Littlewood's Law and the Global Media

Selection effects in media become increasingly strong as populations and media increase, meaning that rare datapoints driven by unusual processes such as the mentally ill or hoaxers are increasingly unreliable as evidence of anything at all and must be ignored. At scale, anything that can happen will happen a small but nonzero times.
topics: politics, psychology, sociology, statistics, philosophy
created: 15 Dec 2018; modified: 18 Feb 2019; status: finished; confidence: highly likely; importance: 5


Online & mainstream media and social networking have become increasingly misleading as to the state of the world by focusing on ‘stores’ and ‘events’ rather than trends and averages. This is because as the global population increases and the scope of media increases, media’s urge for narrative focuses on the most extreme outlier datapoints—but such datapoints are, at a global scale, deeply misleading as they are driven by unusual processes such as the mentally ill or hoaxers.

At a global scale, anything that can happen will happen a small but nonzero times: this has been epitomized as “Littlewood’s Law: in the course of any normal person’s life, miracles happen at a rate of roughly one per month.” Hence, there will be enough ‘miracles’ that all media coverage of events can potentially be composed of nothing but extreme chance chances, even though it would seem like an ‘extraordinary’ claim to say that all media-reported events may be flukes.

Given this, it is important to maintain extreme skepticism of any individual anecdotes or stories which are selectively reported but still claimed (often implicitly) to be representative of a general trend or fact about the world. Standard techniques like critical thinking, emphasizing trends & averages, and demanding original sources can help fight the biasing effect of news.

Noticing an anomaly in the narrative, in March 2017 noted a followup to a news story which got much less press than did the original news story:

Remember how everyone was talking about how Trump must have among his supporters? And remember how some of the incidents were traced to an anti-Trump socialist working at a leftist magazine? Well, the rest of them seem to be the fault of . The Atlantic has of the whole affair.

(His autism/brain tumor defense did not succeed, and he was ultimately .)

Littlewood’s Law

This is an interesting one because it illustrates a version of of Miracles”: in a world with ~8 billion people, one which is increasingly networked and mobile and wealthy at that, a one-in-billion event will happen 8 times a month. Littlewood’s law is itself a special-case of “the Law of Truly Large Numbers”:

The Law of Truly Large Numbers. Succinctly put, the law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen. The point is that truly rare events, say events that occur only once in a million [as the mathematician Littlewood (1953) required for an event to be surprising] are bound to be plentiful in a population of 250 million people. If a coincidence occurs to one person in a million each day, then we expect 250 occurrences a day and close to 100,000 such occurrences a year.

Going from a year to a lifetime and from the population of the United States to that of the world (5 billion at this writing), we can be absolutely sure that we will see incredibly remarkable events. When such events occur, they are often noted and recorded. If they happen to us or someone we know, it is hard to escape that spooky feeling.

Human extremes are not only weirder than we suppose, they are weirder than we can suppose.

Politics

Hate crimes, and Anti-Semitic attacks are pretty rare in any absolute sense in the USA (a country of >325m people), so it doesn’t require a common cause to account for such rare effects. A surprising number of hate crimes turn out to be hoaxes, perpetrated by a member of the targeted group; it might seem crazy for, say, a black person to fake a burning cross on their lawn or a hanging noose, but apparently every once in a while, a black person has sufficient reason to do so. The problem is, there need not be any sufficient reasons. In accounts of con artists, one of the most consistent themes is how understandable their schemes are when you appreciate how much good faith we assume and take on faith, and how otherwise miserably & pathetically understandable they and their malice is (borrowing & stealing wealth & power to fill the empty void within themselves); while in accounts of forgers, hoaxes, and fabricators, the most consistent theme is that the investigator, after exhausting all avenues, examining all minor contributing factors, unconvincingly laying out all sensible motivations like career advancement, frequently interviewing them at length only to be baffled by deflections, and lies, is finally left in silence. Why did they do it? No one knows.

