Why Do Roman Historians Make Their Villains So Persuasive?

Imagine you’re watching Mulan, the Disney movie. The movie proceeds as usual: a Hun invasion, some musical numbers, the devastation of the Chinese peasantry, et cetera. Then comes the first meeting between Mulan and Shanyu on the snowy slopes of Tung Shao Pass. Imagine, however, that instead of growling and playing up the barbarian angle, Shanyu gives a speech to Mulan. A long one, like five solid minutes.

He was provoked, he says. The Chinese built their wall straight through his people’s ancestral pastureland. The Chinese army fortified the good grass and drove his people into the scrublands to starve. He tried to negotiate, but the Chinese garrison killed his ambassador and fed his corpse to the palace geese. Whose conduct, Mulan, is more barbarous?

Mulan replies by firing her rocket at the pass and burying Shanyu (and his army) under a thousand tons of snow. The Huns are defeated, and with them, their inexplicably compelling arguments. Hooray! Cue musical number. The end.

This, to a modern audience, would be a deeply weird cinematic choice.

Why would you give your villain borderline unimpeachable motivations, ones the audience might actually agree with? Why would you give him five solid minutes to explain himself, and never have your protagonist refute him?

Roman historians do this so routinely it might be called a style.

I first encountered it in Tacitus’ biography of his father-in-law, the statesman and general Agricola. Tacitus, a historian and politician of imperial Rome (and a committed imperialist), admires Agricola without reservation. So it is rather odd that Tacitus writes the greatest anti-imperial speech in world literature and puts it into the mouth of Agricola’s primary enemy, the Caledonian chieftain Calgacus.

We, at the furthest limits both of land and liberty, have been defended to this day by the remoteness of our situation and of our fame. … But there is no nation beyond us; nothing but waves and rocks, and the still more hostile Romans, whose arrogance we cannot escape by obsequiousness and submission.

These plunderers of the world, after exhausting the land by their devastations, are rifling the ocean: stimulated by avarice, if their enemy be rich; by ambition, if poor; unsatiated by the East and by the West: the only people who behold wealth and indigence with equal avidity. To ravage, to slaughter, to usurp under false titles, they call empire; and where they make a desert, they call it peace.

(Tacitus, Agricola 30)

This last line is famous enough that you may have heard it before, often paraphrased as “The Romans make a wasteland, and call it peace.”

Agricola then replies with some competent but forgettable pre-battle oratory, and the Caledonians and Romans fight. Agricola wins a glorious victory.

This is very strange, doubly so since Calgacus’ speech was entirely invented by Tacitus. I should clarify: inventing speeches per se is common—nearly all reported speeches in ancient history are fictional.1 It is instead strange that the invention is this good. Calgacus’ speech is better than the supposed protagonist’s.

This recurs constantly in the Latin sources. Julius Caesar lets his German opposite, Ariovistus, eloquently call him a hypocrite right before Caesar crushes him in battle. The historian Livy speaks for Hannibal, the greatest enemy Rome ever faced. What’s going on and why is this so common?

This question occurred to me as I was translating the first oration of Catiline, the villain in the Roman historian Sallust’s masterpiece, the Bellum Catilinae.2 I had the odd experience of thinking that if he gave that speech to me, I’d probably follow him too. But as I considered, I realized this wasn’t just a Sallustian quirk, it was a broader pattern in Roman historiography—the suspiciously persuasive antagonist whose arguments are a little too good, a little too pointed directly at the Roman establishment.3 Thus the genesis of this essay, a case study of the phenomenon in Sallust’s Bellum Catilinae, now in two parts.

I split the essay into two related questions. Part I will try to answer why Sallust writes such good speeches for his enemies. In the process, we’ll cover Sallust himself: his life, his style, the first half of his magnum opus, and I’ll examine his personal motivations for making his Catiline into such a persuasive monster. Part I will thus touch on why the speeches are good, but not so much as to why Sallust put them in there. Part II, then, will cover the second half of the book, show a bit of my translation process, and attempt to explain why Sallust writes history at all.

The Rotting Republican Edifice

Catiline’s conspiracy is famous, not so much for its impact on history, which was minor, but for its participants and its literature. Julius Caesar and Cato the Younger were senators at the time, and Cicero, the greatest orator Rome ever produced, was consul. The literature of the conspiracy, Cicero’s Orations Against Catiline and Sallust’s Bellum Catilinae were used as Latin textbooks for two millennia thereafter.

The backdrop to the Bellum Catilinae is a Republic in decline. After the Roman Republic defeated Carthage in 146 BC, the ancient, martial, Republican virtues rotted from disuse. With no one left to fight, Rome’s politicians spent their energies screwing each other, then screwing each other over, all the while screwing the provinces out of every sesterce they could.

The worthy and the worthless alike all wished glory, honor, and power for themselves; the worthy strove by the right path, but the worthless, proper methods failing them, turned to tricks and lies. Greed births love of money, which no wise man covets; that vice, like an evil poison, unmans the body and manly spirit. Love of money is infinite, insatiable, diminished by neither plenty nor poverty. (Bellum Catilinae 11)

Sallust blames Rome’s triumph over Carthage for beginning the decay, but accuses Lucius Cornelius Sulla of accelerating it. Sulla was Caesar before Caesar: he marched on Rome (twice), thereby starting a civil war, made himself dictator, rewrote the constitution in order to “save” the Republic, then retired from public life. He did not save the Republic, and his spiritual successor, Julius Caesar objected only to his last act, saying that “Sulla didn’t know his ABCs, to lay down the dictatorship.”4

Sulla’s veteran soldiers, though, “lavishly squandered their spoils, and remembering the rape and victory in the civil war,” wanted more. (Bellum Catilinae 14, 16). Half the aristocracy was broke and desperate too, the urban poor of course had nothing, and so had nothing to lose. The Catiline Conspiracy was a byproduct of this long decline and general disaffection, and the last great prelude to the fall of the Roman Republic.

In 63 BC, Lucius Sergius Catilina (Catiline), a patrician from an ancient family, lost his chance to be consul for the second time. Drowning in debt from the truly epic bribery required to run for office in late Republican Rome, he reinvented himself as a radical populist and tried to overthrow the Republic.

For his conspiracy, he first collected the young, impressionable men, “ensnaring their soft and malleable minds by trickery.” The “trickery” included, according to Sallust, lavishing the young men with prostitutes (understandable) and expensive dogs (less understandable). He gathered to himself all the indebted, the dispossessed, and the disgruntled veterans of the civil war. He, unusually, also recruited women, “who at first had supported their enormous extravagance by debauchery, but by the time their age placed a limit on their extravagance, had accumulated enormous debts.” (Bellum Catilinae 24)5

His conspiracy was exposed before he succeeded in seizing power. Catiline’s allies in Rome were arrested by the Senate and executed without trial. Cicero presided over the killings, and depending on who you asked, they were either the apex of his career or the blackest stain on it. Catiline himself died hard—the state was forced to raise an army to put down his coup attempt, which by then had turned into a full-blown revolt, and won only a Pyrrhic victory against his army.

Caius Sallustius Crispus (Sallust)6, wrote the Bellum Catilinae about twenty years later, during the aftermath of Caesar’s assassination, as the Republic gave way to the autocracy of Octavian-cum-Augustus.

As a history in the modern sense, Bellum Catilinae is narrow—competent, but not great. Like a Greek play, the decline of Roman virtue is the single, radiant theme that limns every line of his history, and Sallust makes no attempt at objectivity. But as a book it is fantastic, and my personal favorite of all the Latin prose I’ve ever read. At least half of this is due specifically to Sallust’s writing style.

Sallust’s Life, Voice, and Grudge Against Cicero

“Sallust, according to the judgment of the learned, will rank as the prince of Roman historiographers.”

—Martial

Sallust was many things: a failed populist politician, a partisan of Julius Caesar during his seizure of power, a brilliant writer, and by near-universal ancient testimony, a terrible person.

Expelled from the Senate on grounds of immorality, he was reinstated by his patron Caesar and made governor of an African province, which he looted in such spectacularly corrupt fashion that Caesar was forced to intervene, again, to save his incurably avaricious ass from prosecution. Without further prospects for advancement, he retired to private life, and decided to write history.7

Sallust was also a contemporary of Cicero. Cicero was famous at the time for breaking the Catiline conspiracy, but Sallust’s history leaves Cicero out almost entirely—no orations printed whatsoever from Rome’s greatest orator in the definitive history of his greatest triumph… yet multiple from Catiline. Why? It’s simple; Sallust hated Cicero, and the Bellum Catilinae is deliberately written as a rejection of Cicero’s style. The result is a highly entertaining history, written in a highly original style called inconcinnitas8, “inelegance,” that will stick in your head and infect your prose thereafter.

Cicero’s style (essentially “default Latin” for the last 2000 years) is famous for intricate constructions and giving students a headache hunting for the verb in the page-long sentence. Sallust’s is the reverse.

Here is Cicero, in De Amicitia, “On Friendship”:

For I do not agree with those who have recently begun to argue that soul and body perish at the same time, and that all things are destroyed by death. I give greater weight to the old-time view, whether it be that of our forefathers, who paid such reverential rites to the dead, which they surely would not have done if they had believed those rites were a matter of indifference to the dead; or, whether it be the view of those who lived in this land and by their principles and precepts brought culture to Great Greece, which now, I admit, is wholly destroyed, but was then flourishing; or, whether it be the view of him who was adjudged by the oracle of Apollo to be the wisest of men, who, though he would argue on most subjects now on one side and now on the other, yet always consistently maintained that human souls were of God… (De Amicitia 13)

Now here is Sallust, on a related theme, opening the Bellum Catilinae:

It befits all men who wish to stand above the animals to strive with all their might, lest they pass their life in silence like cattle, whom nature formed groveling, belly-slaves. …

Obedient to virtue are those men who plow, who sail, who build, but many mortals, slaves to stomach and sleep, unlearned and uncultured, pass through life like strangers. To those, against nature, the body is a delight and the mind a burden. Of such men, I count their life and death the same, since nothing is recorded of either. (Bellum Catilinae 1, 2)

Sallust avoids the (effeminate and weak) vocabulary of Ciceronian public life and the debased modern Republic: honestas, humanitas, et cetera. He uses deliberately archaized spellings: lubido for libido, maxumum for maximum and so on, to appear closer to the (manly and tough) ancient virtues—and to his (manly and tough) hero Cato the Elder. According to the grammarian Suetonius, he hired a man specifically to collect archaic words for him.9 Is Sallust’s austere, deliberately unadorned style reflective of his stern Roman Moralist character, who hearkens to the ancient virtues, or is it a reaction to the effete, detestably modern Ciceronian stooges that tried him for corruption and kicked him out of the Senate? Well, if we believe Jerome, Sallust also married Cicero’s ex-wife, so I leave the judgment up to you.10

As a Latin stylist, Sallust is compressed, expressive, ametric, and brilliant. The prose is superb, and can coin a one-liner like no author I’ve ever read. Sallust translates naturally into English, as if inconcinnitas were the default style of our prose, at least post Hemingway.

Cicero at his best is lyrical:

Quo usque tandem abutere, Catilina, patientia nostra?11

How long, Catiline, will you abuse our patience?

Sallust at his best is syncopated, arresting:

Namque pauci libertatem, pars magna iustos dominos volunt.12

Because few prefer liberty, the greater part desire only fair masters.

Sallust deploys this voice most memorably on his villain.

Catiline’s Youth: Unspeakable Outrages with Noble Maidens, and Other Youthful Indiscretions

Sallust describes Catiline thus:

Lucius Catilina, born of a noble family, had great force of mind and body, though that mind was twisted and evil. From his youth he reveled in civil war, murder, rape, and discord, and he spent his youth entirely in pursuit of them. His body endured hunger, cold, and insomnia better than you would believe. In character, he was reckless, sly, fickle, pathologically dishonest, always striving after foreign things, extravagant with his own wealth, yet burning with desire for more. Eloquent enough, but unwise. His monstrous and insatiable temper always yearned for that which was too far out of reach. (Bellum Catilinae 5)

His impure mind, hateful to men and gods, could rest neither in wakefulness nor in sleep, indeed, his conscience ravaged his agitated mind. Thus his pallid color, ugly eyes, sometimes rapid, sometimes slow gait. In his face and countenance the madness was apparent.

While still a youth, Catiline engaged in many unspeakable outrages, with noble maidens, Vestal virgins, and others in a manner contrary to law, both human and divine. Later, he was seized with lust for Aurelia Orestilla, in whom no good man ever praised anything save her figure. And though she hesitated to marry him, fearing his adult son, it was believed that Catiline murdered his son, thereby vacating his house for their criminal wedding. (Bellum Catilinae 15)

To be fair, the catalog of crimes is likely less revealing about the character of Catiline than of the Roman subconscious—Catiline was not a saint, but practically every politician, beyond a certain point in Roman history, is accused of something similar. Nevertheless, this description would scare a Cyclops, and is decidedly not sympathetic to Catiline, the man.

Catiline’s Politics, with Which Sallust Sympathizes

Now what of his conspiracy and its motives? Most secondary sources will say that Catiline, twice passed over for the consulship, simply tried a coup. Sallust says this too; he says that after the madness of the Sullan dictatorship poisoned the atmosphere, Catiline is seized by a procession of lusts: for power, for loose women, for the recovery of his family fortune, that thus degrade his character and kindle the megalomania.

But Sallust also allows Catiline himself to tell us “in his own words” his entirely different motivations, in the following speech that was the genesis of this book review:

My mind has been restless, as you have each heard. My spirit burns hotter each day, when I consider what the future may hold, if we do not act to avenge our lost liberty.

Ever since the Republic was lost to the power of the few, it is for their benefit, always, that kings and tetrarchs are made tributaries, that peoples and nations are weighed down by taxes. Outsiders—whether hardworking, worthy, noble or common—we are the despised mob, without influence, without authority, and bound by obligation to those who should fear us, were the Republic healthy. All influence, power, honor, and wealth goes to them or their cronies; to us is left only dangers, rejection, trials, and poverty.

How long will you suffer this, O bravest of men? Is it not better to die on your feet, displaying your virtues, than to live a miserable and shameful life, lost to dishonor, while other men’s arrogance mocks you?

Assuredly, victory is within our grasp—I so swear by the faith of gods and men. We are in the prime of life and our spirit is strong, while they, rich, slip into dotage. We must only begin this great task, and the rest will follow!

And who among mortals, whose minds are yet virile, could endure the tyranny of money, which they squander building upon oceans and leveling mountains, while we lack our portion, indeed, lack the bare necessities of life? Should they double their houses, and nowhere permit us to kindle our own meager hearths? They splurge on tablets, tokens, and fancy metalwork, they tear down new buildings just to build others, and extract money in every way, so that even with the greatest profligacy they are unable to exhaust their wealth. Yet we are left with destitution within the home, bitter debts without, present misery and a yet more hopeless future: what then remains save the wretched breath of life?

Will you not be awakened? Lo! That which you so, so often longed for, your liberty, not to mention wealth, your due, your glory, is before your eyes! Fortune gives all such prizes to the victor. Let the age, the moment, the dangers, the poverty, and war’s glorious spoils exhort you, yes even more than my speech, to action! Use me, then, either as your commander or your soldier! I recuse neither my spirit nor my body from your service! These plans I hope I shall make real as your consul, together with all of you!

Unless, of course, my eyes deceive me, and you are content to be slaves rather than to rule. (Bellum Catilinae 20)

As I read his speech, I found myself agreeing with the monster. I’d join the monster’s conspiracy after a speech like that. It’s clearly written by a master orator, one from a culture that prized rhetoric above nearly all other skills. But that master orator is Sallust, because Catiline didn’t say any of this!

So why make Catiline so persuasive?

First, I think Sallust just loves the game.

Sallust is a disgraced Roman politician-turned-historian, who wants to be seen displaying skill, displaying arete, in his chosen craft. It would be beneath a Roman historian to write a bad speech, especially since the speeches comprise fully 25% of the text by word count. Sallust loves the game, and it would be profoundly embarrassing to play it poorly.

But, one may counter, Sallust could have reserved his amazing speeches for Catiline’s opponents, and only competently written Catiline’s. Ditto for Tacitus and his speech for Calgacus. Why do the reverse?

It could be that, for patriotic Roman speechwriters, a stirring call to arms is the equivalent of a five-paragraph essay. Perhaps marshaling the arguments of the enemy demonstrates technical mastery better to Sallust’s contemporaries.

But Sallust might also straight-up agree with Catiline on the merits.

Sallust was a partisan of Julius Caesar during his revolution, and Caesar was a popularis, literally, a populist. Thus, Catiline, Caesar, and by extension Sallust, have very similar politics. They share certain ambitions, too; in many ways, Catiline is a failed Caesar, and his conspiracy an aborted crossing of the Rubicon.13

Sallust’s popularis sympathies leak into his history as philosophical asides. Here’s what he has to say on the motivations of the poor, the plebeian class, for joining the Catiline conspiracy.

The entire population of plebs, desiring the cancellation of debts, approved of Catiline’s endeavor. And this indeed, following their own customs, he seemed to do. Because, as always in states, those who have nothing envy the respectable, extol evil, hate tradition, and burn for new beginnings. Hating their own circumstances, they wish to change everything, they nourish sedition and discord without a care, since destitution is easily obtained without injury. (Bellum Catilinae 37)

It’s remarkably modern. Sallust does not say that the poor are intrinsically evil or base in nature. Instead his analysis is materialist. The poor are motivated by their poverty to desire change, any change, reasoning that it must be better than this. Destitution is indeed easily obtained without injury—for wealth and honor, one must bleed.

And if we re-read Catiline’s speech and look at the specific grievances, it seems obvious that this speech, given to a happy, healthy, and free crowd, would inspire precisely nothing. He speaks in the tradition of populists everywhere. The few, the elites, despise you and have all the money. They trample your rights and waste the wealth that should be yours in frivolity. And why should we let them?

Sallust thinks the grievances of Rome’s poor are real; he sympathizes with their plight, and writes a killer speech on their behalf. It’s a strange position to be in, condemning the man Catiline but endorsing his motives. To paraphrase Appian, perhaps Sallust thinks that Catiline had “a most excellent design, too violently pursued.”

Now normally, I’d end the essay here. This is a) probably true, and b) a reasonably satisfying explanation for why Sallust writes such convincing speeches for his filicidal, pathologically dishonest, ugly-eyed villain.

But I think there’s a lot more going on under the surface. Sympathy for the devil is very modern, and it doesn’t quite jive with the later speeches in Sallust. Sallust devotes fully 25% of the text to speeches, split equally between:

• His patron, Caesar, whose motives and character he fully endorses,

• Cato the Younger, his patron’s sworn enemy, a reflex of the ancient Republican virtues, whose character Sallust deeply admired,

• and Catiline, the villain, whose character he deplores but whose motives are sympathetic.

Sallust’s own motivations for writing this history are complex, and I think to best understand, we should examine the cultural context in which Sallust is embedded, so very different from our own. We’ll pick this thread up in Part II, where I posit, among other things, that Sallust is a bard in the grand Indo-European tradition, reaching, like the rest of us, for immortality.14

Everyone who’s ever been in a lab setting, asked to measure some physical constant, has had the following problem: you did the experiment perfectly—you know you did—and your measurement is off by, like, 50%. Something way outside your tolerances. You may vividly recall your very first lab class back in undergrad, where you measured g to be 15 m/s^2, despite the fact that you had a timer, a laser, a photodiode, a calibrated slope for the ball to roll down, and knew the right answer from the start.1

Why is this? Are you simply bad at your job? No, of course not. It’s actually because of the gremlins that live in your equipment.

Yes, the true underlying reality—especially when you’re in a lab—is that gremlins live inside your instruments, and they are actively malicious, and they want your experiment to fail. They are watching you, always, and they are cleverer than you are. That’s what makes your job so difficult, and every paper published is a small victory against the gremlins in a larger war you will almost surely lose.

Why do you think power cycling fixes so many computer problems? It has nothing to do with rebooting; it’s because the gremlins live in the power line and slowly infiltrate your system while it’s on. Once they’re in, they feast on 120 V / 60 Hz, and reproduce. Cutting the power temporarily kills them, so you’ve got a short window of time after reboot before they filter back in. I’ve heard stories that European gremlins eat different voltages than American ones—this is why you cannot plug a European device into an American socket, the different factions fight it out in the transformer and make a wasteland of your instrument.

Gremlins have mystical properties. They are anti-statistical. They resist normal error propagation, indeed, they actively correlate the measurement errors. That’s why you measured g to be 15.

Another interesting feature, especially of electronics gremlins, is that they like to smoke—Cuban cigars, I believe. This is why, when your electronics fail, the magic smoke escapes. It’s not you that burned the op amp. That was Dave, the head gremlin, smoking his morning cigar, and getting a little excited. Nothing you could’ve done, really.

The postdoc who trained me, an eminent gremlinologist, clued me in on likely locations to look if you suspect you’ve been overrun.

• Windows Update.

• Antivirus software.

  • These actively generate gremlins, like a diabolical nursery.

• Python virtual environments.

  • These can sandbox different gremlin factions, to prevent interaction effects, but the gremlins are smarter than you and always find a way to infect your global install.

Your principal investigator, or indeed any important observer, an investor, your boss, exerts a gremlin-attracting field. I’ve been working hard on an effective field theory of this, so that I can quantify how much, exactly, the presence of your boss will sink the gremlins into your equipment, and how short your window of time is before the gremlins cause a heretofore unseen malfunction in your otherwise perfectly engineered device. This is the real observer effect—forget quantum mechanics, that’s not real anyway. It’s gremlins all the way down.

Unfortunately, I haven’t yet managed to get my GUT (Gremlin Unified Theory) to work. But I’m pretty sure I know why…

Anything will lase if you hit it hard enough.

-Arthur Schawlow

What do you think of, when you picture a laser? Maybe a laser pointer? Val Kilmer from Real Genius? The Death Star?

Here are some things you probably do not picture:

• Peacocks

• Mars

• Gin and Tonic

• Cancer

And yet, all of these things will lase, if you hit them hard enough.

Lase is a serendipitous verb. LASER,1 an acronym for Light Amplification by Stimulated Emission of Radiation, accidentally forms a perfect agent noun and implies a verb; just as the driver drives and the conductor conducts so does the laser lase.2

The dictionary definition gives us:

A laser is a device that emits a highly concentrated, narrow beam of coherent light through a process called Light Amplification by the Stimulated Emission of Radiation. Unlike standard light sources (like light bulbs), lasers produce a single color, with light waves traveling in phase and in a straight line.

This is not fully correct. So then, what does it mean, to lase?

What is a laser?

When distilled to its essence, a laser requires three things: the pump, the gain medium, and the cavity. This is a topic close to my heart, born from the hours I spent explaining that an Optical Parametric Oscillator (which I was building) was not a true laser to people who couldn’t care less.

Assuming you, reader, could care less, we need a little background in atomic physics.

Stimulated Emission

Einstein is underappreciated. In 1917, about a year after General Relativity, a decade after Special Relativity, and exactly 100 years before I entered graduate school (coincidence?)3 he published yet another paper that founded an entire discipline: On The Quantum Theory of Radiation.4

I took the opportunity to read it for this post, and it is surprisingly clear (with only two references, one citing himself, lol).

In it, he introduced a completely novel process in atomic systems, which complemented and consummated the two known processes: absorption and spontaneous emission. We now call this stimulated emission. Take a look at the figure below:

Assume we have an electron with two possible states, with energies of 0 (ground) and 𝜖 (excited). Now consider the following three processes.

Absorption, |g> + γ → |e>

The electron is initially in the ground state, a photon comes along with exactly 𝜖 energy, it is absorbed and the electron hops up to the excited state.

Spontaneous Emission, |e> → |g> + γ

The electron doesn’t stay excited forever—it eventually decays and in doing so, emits a photon of energy 𝜖. However, this process is not triggered like absorption, it occurs at a random time and emits the photon in a random direction.5

The random direction didn’t bother Einstein so much, because that has a clean analogy to classical physics (isotropic emission on average). But the random timing of the emission bothered him, and was one of the first hints of the irreducible randomness of quantum mechanics, not formalized until four years later.

Stimulated Emission, |e> + γ → |g> + 2γ

Einstein’s genius was recognizing that a third process was required in order to explain Planck’s radiation law at a deeper level. An electron in an excited state (not ground!) sees an incoming photon of energy 𝜖, jumps down to the ground state and emits two identical photons. The emission of an extra photon is stimulated by the incoming photon.

This process is an amplifier in a way that no classical process can be. One photon comes in, and two identical photons come out: same frequency, same direction, and quantum mechanically indistinguishable.6

Can you, perhaps, cascade this process? Produce a gigantic mass of identical photons? Burn a hole in the wall with it? Yes you can, and it requires three elements to do it, our gain medium, pump, and cavity.

The Gain Medium

In our toy examples above, we posited a two-level system, with ground and excited state, but told you nothing about it. In reality, this is the gain medium, an atomic system that has discrete, accessible energy states for its electrons. A lot of the time, media that fluoresce well are also good laser gain media.

The gain medium sets many of the fundamental properties of the laser, most relevant is the set of allowed wavelengths, and so usually, the laser is named after the gain medium.

They generally fall into six categories.7 These categories live mainly in my head, so feel free to quibble with my categorization in the comments. This terminology tends to be what separates the expert from the novice—anyone who’s anyone pronounces Ti:Sapphire as “Tie-Saph.” I have helpfully included a pronunciation guide for the aspiring laser physicist or forensic linguist, with the unambiguously correct pronunciations of each.

1. Solid-State Lasers

Theodore Maiman invented very first laser in 1960—a Chromium-doped Ruby crystal pumped by a flashlamp. Ruby lasers have since fallen out of fashion in favor of better gain media, like Nd:YAG8 or Ti:Sapphire. Nd:YAG and Ti:Sapphire are the short-pulse classics. Both lase in the near-IR, at 1064 nm and ~800 nm, respectively.9

Ti:Sapphire lasers are an absolute pain to tune up in practice, but still the go-to choice for single-digit femtosecond oscillators because of its ridiculously broad gain bandwidth.

Ytterbium-doped Crystals, usually Yb:YAG/Yb:KGW are the modern default for most ultrafast near-IR laser systems, due to its super low quantum defect (very, very power efficient), and super simple energy level structure. They usually lase near 1030 nm, and when frequency-doubled, produce my favorite shade of green.

2. Gas Lasers

These use discrete vibrational or rotational energy levels of free gas molecules as the lasing medium. The CO₂ laser is the most representative in its class, the old workhorse of machining lasers. 10.6 um wavelength, usually. The other prototypical example is the Helium-Neon, or HeNe laser (usually 632 nm).10 It’s the experimentalist’s best friend, the beautiful bright red that aligns all your optics, with tubes that give you a really nasty shock if you don’t ground them properly. Ask me how I know.

3. Dye Lasers

Literally made of a liquid dye, though less used now on account of sometimes being extremely carcinogenic. The most common are Rhodamine (orange), Fluorescein (green), and Coumarin (blue). Coumarin comes from cinnamon and other plants, though I seriously doubt anyone has cold-pressed enough cinnamon to make a dye laser that way.

4. Excimer Lasers

Properly a subtype of gas lasers, an excimer is an “excited dimer” but almost always lase in the ultraviolet, so form a different category in my head. The most famous are probably ArF & KrF, which set the Deep-UV 193 nm wavelength standard for photolithography in the 1970s and are still going. Also common in LASIK surgery.

5. Fiber Lasers

A doped fiber optic, by far the most common type of laser now, for machining, telecom, whatever, because they’re so robust. Some common types are Erbium:Glass (the telecom standard 1550 nm amplifier), Nd:Glass, Yb:Glass, literally put any rare earth in a special type of glass (see for example ZBLAN fibers, one of the very few things that would be 10-100 times better if made in space).

6. Semiconductor Laser (Diodes)

The final category, less named for their gain medium than their architecture, are basically souped up LEDs. Exempli gratia:

• Quantum Cascade Lasers

• Vertical Cavity Surface-Emitting Lasers (VCSEL)

  • These are the Face ID Laser—if you look carefully at an iPhone in a dark room, you can see it flash!

• Heterogeneous Integration Diodes with horrible acronyms like AlGaAsSb

The Pump

The stimulated absorption process requires a single excited electron. But gain requires many electrons in the excited state, and the pump provides the energy for it.

The goal of pumping a gain medium is to achieve population inversion,11 where you have more excited state electrons than ground state electrons, and you get net amplification of incoming photons.

There’s a bunch of ways to do it—we just learned one of them, you excite an electron with a photon of energy 𝜖, and it’ll jump up to an excited state with energy 𝜖.

There’s some nuances to this, though.

What do you think happens if you have a two level system, and you shine a continuous stream of energy-matched photons on it? Can you achieve population inversion like this? Think about it for a minute before scrolling down.

I’ll give you a hint, one of Einstein’s contributions in that seminal 1917 paper was showing that the absorption and stimulated emission probabilities are equal for any atomic transition, and they both involve a single incident photon…

The answer is: No, you cannot achieve steady-state population inversion with a two-level system.

Why? As your incoming photons get absorbed by the ground state electrons, the excited state gets more and more populated, and the reverse process, stimulated emission, gets more and more probable. Since the absorption and emission probabilities are equal, this process asymptotes to 50% excited, 50% ground state occupancy, with no net photon gain.

But you can, of course, cheat and use a three or four-level system, pumping on one transition and lasing on another. This is easier to explain in pictures than in words, so look at the figure below for a typical example of three and four level systems.

