Wild, Wild Numbers

What does it mean that mathematics began as a cult?

Imagine a world without physics, without calculus, without the concept of gravity or even the number zero. To this world, add a formula, a^2 + b^2 = c^2, that reveals a pattern present in every right triangle in the world, a pattern that previously eluded human perception. Calling it the E = mc^2 of its day would be an understatement. Before Einstein and Newton and Euclid, it was not only the formula of the moment, it may have been the only formula this world knew, period. The Pythagorean Theorem, as it has come to be called, was the most advanced mathematical principle available to the early Western world.

The novelty of the Theorem was perhaps rivaled only by the strangeness of its namesake, Pythagoras. Born in 570 BCE on the island of Samos in ancient Greece, Pythagoras’s influences extend to mathematics, architecture, music, astronomy, and philosophy. He is credited with discovering the first formal system of musical tuning (Pythagorean tuning), the concept that the Earth is round, the geometrical precursors to the Roman basilicas, and the planet Venus. When, a century after Pythagoras’s death, Plato wrote of “philosopher-kings” who would rule with perfect wisdom over their subjects, he may have had Pythagoras in mind.

Pythagoras was also the charismatic head of a prolific cult, the Pythagoreans. They invented both the term “mathematics” and the discipline of mathematics itself. Practicing it as a form of mystical numerology, they worshiped a geometrical symbol called the tetractys, believed in prophecy of the future, and forbade such actions as gathering in groups larger than 10. Far from garnering universal admiration in the ancient world, Pythagoras and many of his followers died in a firefight against alarmed locals in a downfall worthy of Waco, Texas.

These days, we see mathematics as the pinnacle of objective human knowledge, with all the sober, rational rigor that distinction implies. When George Orwell wished to paint a totalitarian society inimical to truth in “1984”, he didn’t have it say “the earth is flat” or “E ≠ mc^2”,  but rather “2 + 2 = 5”.  We gloss over the fact that the origins of mathematics look anything but rational, and that the earliest mathematicians were unabashed mystics—even by the standard of an age that believed in gods and goddesses that occasionally roamed the earth. Is their understanding of mathematics primitive, or is ours bloodless—or downright neurotic?

To this day, the legend of Pythagoras is shrouded in the twin distortions of genius reverence and revisionist skepticism. The contemporary political philosopher Bertrand Russell said of him, "I don't know any other man who has been as influential as he is in the school of thought". His contemporary Xenophanes, on the other hand, satirized Pythagoras as a sort of Doctor Doolittle who purported to speak to dogs and recognize them as departed friends. Every hue of charlatan and saint has been used to color him.

Decisive record-setting on Pythagoras isn’t helped by the historically hazy period he lived in. His fellow Greek philosopher Socrates wrote down none of his writings directly and is primarily known to us through his student Plato, the first Greek philosopher to move away from a purely oral tradition. Pythagoras was born a hundred years earlier and produced neither extant writings nor enthusiastic scribblers among his disciples. 

Worse still, Pythagoras’s protean obscurity was perfect for the self-serving interpretations of ancient scholars and polemicists. He could be whatever you wanted him to be. Plato made him a figure worthy of philosopher-king status to promote the idea of a perfect republic. Future anti-Platonists attempted to credit him with all of Plato’s ideas. Anti-Christians of the 3rd Century CE even set him up as the true messiah to spite Christ. His cult appeal lived on for millennia among occultists and freemasons inspired by his mathematical-geometrical mysticism.

Consequently, nothing about his record is guaranteed. The Pythagorean Theorem that bears his name may not have been an original discovery, for one. Many scholars believed he cribbed the formula during his travels around the Mediterranean and Egypt. He may not have even done that much. If you believe Pythagoras himself, on the other hand, he derived all his doctrines by interpreting his own dreams. Or he discovered them while traveling to the underworld. Or, maybe, he actually said none of this.

This kind of historical ambiguity isn’t especially satisfying. In Pythagoras’s case, scholarly rigor also denies us the juicier tidbits of his story. So let’s put it to the side, and consider the holy-man picture of Pythagoras we largely get from Aristotle. Aristotle doesn’t make much mention of Pythagoras’s intellectual achievement, but he does describe him admiringly as an impressive wonder worker:

  • He is an expert on the fate of the soul after death, and the cycles of reincarnation it passes through—a process known as metempsychosis (also probably stolen from the Egyptians).

  • He had a thigh of gold and could be in two places at the same time.