If someone said, “I don’t really believe these anti-semitic hoaxes are real in the sense of a bunch of anti-Semites have been emboldened by Trump’s election, I think there’s something else going on, like maybe an employee made them up to drum up donations”, you would probably think that was excuse-making; if they had said, “I don’t believe them, maybe they’re actually fake because some schizophrenic or crazy Jew with a brain cancer & a flair for VoIP pranks did them all themselves”, you would definitely think they were desperately coming up with excuses & denying facts, and to not put too fine a point on it, that they should be ashamed of themselves for such a lack of intellectual honesty & flagrantly partisan bias.

Yet, there you have it! It is apparently a real thing, that a (self-hating?) Jew halfway across the world in Israel decided to spend all his spare time hoaxing over the Internet dozens of Jewish institutions with hate-crimes in the US post-Trump-election in part because he is an anti-social & autistic criminal, who may be driven in part by a brain tumor causing a severe personality disorder. It sounds absurdly implausible and made up—yet, among ~8 billion people, there turns out to be at least one evil brain-tumor phreaker Jew, and we all got to hear about his handiwork. “My, Earth really is full of things.”1 (One of the other culprits for the anti-semitic bomb threats, incidentally, was .)

Or consider the by Nasim Najafi Aghdam, unusual for being a mass shooting perpetrated by a woman, but also bizarre in that by the self-described “first Persian female vegan bodybuilder”2 was apparently YouTube removing ads from her pro-veganism & exercise videos popular in Iran. Or how about that English kid who Or the collectively.

Technology

Industrial accidents are similar. In industrial accidents, post-mortems often detail a long series of unlucky chances and interacting failures which all combine to lead to the final explosion. the ‘swiss cheese model’ imagines each layer of systems as being like a slice of Swiss cheese and only when the holes of 6 or 7 layers line up, can anything fall through: The systems were always failing to some degree, but are so redundant that a total failure is avoided, until it happens, and one marvels that 7 different things all went wrong simultaneously. Precisely , planes no longer crash for boringly plausible reasons like “the propeller fell off the plane” or “the pilot couldn’t see the ground in the fog”, and the remaining aviation incidents now tend to be astonishing in some way; the required not just a suicidal pilot who wanted to take a whole plane with him but also abuse of post-9/11 security mechanisms intended to prevent hijacking airplanes & crashing them, or the remarkable idiocy of the co-pilot of , or… whatever it was that happened to . In technology, software engineers who work on global-scale systems (sometimes called “hyperscalers”) are forced to confront the fact that at scale just about anything that can happen will happen eventually—only very rarely, to be sure (otherwise they’d’ve been fixed long before) but a nonzero number of times, and that may be enough to trigger a new failure mode and damage or even collapse computer systems (which remain rather fragile compared to all other systems). These anomalies triggering bugs make fun war stories, but also make a more important point about reality exceeding the imagination of designers, when systems fail in ways or datapoints arise that people (“what do you mean, a can have anywhere from 1 to 48 bits‽”).

Science

Think about scientific papers. Imagine the ideal scenario in which models are always correct, all plans are pre-registered, etc. Because of the massive exponential expansion of the academic-industrial complex worldwide, there’s something like 1 million papers published each year; assuming (unfortunately) fairly normal research practices of testing out a few configurations on a few subsets and using a few covariates and to decide on statistical approach, each paper has the equivalent of hundreds or thousands of NHST tests; thus, it is entirely possible to legitimately see a p=(1 in 1 billion) or p<0.00000005 just when the null is true (), and if you consider just the most recent set of papers from the past decade or so, you could see p<0.0000000005. All with the null hypothesis being true. Of course, in practice, things are far worse than that. Throw in the low but non-zero base rate of fraud, questionable research practices, incorrect parametric modeling assumptions, endemic publication bias, odd phenomenon like the “lizardman constant” in surveys (where a tiny fraction of respondents will always just answer at random or give the troll answer), etc, and there’s a point at which no matter how many studies there are on a particular effect, you still don’t have particularly strong belief in it because the data may simply be measuring ever more precisely the level of crud in that field rather than the substantive effect you want interpret to it as (, but for biases).

