the spheres rotated. In one depiction, the Earth and Counter-Earth move most rapidly, rotating around the Central Fire once every 24 hours, or perhaps more frequently for the Counter-Earth, then the Moon rotating roughly every month, the Sun rotating every year, and so on with the more distant spheres rotating more slowly until arriving at the fixed stars. Now here we have a problem because in this formulation the fixed stars don’t really rotate at all, yet in order to get the right number of tones they have to move at least a little in order to produce a sound. In principle they would move a little bit due to the precession of the equinoxes, but the Pythagoreans almost certainly had no knowledge of this phenomenon at this early point in time. Maybe they just assumed some slow rotation. In another formulation, the Earth moves very slowly, and then the Moon rotates faster, and the Sun faster still, until you get to the fixed stars, which rotate around the Central Fire once every 24 hours.

So in the first model, the Counter-Earth produces the highest tone, the Earth the next highest tone, then the Moon and the Sun, and so on until you get to the fixed stars which produce the lowest tone. In the second model where the stars rotate every 24 hours, they produce the highest tone, and the Counter-Earth the lowest.

Regardless, we still have one problem, which is that there are 10 celestial bodies — remember we had to add the Counter-Earth to get to that magical number 10 — but there are only seven tones in the heptachord. We can get rid of one right off the bat. The Central Fire is at the center of the universe and doesn’t move, so it doesn’t produce any sounds. But this leaves us with nine celestial objects and only seven tones to give them. It seems that the Pythagoreans associated Mercury, Venus, and the Sun with the same tone, because on average, they all have the same angular velocity. So now everything lines up and every object has a tone.

Here you might want to set your podcast playback speed to real-time.

The seven tones, then, of the celestial bodies, would sound like this:

But of course, the celestial objects do not play one note at a time, but all at once. So really the music of the spheres would sound something more like this:

But, of course, much louder than your headphones can produce.

Well, I have been going for some time now, and while there’s more to say about the Pythagoreans, by this point we’ve hit the high notes: their numerology, their astronomy, and their music theory. This episode was a little unusual in that, while I’ve tried to go more or less chronologically, by virtue of their secrecy, it’s hard to separate out the thought of the Pythagoreans into the contributions of individuals and so they all get lumped together even though their movement spanned several centuries. But in the next episode we’ll return to the main thread of pre-Socratic astronomy by turning to Xenophanes, Parmenides, and the Eleatic School. I hope you’ll join me then. Until the next full moon, goodnight, and clear skies.

Additional References

  • Burkert, Lore & Science in Ancient Pythagoreanism
  • Riedweg, Pythagoras
  • Pomeroy, Pythagorean Women