Usually, you have a pump source that excites the electrons into an upper level with a very short lifetime. That upper-most state rapidly decays into a middle state (confusingly called the ‘upper state’), whose transition to the ground is the lasing transition. The excess energy is lost as heat.12

The pump source can be any source of light. Flash- or arc-lamps are common (or used to be), but since specifically targeting the pump transition with photons of exactly the right energy is most efficient, most modern lasers actually use other lasers as the pump! But wait, how do those get pumped? Infinite regress problem?

No, usually, the pump laser is a semiconductor diode laser, and to pump those, you just ram as much electrical current through the diode as you possibly can, and a lot of those “hot” electrons stay in the upper state.

The Cavity

The final piece of the laser, often missing from otherwise good explanations, is the cavity. Real materials have many states, not two. In fact, real materials can lase over a very wide spectrum of electronic transitions. For an example, see the gain bandwidth of a Ti:Sapphire crystal (did you say it right?) below. It’s huge!

That’s why you need the cavity. The cavity is a positive feedback loop that selects a particular lasing wavelength (or set thereof) from the gain bandwidth, and suppresses the others. Usually, the cavity is a set of mirrors.

How does it work? In brief, the electric field of a laser cavity must exactly replicate itself on every round trip. This provides positive feedback via constructive interference.

Let’s sketch this process through the below thin-disk laser cavity:

We start at the first mirror with some light field. It has a magnitude and phase.

The light travels along the length of cavity, through the gain medium and outcoupler, and the phase of the light field advances by 𝜙 as it propagates.13 If the total round-trip length of the cavity is L,14 then the phase advance is 𝜙 = 2π * L/λ, where λ is the laser wavelength.

The phase advance is essentially the total round-trip cavity length, measured in units of wavelength, L/λ. Thus, for the field to perfectly repeat itself, the cavity needs to be an integer number of wavelengths long.

• 𝜙 = 2π n —> L = nλ

So for example, you could choose between a 700 nm laser line and 800 nm by changing the length of the cavity to match.

Additionally, the gain in the cavity must balance the loss in the cavity (including the out-coupled light, any mirror losses, etc.) at the lasing wavelength.

• Gain(λ) = Loss

You increase the gain by increasing pump power, until Gain = Loss and you hit threshold. That’s lasing.

Putting it Together

If you have those three things: A gain medium to provide the stimulated emission, a pump to provide the energy source, and a cavity to provide the feedback, you have a true laser.

You can have lots of other things that look like lasers, but aren’t. An LED and a laser diode are very similar, but the LED lacks the cavity. And indeed, if you slap an LED in a cavity you get a (bad) laser.

A Zoo of Lasers

Thus we return to the premise of this article, with a little more understanding. If you have an atomic system with a bunch of states, and you pump it sufficiently hard, and there’s some sort of feedback mechanism, it will probably lase. I’ve collected a bunch of my favorite examples below, please enjoy!

Some of the animals in the zoo stretch the strict definition of laser, so I’m including a “Spectral Purity” score, where 1 means “yeah, this is a laser only in the loosest sense,” and 4 means “this meets the most pedantic definition of laser.” I reserve the right to be inconsistent and heavily biased in my scoring.

Nuclear-Pumped Lasers

Nuclear-Pumped X-ray Lasers were a part of the Star Wars, or Strategic Defense Initiative under President Ronald Reagan.

There’s actually two kinds, one that uses a nuclear reactor to generate high energy ions that pump a laser medium by various means.15 The main application discussed in the linked review is to shoot down asteroids, by evaporating part of their surface with the nuclear-pumped laser, and using the resulting gas/plasma jet of ablated asteroid material to alter their trajectory.

The other kind is more what you’d expect from a program called Star Wars, a nuclear bomb-pumped x-ray laser. There’s a really great Optica article on this, but apparently the results of the tests were only ever leaked, not published, and left observers “skeptical.”

The article claimed that the X-ray laser emitted peak power of several hundred terawatts at 1.4nm. Those figures have never been officially confirmed, but Livermore said in 1990 that the wavelength was the shortest ever emitted by a laser. The article did not identify the laser transition or test material, but Chapline told me the sample was “an organic pith material” from a weed growing on a vacant lot in Walnut Creek, Calif.

I love the banality of the origin story.

Congress had asked the weapons labs if they could do something with nuclear-weapon technology besides designing warheads, and the two decided that pumping an X-ray laser would be an interesting idea.16

Spectral Purity: 4 for reactor-pumped, 3 for bomb-pumped (it’s more an amplifier than a laser).

Human Cells, Bones, Cancer

Cellular or biological lasers come in two flavors: random and engineered.

Random first: In bone tissue, the natural disorder of the hydroxyapatite mineral structure forms random optical cavities. When soaked in a fluorescent dye (e.g. Rhodamine), and pumped with a suitable source (e.g. a Ti:Sapphire), the various randomly oriented cavities will lase. It’s potentially useful, though this strikes me as more of an excuse to make bone lase than a useful technique:

Random cavities are strongly related to the structural properties of random media, [thus] the emission spectra can also be used to extract structural information of the samples.17

The engineered biolasers are only barely biological. You engineer a cell to express some fluorescent protein, and place it in an external optical cavity. The gain medium is engineered, the cavity isn’t biological, but it does lase and it does involve cells. It is, like random bone lasers, diagnostic of the cell. This technique also produces papers with extremely buzzword-dense titles guaranteed to excite a professor, e.g. Antigen-independent single-cell circulating tumor cell detection using deep-learning-assisted biolasers.

The random biolasers get a Spectral Purity score of 3/4 for the random natural cavities, and the engineered biolasers a 1/4. They count, but only barely—you really can’t just pump a cell and expect it to lase.

Peacock Feathers

Both peacock feathers and butterfly wings are examples of natural photonic crystals, which are, in essence, extremely general tunable mirrors.18

So, someone had the bright idea to dip some feathers in rhodamine, pump it with 532 nm light, and it did indeed lase. Much to the authors’ surprise, and my own disappointment, the extremely cool natural photonic crystals were not responsible for the lasing. Nevertheless, they made a peacock feather lase. I expect future work from the authors to try again on the whole peacock.

Spectral Purity Score: 3/4. They had to dope it with a real laser gain medium, but the cavity was natural.

Jell-O

Theodor Hänsch and Arthur Schawlow wanted to make Jell-O lase. Their first twelve attempts with commercial Jell-O were unsuccessful—the food dyes were too absorptive. But they eventually succeeded by doping unflavored Knox-branded gelatin with rhodamine, fluorescein, umbelliferone (?), and coumarin. Each of these pseud-O-Jell-Os lased when pumped by a nitrogen laser.

Schawlow’s sense of humor comes through in the paper—he has two great notes.

• “Since fluorescein is nearly nontoxic, this may well be the first edible laser material.”

• If the pump beam remained stationary on the Jell-O, lasing faded after about 100 shots from local bleaching, but jiggling the Jell-O restored it.19

Sadly for Schawlow’s purity score, they had to make their own special Jell-O with strong fluorescent dyes. They also didn’t use a cavity, but the purpose was to demonstrate the gain medium, and the edges of the Jell-O are reflective enough to make a weak cavity, and I like Arthur Schawlow, so I’m giving this a 3/4.

Edible Lasers Generally

A more modern version from Anwar et al.20 does a much more systematic job of looking into edible lasers. They explore combinations of edible microcavities made from oil droplets or silver leaf, edible fluorescent dyes, combine them, and put them in peach compote.

One of the applications they propose is really innovative, using multiple distinct laser lines as edible bar codes for fingerprinting food.

Spectral Purity for Anwar et al: 4/4, best example on this list, albeit not naturally formed.

Gin and Tonic

This one sent me on a bit of a journey.

I’d heard the legends of Gin and Tonic lasers in grad school (and I love a good Gin and Tonic), but I couldn’t find any solid references in journals. What kind of scientist does this and doesn’t publish it?

I was able to find some references to a book by Stephen Wilk, reviewed here, who “traces the story to an experiment at Eastman Kodak labs, where two researchers found that very fast flashlamps could indeed excite blue laser pulses from an unidentified brand of tonic water.” But this experiment had no gin! So, I turned to Claude Deep Research.

The gin and tonic laser legend is actually conflating two experiments, the Eastman Kodak labs experiment above, and a paper from NIST’s predecessor the National Bureau of Standards, available here.21

The paper is so short I’ll just copy it in full. Read the last line.

Spectral Purity Score(s): 2/4. Lies! And a gaping hole in the literature…

The Planet Mars and this One Specific Nebula

The Martian Atmosphere is a natural CO₂ laser.

The effect was first definitively proved in 198122, by measuring the emission spectrum and showing it’s about 10^9 times brighter on the core CO₂ line than expected. They get a tiny, 10^-3 gain, and it’s single-pass, so not technically a laser, but a laser amplifier, and not a very good one.

However, another paper (Soviet this time) find that Mars’ atmosphere must lase given a certain temperature profile through the atmosphere, then infer from temperature measurements that not only is Mars always lasing on the daytime side, but Venus is too, but only when “internal gravity waves” (?) perturb the atmosphere.23 They estimate a gain of 2% through the Martian atmosphere, way above the 10^-3 of the earlier paper. Actually, with a decent cavity, that’s not bad for a natural laser—the single-pass gain for an artificial CO₂ laser is of order 0.2% per cm.

The Ant Nebula is apparently also lasing, which is somehow evidence of a double star system, because otherwise the conditions in the gases aren’t right, something something. Astronomers have absolutely crazy chains of evidence. Off-topic, but my favorite example of astronomer bootstrap reasoning is Terrence Tao’s talk on The Cosmic Distance Ladder. Fantastic talk on how we learned to measure distances between stars.

Spectral Purity Score: 3/4, a true laser amplifier in want of a cavity.

Things that Will Not Lase

To wrap up this article, it is sadly necessary to mention that Schawlow is not quite right. Alas, some things simply will not lase, and there’s a general principle involved. Several actually, but I’ll touch on two.

Gain is linear with pumping strength. Pump power → population inversion → gain coefficient. So, if you have some sort of lossy effect that is superlinear with pump power (Auger recombination, for example, where the energy just bleeds off uselessly as heat), the thing won’t lase. Surprisingly, silicon is one of these materials, despite its ubiquity.

There’s another, more fundamental constraint too. The upper-state lifetime, that is, the amount of time an electron will stay excited before spontaneously emitting a photon and dropping down, is proportional to 1 / f^3, where f is the lasing frequency:

Thus, the pump power, to sustain inversion of N electrons in the upper state, is in dynamic balance, and scales like the total energy of pump photons over upper-state lifetime

This single fact explains why near-IR lasers require milliwatts, visible require watts, UV require kilowatts, and X-ray lasers require nuclear weapons. This is a major impediment towards the holy grail—the gamma ray laser (graser).24

And so there we have it. A surprising number of things will lase if pumped… but for silicon, and gamma ray lasers, you really can’t hit them hard enough, or at least not yet.

Behold, there dwelt a man, who kept in his house a small creature, a cat of curious and greedy nature. Among the many toys which the man possessed for his cat, there was one which the cat desired above all others, whose name was Fat Worm.

Now Fat Worm was no common toy, but rather had a body of many colors and surpassing softness, and it held a power over the cat which no other toy could match. The man would often dangle Fat Worm for the cat, whose heart burned with desire for Fat Worm.

And the man would often engage in contests with the cat for possession of Fat Worm. And the man said unto himself, “Surely I shall defeat this small cat in contest, for I am mightier in all ways than this little one. Indeed, my arms are stronger, my mind is keener, and my thumbs are more opposable, the better for grasping Fat Worm.”

But lo, in each contest it came to pass that Fat Worm would always fall into the mouth of the cat. And the man was astonished, and thought to himself, “How does this come to pass? For I am stronger of limb, keener of mind, and greater in size than this humble beast! Yet in each contest, Fat Worm eludes me, and falls into the greedy mouth of the cat!”

And seeing the man’s bewilderment, the LORD spoke thus unto him:

“Dost thou see how the least of my creatures conquereth thee? Not by strength of arm, nor wisdom, nor even the versatility of thumbs is victory won, but by the strength of its desire doth the cat master thee, and takes Fat Worm as its prize.

Though thou art stronger of limb, thou dost not desire Fat Worm as doth the cat. For thou, being man, hast manifold aims, and thy mind is split among them, but the spirit of the cat is consumed by desire for Fat Worm.

By night it dreameth of Fat Worm, by day it plotteth how it might seize Fat Worm. Indeed, even while thou sleepest doth the cat keep vigil by the bathroom door, where thou concealest Fat Worm. Thus doth it come to pass that the cat defeateth thee.”

And the man pondered these words, and saw the truth in them.

Originally written as a Latin composition exercise, in the style of the Vulgate Bible.

Latin in Modern Fantasy

I can think of a few popular, modern novels that use Latin-inspired naming conventions.

Harry Potter is an obvious example—all the spells are in Latin or pseudo-Latin. The summoning spell accio is the first-person conjugated verb meaning “I summon.”

The characters names are Latin-inspired, some more on the nose than others. Severus Snape is severe, Dolores Umbridge combines pain, dolor, with “umbrage.” These names work. They’re suggestive without being too on-the-nose. They sound appropriately wizardly.

Remus Lupin, however, is pushing it. Lupin is clearly from “lupine” < lupus, meaning “wolf.” Remus is also the brother of Romulus, the mythical founder of Rome, and according to legend, they were both suckled by a she-wolf as babes. So a character named Remus Lupin, to a Roman, would sound like you named your son “Wolfy McWolf.” Guess what, Lupin is a werewolf. Nominative determinism strikes again.

The Hunger Games takes a different tack, giving characters from the Capitol classically Roman names to signal both civilization and decadence. Seneca Crane, Cinna,1 President Coriolanus Snow. The names evoke Rome, but they’re not secretly describing the characters. I like them!

The Will of the Many, though, is something else entirely. James Islington’s names are too descriptive, to the point of spoiling the plot. I do not exaggerate—the character names are effectively the entire plot summary.

Meet the Hero, WILL ALONE

We begin with the main character, named Vis. Vis is a Latin word, meaning strength, force, vigor, power, energy, virtue… or perhaps “will.” That’s sort of fun, the title is a bit of a pun!

Vis, being an orphan, is given the foundling name Solum, which is Latin for “alone.”2 Again, this is perfectly fine, and establishes that Latin words are meaningful within the fiction—orphans are surnamed “alone.” Great! Naming your character WILL ALONE is very classic hero’s journey.

Vis, being the hero, is of course very special. For one, he speaks Vetusian, the (now dead) ancient language of the world. Vetus, also correct Latin, is the root of our word “veteran,” and means “old, aged, advanced in years.” Okay, so the ancient language of this world is “Old-ian.” A little uninspired, but we can live with it!

But herein lies our first problem. You see, in the book, we get an example very early on of spoken Vetusian, the word “nonagere.” Non agere is Latin for “to not act,” or perhaps in context “do not react.” This establishes that our main character, Vis, speaks this world’s version of Latin. Keep that in mind. It will become profoundly, bafflingly important later.

No, I Am Your Father

Early on in the story, our hero is taken in by a mysterious older man/father figure named Ulciscor, who is a driven and tormented man, with a single-minded focus on a mysterious goal that is not revealed until near the end of the book, and it is at this point I begin to suspect James Islington is trolling me.

You see, ulciscor is also correct Latin, but it is not a noun. It is the correctly conjugated first-person indicative of the verb ulcisci, meaning “to avenge.” Vis’ mentor is named, literally, I-AVENGE. Gosh, I wonder what core motivation this enigmatic character named I-TAKE-REVENGE could possibly have?

This man of mystery, Mr. I-AVENGE sends our hero WILL ALONE to be trained in combat by a woman, Lanistia. Hey, guess what lanista means in Latin? Yes indeed, a lanista is a trainer of gladiators, but at least this time Islington added an “i.” So Vis isn’t quite being trained by a woman named TRAINER—in English, the feel would be something closer to TRAINIER.

The Whole Cast is In On It

While Vis is at school, and throughout the plot, he meets a scintillating cast of characters, with equally descriptive names. I’ll translate them for you.

• Vis’ clever but short friend Callidus Ericius, or “cunning hedgehog,”

• Idonia, who I frankly don’t remember at all, but her name comes from idonea, meaning suitable or appropriate.

• Emissa, whose name is the past passive participle of “to send,” reading “she who has been sent?” This one could legitimately just be an accident… or perhaps she is a spy. She is the love interest, I believe.

• The rebel and villainess Sedotia who is terrorizing the empire as a member of the group Anguis (snake). I’ll give you one guess what Sedotia means. (Hint: It’s from seditio, which means, stunningly, sedition or rebellion)

But then, we meet Ulciscor’s brother, and transcend parody to become high art.

The (Plot) Hole in Vis’ Brain

You see, Ulciscor, I-TAKE-REVENGE, has a brother. And we eventually learn that Ulciscor’s all-consuming quest for revenge is driven by the murder of his beloved brother.

The dead brother’s name is Caeror.

Now “Caeror” is not a Latin word. But if you change exactly one letter, you get caedor, first-person singular, present passive indicative of the verb caedere, which means “I am slain.”

The plot is literally a Latin grammar lesson. To summarize: our protagonist, the orphan boy WILL ALONE, is found by a mysterious man called I-TAKE-REVENGE, trained for combat by TRAINER, and goes to school where he befriends the short but clever CUNNING HEDGEHOG and prepares to fight the rebel leader SEDITION. We learn that I-TAKE-REVENGE has a brother I-AM-SLAIN, who was cruelly murdered by someone who I fully expect to be named “Neco” (I-MURDER).

James f*cking Islington even conjugated the dead brother’s name in the passive voice!! This cannot possibly be an accident.

But it’s worse that even this, because the book repeatedly tells us that Vis speaks Latin/Vetusian/Old-ian/whatever. It specifically tells us that he is special because he speaks Vetusian. The characters within the story demonstrably understand the meanings of these words—they named the orphan kid “alone” on purpose! And he doesn’t notice his adopted dad is named I-AVENGE?? Did Vis just skip the chapter on common deponent verbs in his Old-ian textbook??

I Am Annoyed, also, Conclusion

Most authors who use this trick modify the Latin words slightly or pull from obscure etymological roots. James Islington does not. These names are just… correct Latin. Correctly conjugated. Islington creates a mystery that is only a mystery to characters who are apparently incapable of understanding their own language. It is literally impossible not to spoil the plot as soon as you know the character names if you speak Latin, like Vis, unless, also like Vis, you have the situational awareness of a rock.

I honestly can’t decide which is true:

• This is an elaborate, absurdist joke for the handful of readers who remember their Latin declension charts.

• Islington has created a new and brilliant form of dramatic irony where the audience is screaming, “HIS NAME MEANS REVENGE!” while the hero just vibes, completely oblivious.

• He just didn’t expect anyone to notice.

But, hey, you know, the book itself was actually pretty good. I’m gonna read the sequel.3

Philosophy Substack has had a little bit of drama recently. Why? One of the most contentious subjects in philosophy, namely, how to properly take an average! Who knew that people were so… mean… on the internet.1 More specifically, Pascal’s wager has sparked some discussion, here quoting and .

For those who don’t know, Pascal’s wager argues that it is rational to believe in God. Why? From Wikipedia:

if God does not exist, the believer incurs only finite losses, potentially sacrificing certain pleasures and luxuries; if God does exist, the believer stands to gain immeasurably, as represented for example by an eternity in Heaven … while simultaneously avoiding boundless losses associated with an eternity in Hell.

If you want to wade into the murk, we have the inciting incident from that perennial engine of Substack discourse, the incomparable Bentham’s Bulldog,2 and of course various responses related to it. There’s also, again from the man himself, a post arguing for Expected Value Fanaticism, an interesting “bite the bullet” sort of argument related to expected values of very-lucrative-but-very-improbable events,3 and then another article defending himself against his critics.

Partially catalyzed by the Pascalian Wars, there’s been a schism in Philosophy Substack, helpfully named by :4

I am obligated to post ’s reply. The Oxford Boys don’t all go to Oxford.

But regardless, the divide is intellectually motivating, hence this post! Fear not, this is not an explicit critique of Bentham’s Bulldog, indeed, I defend him (though a partial critique is, perhaps, implicit). Instead, I’m going to hijack this drama about the use and abuse of the mean to explore two fundamental ways that expected value—the foundational concept of these arguments—can break down.

First, we’ll look at what happens when we permit the system to evolve over time. For many real-world processes (Russian Roulette, gambling, posting on social media) the average outcome for a group of people is not the same as the average outcome for a single person over time. This is the realm of ergodicity, and confusing the two types of averages is a recipe for ruin.5

Second, we’ll return to a one-shot wager like Pascal’s and see how expected value can fail even without time, in a thought experiment I call “Cauchy’s Mugging,” where the rational choice depends entirely on whether you value the mean or the median.

But we’ll get to all that in a minute after we talk about averages more generally—I prefer to bury the lede.

Many Ways to Average a Set

Consider a set of random numbers, e.g. x = {x_1, x_2, x_3, …} What’s the average of this set?

There’s a lot of ways we can answer this question. Fundamentally, when we take an average of a set, we want to extract a single number that minimizes the error of representing the set.

For each element x_i of the set, or vector, x, the average μ is the unique number that minimizes the L_p norm of the error. The L_p norm gives a formal notion of the “distance” between two numbers:

You raise every element of the set to the pth power, sum them, then pth root that sum. The error is simply the difference of every element in the set with some value:

And then the p-average μ_p minimizes that error. For p = 2:

We see the 2-average is the familiar arithmetic mean. This is a very special average, and it is by far the most frequent type that is given the title of “average” with nothing else specified.

Other p-averages

What about other values of p? Do they furnish us with other familiar quantities? Yes!

The L_1 error is minimized by the median, or the 1-average:

The median is the unique number that partitions the set into two equally sized halves, the “middle” number. It’s a really good choice for the “average” of a set when you want to reduce the importance of outliers. Let’s say you’re , and deeply concerned about pickup-line-to-jail expectations being representative of the average experience in picking up women:

We really don’t want Bad Pickup Lines Georg throwing off our intuition, or we’d never get a date. The median-as-average great for this, it functionally ignores Georg. It also happens to be a good choice for talking about “average income,” if you don’t want to be skewed by the ultra ultra rich people.

Is there a 0-average? There is indeed, the mode! Mean, median and mode correspond to the L_2, L_1, and L_0 notions of “average.”

Surprisingly, p = ∞, and even p = -1 also give sensible and useful averages. The ∞-average is simply the largest number of the set, while the (-1)-average gives us the harmonic mean:6

Every single one of these p-averages, despite their different implications for various philosophical and scientific arguments, is a valid “average” or “expected” value of a set. We tend to default to the 2-average for some nice properties (like minimizing the value of π), but in many more situations we care about things like “favoring sparsity” in training an ML model and are forced to use an L_1 norm and a 1-average.

For the rest of the post, we’re going to define our notion of distance as conforming to the L_2 norm. This is going to be the first of several buried premises that are easy to miss in casual argument, and I want you to remember that, because it’s a motif here.

Quick Summary of L_p norms

The Ensemble Average

Let’s introduce the weighted average of a set f = {f_1, f_2, …}, where the L_2 norm is implicit, but each element of the set gets a weight or a multiplicity.7

We can further promote the discrete label i = {1, 2, 3, …} to a continuous variable α, and take a continuous ensemble average over any function of α, f(α). This is a very, very common type of average, and one of the subjects of this post. Its importance also gives it a special notation, <f>.

w(α) is very often a (normalized) probability distribution, and plays two roles—first, it provides the same normalization feature as that 1/N in the discrete case, and second, it allows us to handle multiplicities, multiple occurrences of the same number in the set. Importantly the ensemble average is used to compute expected values in probability theory, and that also gets special notation:

And finally, the tricky one…

The Time Averag—no, Russian Roulette

The time average is a different category of average, and to see this, let’s talk about Russian Roulette.

Here’s the game: One chamber of a six-gun is loaded, five are empty. The “dealer” attempts to shoot you. If you live, you get a billion dollars! If you die, the dealer takes the money back from your unfortunate corpse. You can play this “game” as many times as you like.

Should you play? If so, how many times?

The first question is one of personal risk tolerance, but the second question is actually underspecified.

What do I mean? Well, what’s your chance of winning one billion dollars? Six chambers, five empty → a five in six chance of winning a billion. The expected value of the game, in money, is the ensemble average of all possible outcomes.

Thus, on average, you will make $833 million on the first round. How about on the second round? The calculation is the same, assuming you live.

The expected value says, correctly, that each round you have, on average, an expected profit of $833 million. … so I should play infinite times? Well, no. Look at the graph.

This example was chosen to make it really really obvious that the expected value of a given round does not fully describe the game. The expected $ value for every round is still positive, but the expected amount of wealth (and life) you will have is not.

This is the St. Petersburg Paradox, explored concisely and well by (not me, the other one) in his article The Tyranny of the Mean. Russian Roulette has the St. Petersburg property, and reveals a fundamental distinction: the average across all possible players at any given round (5 of 6 survive) differs from the typical trajectory of one player across many rounds.

Most interesting examples in physics where time-dependence messes with our timeless intuitions are not nearly this obvious, and so we need some formalism.

The Time Average

Introducing the notion of time8 forces us to distinguish not just two but three categories of related averages: the timeless a priori expectation value, the ensemble average of the trajectories after some time t, and the path-dependent time average of a single individual.

For some collection of functions of time:

We can take the a priori expectation value of the system at t = 0.

We can also take a time average over the function’s history, which we’ll denote with an overbar:

And we can also take a “trajectory average,” an ensemble average including the time evolution of the system, evaluated at some point τ.

The trajectory average is that red line in Russian Roulette, that initially rises, then craters as most of the participants enter their “absorbing” state (that is, die).

Ergodicity and Stationarity

Let’s take a moment to re-emphasize this distinction. We have a set of fixed rules for a non-stationary process, and thus, we can correctly describe the process with three kinds of average.

• To discuss the optimal decision in any given round (assuming you exist to play), we choose the a priori expectation value, the God’s eye view of the game.

• To discuss the average state of a particular player, we use the long-time average.

• To discuss the evolution of the system over time, we choose the trajectory average, an ensemble average over the system evolution

Whether or not these distinct types of average coincide is one of the most fundamental properties of any system.

What happens when the a priori expectation value and the trajectory average coincide? The system is then in statistical equilibrium, where every moment in time is equivalent to all the others—we call this system stationary.9 A single round of Russian Roulette can be considered stationary: the ensemble of players’ expected average profit is always the same, regardless of when the round is played.

What happens when the long-time average of a single path converges to the ensemble average of all possible paths? Then, the system is called ergodic.10

A very, very, very consequential fact about the universe is that many physical systems and processes are ergodic (maybe even most!), but not all of them.11

Ergodicity

A lot of physics assumes ergodicity, and all of statistical mechanics does. Statistical mechanics is the physics of really, really large numbers of objects (like atoms in gases, baseballs, solar systems).

Statistical mechanics is the foundational math of classical thermodynamics, concerned with questions like “what is temperature and how does this relate to the speed of individual particles in a gas?” Or “what is entropy?” Its fundamental objects are the macrostate, the bulk properties of a system, and the microstate, the specific configuration of atoms within it.

It’s very useful to think about ergodicity in terms of states: any particle in an ergodic system eventually and uniformly explores every microstate (the possible locations and momentum values for the particle) accessible to the system.12

A consequence of this fact is the time-average of any single particle’s trajectory equals the ensemble average of many randomly initialized particles.

Whenever you actually do statistical mechanics, you assume that for an isolated system in equilibrium, all accessible microstates are equally probable. It makes the problems tractable, but this is only valid if the system is ergodic.

Maxwell’s Demon

Let’s look at how some bedrock results break when your system is not ergodic.

It’s a well known and easily observed fact that if you place two gases, one hot and one cold, in a sealed container, the gases mix, and they eventually come to equilibrium at an intermediate temperature. The Second Law of Thermodynamics says that this will always happen, and forbids the reverse process: a gas mixture spontaneously separating into a “hot side” and a “cold side.”

But Maxwell’s famous thought experiment about a demon appears to disprove the Second Law.

Imagine a very small demon who controls a door separating two chambers,13 containing statistically identical gas mixtures. The demon watches carefully, and whenever a fast-moving (hot) particle approaches the door from the left, he quickly opens it, letting the hot particle pass to the right chamber. He does the reverse with cold particles. Over time, the demon causes one chamber to warm and the other to cool! This decreases the total entropy of the system, seemingly without doing any work, and violates the Second Law of Thermodynamics.

How do we resolve this paradox? We could posit that information by itself carries energy. The demon (who is part of the total system) must act conditionally (open door if hot particle is currently approaching door), and the act of processing that information may dissipate some energy. This is Landauer’s principle: a single bit of information carries 0.018 eV14 of energy at room temperature.

There is a lot to be said about this thought experiment, but one of the things I want to point out is that this system is manifestly not ergodic and our intuitions about how it should behave from statistical mechanics need not apply. Without the demon, gas particles in the box form an ergodic system—given enough time, any particle will explore all accessible regions of phase space, and the Second Law is obeyed. The demon breaks both the Second Law and ergodicity within the gas chamber by creating what are effectively absorbing barriers based on particle velocities.

The information-conditioned action of the demon breaks the ergodicity of the system, and a bedrock result fails to hold.15

Other Non-Ergodic Systems

What other sorts of bedrock results fall apart when you lose the buried premise of ergodicity? Well at least according to Ole Peters’ Nature paper… all of economics. He argues that economics as a field uses a priori expectation values, ensemble averages across many economic actors, to predict individual outcomes. This is wrong, because life is non-ergodic, and many paradoxes and examples of “irrationality” in the literature disappear when you use time averages.