Media

Can we trust film or photographs because they look real? “After all, no hoaxer would be able to or be able to afford to make such a realistic video”, right? Of course not. Not because of “Deep Fakes”, but because humanity has devoted itself with extreme assiduity to churning out millions of highly sophisticated ‘fake news’, applying its utmost ingenuity and considerable resources to… making fictional depictions of fake events, such as Hollywood movies. Many hoaxes or fakes are of high quality simply because they are recycled from commercial media, special effects, mockumentaries, etc, which have the highest standards and often are deliberately designed to erase any hints of being fiction. To give an example, likely hundreds of thousands of people were convinced by a video of a school cafeteria spiked with laxatives, with students soiling themselves; after all, the prank’s so realistic, with its cellphone footage and so many different students affected by vomiting/pooping, certainly no random Internet troll with Photoshop could possibly have faked it—and the hoaxers didn’t, because it was from a . Which series? Well, one you’ve almost certainly never heard of (much less watched), inasmuch as thanks to Netflix & other trends there are now in the USA alone. No one could ever have heard of more than a minute fraction of these US series, but every year there is more accumulated high-quality fictional video available to be weaponized. Fortunately, a laxative prank does not matter, but imagine at some point a bright-eyed young liberal director decides to make a mockumentary of the Trump administration, complete with ‘pee tape’? (How about ) Nor does there need to be a ‘hoaxer’, per se: these can be emergent (a “stand alone complex”?)—perhaps someone saw a clip and didn’t notice the metadata, or posted it with no metadata and a viewer assumes it’s real and reshares it, and that is how the viral hoax comes into being. When it comes to media, “three men make a tiger”.

Tails at Scales

“‘What did the Men of old use them for?’ asked Pippin, delighted and astonished at getting answers to so many questions, and wondering how long it would last. ‘To see far off, and to converse in thought with one another,’ said Gandalf….In the days of his wisdom Denethor would not presume to use it [the palantír] to challenge Sauron, knowing the limits of his own strength. But his wisdom failed…He was too great to be subdued to the will of the Dark Power, he saw nonetheless only those things which that Power permitted him to see. The knowledge which he obtained was, doubtless, often of service to him; yet the vision of the great might of Mordor that was shown to him fed the despair of his heart until it overthrew his mind.”

Gandalf, The Return of the King

As time passes, it becomes increasingly hard to believe rare events at face value, and one has to simply “defy the data”. Sure, that video looks real, but it probably isn’t; it’s bizarre that anyone would run all those bomb hoaxes, but maybe someone did and it wasn’t a vast anti-semitic terrorism wave; and maybe the co-pilot just decided to crash the plane and it wasn’t an ISIS attack after all. At some point, you may have to simply start ignoring all anecdotes & individual datapoints because they border on zero evidence and a priori may simply be fake.

This is life in a big world, and it’s only getting bigger as the global population grows, wealth & leisure grow, and technologies advance. (If you thought humans could think & do weird things and fail in weird ways, just wait until everyone gets their hands on good AI tech!) There are billions of people out there, and anything that can go weird, will. The “Everything not forbidden is compulsory.”

Epistemological implications

Nevertheless, !

Because weirdness, however weird or often reported, increasingly tells us nothing about the world at large. If you lived in a small village of 100 people and you heard 10 anecdotes about bad behavior, the extremes are not that extreme, and you can learn from them (they may even give a good idea of what humans in general are like); if you live in a ‘global village’ of 10 billion people and hear 10 anecdotes, you learn… nothing, really, because those few extreme anecdotes represent extraordinary flukes which are the confluence of countless individual flukes, which will never happen again in precisely that way (an expat Iranian fitness instructor is never going to shoot up YouTube HQ again, we can safely say), and offer no lessons applicable to the billions of other people. One could live a thousand lifetimes without encountering such extremes first-hand, rather than vicariously.

This is not due to whipping boys like “social media” or “Russian trolls”—all of this would be a problem regardless. The media can report with perfect accuracy on each (genuine) incident, but the mere fact of reporting on them and us learning about such vanishingly weird incidents is itself the problem—we can’t put the proper psychological weight on it. This is not just a selection bias3, it is a selection bias which gets worse over time.