  • He once faced down a poisonous snake and won—by biting the snake to death.

  • He dressed in all white (cult leader fashion dies hard).

  • He might have been a reincarnation of Apollo, or else a reincarnation of the bard Orpheus.

  • He could talk to rivers.

  • He persuaded a bear to stop killing.

  • He forbade the eating, and perhaps even touching, of beans, because he believed that reincarnated souls returned to the earth through beans.

  • He also persuaded an ox to stop eating beans, by showing the ox the power of the Pythagorean theorem.

Why is there not a movie about this guy? 

Then there is the Pythagorean cult itself. Strict dietary regimens, rigorous self-discipline, and bizarre prohibitions all figure prominently. You’d need to maintain a strict five years of silence before you could be admitted as a full Pythagorean and meet the man himself. During that time, your diet would be largely vegetarian, rare for the era. (Prior to the coinage of the term “vegetarianism”, vegetarians were known in English as “Pythagoreans”.) You’d follow imperatives such as, "One must put on the right shoe first" and "One must not travel the public roads". You could be permanently expelled for revealing secrets outside the monastery-like community.

What were the juicy secrets that were the promise of life as a Pythagorean? The information revealed to you depended on which of two hierarchical groups you belonged to, the acusmatici and the mathematici

The acusmatici, the lesser of the two, were the possessors of such truths (acusmata) as "thunder is the threat of punishment for those in Tartarus [the underworld]" and "earthquakes are a meeting of the dead". They were the silence-keeping rubes of the bunch, in other words, that Pythagoras didn’t have time for.

The mathematici constituted the inner circle, where the true secrets of Pythagoreanism were taught directly by Pythagoras. As you might expect, these secrets were often mathematical—from meditations on Pythagorean geometry, musical octaves, and the mathematical harmonies made by the moving planets, to more mystical numerological contemplations. Their world, and its truth, was mathematics—everything, for them, was made of numbers.

In a way, we modern people might find this last dogma the least objectionable. We, after all, see mathematics as, if not actually making up everything, at least structuring everything. Our everyday, earthbound world adheres to Euclidean geometry and Newtonian mechanics. Our abstract mathematics determine the concrete world around us. We’re scientific and mathematical as much as we were once religious.

However, when the Pythagorians say “everything is number”, they mean everything, even the hazy stuff. Our mathematics reigns supreme over our material world. The Pythagorians’ mathematics reigns, too, over the spiritual and conceptual world—matter, creation, the four seasons, the masculine, the feminine, marriage. One, or the “monad”, is at the origin of all things. Two, or the “dyad”, represents matter, as well as the masculine. Three, the “triad”, represents balance, and the feminine. And so on. Ten was considered the perfect number, which also made a perfect symbol of the tetractys. A four-leveled pyramid, it counts 4 points at the bottom, 3, 2, then 1 at the top—10 points in all. This tetractys was the object of fascination and obsession by the Pythagoreans, who dedicated prayers like this to it:

Bless us, divine number, thou who generated gods and men! O holy, holy Tetractys, thou that containest the root and source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-born, the never-swerving, the never-tiring holy ten, the keyholder of all.

For the Pythagoreans, the tetractys appears to simultaneously conjure the mystical holiness of the Christian Holy Trinity, the engrossing mental challenge of a sudoku puzzle, and the entrancing mystery of whether that dress was blue and black, or in fact white and gold. A meme that stayed fresh for the whole of the 6th century BC. Nice to live in a simpler time, huh?

It's hard to capture the awe that mathematics inspired in the ancients. As schoolchildren, we're forced to apply monotonous formulas to countless problems across years of mathematical study. The mere presence of an equal sign is enough to send shivers down the spines of many mathematically-allergic adults. Our adult arithmetic life is more likely to consist of counting out loose change to buy a candy bar than in encountering patterns that suggest profound truths about the cosmos. With the rise of credit cards and the decline in cash, we often get to dodge even those trivial mathematical moments.

A point of departure for understanding the Pythagoreans, then, would be to understand how they experienced their world’s first formula. As an incantation? A command from a god? A profound grasp of reality, as when you suddenly feel in nature or in thought—or when you simply take a moment to breathe—that everything feels more real than it did minutes earlier?

We might start with the knowledge that they were a cult. “Cults” in the Greek world were already everywhere in Pythagoras’s day. These cults lacked the outcast connotation the word has today, and usually referred to ascetic orders of priestesses devoted to Greek gods and goddesses like Apollo—part of a mainstream of Greek religion and culture. 