He illustrates with, what else, a game. Let’s say you’re in Vegas:

[You] toss a coin, and for heads you win 50% of your current wealth, for tails you lose 40%.

Is this a good bet? Let’s calculate the expectation value of your wealth x in any given round.

Hell yeah, we gain 5% of our net wealth per flip! My wife would kill for this investment opportunity! Hopefully though, primed by all that talk about ergodicity, we wait before taking this bet: this is a non-ergodic system and the expectation value is going to mislead us in isolation.

Let’s instead calculate the trajectory average. For a player who plays N times, with N_T tails and N_H heads, we have the wealth at flip N

It’s pretty clear that as N → infinity, N_H = N_T = N/2, and we find that, per round for an individual, the next round is likely to decrease and not increase your wealth.

Ole Peters illustrates with a familiar looking trajectory average graph.

Now as a physicist, I’m naturally inclined to snicker at fields that use less math than mine and call them non-rigorous and fundamentally flawed. But of course, one must remove the giant renormalization-shaped log from one’s mathematically rigorous eye before pointing out the speck in the economist’s.16

But there are a lot of non-ergodic processes, to which we should not expect our intuitions primed on stationary and ergodic processes to apply.

• Evolutionary biology

• Climate models with unrecoverable “tipping points.”

• Glasses with metastable states—ever heard the fact that glass is a liquid? What they actually mean is that glass is actually a metastable non-equilibrium state. Less pithy, but correct.

• Publication bias in science

• Social media networks

Markov Chain Monte-Carlo and Ergodic Imposition

I want to take just one more little digression from the main thrust of the article, and talk about how imposing ergodicity in a system lets you decode certain types of ciphers.

The Metropolis algorithm is a Markov Chain Monte Carlo (MCMC) method (shameless plug for my previous article on MCMC) for “obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult.” Yeah I don’t love this definition either, so let’s explore it with an example, following Max Rohde on his blog.

Consider a message encrypted with a simple substitution cipher, where each letter of the alphabet is randomly mapped to another.

We could try to brute-force this cipher, but there are 26! (factorial) possible ciphers, and that would take forever.

Instead, we’re going to use the Metropolis algorithm.17 We define a function sim() that takes our proposed decrypted cipher as an input and produces a number proportional to its “similarity” with English text.18 We then execute this loop:

1. Randomly swap two letters in the cipher map

2. Compute the ratio r = sim(new decryption)/sim(old decryption)

3. If r > 1, accept the swap.

4. If r < 1, accept the swap with probability r and reject it with probability 1-r

Despite its incredible simplicity, this algorithm works! See Rohde’s blog for a cool video of it working. But that last step 4 is a bit weird, right? Step 3 makes sense: if we get a better decryption, keep it. But step 4 says to accept worse decryptions too, with some probability. Why?

I mean the title of this article gives it away a bit, but the answer is that step 4 enforces ergodicity on your solution space.

The Metropolis algorithm, fundamentally, provides a well-defined method for exploring some solution space, and its convergence depends on the solution space having both a stationary distribution to converge to and the ability to explore the entire space from any starting point. That’s what step 4 is—if you start from a random starting point in cipher-space and only accept strictly better swaps, you get stuck in local maxima or infinite cycles. By adding that probabilistic acceptance step at the end, you guarantee the possibility of exploring the entire space. That theoretical possibility is sufficient to guarantee global convergence. The randomness is necessary too—by adding a nondeterministic factor, you avoid deterministic cycles.

I just love this fact. The Metropolis algorithm only works if the method for determining which decryption is “better” has a random error term.

Expected Values and Cauchy’s Mugging

Let’s return now to the use and abuse of expected value and Substack drama, dropping the ergodicity thread for a moment. We’ll pick it back up, don’t worry. For this final section, we’ll use (sorry Liam, you just popped up on my feed) as a jumping off point, who makes the following statement:

Now while I agree makes mistakes, this is not one of them.

The items in the rationalist toolbox (expected value calculations, conditional probabilities, Bayes Theorem, etc.) are only defined with finite inputs. You simply cannot use them to evaluate infinite numbers. The math just doesn’t work like that!

This is false. The expected value of a random variable is calculated as we saw above, as a probability weighted average over all possibilities.

Now the probability measure P cannot be infinite (Liam is correct about this). But the expectation value is perfectly well defined if x_i happens to be infinite.

The expected value can even be infinite even if all outcomes are finite! Take, for example, a game with payoff 2^k with probability 1/2^k. Then the expected value of the game X is infinity.

But wait, doesn’t that… feel wrong? That’s because the actual debate, which Bentham’s Bulldog correctly engages with, is whether it is rational to act based on that infinite value. He says yes. Others say no.

This is a much more subtle disagreement and it depends on, literally, what you value. To illustrate, let me propose an unusual form of Pascal’s wager, in which my god is Cauchy.

Not the mathematician, the distribution.

Cauchy distributions have fat tails and weird properties. They show up all over physics, and have the following form:

The Cauchy distribution, despite being normalized has no arithmetic mean. It realllllly feels like it should, but it doesn’t. Do the integral yourself and check.

It does, however, have a median, and so let’s look at a version of Pascal’s mugging, which I call Cauchy’s mugging, where we are accosted by a crazy holy man in a back alley who happens to look very much like Augustin-Louis Cauchy.

Cauchy Jesus informs us that we face a momentous choice, between two Paths. Path A is a comfortable, secular life with a guaranteed 100 units of utility. If, however, we choose to worship Cauchy’s God and take Path B, we will get a fantastic reward in the afterlife, whose utility is drawn from a one-sided Cauchy distribution, which he helpfully provides as a handout, reproduced in the figure below.

Now we’re fairly skeptical folks, and we don’t really believe Cauchy Jesus, but we’re not so incredulous as to put literally 0% credence on his story. Let’s say we give it 1% chance (or indeed, any finite chance). Our Pascal table looks like this.

But we, as good utilitarians, have a problem. The expected utility for a secular life is 100 in either Path, but the expected utility for following Mr. Cauchy is

Should I… worship the Cauchy God then? If you’re actually committed to making decisions based solely on the L_2 expectation value, then yeah, I guess so. Seems a bit unintuitive, but biting this particular bullet is “being a fanatic” à la Bentham’s Bulldog.19 But what if instead of optimizing for mean utility you optimized for median utility? Well the median of the Cauchy distribution is not infinite. Cauchy has helpfully plotted the median utility of the one-sided Cauchy distribution in his above figure, and it’s just 1 util. So we, on “average,” only get 1 util, with a 1% probability?? This is a terrible deal! Get thee behind me, Cauchy!

Conclusion

I’m really not trying to score any philosophical points on Pascal’s Wager here. Instead, I hope this litany of strange examples has shown that “expected value” is a term with several baked-in assumptions that aren’t always valid or useful. The average, or expectation value of a set, is a tool to condense the information of the set, not the set itself.

We’ve seen that it can mislead when mixed with time. An ensemble average over Russian Roulette promises riches and small chances of death that are inaccessible to any single individual unlucky enough to play.

We’ve seen it can literally fail to have a value when confronted with fat tails. In a simple, one-shot decision like “Cauchy’s Mugging,” the L_2 expected value can be infinite and undefined, offering no guide to action. In that case, perhaps the L_1 average, the median, provides a robust tool with a clear path forward. Or hell, the mode, or the harmonic mean—it depends on what you value and that’s precisely the point.

The central lesson is this: before you calculate an average, even if you’re a philosopher (I joke, don’t @ me), you must first understand the nature of the system you’re analyzing. Is it ergodic? Do its outcomes have a well-defined mean? What exactly am I optimizing for? Always, always be wary of doing math with English. That way lies madness, suspicious infinities, and Substack drama.

13

For example, this one:

19

See Bentham’s apology (in the Socratic sense) for extortion and Pascal’s Mugging:

Bentham's Newsletter

Addressing My Critics on Pascal's Wager

Two people have written replies to me on Pascal’s wager: Richard Hanania and Dylan. I enjoyed Richard’s piece, and felt annoyed by Dylan’s piece. I thought it would be worth explaining why they are both wrong…

Read more

a year ago · 23 likes · 130 comments · Bentham's Bulldog

First of all, I am overwhelmed by the positive response to my article—I'm still a little confused why what is functionally a measure theory lecture is my most popular post. Y'all are nuts, thank you so much.

Secondly, Bentham's Bulldog has replied to me in many places! Scott Alexander too, and doesn't that make me feel like I met a celebrity? Obviously they think I'm wrong, but among other things, BB has challenged me to engage with the actual philosophical literature. So I have! Or more accurately, I used Gemini’s Deep Research tool (which is unbelievable by the way) to give me a comprehensive summary of the state of the art in "the measure problem" of fine-tuning. It produced a 32-page document with 55 references, and frankly I read it in one sitting because it was really fascinating. The LLMs lack a true voice (unless “fun police” is a voice, looking at you Gemini), but they are extraordinarily clear writers.

Anyway, since the discussion surrounding my post was so interesting, I thought I would capitalize on the mom— sorry, I meant to say summarize the discussion for all the interested readers! I am going to shamelessly steal Scott Alexander’s style of “Highlights on X” follow-ups, because blatant theft is the highest form of flattery.

This article will be roughly three parts, I’ll summarize some interesting points from the literature review, I’ll respond directly to Bentham’s Bulldog’s article/comments, and then I’ll highlight some particularly interesting/insightful comments from the discussion.

0.5. Brief Recap and Bentham’s Challenge

Briefly, the fine-tuning argument holds that due to the true fact that the universal constants we observe in physics (speed of light, cosmological constant, Higgs boson mass, etc.) fall in a very narrow life-permitting range, that this is strong evidence of an outside “fine-tuner,” i.e. God. Bentham’s Bulldog made this argument in, among others, his article The Fine-Tuning Argument Simply Works.

In my previous post, linked above, I argue that the fine-tuning argument that God exists is ill-posed and the mathematics behind it meaningless. In essence, when Bentham’s Bulldog does a calculation of the Bayesian probability of the creator, he uses a whole bunch of undefined terms and the result one gets could be literally any number. Bentham’s Bulldog naturally disagrees:

This is an interesting article--basically raising the measure problem--but I find it weirdly triumphalist. In that original article I did not address the measure problem but I have elsewhere (here for example). You shouldn't be totally confident that the measure problem wrecks my view if you haven't read what I—or people on my side—say about the measure problem!

I give a lot of more specific objections in the linked post. But crucially we can *obviously* say that things are improbable even from an infinite set. So, for instance, there's an infinite set of things that the nearest aliens might be thinking about presently. An infinite number of them involve tacos. But still, I would guess with high confidence that the nearest alien to me is not thinking about tacos

Let’s tackle (partially, anyway)1 the first of his objections, that I don’t engage with him or the philosophical literature on this.

1. The Literature

Physics and Cosmology

So the first thing that really jumped out at me is that the measure problem is way more broad than I thought. Forget God for a moment, the problem of measure is just generally a problem for theories of the multiverse. As Gemini writes:

Eternal inflation posits that our observable universe is but one "pocket universe" or region within a vastly larger, perhaps infinite, multiverse, where new universes are constantly budding off. These pocket universes may possess different physical constants or even different effective laws of physics.

The measure problem is the challenge of defining a consistent and unambiguous way to calculate the relative probabilities or frequencies of different types of universes or events within such an infinite ensemble.

I hadn’t really even considered this, but obviously this problem exists in pure physics!

Alan Guth gives a memorable example of why this is a problem—what’s the ratio of two-headed to one-headed cows in the multiverse?

“In a single universe, cows born with two heads are rarer than cows born with one head,” he said. But in an infinitely branching multiverse, “there are an infinite number of one-headed cows and an infinite number of two-headed cows. What happens to the ratio?”

This is a really good question, the answer, of course, is that it depends how you define your universe-generator, i.e. your probability space/measure.

But defining a “typical” or “probable” universe poses similar challenges for the anthropic principle.

Let’s say we’re just one of many universes, but ours is the only one that supports life. Well then of course we find ourselves in this universe, we literally can’t exist in the others! That’s the weak anthropic principle. There’s a strong anthropic principle, which posits that the universe must exist this way, that sounds a lot like a design argument to me.

This is all highly related to Nick Bostrom’s “Strong Self-Sampling Assumption,” which states that we should reason as if we are randomly selected from the class of all observers. But… what class? The measure problem again!

Let’s give some more interesting examples of how different measures affect your cosmology.

So you’ve got some infinite collection of universes, generated according to some process, and you need to calculate relative probabilities. The problem is that in an eternally inflating spacetime, any event that is physically possible will occur an infinite number of times, rendering simple frequency counting useless. Thus, you “regularize” by taking a finite subset of the multiverse, then taking the limit to infinity. There are several choices of cutoff:

• Proper-time cutoff: Counts events or observers up to a fixed hypersurface of proper time, then lets this time go to infinity.

• Scale-factor cutoff: Uses the expansion scale factor as the time variable for the cutoff.

• Stationary measure: Compares processes based on the time since each process individually reached a stationary state.

• Causal diamond measure: Considers the finite four-volume accessible to a single observer from the beginning to the end of their possible observations.2

I’ll just give one example: The proper time cutoff yields an overwhelming bias towards “young” universes. This implies we should all be “Boltzmann Brains,” random thermodynamic fluctuations that simulate life and memory, because these are more probable in a young universe than an old one. Sean Carroll doesn’t like this, because any theory that implies you’re a Boltzmann brain should make you skeptical of your ability to reason coherently about theories. Other cutoffs give you other biases.

Chris Smeenk calls this a "predictability crisis"—eternal inflation literally cannot make testable predictions without solving the measure problem.

I want to raise two more interesting physical arguments. One of them is Max Tegmark’s Mathematical Universe,3 and another is in a paper, which I need to actually read, not just skim, called Algorithmic Theories of Everything that Onid sent me. This proposes that universe histories are sampled from probability distribution based on their algorithmic complexity—universes with shorter descriptions are more probable, and weakens the requirement for universe histories from Turing computable to formally describable.

Okay, well, what do philosophers say?

Philosophers

Pretty much everyone agrees that the universe sure looks fine-tuned to us. To explain this, philosophers like to use either abduction (extrapolating from evidence to hypothesis, sometimes called Inference to Best Explanation), or more formally, Bayesian reasoning, which I argued is ill-posed. Gemini highlighted six explanations

• Design/Theism: It was fine-tuned by God.

• Multiverse: There are many possible universes and we live in one that permits it because of the:

• Anthropic Principle: We observe only what supports our existence.

• Deeper Laws: A more general principle makes fine-tuning seem natural.

• Simulation: We live in a simulation fine-tuned by someone not God

• Brute Fact: Look man, we live where we live, get over it.

I love the way Gemini put this: “The philosophical debate surrounding these explanations often reveals underlying differences in epistemic values and criteria for what constitutes a satisfactory explanation.” Oh my god, yes.

In one sense, the “Brute Fact” explanation is extremely attractive—posit nothing, we live where we live. This is a non-problem. You can see the appeal, in a way. I still like “deeper laws,” personally. Or multiverse, in some way, but isn’t that just an instance of a deeper law?

Now it gets a little wacky, but I love the attempts at formalism! Philosophers propose a few alternate probabilities that differ from standard (Kolmogorov axiomatic) formulation of probability.

• Finitely-Additive Probability (FAP)4, of which the first article I found explaining it is called “LOST CAUSES IN STATISTICS I: Finite Additivity.” This version of probability drops “countable additivity” in favor of “finite additivity,” essentially arguing that P(A or B or C or …) = P(A) + P(B) + P(C) + … can’t apply if there are infinitely many terms.

• Non-Archimedean Probability (NAP) Theory: Extends probabilities to include infinitesimals.

• Non-Normalizable Quasi-Probability: Drops the normalization requirement.

• Truncation: This one is the simplest—instead of relaxing the axioms of probability, just truncate the space you work on. This is subject to the “but why that truncation” problem.

I think this is really interesting philosophically, though you’d have to give me a really good justification why it’s a better idea to drop very reasonable axioms of probability theory just so that fine-tuning arguments for God work, when in my opinion the “deeper laws” objection works better anyway. Either way, one can have a debate on this, because, and this is key, every philosopher in the literature summary confronts the measure problem as though it’s fatal without some sort of workaround. I’m actually fairly happy I was challenged to do this, because it makes me pretty happy with my critique. My core objection is physics, not philosophy, and it is correct per se. Bentham’s Bulldog was pointing out that philosophers have various arguments why we can get around it and still reason with solid foundations.

2. Directly Responding to Bentham’s Bulldog

First, a mistake from me.

Three commenters, from the article and the reddit thread5 pointed out that one of my arguments was flawed. This one, specifically.

“But by construction, you get P(fine-tuning∣¬God) = 0 due to selecting a value from a set of zero measure (in this case, selecting a value from a continuous range). By the premise of this whole argument, then God too must select, whether randomly or not, from a set of zero measure. The probability of God selecting any particular value, P(fine-tuning∣God), is therefore also zero.”

Why does defining the God hypothesis more precisely not end up with that "1/0 \to \infty" situation, as opposed to undefined?

—u/[Anonymous] (reddit)

This remark and the associated note 17 is wrong; by assumption God is not choosing randomly.

—u/Curates (reddit)

Also, it’s a minor point, but you mention that God could not select from a non-measurable set, but this isn’t true. God would, presumably, act non-randomly, and one can definitely non-randomly select from a non-measurable set (for instance, selecting 0 from the Cantor set). Since God is presumably defined as omnipotent, omniscient, and wanting to create life, P(fine-tuning | God) would be 1.

—Onid (Substack)

These three are right and I am wrong.

The point I was trying to highlight was that even if you manage to define naturalism on some continuous interval, P(God selecting our constants from any continuous interval) is still 0, because he might just as well have chosen something else within the life-permitting interval.

While this is technically true for a God who selects randomly, it’s quite simple to counter this argument by simply stating that God doesn’t choose randomly.

I'd have been better off not even attempting to posit a God selection mechanism at all. That’s getting into theology which is (clearly) not my strong point. Onid, however, delivers my rejoinder for me:

Obviously P(fine-tuning | not God) is still completely undefined, so it’s not like this changes anything.

—Onid

Yes, this is right. I'd pretty happily accept a version of the fine-tuning argument that defined both of these things somehow and then made a Bayesian argument. It would have to look like:

Assumption 1: God is this way, here's my justification.

Assumption 2: Naturalism works this way, multiverse budding, whatever, and here's P(our constants | my multiverse).

This is 100% valid and I have no (math) problems with it. I would probably quibble about your justifications or argue with you about why that specific God, but you'd be good on the math.

As an aside, Onid also said something very kind that made me very happy.

Anyway, your point about the monster group was brilliant. I hadn’t heard that one before. —Onid

Thank you!! I was very proud of this analogy, because it’s authentically mine and I feel like it works really well to demonstrate why something really odd can be true but derived from a really simple and uncontroversial principle. Looking forward to your article on SIA!

Bentham’s Bulldog’s Specific Objections

His Comments

Ok let’s look at his specific objections to my article.

But crucially we can *obviously* say that things are improbable even from an infinite set. So, for instance, there's an infinite set of things that the nearest aliens might be thinking about presently. An infinite number of them involve tacos. But still, I would guess with high confidence that the nearest alien to me is not thinking about tacos

The whole point of my article was no we cannot obviously say things are improbable from an undefined set. He’s eliding infinite with undefined. Some infinite sets are totally fine, most are not and you must pick one or I will yell at you. This objection misses the point entirely.

The second objection is better. Let’s talk tacos, because that one generated a ton of discussion and I think I can contribute.

Well-Defined: Aliens Holding Tacos

I originally tried to answer this question with aliens-thinking-tacos, please see the footnote for my formalization attempt,6 but I want to thank u/kaa-the-wise on reddit for pointing out that aliens thinking about tacos has the “what is thinking, anyway” question embedded in it, and runs into the hard problem of consciousness.

So let me modify: aliens holding tacos. Same essence of improbability, no consciousness. Hopefully Bentham’s Bulldog is okay with this.

My response to this is quite straightforward: When we ask "what's the probability that the nearest alien is holding a taco?" we're operating entirely within known physics, and we can quite reasonably formulate a probability space to operate on. We need some statistical mechanics background though.

Very Brief Stat Mech

Statistical mechanics is a branch of physics concerned with the problem that there are just too many atoms. We’re sure that the physics of atoms ultimately gives rise to the physics of e.g. baseballs, but there’s ~10^23 atoms and we can’t write down the equations of motion. So statistical mechanics provides a rigorous framework for taking ensemble averages of many, many atoms, and pulling out useful average quantities, like temperature, entropy, and so on.

Let’s say I have N particles in a 1-dimensional box of side length L, the positions and momenta of these particles is then (x1, x2, x3, ..., p1, p2, p3...). This is a 2N-dimensional “phase space.” The particle states at any point in time occupy some configuration in this phase space. Phase space is continuous, and so there are an infinite number of states, yet we can very reasonably and meaningfully discuss densities of those states, because we can attach a sigma-algebra! This sigma-algebra is the Borel Set, which I briefly mentioned in my article, but essentially consists of countable unions, intersections, and complements of open intervals. It’s a measurable set, and thus we can assign a measure to it (usually Lebesgue). For example, we can discuss the "density" of particles over some interval dx or the system being in some region d^N x d^N p of phase space.

How does this help us with aliens holding tacos?

Well, this is nothing more than a configuration of atoms on the phase space of the universe! Aliens-holding-tacos is some subset of the possible configurations of the Borel set on universe phase space, and (probably) a tiny tiny tiny fraction of it. We can perfectly well use Bayesian reasoning on this and get answers that align with our intuition. The nearest alien is very probably not holding a taco, though if they were, first contact is going to be amazing.

I want to emphasize this, because there was a lot of confusion on this point. Infinite or continuous vs. discrete and/or finite is not the appropriate distinction for having sets of things that we can reasonably talk about—it's describability, it’s the sigma-algebra, it’s the measure.

I should note that this is basically also my response to Scott Alexander’s challenge as well, and I’ll get to that later.

Bentham’s Bulldog’s Article

A third objection is called the measure problem.

…

Now, this objection has the problem of stupendously misstating the physics behind much of fine-tuning. Regarding the cosmological constant, for instance, physicists don’t assume that each value of the constant is equally likely, but instead note that many different values affect the cosmological constant. These values are determined at small scales, but almost exactly cancel out at large scales, so that the universe doesn’t expand either too quickly or too slowly. This is a coincidence on the order of the number of seconds humans have spent happy minus the number of seconds humans have spent unhappy equaling just twelve seconds. The values going into the fine-tuning argument are about 10^120 times greater than the value of the cosmological constant, and when one allows these values to cancel out, they get a slight positive cosmological constant.

This objection is simple to dispense with, it does not actually refute the measure problem, instead it restates that fine-tuning is surprising. This is true! It being true does not give you a well-defined space to work in. That’s what I mean about slightly missing the point, though it’s unfair of me to say that this article, written prior to mine, “missed the point” of my article. Nevertheless.

It also misstates probability. The value of the constants can only be calculated across scales consistent with the basic standard model, and only a tiny slice of values consistent with the standard model produce complex structures. This would be like, to use an analogy from Robin Collins, hitting a tiny red dot on a dartboard surrounded by a white area. Even if most of the dart board is shrouded in darkness, this is very improbable! It doesn’t matter the total percentage of the dartboard that is occupied by red, because the surrounding region is almost all white—hitting the tiny red dot is thus astronomically improbable.

Okay here this is wrong. “The value of [the constants for a hypothetical set of universes]” can absolutely not be calculated “consistent with the basic standard model” without a whole boatload of assumptions that are by no means assured. Well-defined space. Note again the analogy used is a well-defined space. Arguments by analogy hurt me.

One can also get their probabilities to nicely sum to 1 by taking a ~log prior across the values of the constants! When they do that, once again, they get a ridiculous degree of fine-tuning.

Here is a little more subtle, he is trying to pick a space/measure! This isn’t a critique of the measure problem so much as an attempt to show it’s invariant under different ones, but that is just not correct, unfortunately. Pretty much every physicist/philosopher in the Deep Research review acknowledged this is hard, and you have to make some very questionable assumptions (though potentially justifiable!). See the literature review

But once again, we don’t need to bother with all that. The objection simply proves too much. If it were right, then the initial conditions spelling out “made by God, with love” in every language wouldn’t be evidence. After all, there are an infinite number of shapes that the initial conditions could conceivably take on. In fact, the math behind the actual fine-tuning is considerably less messy than the math behind figuring out the odds of the initial conditions spelling out “made by God, with love,” as there are probably infinite different ways the initial conditions could spell that out, and no particularly elegant measured over the values that would render that improbable.

Aha, this is the same objection as aliens-holding-tacos above, and has the same justification. This is well-defined, just completely intractable to exactly calculate. Infinite ways does not equal ill-defined, as the very brief stat mech overview showed us.

He also has some objection to the “deeper physics” postulate that I personally like, but that’s handled well enough in my previous article, I think, where I attempt to prime your intuition that actually, really really weird unintuitive stuff comes from basic obvious seeming principles sometimes.

So in summary, none of Bentham’s Bulldog’s defenses refute the core critique. He convincingly argues that fine-tuning is surprising, but offers us no way to do probabilistic calculations with it. He further offers an intuition pump about aliens and tacos, or equivalently unlikely scenarios that are well-defined, once again demonstrating the peril of arguing by analogy.

The core ambiguity and the problem of measure remain intact. If I were him (taking some charitable liberties with his position) I could posit something like a Borel set on finite-parameter-range multiverses where our universe is a tiny slice, and thus improbable, then use this to argue for theism. That’s emphatically not the only valid choice, we could easily have deeper laws, or a wacky underlying set (Vitali universe?), or literally anything else, but this would at least be a (somewhat arbitrary, but nevertheless) valid mathematical statement.

3. Highlights from Comments

Attempts to Well-Pose Fine-Tuning

Scott Alexander wanted some clarification on whether the fine-tuning argument works if you pose it a little more formally.

Suppose that handwavy/intuitively, the constant could be anywhere between -infinity and infinity. But in order for hydrogen to exist, the constant must be between 0 and 10. And in order for life to exist, the constant must be between 3.790 and 3.792.

Now you could say something like "of all the universes where hydrogen exists, only 1/5000 have life, this is Bayesian evidence for fine-tuning" and this would be fine. As far as I know, Bulldog could say this - the hydrogen example is made up, but probably something like it is true.

Would this rescue fine-tuning? And if so, can't you then say "and in fact, since the set of universes with hydrogen is a tiny subset of all possible universes, the real value must be incomparably larger"? -u/ScottAlexander

Now I wrote my own response, but I, u/kaa-the-wise, and u/Express-Smoke1820 basically said the same thing, and they were less long-winded, so I cede the floor.

No, this very much would not be fine. It is inconceivable what the range of "all the universes where hydrogen exists" is, and what the probability measure on them is. 1/5000 is pulled out of thin air and is just a shiny mathematical wrapper for a feeling.

…

To give this proper grounding, you have to define both the space and the probability measure, also inevitably defining the limits to the model you are offering, because these can serve as the basis for potential disagreement.

I further suspect that one motivation of not doing so is to make one less vulnerable to such disagreement. —u/kaa-the-wise

A sharp response in both senses. Perhaps a bit harsh at the end, the mistake is subtle, after all, but absolutely correct. u/Express-Smoke1820 concurs, with a slightly different flavor:

The dangerous part here is “intuitively, it’s obvious that at the very least it could be anywhere between 0 and 10.”

“Anywhere between 0 and 10” is a well-defined probability space, but there is no non-arbitrary reason to draw the line at 10, or 100, or 30 trillion, or to restrict the set to rational or even to real numbers. So, to the question “what are the odds we get a value between 3.790 and 3.792 out of all possible values,” it might seem like the answer is a hard 0 for any finite interval, but it’s actually not even that - “values” simply doesn’t refer to a measurable set (from the article: “Real numbers? Positive only? Complex? We have no idea, and exactly one example of a universe to guess from.”). This is a problem, not just because it sets out no discernible sample space, but also because it defines no σ-algebra we could use to work with any such space. —u/Express-Smoke1820

They also raise a further objection, which I found thoughtful and interesting.

And (though this goes beyond the scope of MEMR’s article) I’d say there is an additional dimension of uncertainty - the set of natural laws other than the ones we’ve got which God could have created, but didn’t. These should also be considered in any explanation of why God chose these laws and tuned them. But the space of alternative sets of laws of physics is so far beyond quantifiable that I wouldn’t even know where to begin. —u/Express-Smoke1820

Me again now:

I think the valid version would have to look something like “according to this theory of the multiverse, we have universes generated with constants in this range. Life only exists in this tiny subset! Wow, how lucky did we get to end up in this one? Hmm…”

That’s a valid argument, conditioned on the theory of the multiverse being right. But we don’t have that.

Requiring This Much Math Stifles Thought

This objection was touched on earlier, but raised by both Bentham’s Bulldog and Scott Alexander. It deserves an answer as a legitimate concern.