Coping

’In the autumn of 1939, and his young Cambridge student and friend were walking along the river when they saw a newspaper vendor’s sign announcing that the Germans had accused the British government of instigating a recent . When Wittgenstein remarked that it wouldn’t surprise him at all if it were true, Malcolm retorted that it was impossible because “the British were too civilized and decent to attempt anything so underhand, and . . . such an act was incompatible with the British ‘national character’.” Wittgenstein was furious. Some 5 years later, he wrote to Malcolm:

“Whenever I thought of you I couldn’t help thinking of a particular incident which seemed to me very important. . . . you made a remark about ‘national character’ that shocked me by its primitiveness. I then thought: what is the use of studying philosophy if all that it does for you is to enable you to talk with some plausibility about some abstruse questions of logic, etc., & if it does not improve your thinking about the important questions of everyday life, if it does not make you more conscientious than any . . . journalist in the use of the dangerous phrases such people use for their own ends. You see, I know it’s difficult to think well about ‘certainty’, ‘probability’, ‘perception’, etc. But it is, if possible, still more difficult to think, or try to think, really honestly about your life & other peoples lives.”4

What can we do in self-defense?

We could start trying to structure our communications in a way which embodies the true proportions, and builds in the weighting we are unable to do.

  • Crime and crime rates are an easy one—falls in the crime rate should get as much space as the total of individual crimes; if a murder gets a headline, then a year with 50 fewer murders should get 50 headlines about the that reduction’s 50 non-murders (because surely avoiding a murder is as good news as a murder is bad news?).

  • Perhaps in one format, discussion could be weighted similar to a meta-analytic weighting of effect sizes: you are allowed to discuss both anecdotes and studies, but the number of words about a anecdote or study must be weighted by sample size.

    So if you write 1 page about someone who claims X cured their dandruff, you must then write 100 pages about the study of n=100 showing that X doesn’t cure dandruff. That’s only fair, since that study is made of 100 anecdotes, so to speak, and they are as deserving of 1 page as the first anecdote.

  • Weighting could be applied to costs & benefits as well: in a discussion of clinical trial design and bioethics of randomized experiments and whether it can be ethical to run a RCT, one could allow discussion of the Tuskegee syphilis experiment (affecting 399 men) but only if one then has proportionately much discussion of the estimates of the number of people hurt by small underpowered incorrect or delayed randomized trials (usually estimated in the millions), which might require some advanced typographic innovations.

  • A “proportional newspaper” might allocate space by geographic region populations, so there’s a giant void with a tiny little 2-line wire item for Africa, while the (much smaller) USA section requires a microscope.

  • What if one wrote movie or book summaries in a strict scaling of 100 words per X minutes/pages, instead of relying on fading memories ? After all, that’s how one has to consume them, at 1 second per second, and what the experience actually is.

    It seems peculiar that reviews will describe hours of material in a few sentences, and then a 30 second scene might get a loving multi-page description and analysis, since that is not how one watches the movie, and that gives a misleading view of the movie’s pacing, if nothing else. What if social media stopped prioritizing recent short items and instead gave visual real estate in proportion to how old something is?

  • Weight by age: If someone is rereading a 50-year-old essay, that should be given more proportionally more emphasis on a social media stream than a 5-minute old Tumblr post.

More immediately, you should keep your eye on the ball: ask yourself regularly how useful news consumption has really been, and if you justify it as entertainment, how it makes you feel (do you feel entertained or refreshed afterwards?), and if you should spend as much time on it as you do; take try to cut back or ignore recent news (perhaps replace a daily newspaper subscription with a weekly periodical like The Economist and especially stop watching cable news!); shift focus to topics of long-term importance rather than high-frequency noise (eg scientific rather than polling or stock market articles); don’t rely on self-selected convenience samples of news/opinions/responses/anecdotes brought to you by other people, but make your own convenience sample which will at least have different biases and be less extreme (ie don’t go off 10 comments online, ask 10 of your followers instead, or read 10 random stories instead of the top 10 trending stories); insist on sources (if you don’t have time to trace something back to its source, then your followers collectively don’t have time to spend reading it)5; read articles to the end (many newspapers, like the New York Times, have a nasty habit of including critical caveats—at the end, where most readers won’t bother to read to); discount things which are “too good to be true”; focus on immediate utility; try to reduce reliance on anecdotes & stories; consider epistemological analogues of like simply throwing out the top and bottom percentiles of data; and pay attention to the trends, the big picture, the central tendency, not outliers.