The highest of these priestesses were oracles, and Delphi was the most famous of those oracles. Her prophecies are probably familiar to you if you’ve dabbled in Greek myths. Famously—and often fatally—ambiguous, these riddle-like prophecies led astray as often as they enlightened. Croesus, the king of the ancient Lydia, for instance, asked the Oracle whether he would be successful if he attacked the Persian Empire. To which she responded, “If you attack the Persians, you will destroy a great empire”. Guess whose empire got destroyed.

Or take Delphi’s most infamous oracular statement—that the future king Oedipus was destined to kill his father and marry his mother, as immortalized in the playwright Sophicles’ “Oedipus Rex”. This prophecy turned out to be unambiguously true, but this didn’t help Oedipus escape his fate, and it probably even caused it. After hearing the prophecy, Oedipus fled his putative parents, unaware that they had, in fact, adopted him. He was therefore at no risk of killing or marrying either of them. As he fled, he killed a man in a quarrel over who had right of way in their chariots—his father. After a heroic ordeal in which he saves the town of Thebes, he wins the queen’s hand in marriage. Unbeknownst to Oedipus, the queen is his mother. Things get worse from there.

In this social and historical context, a mathematical formula would possess a unique power and appeal. A formula outdoes even oracular prophecy for its stylistic pithiness, but with radically more reliable outcomes. An oracular statement was interpretative, fraught with tragic peril. A formula is sure, certain, fixed—providing patterns and predictability in a world where powerful soothsayers just muddied the waters even more.

In all this, the formula was the perfect vehicle for enterprising ancient cults. Cults then, like cults now, share a despotism born of charismatic leaders, the unquestioning loyalty of their devotees, and the devotees’ commitment to the community of believers, for life. What a devotee gains in return is the promise of a revelation unattainable by the uninitiated. This revelation can't just be any mere truth or fact. It has to be the truth of truths, the truth that explains everything. It must be the way we must live our lives, the way we see the world around us. It must be a brand new lens. a^2 + b^2 = c^2, and everything that follows from that, for better or for worse.

The radical reductiveness of a cult’s view of the world gives it both its seductive power and laughable cartoonishness. Silicon Valley startups strive to be likened to cults because it implies that their mission has taken on such a focused zeal that their employees would bring an obsessive dedication to their work. At the same time, actual cults are creepy punchlines or fanatics for beliefs the rest of us recognize as utter delusion, ready-built for parody in South Park episodes.

The end of Pythagoras’s story epitomizes the contradictions in the connotations of “cult”. No good cult story is complete without its end, and in the Pythagoreans’ case there are two endings to choose from.

In the first: Pythagoras and his students were targeted by hostile local inhabitants, who attacked the residence where the Pythagoreans lived in Croton. In the midst of their flight, they were trapped by a bed of fire that blocked their escape. His students devotedly laid down a path for him across the fire, allowing Pythagoras to flee to safety. He survived, but in his grief at the horrific deaths of his martyred students, he committed suicide himself.

The second: he almost escaped the attack on Croton without any such sacrificial assistance on the part of his students. However—remember the no-touching-beans prohibition? Well, he tragically came to a fava bean field. Unable to run through it, which would have been a violation of his teachings, he was trapped, caught, and killed.

The first story—tragic and heart-wrenching—is the material of myth. The second—comedic and buffoonish—the material of caricature. Pythagoras remained ever the mercurial cult-leader, a contradiction and mystery even unto his death—one ending worthy of Sophocles, the other of Mel Brooks.

For a period of my adulthood, I found myself frequently distracted by numbers I'd find in everyday life. I'd see numbers while walking down the street, then add them up, then add up the numbers I'd just added up. I'd repeat this exercise without really being in control of it, finding myself doing it quasi-consciously with license plates, road signs, and address numbers. Wherever else I was confronted by regular patterns of numbers, no more than 4 or 5 digits in length, I'd fall into this mental tic. License plate number JY6493. 6 + 4 = 10, 9 + 3 = 12. 10 + 12 = 22, 2 + 2 = 4.

This pattern recognition added very little to my life, and I ultimately didn't parlay this into a brilliant career as a math prodigy or poker pro. I more often than not found it annoying—couldn't I do something more useful or creative with my idle brain? Write a story in my head, or figure out a problem I'm having on a work project? But in spite of that, and even though I am not in the slightest a 'math person’, personally or professionally, my brain glued itself to such tasks.