[A]s a less legible objection, I'm nervous that work like this is taking something that's obviously true - it's really weird that these constants are in the tiny realm suitable for life - and merely raising enough objections that you're not allowed to talk about it unless you have ten years of postgraduate math. I think in order to convince me, you would need to find a way to intuitively show me how this actually defuses the surprisingness rather than just lets you pounce on anyone who tries to discuss it rigorously. —u/ScottAlexander

I do slightly feel like I'm getting Eulered. —Bentham’s Bulldog

I had to look this one up. From context, it sounds like “Eulering” someone is throwing math at them they don’t understand until they give up, but I my guess is it probably comes from Scott’s Slate Star Codex article Getting Eulered.

Diderot was quite the clever debater, and soon this scandalous new atheism thing was the talk of St. Petersburg. This offended reigning monarch Catherine the Great…

[S]o she asked legendary mathematician Leonhard Euler to publicly debunk and humiliate Diderot. Euler said, in a tone of absolute conviction: “Monsieur, (a+b^n)/n = x, therefore, God exists! What is your response to that?” and Diderot, “for whom algebra was like Chinese”, had no response. Thus was he publicly humiliated, all the Russian Christians got an excuse to believe what they had wanted to believe anyway, and Diderot left in a huff. —Scott Alexander

Scott has some very nuanced views about this in his article. He tries to thread the needle of not letting people throw math at you to shut you up, vs. acknowledging that math is sometimes very important to make good arguments.

I suppose I have two thoughts on this. The first is that this is a valid criticism of my article! I did use a lot of math, and I did so intentionally. Now I did the best I could to make the article didactic, to really teach people something important and nontrivial about measure theory. While of course still mixing in the Latex and not writing a textbook. But nevertheless, if you have to read a lecture on measure theory to understand it, is it still a good argument, or just Eulering?

I am sympathetic to the notion of being Eulered, especially when it happens to me, but except in egregious cases, I think I reject this concept outright unless used in bad faith. Sometimes arguments just are complicated and technical, and the “problem of measure” is not coherently debatable without some understanding of measure. Especially when you invoke math in your argument, you must, at some point, be ready to do the math, and do it right.

It is, however, incumbent upon those who know more math to attempt to teach it to those who don’t. Especially when making a complicated technical argument.

Just as a note, I’m trying not to come off too elitist here so I’ll say this: I love teaching! I sincerely try to do a good job at explaining the math, and you all have been very kind! Please, ask me for clarification, I will do my level best to explain any point I make at any level of sophistication you ask: five-year-old, postgraduate, whatever. I reserve the right to do much worse on the philosophy points, though, I’m already a little outside my lane here.

Is the Measure Problem a Fully General Dismissal?

Scott worries more along the same lines.

Also, isn't "there could be some unknown physical law causing this" a fully general dismissal for all possible evidence? Suppose that SETI detected a message saying "HI WE ARE ALIENS" in perfect English. It's always possible that there could be some unknown physical law that causes this to happen. Bayes' Theorem just says that the it's-really-aliens hypothesis called exactly this outcome, whereas "there are unknown physical laws" could cause approximately anything - and therefore we should greatly increase our confidence in aliens. In the same way, "someone is optimizing for life" predicts fine-tuning within this range, and "there are unknown physical laws", while they could explain this, could equally well explain any other value of constants.

This is more or less the same as Bentham’s Bulldog’s argument regarding aliens, tacos, etc. and has the same answer. We have computationally intractable but nevertheless well-defined answers to questions that work with known physics. It’s perfectly fine to use Bayes’ rule.

Other Nice Comments

There’s a whole field of philosophy called formal epistemology that studies how to apply rigorous probabilistic methods to philosophical questions like those raised by fine tuning arguments. They have developed tools to address problems you’re concerned with, they’re more sophisticated than you might think. —u/Curates

This is honestly just really interesting, and I knew nothing about this, but the Stanford Encyclopedia of Philosophy has an article on it.

Too much math for me here but I wonder if it boils down to what I think: When people say something like "If such and such constant was 0.0001 different, reality wouldn't work!" There's a buried premise that 0.0001 is 'small'. Small compared to what? The decimal point feels completely arbitrary here. We can dive in infinitely and stick the decimal point way further back, now the number can be hugely different and have no discernible impact.

I call this the Rough-Tuning Rebuttal.

But my physics isn't so hot either so maybe I'm wrong. —u/Ontheflodown

This is maybe not exactly right but it’s very close and very concise.

I’ve raised these same concerns in comments before as well, and I think there are even larger concerns.

1) Bertrand’s paradox. Even if you did have a well defined sigma algebra, its not clear how to pick a uniform prior (in the language of your post, you can have the sigma algebra, and the event space, but picking P is not necessarily easy)

2) Universes with different laws of physics, or uncomputable laws of physics, or individual universes which are too large to be measured make assigning a probability space impossible.

3) The space of all *logically possible* universes is way way way way way bigger than universes permissible with ZFC. It is not even clear if it makes sense to apply probability theory, which is built upon ZFC, to the larger universe of *all logically possible universes*. Logical possibility is unfathomably big. Im not even sure if “all logically possible universes” is coherent (modal logic notwithstanding) —SorenJ

To point 3) here, I think SorenJ is touching on a good point. We generally think of math as being unitary and absolute, but actually, our mathematics is formally based on a set of axioms called Zermelo–Fraenkel set theory, and if you include the axiom of choice, that makes ZFC. There are other formulations! ZF, for example (no axiom of choice). I think this is important to remember, perhaps not so much for its utility, but as perhaps a reminder when you stretch your intuition too far that the space of everything possible is unfathomably vast.

To point 1), Bertrand’s paradox is just extremely interesting, a classic paradox in probability, and I can’t believe I didn’t talk about it in my article, so I’ll mention it briefly here.

Let’s say you have an equilateral triangle inscribed in a circle. You choose a random chord from that circle, what’s the probability it’s longer than the side length of the triangle, P(>△)?

As it turns out, this problem has no answer—it depends how you construct the chord.

• Random Endpoints: You could choose two random points on a circle and draw a chord between them.

  • If you do this, then P(>△) = 1/3.

• Random Radial Point: You could cast a ray from the center to the edge, and choose a random point on this radial line, drawing the chord perpendicular to the radius at this point.

  • If you do this, P(>△) = 1/2.

• Random Midpoint: Choose a random point in the circle, and draw a chord with that as its midpoint.

  • If you do this, P(>△) = 1/4

Which is right? It depends on your construction! This doesn’t say anything we don’t already know, it’s just the most famous illustration of why defining the problem correctly is very important if your probability is to mean something.

On the Self-Indication Assumption

Onid in the comments had some very interesting thoughts about the self-indication assumption. He’s currently writing an article about it, so I won’t preempt him here, but I’m looking forward to it!

On Being a Chatbot

A commenter on reddit, with a since-deleted comment, asked me whether I had prompted a chatbot to imitate the style of Scott Alexander for my previous article. No, I did not, and that’s high praise, thank you!

But in the interests of transparency, since I do frequently use LLMs as a writing aid, I thought I’d paraphrase my response below. Essentially, my strategy with LLMs is to try to leverage their strengths and avoid their weaknesses.

Strengths

• Deep Research for literature review is just amazing and I can’t recommend it enough. For deeply technical subjects it is no substitute for reading the papers, but for something like this article? Excellent substitute.

• Beta Reader: I find LLMs incredibly helpful for essentially beta-reading my posts. I’ll prompt it with “Hey, can you read this for clarity and grammar, and please point out any logical holes in my arguments?” It’s shockingly good at this. In the previous essay, it caught a couple errors in my math, some flaws with how I was arguing, etc.

• General Explanation: Let’s say I’m having trouble with the Vitali set construction (and I absolutely was), asking LLMs to break it down for you is just so helpful, as they tailor the explanation to your level and background (if you tell them to). You should, of course, verify with the formal definition after you understand, but formal definitions are usually light on the intuition, and LLMs really shine here.

• Outlining: My previous post was looooong. I sometimes have a lot of thoughts that I’ll write down in a notepad as a stream of consciousness, then ask the LLM to “organize my thoughts into a coherent and flowing structural outline.”

Weaknesses

• Voice: Except for the occasional really good one-liner, I hate how LLMs write on a sentence-to-sentence level. They often write like an undergraduate five-paragraph essay that’s been sanitized by an HR department—Gemini ends every question I ask it with a section literally entitled “Why This Matters” and must have a profound wrap-up section for every essay. I am so glad I learned to write before these things came out, or I would almost definitely converge on this writing style by sheer inertia, and I hate it.

• Novel math/arguments/ideas: Ask them to regurgitate a proof, or make “the strongest version of standard argument,” and they’re great. Ask them to solve a new math problem, one that’s kinda open-ended? I’ve done this at my work a bunch and they’re just terrible at recognizing when a problem is intractable, stopping, re-evaluating, and trying something else. They have no sense of the ground truth, and are very prone to charge off in a single direction and iterate by random walk. Hopefully this gets fixed in the future, but for now, don’t use them for this.

I have a lot of other thoughts on what LLMs are good for and not good for, but remember not to let them substitute for your own thought! They’re a powerful force multiplier, but they’re also a mirror, and you must be careful not to fall into it.

Conclusion

With all this said, I suspect this will be my last post on the measure problem for at least a little while. So to all of you who subscribed to my blog thinking it was a measure theory textbook, I apologize, it was a bait and switch! Instead, you’re likely to get more posts like this one:

EDIT: Addendum, some math errors from me in the original post.

It was inevitable that an actual mathematician, or at least someone better than I am, show up and find my errors.

Small quibble: I don't understand your mention of AC. If we take on not-AC we can prove that all sets are measurable.

Here I was using the axiom of choice to draw an analogy between unintuitive consequences and simpler principles that imply them. He is right that I am implicitly assuming the axiom of choice in my examples and that I don’t have to do this, though choosing or not choosing the axiom of choice for your universe model is of course a choice. An axiom of axiom of choice?

Bigger quibbles: What do you mean P(God|measurable sets)? The set of measurable sets is not an element of the algebra, it is itself the algebra.

This is sloppy notation on my part and intended as rhetorical flourish, not a formal statement of probability. The set of measurable sets is absolutely just the algebra— this should say something like P(God | “the surprising fact that measurable sets exist and physics occurs on them”), which is completely undefined. I acknowledge the irony of imprecision in a post about making imprecise math statements.

You claim there's no natural choice of algebra. The natural choice of algebra is the Borel algebra over a Euclidean representation of the parameter space.

Elijah suggests the Borel algebra over a Euclidean representation of a fine-tuning parameter is a natural choice. I agree this is true if the sample space (Ω) is well-defined. My core argument, however, is that it isn’t, and so any subsequent choice of σ-algebra is of course also not well-defined.

Then you say that the measure theorist says the probability of picking any particular value from a continuum is zero. This is wrong. Singletons are Borel in R^n. Zero-inflated priors, for example, are bread and butter Bayesian statistics

—Elijah Spiegel

Elijah also points out that the probability of picking a single value from a continuum is not always zero. This is true, you can have “point masses” with nonzero probability and they don’t break your theory and are allowed in Borel sets. The “zero-inflated prior” is essentially having an interval of possible effect sizes and tacking on {0} to that interval to represent “no effect” with nonzero probability. My original statement implicitly assumed an absolutely continuous probability measure. One could mathematically assign a point mass to “our universe's constants,” but of course why would you do that. Nevertheless, he is correct.

Generally speaking, he is right about his small and large quibbles. I think his points show the danger in trying to essentially write a measure theory textbook as a blog post, I have some elisions and errors between the concept I'm pointing to and a fully correct definition, as well as me straight-up forgetting about point masses being allowed in Borel sets, all of which is entirely my fault. This is my bad.

However, I don’t think any of the points he makes are fatal to the overall argument that your sample space of possible physics is just not well-defined. Instead, it just illustrates how hard this problem is in general.

Let me briefly introduce the argument, and we’ll get to the specifics later. The argument holds that the specific values of physical constants, like the cosmological constant or the strength of gravity, fall in a very narrow range necessary for life to exist (they are “fine-tuned”). This is quite surprising, and strong evidence for a creator deity fine-tuner.

In The Fine-Tuning Argument Simply Works, he claims to have examined all objections to fine-tuning and found them wanting. He concludes that fine-tuning provides “ridiculously surprising and crazy strong evidence for God.” He claims that “there are lots of objections to [fine-tuning] but none of them work.” His arguments are wrong, I will demonstrate why, and we will learn a little measure theory.1

Actually, since the whole theme of this article is precision, that doing “math via English” is perilous, let me be more clear.

Bentham’s Bulldog argues that we should believe God exists, based on the specific values of natural constants (e.g. mass of the Higgs boson, speed of light, cosmological constant). He holds that the values of these constants are fine-tuned to support life in a surprising fashion, and that the best explanation for this is that a designer (i.e. God) fine-tuned them to do so. He deploys Bayesian probability (also called “subjective” probability, inverse probability, etc.) in support of this argument. He is not the only one, here is the Stanford Encyclopedia of Philosophy’s entry detailing proponents and formulations of this argument, which are more or less identical.2

Bentham’s Bulldog makes invalid arguments. The fine-tuning argument as stated in terms of Bayesian inference should neither increase nor decrease your belief in a creator, because it is mathematically meaningless. Not difficult, not imprecise—it is meaningless. It can only be rescued by abandoning the rigor of Bayesian inference (which is called something nebulous like “epistemic probability” in the Encyclopedia).

I make no claims about whether God exists.3 If you want theology, you should talk to a priest. Instead, let’s talk about a pervasive problem in (popular and professional) philosophy: attempting to do mathematics through English prose.4

Fair Warning: This post has a lot of math5 and a lot of footnotes, most of which are snippy comments, but some of which are interesting.6

1. A Tale of Two Urns

Problem 1: Well-Defined Balls in Urns

Let's start by doing Bayesian inference correctly, on a well-defined problem. I will also introduce some measure theoretic language to aid us.7

Problem Statement: You're presented with an urn containing red and blue balls. I tell you it's either type “even” (50% red, 50% blue) or type “uneven” (10% red, 90% blue). You draw one red ball. Which type of urn is it more likely to be, even or uneven?

This has a clean answer and a good method of inference. We apply Bayes’ theorem with equal priors.

Bayes' theorem is the following statement:

In English, the probability of A occurring, conditional on B being true P(A|B), is equal to the probability of B occurring, conditional on A being true P(B|A), times the ratio of the independent probabilities P(A)/P(B).8

Here's the correct Bayesian calculation:

• P(even) = P(uneven) = 1/2 (no prior information favoring either)

• P(red|even) = 1/2 (by definition)

• P(red|uneven) = 1/10 (by definition)

• P(red) = P(red|even) × P(even) + P(red|uneven) × P(uneven) = 3/10

We now calculate the Bayes Factor K for these hypotheses. The Bayes factor is a ratio of the likelihoods of observing the data under two competing statistical models, and represents our “update” to our beliefs about the problem. It has the following definition (applied to our problem).

For our scenario, we have K = (0.5 / 0.1) = 5.

Our prior odds, before observing the data, are:

And our posterior odds, after observing the data, are updated by the Bayes factor:

This Bayes factor of 5 means observing a red ball increases the odds of “even” versus “uneven” by a factor of 5. With equal prior odds (1:1), we now have 5:1 posterior odds, giving us P(even|red) = 5/6 and P(uneven|red) = 1/6.

Problem 2: Ill-Posed Balls in Urns

Now let’s do it again, on a slightly different version of the problem.

Problem Statement: You're presented with an urn. I tell you nothing about this urn. You reach in, and draw one red ball. What is the probability that it contains 50% red balls?

If you feel uncomfortable, good. You should. This isn't a hard problem or an imprecise problem. It's not a problem at all. It's a grammatically correct but meaningless arrangement of mathematical symbols, it is ill-posed. But isn’t it surprising that there’s an urn there with a red ball? Does that point to a deeper explanation? Perhaps so! But then again, perhaps not, and you definitely cannot assign probabilities to it without knowing what mixtures are possible or their distribution.

At this point, you may already see the shape of my objection, if so, feel free to stop here, my objection is indeed that fine-tuning is “Problem 2” dressed as “Problem 1.” For the rest of you, you were promised measure theory (and it’s odd that, having been promised measure theory in a blog post, you are still reading). Let’s go.

Note: Feel free to skip this somewhat technical section on measure theory, or just skim it. Frankly, I wrote this section as much to reify my own shaky understanding of it as to explain it to you, dear readers. We resume the polemic in section 3.

2. What is Measure Theory and Why do I care?

Probability: The Threefold Path

A meaningful statement about the probability of some event requires three things. A sample space Ω, a σ-algebra Σ, and a probability measure P. Together, (Ω, Σ) form our probability space and the measure P allows us to make likelihood calculations on that space.

The sample space is an intuitive concept—the set of possible outcomes. The σ-algebra Σ is the set of collections of outcomes to which we can reasonably talk about a probability. For example, the σ-algebra Σ necessarily includes the entire sample space Ω, because P(Ω) = 1, as well as the empty set ∅, because P(nothing happens) = 0. It also includes any subset of Ω we’d like to assign a probability.

We’ll set aside measure for the moment, and use a six-sided die as a concrete example to better understand these concepts.

The sample space is fairly obvious. We require a set of “events” that can occur, to which we want to assign probabilities. Crucially, we must be able to define this set. For a fair d6, it’s pretty simple, it can roll any of Ω = {1, 2, 3, 4, 5, 6}.

The σ-algebra Σ is also reasonably straightforward. For our d6, and indeed for any finite sample space, the power set (the set of all sets) usually suffices. We can, for example, reasonably assign probability to the set {2, 4, 6}, the probability of an even roll on a d6.

For more complex, or infinite sets, a standard σ-algebra on a sample space is the Borel set B, which is constructed from subsets (specifically simple open or closed intervals) of the sample space Ω by iterating (countably many times) a few well-behaved operations: union, intersection, and complement.

Looking ahead, when we start assigning probability densities to sets, we will want “nice” sets where we don’t run into any contradictions—Borel sets are precisely these “nice” sets.

Finally, of course, we require a measure.

The Problem of Measure

Probability requires measure theory to be well-defined. I first want to emphasize that “measuring,” that is, assigning a number to the intuitive notion of the “size” of some collection of objects, is not trivial, in fact most sets are impossible to measure. Not difficult to measure, impossible to measure. This “problem of measure”9 illustrates why we ought to care about some level of rigor in probability.

Now generally speaking, the basic strategy of measure theory is to start with an object that we know we can measure, and build progressively more complex theory on this backbone. Usually, we take a collection of open intervals which we can measure (An interval [a,b] on the real line has length b-a), minimally cover our target set with this collection, and then add up those intervals. That’s the Lebesgue measure. But that doesn’t always work, in fact, it fails on most sets, and works only on so-called “measurable sets.”

Personally, I always feel that a concept of what something is feels most real when I see an example of how it can fail. So to clarify the idea of a measurable set, let’s look at the canonical example of an unmeasurable set, the Vitali set.

The Vitali set is built like this:

1. Take the rational numbers (Q) and shift them by a real number r. This creates a coset of numbers V_r = Q+r.

2. Do this for every real number r to get a collection of cosets, some of which may be identical (V_r1 = V_r2).

3. Now from each unique coset V_r you pick one number v in [0,1], forming a set V.

   Edit: I made a slight error in the original draft here, I forgot to specify v ∈ [0,1].10

The collection of those numbers V is the Vitali set.

Consider: if you shift V by any rational number V + q, you get a completely non-overlapping copy. Yet all these shifted copies, taken together, perfectly tile the interval [0,1]. Ok, well why isn’t it measurable?

Let’s try to assign the Vitali set a measure m. Since shifting it left and right shouldn't change its size (translation invariance), every shifted copy also has measure m. But then:

• If m = 0, then [0,1] is covered by countably many measure-zero sets, so [0,1] has measure 0. Contradiction: we know [0,1] has measure 1.

• If m > 0, then countably many copies each of measure m sum to infinity. But they all fit inside [0,1], which has measure 1. Impossible.

Neither the m = 0, nor m > 0 options work for the measure of the Vitali set, so we declare it non-measurable. This type of set is actually not rare at all. There are, in fact, infinitely many classes of unmeasurable sets, and indeed there are more unmeasurable sets than measurable sets.

Remarkably, despite the fact that almost all sets are unmeasurable, the sets we tend to encounter are measurable:

• All open and closed sets

• All countable sets

• Countable unions and intersections of measurable sets

• Complements of measurable sets

The Probability Measure

You now understand why “measuring” probability isn't trivial. Indeed, the measure must, you know, exist in order to make any meaningful statements about its size or the probability of God.

The probability measure P, rigorously speaking, satisfies all the normal measure properties (non-negative, etc.), but is also normalized, meaning that P applied to our whole sample space Ω is 1. When Bentham's Bulldog assigns probabilities to universe configurations, he's implicitly claiming a measure exists on an undefined space. Now recall the existence, indeed the preponderance, of non-measurable sets.

At the very least, right now you should be doubting whether your intuition, built from interacting with physical objects in 3D space, is especially reliable in abstract set theory (probability, unfortunately, is abstract set theory).

It’s odd, don’t you think, that if you pick a random subset of the real line, it is guaranteed to be unmeasurable with probability 1? Yet all the sets we use for, well, anything, are always measurable? Is P(God|measurable sets) therefore high?

Well, no. This is simply the natural consequence of the axiom of choice. There is no need to invoke an explanation beyond the axiom—it’s actually a single principle with very unintuitive consequences.11

Probability Density and Cumulative Density Functions

I also ought to introduce the concepts of probability density and cumulative density functions. They’re quite intuitive, so I’ll only bother for continuous random variables:

1. A probability density function (pdf) is a function f(x) where the probability of a continuous random variable X having the value x is proportional to f(x), and f(x) is normalized.

2. The cumulative density function is the probability that a continuous random variable X takes a value less than or equal to x, written as P(X ≤ x). It’s the integral of f(x).

There’s one more thing I want to note before ending the deep theory—you see that term up there, P(God|measurable sets)? This is the first time in this article where that P has a well-defined meaning. That P is the probability measure, finitely additive, normalized, etc, defined over a measurable sigma algebra on some sample space.

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3. Your Intuition is Terrible: The Cantor Set

A central idea of the fine-tuning argument is that the physical constants that describe our world have a low probability of being chosen randomly—they are fine-tuned. Indeed, Bentham’s Bulldog says

Because there are at least 10^120 values of the cosmological constant alone, and nothing special, conditional on naturalism, about the finely-tuned one that it happens to be, the odds of fine-tuning by chance of just the cosmological constant are 1/10^120.

Not so fast, says the measure theorist, the probability of selecting any particular number from a continuous set is zero, only intervals are meaningful!12

Now a clever philosopher who’s a bit knowledgeable about physics might counter “But wait! The Planck scale provides a natural discretization, so we’re not choosing from a continuous set, but picking a value from a set of discrete values!”

Oh no, responds the measure theorist. A discrete set of evenly spaced real values is in one-to-one correspondence with the integers, and there is no uniform probability distribution over the integers.13 You literally cannot select a random value. You must choose a finite set of them first, and your choice is arbitrary.

And so on.

The Cantor Set and The Devil’s Staircase

Allow me to show you a very cool set, a very simple fractal actually. It’s cool, it helps show why you can’t trust your intuition about measurable sets, and it’s cool.

1. Begin with the interval [0,1] on the real line

2. Remove the middle third, leaving [0, 1/3] ∪ [2/3, 1].

3. From each remaining piece, remove its middle third.

4. Repeat forever.

What's left—the Cantor set—has measure zero (we removed intervals totaling length 1/3 + 2/9 + 4/27 + … = 1). Yet it contains uncountably many points, as many as the original interval. It is dust that takes up no space but is everywhere dense in itself.14 We can even define valid probability distributions concentrated entirely on this measure-zero dust! I don’t know about you, but this is not intuitive to me at all.

4. Fine-Tuning and Why I am Annoyed

Okay, we finally get to the meat of the essay. What is the fine-tuning argument, as stated by Bentham’s Bulldog?15

I’ll actually let him state a lot of it, because he does a good job laying out the premises.

The Fine-Tuning Argument According to Bentham’s Bulldog

Premise 1: “The parameters of the universe are finely tuned in the sense that they fall in a very narrow range needed for life.”

Note that he distinguishes three types of fine tuning. Four, actually, if you include his mention of Robin Collins’ work on fine-tuning for discoverability.

1. Nomological Fine-Tuning (the laws themselves): Most conceivable physical laws would produce either trivial patterns or random chaos. Laws capable of generating complex, stable structures represent a vanishingly small subset of logical possibilities.

2. Constant Fine-Tuning: Physical constants fall in narrow life-permitting ranges:

   • Cosmological constant: fine-tuned to 1 part in 10^120

   • Strong nuclear force, electromagnetic force, quark masses, etc. all require precise values for complex structures.

3. Initial Condition Fine-Tuning: The universe's low-entropy starting state represents 1 part in 10^10^123 of possible configurations.

4. Fine-tuning for discoverability (Robin Collins): The values of physical constants are optimized for scientific discovery by humans.16

Premise 2: This fine-tuning is highly improbable given naturalism/atheism.

“If there is a God, then the universe being finely tuned makes sense. God would want to create a universe capable of sustaining life.”

Premise 3: This fine-tuning is expected given theism (God would create life-friendly conditions).

“The odds of fine-tuning conditional on naturalism are extremely low.” By naturalism, let’s just take this to mean not-God.

Note that this is a probabilistic argument, and that Bentham’s Bulldog explicitly endorses Bayesian epistemology, so I don’t think I’m being uncharitable as reading this as Bayes’ theorem here.

Trying to Formalize Fine-Tuning, and Failing

If I may, let me just restate the argument, as I understand it, with math. His argument is precisely a statement that the Bayes Factor for fine-tuning is large under theism and not naturalism.

We expand via Bayes’ theorem to get our posterior odds…

So indeed, were this calculation well-defined, we should update our beliefs in favor of theism!

But wait! Stop right there! Can we do this at all? Primed with that interminable section on measure theory above, we answer: no, we cannot. Remember that probability has a well-defined meaning; that P symbol is a measure, not the English phrase “the probability of”!

Now, the priors, P(God) and P(¬God), are perfectly well-defined—they're your initial degrees of belief, and you can assign what you like. The problem is the “likelihoods,” particularly P(finetuning∣¬God). To define this likelihood—the probability of fine-tuning given naturalism—you must specify a probability space. That means:

• A sample space Ω (all possible values the constants could take)

  • Real numbers? Positive only? Complex? We have no idea, and exactly one example of a universe to guess from.

• A σ-algebra Σ (which subsets we can measure)

  • Same problem.

• A probability measure P on Σ

  • If the constant can be any real number, there's no uniform distribution over ℝ. If we restrict to [0, M] for some M, why that M? Why uniform rather than log-uniform or Gaussian?

We have none of these. The Bayes Factor derived from it is meaningless.

Now there’s a possible objection to this. If P(fine-tuning∣¬God) = 0 and P(fine-tuning∣God) > 0, then the Bayes Factor K would be infinite.

But by construction, you get P(fine-tuning∣¬God) = 0 due to selecting a value from a set of zero measure (in this case, selecting a value from a continuous range). By the premise of this whole argument, then God too must select, whether randomly or not, from a set of zero measure. The probability of God selecting any particular value, P(fine-tuning∣God), is therefore also zero17 and the Bayes Factor is undefined.

An Aside on Theology

This discussion of math is entirely apart from the problem of whether P(fine tuning|God) has any theological meaning at all, as it requires us to know what God would do given some motivation. Bentham’s Bulldog argues that fine-tuning implies an omnipotent selector, but shouldn’t an all-powerful, all-math-loving being want his universe to emerge naturally from a deep symmetry?

From that perspective, our apparently 'fine-tuned' and perhaps 'contrived-looking' universe might even be considered less likely under such a conception of God. To assert a high P(fine-tuning|God) is not a neutral step; it embeds significant, debatable, and ultimately non-quantifiable theological assumptions about divine motivations and aesthetic preferences.

Should we therefore assign P(fine tuning|God) = 0 because our universe appears fine-tuned and ugly? I have no idea, neither do you, and stop assigning probabilities to theologically opaque propositions as if they were well-defined events.

Briefly, the Self-Indication Assumption

Let's briefly talk about what I think is a very similar mistake in his other writing. I won’t go into too much detail, because it’s a bit outside my bailiwick, but I think it’s illustrative of the same mistake. Bentham’s Bulldog is fond of the self-indication assumption (SIA), which is a somewhat technical argument that you should condition your likelihoods on the fact that you exist.

Specifically, he argues for the “Unrestricted” version of SIA (USIA), which cares about absolute numbers of people, not just the share of possible people who exist.

1. God, being perfectly good and omnipotent, would create the maximum possible number of people. He argues this number is “unsetly many” (too large to be a mathematical set) - at minimum Beth 2, but likely much larger.

2. Even the most generous naturalistic scenarios (infinite multiverse, infinite universe) can only produce aleph-null people - the smallest infinity.

3. Beth 2 is infinitely larger than aleph-null. There are vastly more ways for you to exist if Beth 2+ people exist than if only aleph-null exist.

Your existence is therefore infinitely more likely under theism than naturalism. Even if you assign theism very low priors, an infinite Bayes factor overwhelms any finite prior probability ratio.

He writes several thousand words of increasingly baroque thought experiments and arguments by analogy defending this position, a lot of which appear convincing, but… I mean, do I have to say it? Bentham's Bulldog sees SIA predicting absurdities under naturalism and concludes theism. I see SIA predicting absurdities under ill-defined probability distributions and conclude that having well-defined probability distributions is important to avoid concluding nonsense. It's Problem 2 dressed as Problem 1, again.