The world is only getting bigger.

See also

Appendix

Origin of “Littlewood’s Law of Miracles”

I try to trace back “Littlewood’s Law of Miracles” to its supposed source in Littlewood’s A Mathematician’s Miscellany. It does not appear in that book, and further investigation indicates that Littlewood did not come up with it but that Freeman Dyson coined it in 2004, probably based on the earlier “Law of Truly Large Numbers” coined by Diaconis & Mosteller 1989, in a case of Stigler’s law.

and other sources on “Littlewood’s Law of Miracles” all attribute it to mathematician (best known for his collaborations with Hardy & Ramanujan). Curiously, no one ever quotes Littlewood’s original formulation but typically a paraphrase by :

Littlewood’s law of miracles states that in the course of any normal person’s life, miracles happen at a rate of roughly one per month.

Paraphrases are often wittier & more memorable than the original, but I do like to see the originals to see what else they said. WP attributes the quote to a Littlewood anthology of essays, Littlewood’s /Littlewood’s Miscellany (1953/1986), without specifying chapter or page number. (Indeed, no writer on Littlewood’s Law specifies chapter/page number when citing either version of A Mathematician’s Miscellany.)

Puzzlingly, at no point in the book, either the or 1986 editions (which appear near-identical), does Littlewood ever define a “law of miracles” or speak of “one per month”.

The relevant essay/chapter appears to be “Large Numbers”, which is a discussion of large numbers such as astronomical units, switching over to probabilities & coincidences. Littlewood goes through a miscellany of calculations intended to show that various unlikely things would be expected to happen in England or the world based purely on probability, and ends with a discussion of integer factoring.

This section is a logical place for him to define “Littlewood’s law”, but he never does. The closest that he comes is the section of the subchapter, “Large Numbers: Coincidences and Improbabilities §12”, where he discusses a statistical thermodynamics question of heat (a puzzle we would probably describe as “how likely is it that a snowball could survive a week in Hell by random thermal fluctuations?”), where he offhandedly describes the necessary enormously-improbable macro fluctuation as a “miracle”. (He ultimately concludes that, if I understand the units correctly, the snowball would have a chance of survival of just 1 in .) The word “million” does not appear, but going back 5 pages to §5, Littlewood offhandedly employs the unit 106 (ie 1 million) as apparently a kind of cutoff for an impressive coincidence:

§5. Improbabilities are apt to be overestimated. It is true that I should have been surprised in the past to learn that [atheist] Professor Hardy had joined the [Christian AA-predecessor] . But one could not say the adverse chance was 106 : 1. Mathematics is a dangerous profession; an appreciable proportion of us go mad, and then this particular event would be quite likely.

…I sometimes ask the question: what is the most remarkable coincidence you have experienced, and is it, for the most remarkable one, remarkable? (With a lifetime to choose from, 106 : 1 is a mere trifle.) This is, of course, a subject made for bores, but I own two, one starting at the moment but debunkable, the other genuinely remarkable…

Searches for “month”/“million”/“miracle” all failing and having reached a dead end with Littlewood himself, I turned back to examine the Freeman Dyson source more carefully in the hopes of a quote or exact page number.

The source for Dyson’s paraphrase of Littlewood is a 2004 New York Review of Books book review “One in a Million”, reviewing a 2004 translation of a French book about skepticism (Charpak & Broch’s , translated by Bart K. Holland).