I might not be so far apart from Pythagoras and the Greeks in experiences like this. Wikipedia, indeed, details a whole numerological system derived from Pythagoras (earning him yet another distinction as “the father of numerology”). In this system, one’s name and birthday get reduced down to numbers, “name numbers” and “birth numbers”, that represent one’s personality, destiny, inner nature, and purpose. My name, Jesse Thomas Germinario, for instance, would have the numbers 13 (Jesse) + 22 (Thomas) + 64 (Germinario) = 99 →  9 + 9 = 18 → 1 + 8 = 9. According to thesagedivine.com, a 9 name number makes me a “Determined Leader”: “A leader with a chip on their shoulder, number 9 is born with innate spirit that others are attracted to.” Sure.

We’re well within our rights to question such mystical pseudo-mathematics—we’ve learned a great deal about rationalism in the many intervening years since Pythagoras died. But we’d also do well to question where that’s left us. Today, nearly every aspect of our lives are mediated by numbers. We go to the store to buy some groceries and spend $70. We itemize the list—one gallon of milk, two dozen eggs, five apples—and navigate the prices of each, right down to two decimal points and sales that might cut that $3.99 gallon of milk to $3.49. The activity at the supermarket takes us 20 minutes in total, plus 15 minutes to get to the store and 15 minutes to get back. We stay on Route 10 for 3/4s of a mile, then turn right on to Route 202 S for 5 minutes. We get home at 5:40 p.m.; it’s Sunday night and we only have a few more hours before the weekend is over.

Above all, we experience ourselves as quantified individuals. I have an age, which defines me as much as any other single characteristic, right up there with gender and race. I have an annual salary that shapes my social worth and determines my economic security. I have a number of years that I've spent in New Jersey growing up, a number of years in Berkeley and San Francisco, a number of months in New York and Seoul, a number of days or weeks in various countries I've traveled to around the world, a number of years in this or that job, this or that school. I have GPAs and SAT scores and class rankings that no longer matter much at all, though at points in my life they singularly determined my destiny. If I was a baseball player, I would have dozens of acronymized stats that determine my skill and status. If I was a prisoner, my name and identity might itself be reduced to a number.

These numbers aren't imbued with anything like the magic and delight a Pythagorean might regard their tetractys. Our numbers can be depersonalizing, obsessive, and outright coercive. When I was 28, I fell into the worst depression I'd ever felt in my life. For months, I found myself continuously drawn to any evidence that I was a tremendous, unsalvageable failure. I was on the verge of 30, and the number felt to me like a cliff. I felt as soon as I passed that threshold, my opportunities in life would disappear. I'd be too old to start something new. Too old to fulfill my dreams. Too old to find love. Too old to find community. I'd look at how much I was making, how many years I'd spent at jobs that weren’t my ultimate passion, how many Facebook friends I had. Counting, comparing—and nothing makes comparison easier than numbers.

In retrospect, of course, this all feels silly now. I didn’t fall off a cliff after 30. The darkness of the experience taught me some of the richest lessons of my life, among them counting my blessings instead of furiously reiterating my worries. But I never reckoned directly with the strange role of numbers in this period of my life. And I wonder now what the opposite of this numerological self-flagellation is—if it looks like the ancient Greek’s ecstasy in regarding revelatory mathematical truths.

Perhaps instead of pitting my 600 Facebook friends against others who have 800 or 1,000, I could marvel at the very fact that there are at least 600 people who exist out there, in the actual physical world, who would have felt comfortable counting me as a friend at some point in time. In “30”, I could see 30 years of real life. 30 years that have actually been filled by the person who identifies themselves by my name and inhabits my body. 30 years of having grown, acted, loved, created, learned, and believed.

We’re well underway into the year 2021, after a 2020 many of us were happy to leave behind. There are many stories you could tell yourself about this year, this number 2021. You can see in it a turning of the page, a new year and a fresh slate, if hardly a blank one. You could see it as our chance at redeeming this decade of the 2020’s that we’re just beginning. You could add up their numbers—2 + 0 + 2 + 1 equals 5, a number which the Pythagoreans believe represented marriage, the sum of 2 (the feminine) and 3 (the masculine). 

Or you could see 2021 as four more actual seasons, with 365 more actual days, with the countless, uncountable, actual moments that these days will be filled with. A year just as Pythagoras and his followers would have experienced it millenia ago. A year just as every future person will experience it millenia hereafter, telling all the stories they’ll tell in turn.