I think the similarity of these two arguments reveals the deeper flaw in his arguments generally—his tendency to treat probability statements as meaningful just because they're grammatically correct.18

5. Bentham’s Bulldog has objections!

But let it not be said that Bentham’s Bulldog is a casual writer, he thinks through the possible objections to his argument quite thoroughly! I argue he does so incorrectly, but that’s rather the topic of debate, isn’t it? I’ll go through my major gripes with his objections.

“Objective” vs. “Subjective” Probabilities and Arguments by Analogy

He first attempts to head off objections like mine by distinguishing between “objective chances and subjective chances.” He essentially identifies objective probability as the frequentist interpretation of long-run frequency in many trials, which he rightly dismisses as not useful with a single universe to extrapolate from.

The “subjective chance” he identifies is pure Bayesian.

To determine subjective chances, you should imagine a rational agent assigning probabilities to the possible outcomes before they know the actual outcomes. In the case of fine-tuning, for instance, a person who didn’t yet know what values the laws fell into would think there was a super small chance it would fall in the finely-tuned range because it could fall in any range.19

Unfortunately, Bayesian probability theory requires well-defined probability spaces and measures just as much as frequentism. This does not help him. Indeed his analogies reveal the problem with his framing of fine tuning, and why I hate arguments by analogy in general.

[I]magine that we discover that several billions years ago someone built a special machine. This machine will spit out a number, determined by some undiscovered laws of physics that can’t be different. The number could be any number from 1 to 100 billion.

Imagine that we know that the person who made the machine just loves the number 6,853… Now imagine that the machine spits out the number 6,853.

Clearly, this should give us some evidence that he rigged the machine. Even though the machine, on the hypothesis that it’s not rigged, is wholly determined by the laws of physics, it’s unlikely that the laws of physics would determine 6,853—there’s nothing special about it. In contrast, he’d be much likelier to pick that number. Therefore, if the number is picked, it favors the hypothesis that it’s rigged.

This analogy has a well-defined sample space, algebra, and measure. This is absolutely perfect Bayesian reasoning, and good job on him. This is “Problem 1” from our red balls in urns scenario. But fine-tuning is “Problem 2.”

Arguments from analogy fall into this trap continuously and I see it everywhere. Your analogy has to actually map onto the problem you’re analogizing, or it doesn’t work! A → B does not imply C → D if you show only that A is similar to C and B to D! You have to show the arrow, and that’s the hard part and the entire discipline category theory.20

Deeper Laws and Monstrous Moonshine

Bentham’s Bulldog fairly quickly dismisses the possible objection to the fine-tuning argument that there may be deeper, more fundamental laws that we don’t know about yet that predict fine-tuning. This is very closely related to the concepts in physics of naturalness (essentially just another name for fine-tuning of constants,21 like the Higgs mass and the cosmological constant) and the hierarchy problem (the observed vast difference in the “strengths” of the weak force and gravity).

Now of course, in his dismissal he gives an analogy about royal flushes in poker that have well-defined probability distributions, which is wrong for the same reasons as above. But additionally, he more or less argues that the Deeper Laws objection just pushes the problem back a level.

There could be more fundamental laws resulting in any specific arrangement. It’s thus monstrously improbable that they’d produce a finely-tuned arrangement rather than the overwhelmingly more populous non-finely-tuned arrangements.

This is where he and I just irreconcilably disagree.

Maybe it’s just because I’ve been reading so much of his stuff writing this article, but let’s look at an analogy.22

The Classification of Finite Simple Groups

Group theory is the area of mathematics most deeply concerned with symmetry, and therefore very relevant for physics. The Classification of Finite Simple Groups is a triumph of modern mathematics, and it says that we can uniquely classify every finite, simple group (group building blocks that are not decomposable into simpler groups) into one of of four classes, the first three of which have infinitely many members.

There are the cyclic groups, the alternating groups, Lie groups, and exactly twenty-six exceptions, called the sporadic groups. Now I want you to imagine you know nothing about math, but lots about fine-tuning, and listen to this argument.

There are these four classes of groups, right?

The cyclic groups are simple to understand, they're derived from the symmetries of rotating (prime) polygons. A pentagon has 5 rotational symmetries, so the cyclic group C₅ has exactly 5 elements.

The alternating groups are the symmetry groups of even permutations of objects. If you have 5 objects, there are 5!/2 = 60 ways to rearrange them using an even number of swaps, so A₅ has exactly 60 elements.

Lie groups are concerned with continuous symmetries, like rotations in 3D space—SO(3) has dimension 3 because you need 3 numbers to specify a rotation in 3D. They each have dimensions and element counts that flow directly from the objects they describe.

Then there's the sporadic groups. There's subclasses here, actually, where some of them have good, less simple, but intuitive explanations. And then there’s the monster.

Monstrous Moonshine

The monster group has exactly 808,​017,​424,​794,​512,​875,​886,​459,​904,​961,​710,​757,​005,​754,​368,​000,​000,​000 elements, and the smallest faithful (complex) representation lives in a 196,883 dimensional vector space.

In a very real sense, this is more “fine-tuned” and less natural than any physical constant—this is one of the fundamental, irreducible building blocks of group theory, it’s exact to the last digit, and it just happens to be of order 8 x 10^53. Most of the other groups we usually care about have orders of, like, six.23

Is this natural? Well in one sense, yes, absolutely. This is a direct consequence of the definition of a finite simple group. If you have a collection of objects with a group multiplication operation, and you require them to be associative, have an identity element, and have an inverse, the monster group is fully implied by this definition. While true, however, this explanation isn’t all that satisfying.

In another sense, of course, this is monstrously unnatural, and I want a better explanation, one that makes the connection seem inevitable. Why do these unbelievably simple axioms necessarily imply that there are exactly twenty six groups out of an infinite set that don’t fit a good pattern, and one of them has exactly ~8 x 10^53 elements?

Well, I could make a very convincing fine-tuning argument out of this. After all, what’s the probability that such a thing would occur by chance? Small, surely, unless God decreed it by fiat?24 One might even be tempted to calculate a Bayes factor…

Fortunately, mathematicians don’t do this. The fine-tuning of your theory is a defect, not a result, and in 1978 a really, really bizarre connection25 was found between the monster group and modular forms from the (seemingly) totally unrelated field of complex analysis, called “monstrous moonshine,” and it won a Fields medal.

What about the objection that this merely pushes the tuning back to the level of choosing axioms? But the claim that these undiscovered fundamental axioms would have to be “chosen” from a “vast space” of mostly non-life-permitting axioms is merely to re-assert the original probabilistic fallacy at a meta-level. The lesson from mathematical structures like the Monster group is precisely that profound specificity and apparent fine-tuning in the outcome can be an intrinsic, necessary, and non-contingent property of a system defined by (perhaps simple) fundamental rules.

So no, I do not accept his criticism that it is “monstrously improbable” when simple laws produce unbelievably unintuitive results. Mathematicians have the advantage of being able to define their axioms, and so we could see, a priori, that the monster was in fact natural somehow—it’s implicit in the group axioms. But physicists must work from the other direction, slowly digging our way down to the axioms of the universe. An appeal to theology is antithetical to the entire intellectual project of physics. It is not victory, but surrender.

To Conclude This Overlong Rant

Our physical models are fine-tuned. This is inarguably true, because they are fine-tuned by us, physicists, to match the empirical results we get from interacting with the world. The work of the physicist is to improve the map, and not to confuse it for the territory.

Let me be crystal clear.

• I am not arguing that a creator does not exist.

• I am not arguing that the universe itself doesn’t appear fine-tuned in a nomological, initial conditions, or universal constants sense.

• I am not arguing that this isn’t deeply puzzling and cries out for an explanation.

But none of this is Bayesian evidence for a creator. Nor is it evidence against one, because it is not Bayesian evidence at all. Instead, Bentham’s Bulldog appeals to probability theory's language and authority without accepting its constraints. He does math with English instead of math, and his arguments are wrong.

The honest position is this: we don't know how to assign probabilities to fundamental physics. We have exactly one universe, incomplete understanding of its laws, and no way to define the space of “possible physics.” The fine-tuning observation may be TRUE—the constants may indeed fall in a narrow life-permitting range for no good reason and there may indeed be a God who made it that way! But the argument that fine-tuning provides calculable evidence for God is not VALID—it relies on undefined probability distributions over undefined spaces.

With that, I rest my case. I hope you enjoyed it.

I recently tripped over Chesterton’s Fence while on vacation in Tobago. I also found lots of interesting information about the cassava root, which, as I have discovered, is both deadly and tasty!

A while back, I read the excellent article Tradition is Smarter than You from Scholar’s Stage, which introduced me to the concept of Chesterton’s Fence, the evolutionary biologist Joseph Heinrich (good Dwarkesh podcast episode with him), and the cassava root. I’m going to follow the article fairly closely, so I would be remiss if I didn’t tell you to just go read it, but the basic thesis is as follows:

1. Humans individually are clever apes

2. Humans collectively are very clever apes

3. Human collectives accumulate traditions that are sometimes adaptive

4. The reasoning/adaptivity of traditions is not always obvious

5. Beware the temptation to overweight #1 (your own intelligence) and underweight #2 (traditions that have no obvious purpose)

The article illustrated its point with a fable and an extended anecdote. From G.K. Chesterton’s 1929 book, The Thing: Why I Am a Catholic, Chesterton writes:

In the matter of reforming things, as distinct from deforming them, there is one plain and simple principle; a principle which will probably be called a paradox. There exists in such a case a certain institution or law; let us say, for the sake of simplicity, a fence or gate erected across a road. The more modern type of reformer goes gaily up to it and says, "I don't see the use of this; let us clear it away." To which the more intelligent type of reformer will do well to answer: "If you don't see the use of it, I certainly won't let you clear it away. Go away and think. Then, when you can come back and tell me that you do see the use of it, I may allow you to destroy it."

Joseph Heinrich provides an illustration of this principle with an anecdote about the cassava root, also known as manioc or yuca. Let’s take an aside for a moment and learn about cassava.

Cassava Plant Uses and Toxicity

The cassava plant produces large, starchy roots, can grow in bad soil and hot weather, and is extremely drought resistant. It is a staple crop for much of the world, used to make, among other things, tapioca pearls, West African Fufu, and “Provisions” here in Tobago (along with other starchy foods). It has two drawbacks — it goes bad very quickly without proper storage methods, and is very poisonous.

Cassava contains two glycoside toxins, linamarin and its methylated relative lotaustralin, stored in the cell vacuole of the root. Linamarin is the primary toxin, and consists of a glucose molecule bonded to, essentially, acetone + cyanide (a glycoside generally is a sugar bonded to a not-sugar).

When the vacuoles inside the cells are broken (by chewing the plant, for example), the linamarin is hydrolyzed by an enzyme in the cell wall into glucose and acetone cyanohydrin, which quickly breaks down into acetone and cyanide. Cyanide is exceptionally poisonous to basically everything, because it interferes directly with cellular respiration/the electron transport chain. Cellular respiration is the term for the process of converting glucose and oxygen into energy, and is extremely fundamental to cell operation, hence cyanide’s extreme toxicity. Cyanide is also apparently a major component of interstellar gas clouds (in cyanogen form), and is used to measure the temperature of those clouds. That’s a new one to me, and a great fun fact at certain types of parties.

Tradition is Smarter than Cassava Roots

Now the poison can be effectively removed from the cassava, but as Joseph Heinrich explains, this is a time-consuming and seemingly pointless process. I’ll quote him at length here, because it’s a great anecdote. Remember manioc = cassava, and I’ve bolded an importance sentence.

In the Americas, where manioc [(cassava)] was first domesticated, societies who have relied on bitter varieties for thousands of years show no evidence of chronic cyanide poisoning. In the Colombian Amazon, for example, indigenous Tukanoans use a multistep, multiday processing technique that involves scraping, grating, and finally washing the roots in order to separate the fiber, starch, and liquid. Once separated, the liquid is boiled into a beverage, but the fiber and starch must then sit for two more days, when they can then be baked and eaten.

…

However, despite their utility, one person would have a difficult time figuring out the detoxification technique. Consider the situation from the point of view of the children and adolescents who are learning the techniques. They would have rarely, if ever, seen anyone get cyanide poisoning, because the techniques work. And even if the processing was ineffective, such that cases of goiter (swollen necks) or neurological problems were common, it would still be hard to recognize the link between these chronic health issues and eating manioc. Most people would have eaten manioc for years with no apparent effects. Low cyanogenic varieties are typically boiled, but boiling alone is insufficient to prevent the chronic conditions for bitter varieties. Boiling does, however, remove or reduce the bitter taste and prevent the acute symptoms (e.g., diarrhea, stomach troubles, and vomiting).

…

Now consider what might result if a self-reliant Tukanoan mother decided to drop any seemingly unnecessary steps from the processing of her bitter manioc. She might critically examine the procedure handed down to her from earlier generations and conclude that the goal of the procedure is to remove the bitter taste. She might then experiment with alternative procedures by dropping some of the more labor-intensive or time-consuming steps. She’d find that with a shorter and much less labor-intensive process, she could remove the bitter taste. Adopting this easier protocol, she would have more time for other activities, like caring for her children. Of course, years or decades later her family would begin to develop the symptoms of chronic cyanide poisoning.

…

Thus, the unwillingness of this mother to take on faith the practices handed down to her from earlier generations would result in sickness and early death for members of her family. Individual learning does not pay here, and intuitions are misleading. The problem is that the steps in this procedure are causally opaque—an individual cannot readily infer their functions, interrelationships, or importance.

…

Perhaps it’s actually rather easy to individually figure out the detoxification steps for manioc? Fortunately, history has provided a test case. At the beginning of the seventeenth century, the Portuguese transported manioc from South America to West Africa for the first time. They did not, however, transport the age-old indigenous processing protocols or the underlying commitment to using those techniques. Because it is easy to plant and provides high yields in infertile or drought-prone areas, manioc spread rapidly across Africa and became a staple food for many populations. The processing techniques, however, were not readily or consistently regenerated. Even after hundreds of years, chronic cyanide poisoning remains a serious health problem in Africa. Detailed studies of local preparation techniques show that high levels of cyanide often remain and that many individuals carry low levels of cyanide in their blood or urine, which haven’t yet manifested in symptoms. In some places, there’s no processing at all, or sometimes the processing actually increases the cyanogenic content. On the positive side, some African groups have in fact culturally evolved effective processing techniques, but these techniques are spreading only slowly.

…

The point here is that cultural evolution is often much smarter than we are. Operating over generations as individuals unconsciously attend to and learn from more successful, prestigious, and healthier members of their communities, this evolutionary process generates cultural adaptations. Though these complex repertoires appear well designed to meet local challenges, they are not primarily the products of individuals applying causal models, rational thinking, or cost-benefit analyses. Often, most or all of the people skilled in deploying such adaptive practices do not understand how or why they work, or even that they “do” anything at all. Such complex adaptations can emerge precisely because natural selection has favored individuals who often place their faith in cultural inheritance—in the accumulated wisdom implicit in the practices and beliefs derived from their forbearers—over their own intuitions and personal experiences.1

Tradition is Smarter than My Family?

Now Heinrich is burying the lede here a little. Taste is actually a pretty reliable indicator of glycoside content in the raw plant.2 Karlton et al. in the linked study showed that among Malawian farmers with a history of cassava cultivation, their ability to predict cyanogenic content correlated strongly with their classification into “cool” (sweet) and “bitter,” and that tasting the tip of the root improved the toxicity prediction. The farmers achieved an r^2 of 0.67 when graphing subjective bitterness on a five-point scale vs. measured cyanogen content, a strong, but not perfect correlation. HOWEVER, cyanide is not a glycoside but a derivative, and it is the glycosides themselves, linamarin, etc. that are actually bitter, not the poisonous cyanide. So taste cannot be a perfect indicator.

All of this brings me to my vacation to Tobago, my wife’s home island, and hanging out (or “ limin’ ” in the local slang) with her family. Well, they served me a lot of cassava, which was very tasty.

While I was eating the cassava (which I noted was not in the form of a mash, grate, or any other product of extensive processing), I asked them about the cassava preparation. Do you have to extensively process the cassava root you eat? Is it poisonous if you don’t, and thus poisonous… right now?

Nope! they said. That’s mostly a myth, all you have to do is peel the cassava, wash it and boil it, and it’s completely fine! Don’t worry about it.

Well, that was a bit concerning. That’s not what the Scholar’s Stage article said at all! In fact, the article warned against more-or-less exactly this failure mode when preparing cassava! A Trinibagonian article I found called Boil Your Cassava Well actually falls exactly into the failure mode described by Heinrich, they:

1. Reference having two types of cassava,

2. Mention the need for careful processing of the bitter type, and

3. Get the processing wrong, suggesting that all you must do is thoroughly boil it!

   1. “It is essential to clean and boil cassava thoroughly before consuming to remove all traces of cyanide.”

According to my reading of the Joseph Heinrich passage, this is absolutely not the case, the intense multiday processes are necessary to reduce the cyanide level below the chronic toxicity level, not just the acute level where the bitter taste is probably a decent indicator.

Have we removed a critical Chesterton Fence? Is my family going to get chronic cyanide poisoning? To my slight embarrassment, I did look around for any signs of goiters (a swollen thyroid gland, a classic symptom of chronic cyanide poisoning) at the dinner table. I found none.

Tradition is Smarter than Me

As it turns out tradition is still smarter than me. Despite having just tried to defer to a very respectable tradition, I deferred to the wrong one.

According to the Trinidad and Tobago Ministry of Agriculture, Land & Fisheries,3 all cassava varietals grown in T&T (for the fresh market) are of the sweet variety. They are in fact completely safe to consume by just peeling and boiling.

Note that my family told me that the necessity of extensive processing was mostly a myth. This is not true in general, but it is true on Tobago, where only sweet varieties are grown for market. Tradition, even in the absence of a correct explanation, is nevertheless correct.4

It turns out that most of the cyanogens in cassava are contained in the skin. For the bitter varieties, enough of the cyanogens penetrate into the flesh of the tuber that simple peeling and boiling isn’t adequate, hence the intensive grating/mashing to expose all of the cyanide derivatives to water. Cyanide, being water soluble, can then be extracted by soaking. But for the lower-cyanogen-concentration sweet varieties, peeling really does remove most of the cyanogens, and for Tobagonian cuisine, that’s all you need!

This brings me to my second question I asked my family — do you have to discard the water? Cyanide is water soluble, and, logically speaking, if boiling in water solubilizes and thus removes the cyanide from the cassava, it must necessarily then poison the water…

Also no. Apparently in Tobago it’s used to make punch creamier. I went back and reread the Joseph Heinrich passage after learning this, and guess what? The indigenous societies studied by Heinrich do this too. I guess the hot water boils off all the cyanide gas? My intuition has served me very poorly indeed.

Tradition is Hard to Defer To

In my ever-expanding quest to avoid writing like an LLM, I will not end this essay with a profound moral lesson or a tight, just-so story. I confess I would love to, but the lovely Tobago beaches have turned my brain into boiled cassava, and I honestly have no idea what conclusions to draw from this, other than that deferring to tradition is a tricky and granular process.

Perhaps the most relevant question, however, is how do you know when to defer to tradition? How do you avoid tearing down the “good” Chesterton Fences, while removing the “bad” ones?

My guess is that most of the time, when tearing down a Fence, the first-order, obvious, immediate consequences dominate. But not always! Opaque chains of causality are the bane of accurate theory.

I had a few unstructured thoughts that I’ll write down for possible future investigation, related to the question “When are Chesterton Fences likely?”

1. Context change — Traditions are highly local, and imported traditions from a different context are less likely to be authentically adaptive. This is pretty much my error in this cassava saga.

2. Interaction effects — Chesterton’s Fences are often causally opaque because they rely on second-order consequences of actions. Complex ecologies and biological systems in general are strongly subject to interaction effects, where changing one part of the system in isolation is impossible.

3. Long time horizons — Your ability to predict consequences of your actions decays rapidly (exponentially? factorially?) as time goes to infinity, and your first-order, local, linear intuitions or calculations may not serve you very well. I associate this failure mode with the “longtermism” school of effective altruism. I’m not just taking potshots at EA for fun, I actually like them quite a bit, but they exemplify this failure mode better than any others I can think of.

4. Since Substack added Latex, and this is my blog, I feel obligated to formalize statement 3. by deriving a Fence Ratio F. Be warned, this is not especially useful.

   1. Let the jth consequence of your actions be c_j(t) over time t, let the validity horizon of your modeling have time constant τ, then c is described by a series expansion where each coefficient a_i represents your intuition or model about the “direction” of the consequence, and let us specialize to a single axis of consequence by dropping the j index:

   2. Your error in predicting the consequences of your actions \Delta c is given by standard error propagation of your modeling errors \Delta a_i:

   3. Thus, your Fence Ratio F is given by \Delta c / c:

      When the Fence Ratio exceeds 1, your prediction error exceeds your prediction, and that’s when we should be wary of Chesterton’s Fence. I think that’s sufficient Moral Profundity for a conclusion, though it’s not a very good formalization. Future work, perhaps.

No post about tradition is complete without Tradition (from Fiddler on the Roof).

The ancient Greeks were, quite reasonably, concerned by this question, because their democracies died all the time. In fact, this happened so much that the most eminent philosophers and historians of the classical period developed a theory that rationalizes the rise and fall of democracies, oligarchies, and tyrannies. In this article, we will investigate whether their theory, called anacyclosis, holds up under scrutiny, and by scrutiny, I mean Monte Carlo simulations of government transition based on historical data from 1,035 Greek city states. But first, some history.

The Poleis of Ancient Greece

The peculiarities of classical Greece make empirical theories of political revolution much easier to imagine than in, say, the Persian Empire, which was a hereditary monarchy for pretty much its entire history. The dominant mode of social organization in the archaic and classical Greek periods is the polis, the city-state. Usually, there’s an independent mother city (Athens, Sparta, etc.) that politically, economically, and culturally dominates its surrounding hinterland. Each city has its own constitution, or form of government, but shares a common Greek culture and language with its neighboring poleis.

This social structure is as dynamic as it is unstable, and there were many political revolutions. The ancient world’s most sophisticated theories of political evolution grew out of this dynamic — they classify government into a few categories based on which group holds power, and posit that they devolve sequentially from higher to lower forms. Let’s take a quick look.

Governmental Types in Ancient Greek Thought

The Inventory of Archaic and Classical Greek Poleis, about which we will have much more to say later, gives a nice introduction to the types of constitution.

In Greek political theory politeiai [political institutions] were divided into types according to how many people constituted and manned the principal organs of government. Basically, there were three constitutional types: the rule of the one, the few and the many. Pindar is the first we know who distinguished between rule by a tyrant, or the wise, or the whole army. About a generation later, Herodotos has a debate about the three basic types of constitution, here described as demos, oligarchia and monarchia. [In the early 4th century BC], Plato called the three forms tyrannis, aristokratia and demokratia.

Linear Evolution in Plato’s Republic

Plato1 made a finer distinction, dividing government into five categories of constitution in his Republic, and additionally giving their sequence of devolution.

Aristocracy (rule by the best) → Timocracy (rule by honor/worth/money) → Oligarchy (rule by the few) → Democracy (rule by the people/mob) → Tyranny (rule by one man).

Plato writes that governments devolve in this order, from best to worst, in a linear fashion, terminating in tyranny. I went back and checked the Republic to see if he makes any claims of cyclical nature, and I don’t think that it does, but the Republic is very hard to read generally,2 so I’m not 100% sure.

Aristotle, a student of Plato, the tutor of Alexander the Great, and a giant of philosophy in his own right, generally agreed with Plato, but distinguished between a good and a bad form of monarchy (basileia versus tyrannis), minority rule (aristokratia versus oligarchia) and majority rule (politeia versus demokratia). His conception, however, was also linear (as best as I understand).

Anacyclosis in Polybius’ Histories

Polybius was a Greek hostage and historian in Rome during Rome’s rise to power, and he improved upon Plato and Aristotle’s framework. Polybius divided government into three categories, each with a virtuous and corrupt form, for a total of six constitutions. They are as follows, from his Histories:

The virtuous aristocracy is corrupted into an oligarchy, which is overthrown by the people as a democracy, which degenerates into mob-rule or ochlocracy. A great leader emerges from mob chaos to create a monarchy, which descends into tyranny before being overthrown by the noble aristocracy, beginning the cycle anew. He called this cycle anacyclosis.

The Anacyclosis Institute (to whom I must give credit for the genesis of this article)3, offers the following comments on this process.

There is good reason to think that Polybius and his predecessors arrived at this theory empirically. After observing the rise and fall of many hundreds of city-states, most of which cycled through several of the governmental forms mentioned above, Greek political thinkers concluded that these transitions from one form to another were not random. Rather, they seemed to follow simple and recognizable patterns. For example, tyrants were frequently overthrown by groups of aristocrats, while popular revolutions frequently overthrew oligarchies and ushered in democratic rule. Interestingly, the reverse of these trends (aristocracies being overthrown by tyrants or democracies turning into oligarchies) were statistically less likely to occur.

Through such observations, Polybius extrapolated the likely complete course of political evolution for an independent state whose lifecycle is not cut short by war or disaster.

Polybius, Plato, and Aristotle essentially agree on the pattern — we go from rule-by-few (aristocracy/oligarchy) to rule-by-many (democracy) to rule-by-one (monarchy/tyranny), with an optional cycle back to rule-by-few.4

Polybius thinks there’s a way out of this cycle. If one combines all three forms of goverment into a mixed constitution, a blend of democracy, aristocracy, and monarchy, one can create a stable system exempt from anacyclosis. Polybius thought that the Roman Republic was the embodiment of this mixed constitution and the reason for its strength and longevity. The Founding Fathers of the United States of America, and John Adams in particular, were obsessed with Polybius, and designed the structure of the United States government to avoid anacyclosis.

So to recap, we have several explicit claims, of which various authors claim subsets.

1. Political evolution follows a predictable pattern of oligarchy → democracy → monarchy

2. This pattern may be linear (Plato) or cyclical (Polybius)

3. The reverse transitions are unlikely/unnatural.

Unfortunately for Polybius, he lacked the tools to quantitatively investigate his theory. Fortunately for us, we are much better than Polybius at linear algebra.

Political Evolution is a Markov Process

Implicit in anacyclosis is actually a fourth claim, the most important claim, that anacyclosis is “memoryless.” In other words, the next type of government depends only on the current type of government: democracies always devolve to tyrannies, independent of what preceded democracy. In the theory of stochastic processes, this property is called the Markov property. We can use the Markov property to evaluate the validity of Polybius’ claim.

First though, we need data. Fortunately, the Copenhagen Polis Centre has done most of the work for us, and compiled An Inventory of Archaic and Classical Poleis,5 a monumental work that compiles the existing data/metadata on the 1,035 identifiable Greek city states of the Archaic and Classical periods (c.650-325 BC). Among the data found in the Inventory is a list of city states and their known government types, ordered by date.

This data is actually all we need in order to pretty fairly evaluate the validity of the theory! For each city state, we can simply extract ordered pairs of government types from this list, and count the frequency with which these transitions occur. Because anacyclosis is a Markov chain (remember that means memoryless), these transition frequencies completely define the system! Note that this method completely ignores staying in the same state as a transition of interest (that requires much more sophisticated data parsing). So this method will probe only when governments change between distinct types.

This will make more sense as we actually construct the transition matrix, and learn how to analyze Markov processes more generally.

An Introduction to Markov Processes through the Inventory of Greek Poleis

A Markov process, named after the Russian mathematician Andrey Markov, is a type of random process. It has discrete states and a notion of time. At each time-step in the process, each state X has a probability P(X→Y) of transitioning to state Y. The Markov (memoryless) property ensures that this probability is the only relevant characteristic of the system.

Markov processes are often represented by graphs that abstract these transition probabilities, like the one below from Wikipedia showing a two-state Markov process.

In essence, a Markov process is the simplest form of a probabilistic state-machine that still has interesting behavior. For any process that can be assumed to be stochastic, or perhaps a system complex enough that its behavior approximates a stochastic process, we can model it is a Markov process and immediately extract nontrivial, useful properties (as we shall see later).

Processes that (approximately) have the Markov property show up everywhere. Weather prediction, stock price prediction, and population genetics are all examples of approximately-Markov processes. In each of these cases, while the real system may have complex dependencies, a Markov model captures enough of the important behavior to be very useful while still being mathematically tractable.

So how would we construct a Markov model for political evolution in Greek poleis? Like any good scientists, the first thing we have to do is create a good visualization of our data, and stare at it.

The Data

Looking at our dataset, we have six distinct types of constitutions listed by the Inventory. I’ll quote briefly from it here.

In the Inventory, when we classify the constitution of a polis, we distinguish between basileia, tyrannis, oligarchia and demokratia, but we ignore variants of the two latter types, and all attestations of basileia belong in the Archaic period…

In a few cases of serious doubt, we have used Mix. to describe a polis with an unidentifiable mixture of characteristics.

The inventory also has another category in the data not listed above, politeia, which Aristotle defined as the “good” form of democracy, but is also the general term for a “polity” in Greek. Both of these last forms, politeia and “mixed” are very rare in the Inventory and slightly confusing.

We should note also that the Inventory says that the term basileia might change meaning over time, as it is only attested in the Archaic period and not the Classical period. Second, that “in actual fact, all polis constitutions were mixed,”6 to one degree or another.

But for a first cut, let’s ignore these complexities, and take a look at the data. First, the total frequency of government types:

We note that the mixed and politeia types are very rare, and not likely to affect our results. Excellent! Let’s ignore them. Second, that if we combine basileia and tyrannis, the constitutions are roughly equal in frequency between the autocratic, oligarchic, and democratic categories. Interesting! We’ll keep the two types of monarchy separate for now.

Next, we can plot the frequency that any constitutional type appears first, or last. If Plato is correct, we would expect to see lots of oligarchies initially, and lots of tyrannies finally. We… might see some evidence for that? We really see more of a transfer between oligarchy and democracy from this graph, and the number of monarchies slightly decreases, but not a lot. I don’t think Plato gets much help from the data here.

Finally, we can plot the frequency of government transition type. We define this naively, taking the sequence of governments in the inventory, and plotting the frequency of each ordered pair, i.e. demokratia, oligarchia, basileia would could as one occurrence each of dem.→ol., ol.→bas..

This last plot is pretty much the key to Markov processes. We can simply reinterpret each column of the above heatmap as a probability of transition between states. Thus, by normalizing each column of the transition frequency heatmap, we get a transition matrix T that defines the process. Because our system has the Markov (memoryless) property, the single-step transition matrix entirely defines the process — it is a Markov process.

How does the model work in practice? Let’s enumerate the types of government as the ordered list:

[bas., tyr., ol., dem., mix., pol.]

Then we can define a one-hot vector v that corresponds to my state, e.g. [1, 0, 0, 0, 0, 0] = the system is in the basileia state. The probability of transitioning to any other state is then given by a vector p, equal to the the transition matrix T times v.

Let’s explicitly construct T for our data. Since “mixed” and politeia are ill-defined and occur so infrequently, I feel fairly justified in simply dropping those columns from the data.

From this matrix, we can then recreate the Markov-process-style node graph.

Assessing Validity

We’re now in a position to partially assess the validity of anacyclosis as it relates to the data, in a first-order sort of fashion.

The first claim is that political evolution proceeds in the order oligarchy → democracy → monarchy. Our data is fine-grained enough that we can split monarchy into its “virtuous” and “corrupt” forms, basileia and tyrannis, and so let’s look for oligarchy → democracy → basileia → tyrannis in the data.

Looking first at the basileia → tyrannis transition, we actually find excellent support for this in the data! Basileia to tyrannis transitions happen about 13 times more often than the reverse. However, there’s a confounding variable. Remember that the Inventory says that basileia is attested only in the Archaic period, so any basileia → tyrannis transitions might be the result of redefinition as opposed to transition. Let’s call this one a partial thumbs up, though.

How about the posited oligarchy → democracy transition? Not so much. These two nodes have the tightest connection in the graph, and the transition rates are essentially even, with oligarchy to democracy being ever so slightly more favorable than the reverse.

Finally, what about democracy → monarchy? It seems as though democracies don’t ever go to basileiai, which certainly doesn’t support the theory of anacyclosis, but again, this could be a definition thing — if we started the chain at basileia, we could have passed through the Archaic period before we got back around, when basileiai had turned into tyrannides. Unfortunately though, the democracy → tyranny transition (29%) is much less common than tyranny → democracy (56%). This gets even worse if we consider basileia the same thing as tyrannis, which has even more asymmetry between the two transition frequencies. So the democracy to monarchy transition doesn’t find much support here, in fact, more the reverse.

In fact, what is the most common cycle? Let’s re-plot the Markov chain where we combine tyrannis and basileia into “monarchy.”

This doesn’t really help us much. The most plausible cycle by far is simply oscillation between democracy and oligarchy, which does not at all fit into the anacyclosis paradigm. It seems we require a more sophisticated analysis to extract the probable dynamics.

Markov Chain Monte Carlo

Okay, well then what does happen to a hypothetical average Greek polis? We can use a Monte-Carlo style simulation to find out.

“Monte Carlo” is a cute name for a very simple technique — if you have the rules of a system and want to understand its behavior, just simulate a whole bunch of random instances of that system and average the results. The simulation method is called “Monte Carlo” because one of the inventors had an uncle who gambled too much in real Monte Carlo.

Nevertheless, this simple technique is extremely powerful. To implement, we

1. Choose an initial one-hot state v

2. Multiply by our transition matrix to get p = Tv, our vector of state probabilities

3. Choose a random state from p, weighted by the probabilities of each state, i.e. if p = [0.1, 0.2, 0.3, 0.4], we have a 10%, 20%, 30%, and 40% probability, respectively, of choosing states 1, 2, 3, 4.

4. Repeat this for n timesteps

5. Repeat steps 1-4 for m simulations

Let’s try this out, keeping the basileia/tyrannis distinction, just for fun.

So keeping basileia as a separate category doesn’t do much, its occupancy fraction immediately hits zero (on average) and never recovers. The most distinct feature by far appears to be the oligarchy-democracy oscillation, settling after 10 timesteps into an even mixture of democracy and oligarchy (remember this is an average, at each timestep, the system can only be in one state). The tyrannis initialization appears to cause the settling to happen faster, but doesn’t differ in the essential trend. We also appear to stabilize at a steady state for any initialization parameter! Perhaps this is the fabled “mixed” constitution that Polybius thought made the Roman state so powerful and stable? We shall formalize this thought later.

Before we do steady-state analysis, we should check for common cycles. Let’s plot the most common cycles we find, dropping the basileia/tyrannis distinction (both are monarchy) for clarity. 1000 more simulations…

The most common cycle by far is the 2-state democracy → oligarchy → democracy cycle. The next most common state is this same cycle, twice in a row!

But wait, if we look down the list at the fourth most common cycle, it’s democracy → monarchy → oligarchy → democracy, that’s anacyclosis! It is the most common 3-state cycle! We found it!

Did we just prove anacyclosis is real? Well, uh, it depends what you mean, I guess.

Instead of answering the above, difficult question, I choose to reinterpret the original theory in light of the data — when Polybius wrote that “anacyclosis consists of predictable, cyclic transitions from democracy → monarchy→ oligarchy,” he clearly must have meant that given the empirical transition probabilities derived from a Markov model of Greek city state constitutional data, a Monte Carlo simulation will show the most common three-state cycle is democracy → monarchy → oligarchy.

I think this is a very reasonable translation of the original Ancient Greek.

Mixed States and the Stable Distribution

Now Polybius was particularly interested in ways out of this endless cycle. How can we find a stable governmental state? Both our answer and Polybius’ answer is the same, and is already hinted at by the results of the simulations. We noted earlier that the simulations, regardless of initialization state, seemed to settle into a predictable distribution of government types, roughly 40% oligarchy, 40% democracy, and 20% tyranny. This was also Polybius’ answer, that a mixed state was a stable point of this Markov process.

Polybius took as his example par excellence the constitution of the Roman Republic, which had popular assemblies (democracy), the Senate (oligarchy), and two consuls with executive power (a dash of monarchy). He felt that this mixture was much more stable than any pure state, and lent Rome its fabulous power.

According to our model, Polybius is absolutely correct.

How can we find the stable state of our Markov process? Well first, we extend the model to allow for mixed states by not forcing our state v to be one-hot. That’s fairly easy. But how do we find the long-term stable state, if there is one?

Let’s think geometrically about our transition matrix. For a 2 × 2 transition matrix, we can visualize its action by seeing how it transforms a set of vectors arranged along the unit circle in the plane.

Under this linear transformation, the unit circle becomes an ellipse. The special directions that remain unchanged (up to scaling) by this transformation are called eigenvectors. These are precisely the principal axes of the resulting ellipse.

Mathematically, an eigenvector v with eigenvalue λ is defined by the equation:

where T is our transition matrix. But this is very helpful for us, because if I now apply T twice, I get

Thus when we apply T repeatedly, each application multiplies the magnitude of v by λ while preserving its direction.

This means:

• If λ = 1, v maintains its magnitude: it's a stable state

• If |λ| > 1, v grows without bound

• If |λ| < 1, v shrinks toward zero

For Markov transition matrices, the Perron-Frobenius theorem guarantees that 1 is always an eigenvalue, and all other eigenvalues have absolute value strictly less than 1, meaning they decay to zero after a long time. When the Markov chain is also ergodic, meaning that you can visit any state from any other state, and you never get stuck in deterministic cycles, this unit-eigenvalue direction corresponds to a unique stable distribution called the stationary distribution.

Any initial distribution will converge to this stationary distribution as we repeatedly apply the transition matrix. We actually saw this in our Monte Carlo simulations — did you notice how, no matter the initial state, we always ended up with the same fraction of oligarchy/democracy/monarchy?

So we can quite quickly find the stationary distribution of our transition matrix by performing an eigendecomposition of our transition matrix. We solve

for all v and λ.

Let’s take our transition matrix where we combine basileia and tyrannis into monarchy.

The eigendecomposition of T yields three eigenvectors.

Which does indeed have a unit-valued eigenvector! To find the stationary distribution, we only need to normalize the eigenvector with eigenvalue 1, by dividing it by its column sum (it’s a probability vector, remember).

We can visualize this final distribution with a bar graph. Bar graphs are the most useless graph type, but they are visually arresting due to large bars of solid primary colors, so I’m making one.

This graph shows the anacyclotically stable distribution of the Markov poleis model. I don’t know if the word anacyclotically will ever really catch on in popular discourse, but I think it really rolls off the tongue. Perhaps instead we should call it the Polybian distribution. It does look remarkably similar to the Roman system, which was deeply suspicious of kingship, but recognized its utility, and hence had two equal consuls in the place of one tyrant, as well as theoretically balanced popular assemblies and an aristocratic/oligarchic Senate.

So that’s it! This is the final confirmation that Polybius was on the right track, and that if he had only been better at linear algebra, he could have quantitatively estimated the proportion of democracy, monarchy, and oligarchy to inject into a politeia to stabilize it against the inevitable anacyclosis, assuming of course that by anacyclosis he actually meant the stochastic Markov process we’ve been working with this whole time, and not the actual anacyclosis that he wrote down, which is given minimal support by the actual data. Easy!

I should note, for future work, that there is at least one major oversight — for any given year in even the most fractious Greek polis, the probability of government transitioning to an entirely new category is small. In other words, I could have structured this process around a timestep being a single year, instead of an arbitrary “government transition time,” and gotten very different looking processes, with the same long-run transition probabilities. Oh well, you always have to leave work for the next researcher.

Addendum: Methodological Validity

Mere minutes after posting this article, I had a thought — is the Greek poleis data set even capable of detecting an anacyclosis cycle in principle?

Let’s say I have a sequence A → B → C → A → …, like our poleis dataset. Then, because of spotty recordkeeping, let’s say I decimate this sample by randomly deleting entries, so maybe I’d get A → B → __ → A → …. Without knowledge of the original sample, there’s a spurious B → A transition in our data!

So the question I want to ask here is, given a sequence composed of a pure cycle S = A → B → C → A …, if I randomly sample this sequence by throwing away all but a fraction f of the data points, can I still detect my sequence above noise?

Formally, let’s ask the question in the following way: I sample a fraction f of the data points from my sequence S, and construct a Markov transition matrix T by naively measuring transition frequency between neighbors in my sampled sequence. How often will I measure that the probability of the original sequence is greater than that of the reverse sequence? This is a sensible definition of “noise” because if we are solely interested in three-element sequences with unique elements, there are exactly two, A→B→C and C→B→A.

This sounds like an interesting analytical problem, but keeping with the Monte Carlo theme of this article and my own laziness, let’s just try it in code. I’ll construct a sequence of length 100 (A→B→C→…), decimate it, keeping a fraction f of the data, construct my matrix T, and then check whether the original cycle probability exceeds the reverse cycle probability.

The results are as follows.7

And for completeness, does this give sensible results when run it on the reverse sequence C → B → A? In other words, what’s the spurious detection rate for a sequence that doesn’t contain the cycle at all?

Yes. This test does indeed give sensible results.

How about a random sequence? What do we expect on average?

Yes, this also makes sense. For a random sequence, we detect that A→B→C is more probable than the reverse about 50% of the time. I think this is a “type III” error, where I correctly detect that the probability of my sequence is higher than the reverse sequence (the noise), but this doesn’t mean anything, because the underlying generator is fully random.

Based on this, I think I can say that the cutoff point where this detection mechanism starts to work is when the probability of detection exceeds ~50% on that curve, so let’s say a sampling fraction of about f = 0.2. That’s not too bad, I think I can consider my method valid enough for a substack article.8

There’s something so validating about hearing an AI-generated NPR podcast host cover articles you wrote and say “woooOOOOOoooow.” It really feels like you’ve made it.

I’ve been playing with NotebookLM recently, which is Google’s new AI note-taking tool. Its basic functionality lets you to upload “sources,” — text files, .pdfs, etc. — aggregate that information, and allow you to ask questions about those sources to a large language model (LLM). Claude, another LLM from Anthropic, has a similar feature called “Projects” that I absolutely love. But one thing that makes NotebookLM really stand out is the “Audio Overview/Deep Dive” feature.

The Deep Dive/Audio Overview feature essentially generates a podcast episode about your sources, and it absolutely nails the NPR podcast style. It is so similar that it’s eerie — There’s two hosts, one male and one female, with classic “American media” accents, who have a back and forth conversation about your topic at the level of a general audience (complete with bad jokes, correctly-timed laughter, and ham-fisted metaphors). The voices are tuned to mimic the exact style of vocal modulation that will be instantly recognizable to anyone who listens to e.g. Planet Money. The AI hosts have all the vocal quirks of podcasting, including overuse of descriptors like “mindblowing,” “wild,” and saying “get this” before introducing a new factoid, and “you’ve got it, you’re catching on” afterwards.

In fact there are actually two new Google tools, Google Illuminate being the other one, that generate audio overviews. Google Illuminate is slightly different in focus and features, as it only allows you to input scientific publications from ArXiv and a few other websites, and allows you to tailor the audio overview to general audiences, experts, beginners, etc. It also allows you to set the tone, and “semi-professional” seems to be the default.

Andrej Karpathy on Twitter showcases this brave new podcasting world by generating, in less than two hours, a series of ten podcasts exploring interesting historical mysteries, like the Baghdad Battery and the Roanoke Colony disappearance. I really liked them.

The audio quality is excellent, and the content (again plagiarizing Karpathy) is mediocre-to-good with “flashes of greatness.” Like most podcasts aimed at a general audience, it does a pretty decent job getting the main idea across, and absolutely refuses to end an episode without making a broad generalization with “profound” implications for the rest of life’s mysteries. Usually “life is complicated, you know? *Cue sometimes-slightly-offputting-AI-laugh*.” In general, I wish they went into a little more technical detail, with a few less metaphors.

I make fun of it, but I absolutely love this feature, and I feel as though I am in the absolute center of its target demographic. After exploring this a bit, I wanted to learn more about direct audio generation, and so I listened to the AI generated summary of a paper on AudioLM, and you know what? I actually get the general idea, terrible metaphors notwithstanding. This tool has a strong tendency to venture into "string theory is like a taco" territory, but manages to get the point across nonetheless.

I foresee a massive boom in podcast quantity.

Now of course a tool like this is just too tempting to try out on your own writing, and I caved to that temptation the second I thought of it. It is so ego-validating to hear an AI call your work “mindblowing,” regardless of how valid that assessment is. It is slightly less ego-validating when the AI hosts lose the plot a little bit and pontificate about how it’s “so crazy that Assyrian spam texts can say so much about our relationship to authority in the modern world, you know?” The training set is clearly poisoned by NPR.

So without further ado, I present to you, Maximum Effort, Minimum Reward’s new podcast! And for extra bonus meta points, I actually had it generate a podcast episode of this very blog post. That one is probably the most interesting (scroll down to the bottom of this post), as I may have hit some sort of strange recursive failure mode. Either that, or “AIs talking about themselves getting stuck in loops” is a common theme of the training data.

Minimum Effort, Audio Reward

A series of AI-generated podcast episodes on the works of Dylan Black.

This one is possibly the best example showcasing the feature. It’s got humor, it explains the topic well. Even the overgeneralization at the end is fairly relevant. Great podcast episode, 10/10, no notes.

This one has a surprise third guest! Right at the beginning, a man responds to the normal male host, instead of the female host! But it’s generally a really good and engaging summary of the blog post. The back-and-forth is done really, really well, and the jokes land well, though this article relies a bit more heavily on syllable stress than most, and the AI has trouble with stress on “tethera, methera,” etc.

This one is generally very good, because they read some of the text verbatim and nail the comedy, but leans way too hard into “fascinating” generalizations about modern life and moral pronouncements. It also describes one of the spammers as “smitten” with Ashurnasirpal, when they are clearly suspicious, not smitten. The quality drops precipitously after six minutes or so. Apparently this technique forces us to “confront our deeper relationship with sority [sic].” I think they meant authority.

The host “was watching The Devil Wears Prada recently.” Uh huh, sure you were. My favorite quote, talking about my article: “It’s more interesting than it sounds, I promise.” Ouch. It also makes a couple, ah, questionable speculations about how the sky would look if our eyes were different, and somehow turns an article about objective first-principles derivation of wavelength spectra into a metaphor about subjective points of view in The Devil Wears Prada. That’s so LLM.

This one had some of the best/worst metaphors. “It’s like trying to nail jelly to the wall! It’s just like, they’re so slippery!” It also has some pretty obvious AI audio quirks.

NotebookLM on NotebookLM

And finally, the ultimate test. What do the podcast hosts of Deep Dive think about their own audio? Can they recognize themselves? Can you, hosts? I’m talking to you directly.

Nope. In fact, it seems to think that the male host is me. It actually says “maybe one day we’ll have an AI guest host on this very show!” (About 10:00). The AI entirely makes up the content of the Ancient Egyptian and Emoji Chain Letters post, invents a fictitious History of Coffee post, and praises it extensively.

And at the two-thirds point, about ten minutes, it “ends” the episode, then continues from a middle point, restating the long digression about creativity and then goes on to talk about the AI generated episode where the AI gets stuck in a loop. It thinks the loop is a digression on creativity, which I suppose is true in a meta-sense. The English language might need more pronouns for AI talking about its own AI-generated content that it hasn’t seen yet. It’s talking about itself getting stuck in a loop, while stuck in a loop.

Compare ~9:00 and ~12:08 to see the problem, or back up about ten seconds to hear the AI’s opinion of its own loop. In the words of the AI:

9:00 “So did it pass that Turing Test on itself? Not really, it was more like it hit a wall with its own programming. It knew the format and the style were similar, but it couldn’t grasp that it was the one creating them. Like it was searching for a ghost in the machine, not realizing it was the ghost the whole time.”

That is in fact exactly what happened, AI. Maybe this is a weird meta-way of passing the Turing Test, plus some NPR metaphors, or maybe this is exactly what NPR podcast hosts in the training set think AI should say about itself.

Aside from this loop issue, the episode is unusually long, unusually boring, unusually incorrect in its facts, and unusually off-topic. I wonder if this is some actual recursive generation failure mode, or whether this article simply doesn’t have the same quantity of text content to generate with.

You can read the previous post in the Bookmarks Bin series below:

I’ve tried to categorize the links, but may not have done a very good job. Enjoy!

Math

1. Trajectoids are 3D shapes that roll along a desired path. Imagine taking a pencil and paper and drawing a random continuous path on the paper. Is there such a shape that, if placed on the start of that path, will trace out that path as it rolls? Yes. The Nature paper and a Youtube video demonstration:

2. Tiling the plane. It’s quite easy to cover a flat, 2D plane using only a single shape. A square, for example. But for many years, mathematicians have been searching for a single shape that can aperiodically tile the plane. Well, they found one. Two, actually, within three months of each other. The Einstein Shape can aperiodically tile a plane (if you’re allowed to reflect it) and the specter can do it without reflection.

3. Gian-Carlo Rota discusses the failures of modern differential equations education.

4. 37 is the median value for the second prime factor of any integer. What?

5. Conditioning on a Collider. Say it with me, correlation is not causation! Spurious correlations show up all the time between unrelated variables, and one of the ways is through a “collider variable,” a variable that is causally influenced by two (or more) other variables in a causal diagram.

6. There is Only One Hypothesis Test, and its sister article, Hypothesis Testing is Only Mostly Useless. The author argues there is fundamentally only one statistical test, with a common framework. This gave me a great mental model to use (and some cautionary tales) whenever I have p-values to calculate or look at.

Science and Engineering

1. The 1859 Carrington Event disrupted global telegraph systems, but newly discovered "Miyake events" are far more powerful solar phenomena. These events, detectable in tree rings, offer precise historical dating tools while posing a potentially catastrophic threat to modern technology and infrastructure.

2. What’s inside a 1 dollar radar sensor? A teardown of a radar chip by a very knowledgable electrical engineer.

3. Smoking reduces all cause mortality? “Not highlighted in the CDC study and unlikely to be recommended, smoking caused the most significant reduction in mortality from all causes (HR 1.65) in both the mild and moderate to severe depression groups compared to the base model (HR 1.78). This was followed closely by physical activity (HR 1.67), which is much more likely to be encouraged.” Hmm. I suppose smoking → happiness → live longer is somewhat plausible…

4. In the same vein, eating ice cream also reduces all-cause mortality. Despite extensive analysis and attempts to “make it go away,” the association between ice cream consumption and lower heart disease risk in diabetics remained robust. Ice cream → happiness → live longer ? Maybe all medical studies are just BS.

History

1. Ancient monkeys made tools?

2. Ancient Egyptian Perfume Recipe

3. The Hyde Park Picture House in Leeds, England is the world's last gas-lit cinema. It’s also a deathtrap, and the renovators wrote an article about their quest to engineer it not to blow up.

4. LLMs translate Latin Demonology manuals.

5. A brutal academic takedown of a popular article on the most mundane of topics. Qing dynasty administrative documents. Was there an “administrative revolution?”

6. The Babylonians used floating point notation. Knuth explores some of their algorithms.

7. George Kennan’s Long Telegram, one of the (if not the) founding documents of the Cold War.

8. The Homeric question. Who wrote the Iliad and Odyssey?

Code and Tech

1. Horcrux encryption. Split your file into seven different encrypted pieces, that can only be reconstructed if you have a predefined subset of them.

2. Fatebook, a website meant to help you predict your output, and its parent site Quantified Intuitions, which has some fun prediction-market style calibration quizzes

   

3. What if Linus Torvalds gets hit by a bus?

4. Overview of hardware concepts that scientists should understand to write fast code, focusing on CPU architecture, memory hierarchy, and optimization techniques. It covers topics like cache usage, SIMD vectorization, branch prediction, and multithreading, using Julia to demonstrate how these hardware features impact code performance.

5. Targeted ads suck at their job. Why do companies want your data anyway?

6. The FCC issued orders to 20 phone companies demanding they show cause why they shouldn't be removed from the Robocall Mitigation Database for submitting deficient or nonsensical “robocall mitigation plans.” This article collects the most unhinged submissions. The FCC received blank pages, a Windows Printer Test, and an “indiscernible object.”

7. Bing can read captchas, but only if you hide them in pictures of lockets.

8. Test your prompt injection skills, by trying to fool an LLM into revealing the secret password with increasingly clever prompts.

9. Internet Artifacts: A virtual museum containing images of important internet milestones, like a topological map of ARPANET and the first spam email.

10. Two Hackers, One Keyboard. Has anyone had the desire to imitate this (in)famous NCIS scene? Now you can.

Society

1. We have used too many levels of abstraction and now the future looks bleak. “A big percentage of so-called experts today only know how to configure tools, but they understand nothing about how things work at the deeper level. This is a real challenge and a big problem for the future.” As always, treat predictions of doom with healthy skepticism, but I do worry about this sometimes.

2. The exam that broke society, the long shadow of the Keju, the Chinese imperial civil service exam.

3. How Jon Stewart made Tucker Carlson. Stewart's well-intentioned efforts to reform journalism, in true monkey’s paw fashion, ended up accelerating trends that created our modern fragmented media landscape that he now rails against.

4. Book Review: Going Infinite. Zvi Mowshowitz reviews Michael Lewis’ book on the collapse of FTX and Sam Bankman-Fried.

5. The evolution of Evony ads. Anyone else remember these ads?

   

6. The trials and tribulations of LBJ’s shower.

7. Things you’re allowed to do, a list of unconventional advice about things you’re allowed to do that you thought you weren’t, or didn’t even know you could.

8. How Effective Altruism lost its way. Eh, I still like them.

9. The trouble with the view from above. States seek to make society "legible" by standardizing complex local practices. Local systems are very different and encode valuable local knowledge but are illegible to outsiders. For example, how’d you get your last name?

10. The Awful German Language. A satirical essay by Mark Twain about the very unusual features of German.

11. The Psmiths’ Bookshelf, the best collection of book reviews on the internet!

12. Monopolies are good, actually (in tech, anyway). This was really a great essay.

I got a cat! Isn’t he handsome? His name is Caligula, Chef Gerald Caligula in full. Gerald because my wife thought it was hilarious to name a cat Gerald, Chef because he keeps opening our knife drawer, and Caligula because of the little white boots on his front paws.

Now, the name “Caligula” has some cultural resonance that tends not to be associated with boots, more with incest and murder, but let me explain.

Caligula, The History

Germanicus Julius Caesar was a Roman general and politician who was most famous for his campaigns in Germania, hence his first name. He and his wife, Agrippina the Elder, spent much of their lives in military camps, where they had a child who would later be the third emperor of Rome. His name was Gaius Caesar Augustus Germanicus, better known as Caligula. You see, caliga means “boots” in Latin, military boots specifically, and while growing up in the camps, the soldiers nicknamed little Gaius “Caligula,” or “little boots.” -ulus is a Latin diminutive, for example feles means “cat,” and feliculus means “little kitty.” The historian Emma Southon, in her immensely entertaining book Agrippina, thinks that a better translation of Caligula might be something like “booty-kins,” that really emphasizes the childish, disrespectful nature of the nickname.

Why do we refer to Caligula by his childhood nickname? Well, partly to differentiate him from all the other Romans with near-identical names, but also partly because he was hated by the Roman nobility, and they called him Caligula in the histories as a form of disrespect.

Caligula is famous as a byword for cruel despotism mainly because the historians Tacitus and Suetonius wrote absolutely salacious accounts of his time as emperor, accusing him of pretty much every depravity they could think of.

Among other things, he’s accused of living “in habitual incest with all his sisters,” turning the imperial palace into a brothel, proclaiming himself a living god and demanding he be worshipped, appointing his favorite horse as a consul and a priest, squandering the imperial treasury on enormous vanity projects, and ordering mass executions whenever it suited his fancy.

Now Emma Southon thinks, and I agree, that most of this is very likely to be propaganda from his enemies. Indeed, it was more unusual in the early Roman Empire not to be accused of incest, as it was something of a trope. Caligula’s disdain for the nobility and popularity with the people is well-attested, which likely drew the ire of the nobility. Irrespective of their veracity, the accounts of Tacitus and Suetonius make for fantastic reading (you’re certainly never bored), hence their popularity through to the present day.

Of course, my cat is guilty of none of these crimes, though he absolutely would murder the birds outside my window, if only he could get his little paws on them.

War Against Neptune

Now I’ve been learning Latin, and it occurred to me that many of the specific stories told about the Emperor Caligula actually sound a lot like cat behavior. Take, for example, the famous story of Caligula ordering his soldiers to “make war on the ocean.”

This is very odd behavior for a human (but perhaps not for a cat), though the interpretation of this as “declaring war upon the ocean” is actually from Robert Graves’ novel I, Claudius — the Roman sources leave Caligula’s motivation ambiguous. Regardless, as part of my Latin composition exercises, I have written several fables that chronicle the res gestae Imperatoris Geraldi Caligulae, and I want to share them with you. So without further ado, I present Bellum Contra Neptunum, the first in a series of adventures of Gerald Caligula, Emperor of the Washing Machine, Sovereign of the Linen Closet.

Annalis Prima Geraldi Caligulae, Bellum Contra Neptunum

I, Gerald Caligula, Emperor of the Washing Machine and Sovereign of the Linen Closet, will tell you the tale of my epic battle with Neptune. I am a cat, most regal of creatures, and I have little white boots on my paws, hence my illustrious name.

Ego, Geraldus Caligula, Imperator Machinae Lavatoriae et Dominus Armarii Lintearii, tibi narrabo de bello titanico contra Neptunum. Feles sum, maxime regius ex animalibus, et albas caligas habeo in pedibus meis, nominor ergo Caligula, clarissimum nomen ad clarissimum felem. 

As Emperor, it is my solemn duty to protect my realm against all threats, seen and unseen. Neptune, Lord of Baths, is the avowed enemy of all feline kind, and only yesterday I discovered that his nefarious influence had seeped into the very heart of my domain. I uncovered a portal to Neptune's realm within the bathroom, cunningly disguised as an innocent "sink."

Ut Imperatori, officium sacrum mihi est meum regnum defendere omnibus ab hostibus, visis invisisque. Neptunus, Dominus Omnium Lavacrum, hostis manifestus et declaratus est omnis generis felini. Heri tantum, eius influentiam nefariam in ipso corde mei imperii insinuatam detexi. Ego portam ad regnum Neptuni in balneo repperi, quae callide dissimulata erat ut foramen.

I was perched regally on the counter when one of my subjects entered and turned a strange silver lever, unleashing a raging torrent of water into the basin that serves as one of my many napping spots. A lesser cat might have fled, but no ordinary feline is Gerald Caligula! I immediately declared war, announcing my intentions with a battle cry that my human subjects erroneously refer to as "yowling."

Nuper, regaliter in mensa insidebam cum una ex subditis meis (nomine Sophia) intraret et innaturalem vectem argenteam verteret. Statim, torrens magna vi effluxit in foramen qui mihi saepe serviebat pro loco dormiendi. Feles minoris animi fugisset, sed tam vulgaris feles non sum! Ego sum Imperator, et statim Neptuno execrato bellum indixi, intentiones meas nuntians fremitu horribili, quem homines falso “vagitare” appellant. 

Ignoring such human foolishness, I leapt to the basin, swatting the watery invader with my mighty paws. My subjects watched, no doubt stunned by my bravery and tactical brilliance. The humans cried, "Gerald, no!" -- clearly a cheer of encouragement. I swatted the silver lever with all my might, and lo! the torrent ceased. 

Nugis hominum neglectis, in foramen insilui, magnis pedibus invasorem aquosum verberans. Subditi mei spectabant, sine dubio stupefacti ob meam audaciam. Clamabant “Noli, Geralde!,” scilicet me laudabant. Iterum argenteam vectem omnibus viribus verberavi, et ecce! Torrens cessavit. 

Neptune's armies had been beaten back, and I gloried in my victory. That's when I spotted them -- a row of colorful wands standing in formation, which I had often observed with suspicion. The humans call them "toothbrushes," but I recognized them for what they truly were: Neptune's tridents, the very manifestation of his power in my realm.

Exercitum Neptuni ipse reppuleram, in victoria mea gloriatus sum. Eo momento, illos animadverti – in acie constitutae erant virgulae coloreae, quas saepe cum suspicione perspexeram. Homines stulti eas appellabant “peniculos dentarios,” sed forma vera nunc mihi se revelabat: tridentes Neptuni, manifestationes ipsae potestatis suae in regno meo. 

With the skill of a master thief, I carefully extracted each trident from its holder. One by one, I carried these dangerous talismans to my royal bedchamber (known to the commoners as “the linen closet”). There, I arranged them in a display befitting my triumph, a warning to all who might challenge my sovereignty.

Arte optimi furis, singulos tridentes diligenter extraxi ex eorum vaginis. Singillatim haec talismania periculosa spoliavi ad cubiculum meum regium (vulgo notum ut Armarius Lintearius), ibique triumphans ea disposui, pro monitu omnibus qui me ipsum evertere temptarent. 

When my subjects discovered my victory tableau, they reacted with awe. Clearly overwhelmed by the magnitude of my accomplishment, they muttered “Oh, Gerald” and “Not again,” which I can only interpret as expressions of deepest gratitude and admiration.

Cum subditi mei victoriae spectaculum detexissent, reverentes horrescebant. Manifeste superati magnitudinem facinoris mei, murmuraverunt “O, Geralde” et “Vae mihi, iterum,” quod non nisi ut exclamatio profundissimae gratiae admirationisque intellegere poteram. 

And so, the Battle of the Bathroom Sink came to a glorious close, and I, Gerald Caligula, stand victorious! Let my display of Neptune's chewed tridents be a stern warning to any other god who dares to invade my realm! For in this household, there is but one supreme deity, and his name is Gerald Caligula, Emperor of the Washing Machine, Vanquisher of Neptune!

Itaque, Proelium Foraminis Balnearis ad finem gloriosum pervenit, et ego, Imperator Geraldus Caligula, victor existo! Spectaculum tridentium morsorum Neptuni moneat omnes deos qui regnum meum invadere audeant! Nam in hac domo unus deus supremus est, et is est Geraldus Caligula, Imperator Machinae Lavatoriae, Invictus Neptuni Domitor!

Multas gratias Lucae agere velim ut hanc fabulam corrigat. Gratias tibi, magister!

That quotation is from the Tale of the Shipwrecked Sailor, the oldest complete story that has come down to us from antiquity. It was written in Hieratic, during the Middle Kingdom of Ancient Egypt, and I read it in the original Egyptian from the comfort of my bed, four thousand years after it was written.

Reading an ancient story is an experience unlike any other. It is a glimpse into the mind of an alien — utterly foreign, yet oddly familiar. A yawning gap of time, culture, and language divides me from the author, but I read his tale nevertheless and I marveled when the sailor encountered the serpent god with eyebrows of lapis lazuli.

I've often felt the pressure to better myself, and I occasionally crack a technical manual or a literary classic in my spare time in deference to that pressure. I've known colleagues whose hobbies are essentially identical to their work — Nothing would depress me more, though I’d probably be better at my job.

Instead, my hobbies are almost militantly useless. I write a blog in which I mathematically quantify the hot-dog-ness of various sandwiches, and simulate spin density waves if they were made of guinea pigs. The only foreign languages I speak are Latin and Ancient Egyptian, into which I've translated The House of the Rising Sun (so useful). The history books I love best are the furthest removed from my own time. In general, my delight in a hobby is inversely proportional to its utility.

But despite, or perhaps because of, the unavoidable pressure to be productive that pervades modern life, I feel that my useless hobbies are not only personally valuable, but essential to a life well-lived, and I think the ancient philosophers tend to agree with me.

Wúwéi, non-action in accord with the natural flow of the universe, was praiseworthy to the Daoist sage Zhuangzi, born 2300 years ago in ancient China. Zhuangzi was renowned across China for his wisdom, and his counsel was greatly desired by the political elites of the time. When the duke of Qi, one of many desiring wise counsellors, invited Zhuangzi to become his chief minister, his messengers found the old sage fishing among the river reeds. Upon receiving this job offer, Zhuangzi did not look up from his rod, and said,

“I have heard that there is a sacred turtle in Chu that died three thousand years ago. The duke keeps it in a casket wrapped in cloth, and has placed it in a temple. May I inquire whether the sacred turtle wanted to be dead and to have its bones venerated by man? Or was its intention to stay alive and crawl around in the mud, dragging its tail?"

"Naturally," replied the messengers, "it hoped to crawl around in the mud, dragging its tail."

"Go home," said Zhuangzi, "I also want to crawl around in the mud, dragging my tail."

Epicurus too understood the value of non-productive pursuits. He was a Hellenistic philosopher who suffered from chronic pain all his life, and perhaps fittingly, developed a philosophy focused on pleasure and pain. To Epicurus, what is true pleasure? True pleasure is not the fleeting pleasures of wealth, rich food, and debauchery, but a restrained, mental satisfaction that lingers, like heat from the embers of a hearth. True pleasure is the absence of pain, and freedom from unnecessary desires.

This pleasure comes with freedom from the desire for wealth, freedom from the fear of death and of the gods, the bond of the tight-knit community, and from pure intellectual exploration — this pleasure is serenity, ataraxia in Greek. The fear of death was simply one more pain to overcome in the life of Epicurus, and so his followers wrote thusly on their tombstones: Non fui, fui, non sum, non curo — I was not, then I was, I am no more, I do not mind. 

Epicurus reminds us that pursuit of wealth does only so much to decrease the pain of life. For indeed, what shall it profit a man, to gain the world but lose his soul?

But for me, there is still more to life than wúwéi and ataraxia, for there is pleasure too in purpose. John F. Kennedy, another great philosopher, spoke to this purpose when he said of the Apollo moon mission that we choose to go to the moon not because it is easy, but because it is hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one we are willing to accept, one we are unwilling to postpone, and one we intend to win.

Indeed, to strive for one’s excellence, for one’s aretḗ, in any field of human endeavor, for no practical benefit whatsoever — this is the highest pursuit of man, his virtue par excellence, and his greatest good. To strive for mastery in a field, regardless of practical benefit and indeed in spite of it, is a noble and fulfilling pursuit.

So why do I waste my time? To exist in non-action, to be free of the burden of utility, to find ataraxia, to strive for something difficult, yet not lose myself in pursuit of material gain, and because ultimately, it is my time to waste.

In the words of Seneca, omnia aliena sunt, tempus tantum nostrum est — All else is foreign to us, only time is ours. And I intend to spend mine generously spiced with those useless hobbies that bring me joy.

Sub-disclaimer: This blog does not endorse the sacking and burning of cities.

Daenerys Targaryen is a Merciful Queen

Let us discuss Daenerys Targaryen, The First of Her Name.

Daenerys is one of the main characters from the TV series Game of Thrones, and a complex character whose meteoric rise and inexplicable fall mirrors the rise and fall of the show’s popularity. She spends most of the show building alliances, taking cities, being generally nice to people and children with only occasional references to slaughtering her enemies and burning their towns. But she stumbles at the last hurdle — at the very moment she finally regains her birthright, the capital city of Westeros and the dominion of the world, her traitorous advisors kill her and give her body to a dragon. So passes Daenerys, daughter of Aerys, rightful Queen of Westeros.

I vividly recall the internet’s (and my friends’) reactions to the sack of King’s Landing in the penultimate episode of Game of Thrones. Almost universally, people were appalled by Daenerys burning the capital and its citizens after the city’s surrender. Even the in-universe characters were appalled! How could she do this? But even while watching it, I thought her actions both:

1. Made perfect sense as a response to rebellion, and

2. Were rather tame by the standards of ancient and medieval warfare.

Hence, I claim that Daenerys Targaryen, the first of her name, is not an appalling monster, but actually rather merciful, and her spineless and craven advisors must have dug up a treasure trove of twenty-first century moral compasses somewhere in the intervening seasons, or they would think so too.

The Siege and Sack of King’s Landing

Just to make sure we’re all on the same page, let’s summarize the events before, during, and after the climactic battle. I particularly recommend this video by Invicta (who actually makes a lot of the same points that I do!).

Belligerents

• Queen Cersei Lannister leads the Lannister forces, including sub-units of Iron Fleet navy, Golden Company infantry. She currently holds King’s Landing, and is in rebellion against Daenerys.

• Queen Daenerys Targaryen leads the Targaryen forces, including sub-units of Unsullied (Eastern) infantry, Northmen, and Dothraki (steppe) cavalry. She is the mostly-acknowledged rightful queen.

Preparation and Pre-Battle Moves

• Cersei strengthens her position by allying with the Iron Fleet and hiring the Golden Company to bolster her defenses.

• Missandei (one of Daenerys’ chief advisors and a personal friend) suggests a “hearts-and-minds” campaign strategy to defeat Cersei, which Daenerys rejects as impractical.

• Jon and Tyrion (more advisors to Daenerys) propose a siege, starving out King’s Landing, to which Daenerys accedes.

• Inexplicably, Daenerys splits her forces into a land contingent and sea contingent, and marches/sails to King’s Landing.

• The Iron Fleet ambushes Daenerys, shooting down her dragon Rhaegal. Missandei is captured.

The Siege

• Daenerys’ army reaches King’s Landing, camps outside the walls, and begins the siege.

• Daenerys and Cersei meet outside the walls, in a sort of parley/negotiation. Missandei is executed by Cersei as a deliberate insult.

• Daenerys’ army begins the siege of King’s Landing, inexplicably (again) without any siege weapons.

• Tyrion urges Daenerys to allow the surrender of King's Landing if the city’s bells ring. Daenerys … agrees? Maybe?

The Battle

• Daenerys launches a surprise attack on the Iron Fleet with Drogon the dragon, effectively destroying it.

• Daenerys and Drogon then target the city’s defenses directly, decimating the Golden Company outside the city gates and the scorpion artillery on the walls.

• Following the aerial dragon attack, Daenerys’ forces breach the city gates and engage the Lannister forces.

Surrender and Sack

• As Daenerys' forces gain the upper hand, the Lannister soldiers start surrendering. The citizens of King's Landing cry out for the bells to ring, which Cersei ignores. The citizens ring the bells anyway.

• Despite the surrender, Daenerys continues her assault, burning much of the city and killing many civilians and soldiers.

Aftermath

• The battle concludes with King's Landing in ruins and a significant portion of the population dead from fire, collapsing buildings, and street combat.

• Daenerys takes control of the ruined city and addresses her forces, promising further conquests. Jon and Tyrion, those pansies, are horrified.

Overall, this is a fairly typical progression of a city siege. To recap for later sections, I want to highlight a very important series of events.

1. Daenerys lays siege to King’s Landing

2. There are some pseudo-negotiations between Cersei and Daenerys, in which Cersei rejects any surrender by murdering an advisor/friend of Daenerys

3. Daenerys’ army breaks into the city by force.

4. The city (though not Cersei) surrenders.

At this point, should Daenerys lay down her arms and accept this surrender? To answer this question, let’s look at the rules of ancient warfare, to see what our ancestors might have done.

The Ancient Rules of War

I will use the Romans as the prototypical example to illustrate the way the ancients conducted war, and what they thought of as “fair play.” Roman history is great for this, as it’s well-studied and involves pithy Latin phrases that make good section titles.

Murum aries attigit is one of the more impressive turns of phrase from Julius Caesar’s Commentaries on the Gallic War.1 It is Latin, and translates literally to “the [battering] ram has touched the wall.”

In ancient warfare, the moment where the first siege weapon touches the city wall is the dividing line between prelude and siege. Sieges are important, because they are hated by both beseiger and beseiged alike — during a siege, most everyone is bored, starving, and dying of disease, sometimes literally for years.

And so before the main assault begins, you, as the city’s ruler, have all sorts of options. Most importantly, you can surrender, with a decent chance to live through the process, if you’re not particularly irritating or dangerous.

But after the ram touches the wall, it’s on. And if the enemy breaks into your city, they can do anything they want to you, even if you surrender, and no one will bat an eye. After all, if you could surrender at any time and be immune to the natural consequences of resistance (mass murder, pillage, rapine, etc.), why would anyone surrender before the siege?

A still more impressive phrase from classical Latin, vae victis translates to “woe to the vanquished,” and is the malevolent complement of “to the victor go the spoils.” This phrase comes from the historian Livy, who tells the history of the first sack of Rome after a siege by the Gauls (ancient, barbarian Frenchmen), in 390 BC.

At last not only food but hope failed [the Romans]… The army insisted that they should either surrender or purchase their ransom on the best terms they could, for the Gauls were throwing out unmistakable hints that they could be induced to abandon the siege for a moderate consideration…

A conference took place between Q. Sulpicius, the consular tribune, and Brennus, the Gaulish chieftain, and an agreement was arrived at by which 1000 lbs. of gold was fixed as the ransom of a people destined ere long to rule the world.

This humiliation was great enough as it was, but it was aggravated by the despicable meanness of the Gauls, who produced unjust weights, and when the tribune protested, the insolent Gaul threw his sword into the scale, with an exclamation intolerable to Roman ears, ‘Woe to the vanquished!’

-Titus Livius, Ab Urbe Condita 5.48.

In other words, the Gauls tried to cheat the surrendering Romans out of their gold with extra-heavy weights, and when the Romans noticed, the Gallic chieftain threw his sword on the Gallic side of the balance, and said “sucks to be you, pay up.” And the Romans did. What else could they do?

Interestingly, Jeremy Armstrong in his book, War and Society in Early Rome, gestures at the idea that this incident was a sort of foundational trauma for the Roman state. When Rome rose again after the sack, it was a mutated society — militaristic, community-oriented, and brutally punitive to vanquished enemies.

I would like to emphasize that though I used the Romans to illustrate my point, these general principles are near-universal before very recent history. The clearest example I know of, where these rules are literally spelled out, comes from the Hebrew Bible, in Deuteronomy.

When thou comest nigh unto a city to fight against it, then proclaim peace unto it.

And it shall be, if it make thee answer of peace, and open unto thee, then it shall be, that all the people that is found therein shall be tributaries unto thee, and they shall serve thee.

And if it will make no peace with thee, but will make war against thee, then thou shalt besiege it.

And when the Lord thy God hath delivered it into thine hands, thou shalt smite every male thereof with the edge of the sword.

But the women, and the little ones, and the cattle, and all that is in the city, even all the spoil thereof, shalt thou take unto thyself; and thou shalt eat the spoil of thine enemies, which the Lord thy God hath given thee.

-Deuteronomy 20:10-14, King James Version

It is with these rules of war in mind that we must evaluate Daenerys’ conduct.

Conduct in Context

We see that far from being outside the norms, or unusually harsh, Daenerys’ behavior is actually quite proper for an ancient ruler.

First of all, King’s Landing was given a chance to surrender peacefully. Cersei said no, in rather dramatic fashion by executing Daenerys’ advisor. Daenerys burned (most of) her city in response. Perfectly fair!

This incident has many historical parallels, but the first one I thought of was the Mongol reaction to a Khwarazmian city’s killing of Genghis Khan’s son-in law, named Toquchar. Here is what the Khan’s son Tolui did to it in revenge for Toquchar’s death:

Tolui arrived at the city on 7 April and the inhabitants, awed by the size of his force, immediately sought to agree surrender terms. Because the killing of the khan's son-in-law had been a grave insult to the Mongols, all proposals were rejected; the assault had begun before the end of the day, with the walls being breached on 9 April and the city captured the next day…

Toquchar's widow supervised the massacre of the entire population of the city, with the exception of 400 craftsmen. Unlike in Merv, all children were killed, and the corpses of the alleged 1,747,000 victims, including all the cats and dogs in the city, were piled in great heaps. The ground was subsequently ploughed over.2

Thus we see that Daenerys’ conduct at King’s Landing is well within the historical norms for her spiritual cousins (the Dothraki being a pseudo-Mongol horde), even relatively tame. I would say she shows remarkable restraint by comparison in leaving King’s Landing a standing city at all, complete with live dogs and cats, despite the holier-than-thou handwringing by her treacherous advisors, Jon Snow and Tyrion Lannister.

Daenerys is, generally, portrayed as generous to her friends and subjects, and horrifically violent to her enemies and those who rebel against her rule. Cersei is undoubtedly in rebellion by this point in the series, and empires the world over have developed various strategies for deterring this sort of thing. Let’s use my favorites, the Assyrians, to illustrate how a rebellious noble might be treated.3

Again, Daenerys acts in concert with her historical parallels, but with less flaying and more fire, befitting her image. Furthering the comparison to an Assyrian monarch, Daenerys has impeccable, bloodthirsty one-liners that would make Ashurnasirpal proud. In fact, let’s do a side-by-side comparison.

Here’s our lovely queen, talking about the Masters of a city called Meereen:

And here’s another spiritual cousin of hers, Tiglath-Pileser I, explaining what he does to cities that do not submit:4

She even has an Assyrian-style name!

The First of Her Name, the Unburnt, Queen of Meereen, Queen of the Andals and the Rhoynar and the First Men, Khaleesi of the Great Grass Sea, Breaker of Chains and Mother of Dragons — Daenerys Targaryen is the very model of strong Assyrian king!

Though really, only one name, and only six titles? Is she a peasant, to be thus degraded?

But… a Targaryen Humanitarian?

Well, perhaps I exaggerate a bit, but I just couldn’t resist the rhyme.

Throughout the series, you could indeed call Daenerys a proto-humanitarian, what with all the time she spends freeing slaves, being pleasant to children, and other nice things (nice for us, anyway, the Romans would not have appreciated freeing slaves, or being nice to children).

But first and foremost, she is a conqueror, and to their credit, the showrunners generally make her act like one. She makes poor tactical decisions, but the logic of power is actually quite sound, and very much in-line with historical norms, even if the eighth season puts an out-of-character twenty-first century spin on the morality of it. Making an example out of King’s Landing to deter future rebellion is a politically savvy and historically accurate choice. It will save more lives in the long run! Maybe!

Of course the Assyrians, after three hundred years of running an empire exactly like this, were so hated that when their subjects inevitably revolted, they literally erased the the Assyrian civilization from the Earth. So I suppose there are some drawbacks to the flaying-based approach to deterring rebellions. Certainly, Daenerys’ subsequent vilification by her former subjects tends to argue against this approach, and it is always worth remembering that history is written not by the victors, but by the screenwriters.

And as if to emphasize that point, after the tragic death of our heroine, her dragon apparently turns into an anti-monarchist partisan, and symbolically burns the Iron Throne.

So maybe we shouldn’t give the writers too much credit for historical accuracy. But we can try!

Can Meryl Streep See Color?

The Devil Wears Prada is a 2003 novel and 2006 movie, featuring a young journalist Andrea “Andy” Sachs (played by Anne Hathaway) who lands a job as an assistant to the domineering/borderline abusive fashion magazine executive Miranda Priestly (played by Meryl Streep). The movie follows the personal struggles of Anne Hathaway at her new job as she struggles to decide whether to

1. be a normal person with friends, a significant other, and a work-life balance, or

2. sacrifice her personal life for her job, and follow in the footsteps of Meryl Streep (Satan).

There’s one scene in particular I’d like to talk about. In this scene, a newly employed Anne Hathaway is taking notes on a meeting involving Meryl Streep and two near-identical greenish belts.

The lady in the black coat says something to the effect of “It’s a tough call [choosing between these belts]. They’re so different!” Anne Hathaway, the audience stand-in, snickers, because the belts are damn-near identical, but The Father of Lies Meryl Streep hears her and delivers a blistering reply involving the blue sweater that Anne Hathaway is wearing. I’ll quote the scene in its entirety, to give you a flavor:

Miranda Priestly (examining a dress): Where are the belts for this dress? Why is no one ready?

Jocelyn (holding green belts): Here. It’s a tough call. They’re so different.

Andy Sachs: (snickers under her breath)

Miranda Priestly (turning to Andy): Something funny?

Andy Sachs: No. No, no, nothing’s… you know, it’s just that… both those belts look exactly the same to me. Y’know, I’m still learning about this stuff, and uh… (giggles uncomfortably)

Miranda Priestly: This… “stuff”? Oh, okay. I see, you think this has nothing to do with you.

You… go to your closet, and you select… I don’t know, that lumpy blue sweater for instance, because you’re trying to tell the world that you take yourself too seriously to care about what you put on your back, but what you don’t know is that that sweater is not just blue, it’s not turquoise, it’s not lapis, it’s actually cerulean.

You’re also blithely unaware of the fact that, in 2002, Oscar de la Renta did a collection of cerulean gowns, and then I think it was Yves Saint Laurent, wasn’t it?… who showed cerulean military jackets. I think we need a jacket here.

Nigel: Hmm.

Miranda Priestly: And then cerulean quickly showed up in the collections of eight different designers. Then it filtered down through the department stores, and then trickled on down into some tragic casual corner where you, no doubt, fished it out of some clearance bin.

However, that blue represents millions of dollars and countless jobs, and it’s sort of comical how you think that you’ve made a choice that exempts you from the fashion industry, when in fact, you’re wearing a sweater that was selected for you by the people in this room… from a pile of “stuff.”

It’s a great, character-establishing scene, and it feels relevant, like it’s saying something important. There’s only one problem. Here’s the sweater Anne Hathaway is wearing:

That’s not cerulean! That’s like, royal blue or something! And I can prove it, with physics!

What is Cerulean, anyway?

The word cerulean comes to us from the Latin word for sky, caelum. The color of the sky, caerulus in Latin, entered English as “cerulean.” Thus, the color cerulean should be the color of the sky. But what is the color of the sky?

Blue, right? When you look up at a clear sky, you see some shade of blue color. Why? The answer was discovered in the late 1800s by The Right Honourable Lord Rayleigh, who, besides having a magnificent mustache, was the first to work out the phenomenon of Rayleigh scattering, the scattering of light by particles much smaller than the wavelength of that light.

Unpolarized light of intensity I_0 of wavelength λ incident on a spherical particle of diameter d and refractive index n will produce a scattered intensity of

to an observer located a distance R and polar angle θ away from the scattering center.

Note the 1/λ^4 wavelength dependence. This wavelength dependence is characteristic of dipole scattering, the first and simplest type of scattering you learn about in physics. We can summarize this as shorter wavelengths are scattered more. Looking at the visible spectrum, we see that shorter wavelengths are more blue.

If you’re looking at the sky, with the sun at your back, the light that makes it to your eye is dominantly light scattered by the atmosphere, of which blue light scatters more than red light. Thus, the sky you observe is blue.

Deriving the Color of the Sky

Going a little further, we can use the expression for Rayleigh scattering to derive the color of the sky on a sunny day.

Since molecules in the atmosphere are randomly oriented, one may integrate the Rayleigh scattered intensity over a bounding sphere enclosing the molecule, and derive an average scattering cross section, an effective cross-sectional area for collision frequency/scattering amplitude.

Now, if we’re discussing Rayleigh scattering in the atmosphere, it is convenient to express this scattering cross section in terms of α, the molecular polarizability, which is a nice property for estimating scattering of elemental gases.

The fraction of scattered light per unit length, as a function of wavelength and polarizability, is then

where N is the number of particles per unit volume.

Earth’s atmosphere is roughly 78% nitrogen (N2), 21% oxygen (O2), and 1% argon (Ar), and so if you know the molecular polarizability of each gas, you should be able to derive the spectrum of scattered light seen by your eye. That’s what we’re going to do — we’re going to calculate cerulean from first principles, and prove that Meryl Streep is wrong about the color of Anne Hathaway’s sweater.

The Plan of Attack

To calculate the color of blue sky, which I will call true cerulean, we’re going to follow this plan.

1. The atmosphere is mostly nitrogen and oxygen, so we calculate an effective scattering cross section for both nitrogen and oxygen gas.

2. Since the atmosphere is ~79% nitrogen and ~21% oxygen (let’s just ignore Argon), we can take a weighted average of these two scattering cross sections to get an average scattering cross section for “Nitrox” gas, which will be our proxy for atmospheric scattering.

   1. The scattering coefficient per unit length is linear in the scattering cross section, so the cross section vs. wavelength will tell us the relative fraction of scattered power vs. wavelength with no further math needed.

3. We then multiply this scattering function by the sun’s emission spectrum, producing the spectrum of solar light that is Rayleigh scattered into our eye. We’re going to ignore absorption and multiple scattering events.

   1. We’re also going to model scattering from water vapor, properly Mie scattering, as wavelength-independent scattering, which I’ll incorporate into the scattered spectrum.

4. Finally, we take this calculated scattered light spectrum, and use a Python package to convert this spectrum into an RGB color that we can display on our computer screen, and compare to Anne Hathaway’s sweater, thus proving Meryl Streep wrong.

Calculating Scattering Cross Section

First, we estimate the scattering cross sections of N2 and O2, the dominant atmospheric gases. Since the atmosphere varies quite a bit in temperature from the ground to low-earth orbit, I first looked at the temperature dependent molecular polarizability coefficients. I found this paper1, in which there is a nice table giving us the molecular polarizability volume for various gases from about 200 Kelvin to 360 Kelvin, perfect for atmospheric temperatures.

A NASA technical note from 19772 provides a helpful table of measured absolute scattering cross sections for various atmospheric gases against which we can check any estimations we make using these values.

Here’s a plot of the scattering cross sections estimated at 300 Kelvin (room temperature) for nitrogen and oxygen gases, with the 1977 NASA data as reference points.

Looks good! Let’s average these two gases into our “Nitrox” effective scattering cross section, and check its temperature dependence to make sure it’s not a problem.

It looks like the effects of temperature are pretty much negligible. Great, let’s ignore it.

Solar Spectrum

Now, in order to calculate the spectrum of light that reaches our eye, we need the spectrum of the sun. I’m going to use the data from the National Renewable Energy Laboratory (NREL)’s page on solar spectra. They actually give us three measured spectra, the “Extraterrestrial,” the “Global Tilt,” and the “Direct + Circumsolar” spectra.

I had no idea what the difference was between these three, so I asked ChatGPT.

1. Extraterrestrial: This is the solar energy that is incident on the Earth's atmosphere outside of the atmosphere, assuming no atmospheric absorption or scattering. It is essentially the solar spectrum that would be observed if you were in outer space and measuring the sunlight before it interacts with the Earth's atmosphere.

2. Global Tilt: Global Tilt represents the spectral irradiance of solar energy that is received on a surface tilted at a specific angle (in this case, 37 degrees) towards the sun, taking into account the atmospheric conditions specified in the data. This is essentially the solar energy spectrum incident on a surface that is oriented to capture sunlight for energy generation. It includes both direct sunlight and diffuse (scattered) sunlight, which is why it's referred to as "global."

3. Direct + Circumsolar: Direct + Circumsolar represents the spectral irradiance of direct solar radiation along with the circumsolar component. Direct solar radiation is the sunlight that arrives directly from the sun without scattering. The circumsolar component includes the direct solar radiation and the portion of scattered sunlight that is within a certain angular range around the sun. This component is particularly relevant for concentrating solar technologies that track the sun's position in the sky.

We’re going to choose the Extraterrestrial spectrum, because that seems like the cleanest, unscattered one. But just for completeness, I tried this whole analysis with each of the different spectra and it made no difference to the final color.

Rayleigh and “White” Scattered Spectra

We’re basically done, all that remains is to multiply our scattering cross section with the solar spectrum, and find out what color that is. I’m going to normalize everything because only the relative scattered power in each wavelength matters.

Additionally, I’m now going to try to take into account water vapor scattering. Water droplets in the atmosphere are much larger than a single molecule (think clouds), and are basically “white” scatterers, meaning they scatter all wavelengths roughly equally (Clouds, for example, are white). Thus, we can estimate the scattered light from a cloud or background haze as just some fraction of the solar spectrum, which is much easier than going through the whole formalism of Mie scattering.

We’ll calculate the scattered spectrum, then feed it into some python code that converts it to a color!

You’ll notice I plotted three spectra here. The blue spectrum is precisely what I said earlier, the product of our Rayleigh scattering cross section with the Extraterrestrial solar spectrum, truncated to the 380 nm — 780 nm range, the limit of what a human eye can detect.

However, to be compatible with the color-estimation code, I had to construct a 5 nm moving average, and that’s what the two “averaged” spectra are. The orange and green spectra, respectively, are the Rayleigh and “white” scattered intensities.

From Spectrum to Color

Now as you may know, most displays use an RGB system for displaying color. If you’ve never looked at your computer screen really closely, try putting a small drop of water on your computer screen. It will act as a magnifying glass and allow you to see the little red, green, and blue pixels that make up your display.

Following this Python tutorial from scipython.com3 I’m going to use something called the CIE 1931 Color Matching Functions to map spectral power onto red, green, and blue RGB pixel intensities.

Now the RGB method of color display is pretty good, but it isn’t perfect. The set of colors which the RGB system can display is smaller than the full set of colors the human eye can see.

The CIE chromaticity diagram below explains the system. The irregular blob is the colors visible to the human eye, and the triangle is the “gamut” of colors permissible in an RGB system. The vertices of the triangle represent the “pure” pixel colors.

In any case, using the color matching functions, we can come up with the actual color of our calculated spectrum and compare it to Anne Hathaway’s sweater. Take that, Meryl.

So, is the Sweater Cerulean?

And now, the moment of truth.

Uh, hmm, you know, the true cerulean I calculate is really quite close to Anne’s sweater color there. Shit, I think Meryl Streep might have been right.

Okay, wait, we can do a little better. Earlier, I explained that water droplets in the atmosphere are pretty close to “white” scatterers (wavelength-independent), and they make up some proportion of the scattered light in your eye. We can average the Rayleigh scattering spectrum with the white scattering spectrum, and come up with a range of colors that can be considered cerulean.

You know, they’re all really close to that sweater color under different lighting, except the last few. The fully “white” scattering kinda looks like Los Angeles on a bad day, or light near the horizon.

Well, uh, it’s a good thing I’m not Meryl Streep’s assistant, because I would have wasted like 8 hours of work time trying to figure out what color cerulean is, only to find out it’s exactly what she told me.

There’s probably a life lesson in there somewhere, but I’m not sure what it could be.

Note: After publishing, I noticed some links don’t work well on a mobile device coming from email, but can be fixed by manually deleting the auto-appended fragments at the end of the URLs, e.g “%2F=&utm_source=substack&utm_medium=email” in the first link.

Miscellaneous

• The Dead Internet Theory: Was the late 2010s internet already entirely peopled by bots? Probably not, but I feel as though this article anticipated ChatGPT in a lot of ways.

• Tradition is Smarter than You Are: Rationality may not be the key to humankind’s success. Instead, look at cultural memory and cultural evolution.

• Book Review: Origin of Consciousness in the Breakdown of the Bicameral Mind: Scott Alexander’s book review of Julian Jaynes’ masterpiece.

• Planetary Scale Vibe Collapse: Related to the above topic, an ethnologist describes witnessing a transition from Jaynes’ “unconscious” tribal state of mind to a “conscious” individual state of mind. Fascinating reading, but take with a grain of salt.

• Evidence of Fraud in an Influential Field Experiment About Dishonesty: Datacolada provides (nearly) irrefutable evidence of fraud in a paper (about dishonesty!) by an esteemed psychologist.

• Investing for Geeks: Short and sweet investing advice.

• Photopea: Free online Photoshop!

• What to Say Instead of Trivially: Synonyms for “trivially” to use in math proofs that will irritate your readers. For example, “Galois died in order to show us that…”

• Make Your Own 30 kW Induction Furnace: Follow these instructions if you want to make a 30 kW induction furnace in your basement. Or don’t, you might die.

• Official Ordination Package of the Church of the Flying Spaghetti Monster

• Holier than Thou: Precision Holes by Drilling, Boring, and Reaming: A very useful and surprisingly interesting discussion of precision drilling.

Language and History

• Fully Justified Metroid Madness: A Twitter Thread from Matt Gemmell about the most impressive linguistic feat he’s ever seen.

• The Latin Library: a massive online compendium of Latin literature, compiled by William L. Carey.

• The Writing of a Skillful Scribe: Learn Hieratic via the story of the Shipwrecked Sailor, the oldest (complete) story in the world.

• Grisly Assyrian Record of Torture and Death: An article detailing the violence in Assyrian royal inscriptions, and an inspiration for my series on Assyrian kings replying to spam texts.

• Long live the King of Kings! Accession ritual in ancient Persia: The rites of kingship in the Achaemenid Empire.

• Having Had No Predecessor to Imitate, He Had No Successor Capable of Imitating Him: Who wrote the Iliad? A blog post reviewing the history of the Homeric Question, by Fantastic Anachronism, the entirety of which is worth reading, really.

• Learn to Read Korean in 15 Minutes: A comic strip that teaches you to read Korean.

Didactic Math/Physics Blogs

• A Bit Correlated: Tomas Polakovic’s blog that teaches you solid state physics via Julia (the programming language) or vice-versa.

• Trizen's Blog: Daniel Șuteu’s blog about mathematics and computer science, especially number theory. He’s also the only person I’ve ever seen post public answers to Project Euler (good for him, I learned a lot).

• Epsilon Squared: Ethan Epperly’s awesome applied mathematics blog.

Single-Concept Math/Physics Tutorials

• How To Solve It, by Polya -- A Summary: The essence of Polya’s famous book on mathematical problem solving in a single page.

• Feynman's Derivation of the Schrodinger Equation: A mathematical companion piece to Richard Feynman’s Nobel Lecture.

• The U.S. Particle Accelerator School: An online repository of courses and materials on every conceivable aspect of accelerator physics.

• Noises and Causes: The different types of power-law noise processes in physics, their origins and characteristics.

• Spectral Analysis of Noise: Related to the link above, a thorough tutorial on the analysis of noise in time-series data

• The Allan Variance Handbook: A handbook of definitions and tables relating to the Allan Variance, a method of analyzing noise and stability in time-series data

• Power Law Distributions in Empirical Data: Clauset et al. “present a principled statistical framework for discerning and quantifying power-law behavior in empirical data.” Looooots of people misuse power-law fits.

• Legendre Transforms for Dummies: If you ever wondered why there were so many Legendre transforms in thermodynamics or classical mechanics, here you go.

• The Triangle of Power: 3blue1brown explains a novel notation for exponentials, logarithms, and square roots that highlights their inherent symmetry.

• Powerlog: The Missing Operation: Epic Math Time describes the property of the “missing” operation in the sequence addition → multiplication → _______ → exponentiation.

Age of Mythology is a fantastic game. Made by the same studio responsible for the much more popular Age of Empires series (Ensemble Studios), Age of Mythology is a real-time strategy game where you take control of one of three ancient civilizations, the Greeks, Norse, or Egyptians (or the Atlanteans with the Titans expansion, and we won’t count Tale of the Dragon). Each of these factions has a rich mythology, and the normal playstyle of real-time strategy games is modified by god powers and myth units, which are fantastical creatures like a Chimaera or a Troll.

This game is incredibly fun to play, especially when I found the infinite god powers cheat code as a kid (it’s “WRATH OF THE GODS”). I loved playing the campaign, using the cheat, and spamming the Meteor god power until I won.

But one of the other interesting aspects of the game was that whenever you clicked on a unit, like a Villager, it would say something in the language of the ancient civilization you were playing. If you were playing the Greeks, the Villager would speak Ancient Greek, if you were playing the Norse, it would speak Old Norse. If you played AoM for any length of time, I’m entirely certain you remember hearing Prostagma! Voulome! alllll the time from the Greek villagers.

Some Greek and Norse Unit Sounds, Translated

The great thing about the Greek and the Norse unit sounds is that enough people speak these languages to basically recognize the words and translate what they’re saying!

For example, this stackexchange answer gives translations (which I have not verified) for the Greek and Norse units. Here’s some Norse unit sounds:

Veiðimaður (vey-thee-mathur) - Hunter.
Fjósamaður (fyoh-sa-mathur) - Farmer. (literally "Barn man")
Timbursveinn (timber-svait-n) - Woodcutter. (literally "timber-lad")
Grjótsveinn (groht-svait-n) - Miner. (literally "rock-lad")
Satt (saht) - Right. (literally "truly")
Hver er þar? (kwer-air-thar) - Who is there?

A Greek guy on aom.heavengames gives some translations of the Greek unit sounds:

Maista - Μάλιστα - Yes sir/as you command
Prostagma? - Πρόσταγμα - Command? (what's your command)
Etimos - Eτοιμος - Ready
Voulome - Βούλομαι - I want (stupid btw :P)
Hesvole - Εισβολή - Invasion
Esto - Eστω -Allright

What about the Egyptians?

Since no one speaks Egyptian, I have never been able to find an equivalent list of the Egyptian unit sounds, with translations.

The problem, of course, is that no one knows exactly how Ancient Egyptian sounded, so finding random people to translate AoM’s interpretation of Ancient Egyptian is very hard.

However, we do have some idea of what Egyptian sounded like, through some cross-referencing with contemporary Akkadian texts, and phonological reconstruction via its daughter language Coptic. Since Egyptian, like many Semitic languages today, only wrote down consonants, reconstruction of the vowel sounds is particularly difficult.

Because of this, most Egyptologists today use the so-called “Egyptological Pronunciation,” which consists of adding an “e” sound between all written consonants. Of course, it’s a little more complicated than that (we treat some of the pseudo-vowels as true vowels for pronunciation), but that’s the essentials. Thus, nfr becomes “ne-fer.”

I’m fairly certain that the makers of Age of Mythology used this simplified Egyptological pronunciation for the Egyptian units, and I’m pretty sure I can understand some of the words. I’m also pretty sure that some of the phrases are just nonsense, but maybe it’s the pronunciation.

Here’s an example of Egyptian gameplay, with plenty of unit sounds:

Decoding the Egyptian Unit Sounds

Combing through the (scant) online resources about the Greeks and the Norse unit sounds, I find that the three major factions have basically the same vocabulary in use for their units.

Following the stackexchange categorization1, there are four categories that are present in both the Norse and the Greek unit sounds:

1. Expressing Readiness

2. Agreeing

3. Job Descriptions

4. Battle Commands

It stands to reason that the Egyptian units should have the same commands, probably with something approaching the Egyptological pronunciation.

Since I have Age of Mythology: Extended Edition on Steam, I was able to log in and test out the Egyptian faction, and see which actions triggered specific sounds. Below, I will give the phrases that I think I understand, and if I think it’s incorrect grammar, I’ll mark it in red.

However, since I doubt the developers/voice actors spoke Ancient Egyptian, I am going to have to try to guess what they were going for with each pronunciation. In other words, take all this with a grain of salt, because there is a lot of room for error.

My Translations

Note: I’ll put a phonetic transcription in quotes, followed by the context in which that sound is played. Underneath, I’ll write the hieroglyphs, transliteration, and translation. Like so:

Readiness/Agreeing

First up, we have the “readiness” and “agreeing” words that units say when you order them to move somewhere or build something.

I’m nearly certain of “grg” and “tiw,” but the others are speculative. A lot of the units also say “akh” when ordered to do anything, but that doesn’t seem to map onto anything sensible. This is the best I can come up with:

Job Descriptions

Next up, we have the Job Descriptions/Resource Gathering sounds.

I’m almost 100% sure of the first two, the last one is probably pure speculation on my part. That’s a Late Egyptian possessive construction, and I do not see how the AoM devs would have known that. And I can’t understand at all what the units say when you make them gather wood.

Battle Commands

And this is the only battle command I could confidently translate (and the grammar is wrong).

The word “attack” here is a noun. Egyptian writing is pretty ambiguous about nouns and verbs, so this could also be a verb of some kind, but my dictionary says it’s a noun. So this exclamation would be like if your commander ordered you into battle, and your war cry was “Battle!”

A Call for Help

In this article, I’ve translated maybe half of the common phrases you hear in-game. But there are still a lot of these phrases that I just can’t make heads or tails of. For example, what the hell is the pharaoh saying?

If anyone wants to take a crack at further translations, please let me know, this has been bothering me for nigh on fifteen years at this point.

The ageofempires.fandom.com website gives phonetic transcriptions of the common Egyptian unit sounds, which may or may not be helpful, and here’s a video containing all the Egyptian unit sounds, some of which aren’t even used in the game:

An Update from George Wing

Update: April 27, 2024.

An exciting new discovery has been made in the field of forensic video game transcription (archaeoludophonology?).

An interested reader, George Wing, has confidently identified the ubiquitous “ee-ned-djer-ek” sound that the pharaoh makes at 0:07 in the Egyptian gameplay video, as well as several others (that appear to be unused in game), but that appear in the Voice Clips & Sounds.

For the remaining identifications, George points out that since these voice files appear to be unused, they lack the relevant context to make the identifications certain — you can’t see what the units are doing when you click on them. Nevertheless, they fit well into the “job descriptions” schema.

And please don’t forget to subscribe if you liked this article!

I have a twofold purpose for writing such an article.

1. I want recommendations! Got any great history podcasts I haven’t tried? Send them my way! Post them in the comments! Whatever!

2. I want to do a little bit of advertising for smaller podcasts that I think are really good. I think they deserve recognition for excellent work.

This is not an exhaustive list, by the way. I am only going to put podcasts on this list that I have personally listened to many episodes, and really, really love. Further, I’m not going to categorize these podcasts, because that’s hard!

Click the title of each podcast to go to its website, or whatever is the closest I could find to their website. I apologize in advance for my lack of talent writing reviews, they’re all really good, go listen to them.

Less Popular Podcasts I Particularly Want to Highlight

I’m putting these up first because I want them to be more noticed than the really popular ones which I’ll put a bit lower.

In the Shadows of Utopia

A deep exploration of Cambodian history, a subject about which I was entirely ignorant, against the backdrop of the rise of the Khmer Rouge, one of the most (if not the most) murderous regimes to have ever existed. Come for the morbid fascination with Pol Pot and the Killing Fields, but stay for the deeply empathetic retelling of Cambodian history as worthy of study in its own right. The Khmer Empire and the colonization of Indochina by the French are key to comprehending the Khmer Rouge, but I found them fascinating entirely independent of twentieth century events, particularly the ancient Khmer empire.

What is Politics?

More anthropology than history, this podcast explores the origins of social structures, with applications to modern society, and makes a strong case for human anthropology being required study for any politicians. Perhaps “applied anthropology” is a good term for this podcast.

A couple highlights:

• It provides a coherent, consistent definition of political right and left (a travesty that this isn’t a standard definition taught in school),

• It did a hilarious and scathing book review of David Graeber and David Wengrow’s new book The Dawn of Everything,

• It threw a bunch of shade at postmodern academic culture, which I enjoyed with all the bitterness of a physicist forced to teach electromagnetism to humanities majors.

Love this podcast.

The Oldest Stories

Okay, full disclosure, I have not really listened to many episodes from this podcast. However, I have listened to his most recent series on Biblical and Historical Israel, and it catapulted this podcast into favorite territory immediately. His (and others’) theories on the origin of the ethnonym Hebrew as the Akkadian Habiru, meaning bandit or outlaw wanderer drew me in immediately, and the careful exploration of the historicity of the Old Testament cemented this series as an all-time favorite. A must-listen for anyone interested in the Ancient Near East.

History of Persia

This podcast really belongs in the “Single Topic Deep Dives” section, but it’s so good I wanted to advertise? I love the history of the ancient world, and Trevor Culley does an awesome job of bringing to life the story of the Persian Empire from the Persian perspective, a refreshing change from the Greco-Roman lens through which ancient history is often viewed. A couple of random interesting things I learned from the podcast:

• At some point, there was some sort of split split in the ancestor religious traditions of Hinduism/Zoroastrianism, and one side chose the Daevas and one side the Asuras.

• Themistocles, who defeated the Persians at Salamis, was probably playing both sides, and ended up as a governor subordinate to the Persian King.

• There was a community of Jews on the Elephantine, an island in the Nile, who got into a bunch of religious spats with the local Egyptians.

Spec Media

This one is a Hardcore History parody, and brilliant at it. The episodes on the Battle of Endor (Star Wars) in the style of Dan Carlin are hilarious. The narrator absolutely nails Dan Carlin’s verbal quirks, and the description of the horror that the stormtroopers must have felt being swarmed by Ewoks made me crack up.

The Timur Podcast

A podcast detailing the life of Tamerlane, Timur the Lame, or simply Timur, a conqueror in the mold of Genghis Khan, only Muslim and arguably more brutal. His life is incredible, and so is this podcast.

Hardcore History and Friends

Dramatic, multi-episode long-form podcasts diving deeply, but not too deeply, into a variety of historical topics.

Hardcore History

This podcast is the best history podcast out there, hands-down. No other podcast has the combination of narrative and voice-acting that Dan Carlin brings. As he himself points out in every episode, he is not a historian, but he does do his research, and if he sometimes simplifies for narrative effect, or speculates a little further than the evidence will take him, well, so did Herodotus.

My two favorite episodes/series: his series on the Mongol conquests and the Celtic Holocaust.

Martyrmade

Darryl Cooper is a podcaster in the vein of Dan Carlin, that is, a talented non-historian with a gift for compelling storytelling. He has made by far the best podcast I have listened to on the Israel-Palestine conflict, Fear and Loathing in the New Jerusalem. His more recent podcasts, particularly on the American Labor Wars (entitled Whose America?) are also phenomenal. And if he sometimes goes a bit off the deep end (see his podcast about egregores, which are basically thought-demons), it’s always worth listening to for the deep and thorough analysis of the thought-provoking questions that Darryl raises.

History on Fire

Daniele Bolelli’s podcast is the closest podcast in style to Dan Carlin’s Hardcore History. His episodes on American Indian history (his specialty) are particularly good (anything about Sitting Bull, Joe Medicine Crow, the Lakota Sioux, etc.). He’s recently come back from a stint behind a paywall, and I’m having fun listening to his “rerun” content.

Conflicted: A History Podcast

A relatively new (?) podcast with a fantastically interesting selection of topics — the series on the Partition of India became an instant all-time favorite series, and the two-episode look into the Praetorian Guard of the Roman Empire and their predilection for murdering emperors and auctioning off the throne was great. I say new (?), because I pretty regularly trawl for new history podcasts, and this came out of nowhere for me last year and slapped me with about thirty hours of really high quality content. I was very pleasantly surprised.

Conversational History (and Comedy)

Two or more people discuss history in a conversational fashion, with back-and-forth between hosts. Funny and lighthearted, even when the topic is serious. Apparently most historians double as amateur comedy duos.

The Rest is History

Tom Holland and Dominic Sandbrook, the historians and amateur comedy duo behind The Rest is History bring history to life in shorter, that is, hour-long podcasts on an incredible variety of subjects. I loved their recent series about the Hundred Years War, their Mystery of the Cathars, and their episode casting Love Island with historical characters.

Lions Led by Donkeys

I think if I had to choose one word to describe this podcast, it would be “sarcastic.” This hilarious podcast relentlessly mocks the failures of military command. In between excoriating basically everyone who has ever held a weapon, the hosts Joe and Liam manage to insert bits of fascinating military history.

Empire

Anita Anand and William Dalrymple are another example of the historian (and journalist) comedy duo that is a surprisingly common trope among history podcasts. Fortunately, they’re my favorite (second favorite? I can’t decide between this podcast and The Rest is History). Dalrymple’s specialty is 18th-19th century India, and it really shows in their episodes on the East India Company. I particularly loved the incredibly bloody story of the Koh-i-Noor diamond. Both hilarious and horrifying, those episodes made me believe that the curse of the Koh-i-Noor is real.

Single Topic Deep Dives

Hundreds of episodes on a single topic, in bite-sized pieces. Multi-year grand narratives of a civilization, usually.

History of Rome

Basically the gold-standard for history podcasting, along with Hardcore History, Mike Duncan takes you on an exhaustive journey through the entire history of Rome, (up to the Byzantines), in an entertaining and informative podcast. His humor is pretty dry, so you’ve got to look for it, but it’s there and really brightens it up. Nothing much to say really, if you’re interested in Rome, this is the podcast for you.

Revolutions

Mike Duncan’s second offering, going through most of the world’s major revolutions. One series per revolution, and my favorites were the French Revolution and the Russian Revolution. Each series is long enough to be its own podcast, and each is done with the attention to detail and dry humor that characterizes Mike Duncan. Seriously, go listen to the French Revolution series, it’s absolutely fantastic.

History of Japan

Last but not least, Isaac Meyer has been podcasting about Japanese history for like a decade now, and he’s really good at it. His podcast is not chronologically ordered, so search his back catalog for topics that stick out as interesting to you. I loved the series on the Birth of the Samurai, and its companion series the Fall of the Samurai.

A Quick Recap of the Hubbard-Demelza Model

Last time, we derived a model for the dynamics of fermionic guinea pigs in pigloos, and I began to really wish that Substack supported LaTeX.

In this model, we have three terms. The first term is known as the “hopping” or “tunneling” term, and describes a pig hopping from pigloo β to pigloo α, while preserving its spin σ. The second term is the self-energy term, and counts the number of pigs, yielding a quantum of energy E for each pig, regardless of state. The third term is the interaction term, which adds an additional energy when two pigs share a pigloo. Pigs generally don’t like sharing pigloos.

In honor of John Hubbard, who first derived this Hamiltonian for solid-state physics and of Demelza, the friendliest guinea pig ever, we’re going to name our Hamiltonian the Hubbard-Demelza model, and solve it exactly in the case of two pigloos. We’ll simulate some more guinea pig dynamics in larger pigloo lattices, but the two-pigloo case is of fundamental importance, because my parents’ guinea pigs have two pigloos.

Simplification in the Pigloo Dimer Case

First we’re going to simplify the Hamiltonian a little bit. You see, that second term in the Hamiltonian that simply counts the quanta of energy for each pig, and captures no interactions between pigs or between pigloos. It’s not a very interesting term, and in fact only causes a constant energy shift without affecting the pig dynamics.

So let’s cross off that second term, and write out the Hamiltonian explicitly for the two-pigloo case, which we’ll call a Hubbard-Demelza Dimer.

This Hamiltonian is actually quite simple. Assuming all guinea pigs are in the ground state and not excited (no fresh produce), there are 4 states per pigloo, and 2 pigloos, giving us 4^2, or 16 total states. I’m not going to enumerate them, but imagine all sixteen of them watching you with their beady little eyes.

Remember that the Schrodinger equation (our goal) is a matrix equation, and we’ve just constructed an abstract representation of our 16x16 Hamiltonian matrix. Now we need the explicit matrix form.

Since we have sixteen total states (our Hilbert space has dimension 16), our Hamiltonian matrix is 16x16, which is pretty large. Fortunately for us, the Demelza-Hubbard model can be partitioned into much smaller, easier to handle subspaces.

Conservation of Pigs

The Demelza-Hubbard model has no term for generating guinea pigs ex vacuo, nor for destroying them.

Consider five sub-cases of our problem, containing, respectively, 0, 1, 2, 3, and 4 guinea pigs. Conservation of pigs implies that the Hamiltonian matrix has no transition elements that couple these subspaces. Thus, our Hamiltonian is block-diagonal, and the pig number states can be treated as evolving separately.

Conservation of pigspin

Each pig has a pigspin of +1/2, an up-pig, or -1/2, a down pig. Our Hubbard-Demelza Hamiltonian has no terms which correspond to spin-flipping — an up-pig remains an up-pig when hopping pigloos. This means the total pigspin S_z is also conserved, and we can partition the Hamiltonian by total pigspin as well.

Subspace partitions

The following table lists all the independent subspaces, and numbers the pig states allowed by the Pauli Exclusion Principle in each state.

Each of these nine subspaces may be solved separately.

Exact Solutions to the Half-Filled Dimer

Let’s look at the largest subspace, the 2-pig, 0-pigspin subspace containing one up pig and one down pig.

Here are the possible states:

A general vector in our system will be represented by

If we apply our Hamiltonian to these basis states, we can explicitly construct its matrix form. The procedure is a bit involved, and is left as an exercise for the reader, but here’s the answer:

Let’s first look at the energy spectrum. The eigenenergies and eigenstates of the system (the stable states and their energies), can be found by diagonalizing the Hamiltonian, and this gives us:

Now what are these states? Roughly, these are the stable states of the system, the ones in which the guinea pigs remain stationary, i.e. the pigloo occupation does not change with time.

For example, consider the first state:

This even superposition of singly occupied pigloos is a stable state. A guinea pig thus prepared will remain in this state indefinitely. It also happens to have zero net energy.

The energies of the other states are not so simple, and are controlled sensitively by the ratio of U to t, called k = U/t here. When U » t, the pig-pig interaction dominates, and when U « t, the pig-tunneling behavior dominates.

As you can see, when U/t → infinity, we have ω+ → infinity, and ω- → 0. We see the blue & green and red & orange states become degenerate, and the singly-occupied pigloos become highly favorable. This is also known as the highly-correlated limit.

By contrast, when U/t → 0, pig hopping energy dominates, the blue & orange states converge, and we have three classes of nondegenerate states. This is known as the tight-binding limit, where guinea pigs are tightly bound to their individual pigloos.

Two-Pigloo Dynamics

We can now solve the Schrodinger Equation by direct integration, and look at its dynamics.

Let’s prepare our system with in an initial condition that is not a stable state, so that we can see how the pig states evolve with time. I’ll choose U = t = 1, an equal balance of pig hopping and interaction energy.

The evolution of the pig states is then given by the solution to the Schrodinger equation above. Here, we plot the probability of finding the pigs in a given state vs. time.

Notice that if we prepare the pigs in the initial state |↑↓,0>, after about 6 units of time, the pigs will be found, with 100% probability, in the state |0,↑↓>. This is a coherent transfer of two pigs between pigloos. Cool!

We can compare the highly correlated limit to the tight binding limit, and see the differences:

The highly-correlated limit behaves as an effective two-state system, and we can excite guinea pig Rabi oscillations. The tight-binding limit shows similar, more complex oscillations involving all four states.

Simulations of N-pigloo systems

The two state model is very cool, and gives us a really good idea of how to control pig dynamics in simple systems. Larger systems become computationally intractable quite quickly, but we can look at an 8-pigloo lattice pretty easily.

Disclaimer: It’s about here that my expertise ends. We’re moving into computational quantum mechanics with nontrivial problems. I am fairly sure I did the following simulation right, but who knows?

Let’s simulate what happens in an 8-pigloo lattice when you initialize four guinea pigs into it. In an attempt to reduce errors, I’m going to use the OpenFermion package from Google, and modify their Hubbard Model Notebook for guinea pig dynamics.

Here is the Google Colab code I’m using.

Half-Filling of 8-pigloo 1D lattice

Okay, so let’s take the 8-pigloo lattice (8 adjacent pigloos) and initialize them with four pigs, two up and two down, in the first two pigloos. I’m using the Hartree-Fock method of initialization from OpenFermion, which, if I’m being honest, I do not fully understand. However, the state is going to look something like this:

As you can see, the pig density (the expectation values of the number operators for up and down pigs) shows two pigs in each of the first two sites. Due to the Pauli Exclusion principle, the two pigs in each pigloo have opposite pigspin.

Now, using a Trotterization method where we split the Hamiltonian up into parts, evolve each part separately, then multiply their propagators, we can simulate the evolution of the pig density. I’m using t = 1, U = 2 here.

The following figure plots the pig density over time, reading left-to-right, top-to-bottom.

As you can see, the initial concentration of pigs in the left-most pigloos spreads over time to become a more uniform distribution of pigs, but it does so as a wave of pigs traveling through the lattice, eventually breaking on the right-most edge of the lattice. This behavior agrees well with experiments conducted in the lab, though sometimes the particles get scared, run off, and have to be lured back with treats.

The Most Important Question

So that’s it! This is the culmination of our foray into quantum dynamics of guinea pigs. We’ve explored the consequences of the fermionic nature of guinea pigs, analytically solved the important pigloo dimer case, and simulated a probability wave of guinea pigs as they spread themselves out among adjacent pigloos.

We’re finally in a position to ask the most important question of all. Can we make a quantum computer from guinea pigs?

Simplifying a bit, universal quantum computation requires 4 elements. If guinea pigs can do all of them, we can make a quantum computer out of them.

1. A Qubit

A qubit is simply a two-level quantum system. You must have a way to represent 0, 1, and any arbitrary superposition of the two.

This is trivial with guinea pigs. We could use their pigspin degree of freedom, for example, with up-pigs representing 0 and down-pigs representing 1.

We could also use a dual-rail encoding scheme, where we label one pigloo ‘0’, and another pigloo ‘1’, and the qubit is encoded by whichever pigloo contains a guinea pig.

2. Single-Qubit Gates

We must be able to manipulate the quantum state of a single pig. Again, this is trivial with guinea pigs, as they’re relatively easy to catch, having very short legs.

3. Two-Qubit Gates

This one is tricky. We need to generate entanglement between guinea pigs for quantum computation. Entanglement is a much-mythologized property of quantum mechanics, where two independent single-particle states become a single, indivisible dual-particle state. This is where all the power of quantum computing comes from!

The simplest way to do this with guinea pigs would be to find some sort of guinea pig beamsplitter and interfere pigs. I am not currently sure how to do this, but it’s an active area of research. Presumably, some form of guinea pig obstacle course with multiple inlets and outlets would do the job.

Larger lattices of such guinea pig interferometers could be used for Bell state preparation.

4. Measurement

Finally, we need some way to read out the guinea pig state with high fidelity. Naturally, disturbing the guinea pigs alters their state, and so this is also quite tricky. Fortunately, the advent of camera phones makes it quite easy to photograph unsuspecting guinea pigs.

In conclusion, yes, it is possible to build a quantum computer with guinea pigs, indeed, it is a much more promising approach than several current approaches to quantum computing! However, given the need of ~millions of qubits for a useful simulation, the overhead in hay and lettuce would make such a computer prohibitively expensive.