Dyson’s review is (as usual for the NYRB) behind an impenetrable paywall but the review was reprinted in 2006 as chapter 27 of Dyson’s collection (ISBN: 1590172167), which is easily accessible, and the relevant sections about Littlewood read:

…The book also has a good chapter on “Amazing Coincidences.” These are strange events which appear to give evidence of supernatural influences operating in everyday life. They are not the result of deliberate fraud or trickery, but only of the laws of probability. The paradoxical feature of the laws of probability is that they make unlikely events happen unexpectedly often. A simple way to state the paradox is Littlewood’s law of miracles. John Littlewood was a famous mathematician who was teaching at Cambridge University when I was a student. Being a professional mathematician, he defined miracles precisely before stating his law about them. He defined a miracle as an event that has special significance when it occurs, but occurs with a probability of one in a million. This definition agrees with our commonsense understanding of the word “miracle.”

Littlewood’s law of miracles states that in the course of any normal person’s life, miracles happen at a rate of roughly one per month. The proof of the law is simple. During the time that we are awake and actively engaged in living our lives, roughly for eight hours each day, we see and hear things happening at a rate of about one per second. So the total number of events that happen to us is about 30,000 per day, or about a million per month. With few exceptions, these events are not miracles because they are insignificant. The chance of a miracle is about one per million events. Therefore we should expect about one miracle to happen, on the average, every month. Broch tells stories of some amazing coincidences that happened to him and his friends, all of them easily explained as consequences of Littlewood’s law.

…If this idealized picture of a telepathy experiment were real, we should long ago have been able to decide whether telepathy exists or not. In the real world, the way such experiments are done is very different, as I know from personal experience. When I was a teenager long ago, parapsychology was fashionable. I bought a deck of and did card-guessing experiments with my friends. We spent long hours, taking turns at gazing and guessing cards. Unlike Broch, we were strongly motivated to find positive evidence of telepathy. We considered it likely that telepathy existed and we wanted to prove ourselves to be telepathically gifted. When we started our sessions, we achieved some spectacularly high percentages of correct guesses. Then, as time went on, the percentages declined toward twenty and our enthusiasm dwindled. After a few months of sporadic efforts, we put the cards away and forgot about them.

Looking back on our experience with the cards, we came to understand that there are three formidable obstacles to any scientific study of telepathy. The first obstacle is boredom. The experiments are insufferably boring. In the end we gave up because we could not stand the boredom of sitting and guessing cards for hours on end. The second obstacle is inadequate controls. We never even tried to impose rigorous controls on communication between sender and receiver. Without such controls, our results were scientifically worthless. But any serious system of controls, stopping us from chatting and joking while we were gazing and guessing, would have made the experiments even more insufferably boring.

The third obstacle is biased sampling. The results of such experiments depend crucially on when you decide to stop. If you decide to stop after the initial spectacularly high percentages, the results are strongly positive. If you decide to stop when you are almost dying of boredom, the results are strongly negative. The only way to obtain unbiased results is to decide in advance when to stop, and this we had not done. We were not disciplined enough to make a decision in advance to do 10,000 guesses and then stop, regardless of the percentage of correct guesses that we might have achieved. We did not succeed in overcoming a single one of the three obstacles. To reach any scientifically credible conclusions, we would have needed to overcome all three.

The history of the card-guessing experiments, carried out initially by at Duke University and later by many other groups following Rhine’s methods, is a sorry story. A number of experiments that claimed positive results were later proved to be fraudulent. Those that were not fraudulent were plagued by the same three obstacles that frustrated our efforts. It is difficult, expensive, and tedious to impose controls rigorous enough to eliminate the possibility of fraud. And even after such controls have been imposed, the conclusions of a series of experiments can be strongly biased by selective reporting of the results. Littlewood’s law applies to experimental results as well as to the events of daily life. A session with a noticeably high percentage of correct guesses is a miracle according to Littlewood’s definition. If a large number of experiments are done by various groups under various conditions, miracles will occasionally occur. If miracles are selectively reported, they are experimentally indistinguishable from real occurrences of telepathy.

Dyson 2004 does not attribute Littlewood’s Law to A Mathematicians Miscellany and gives no source at all. One might guess that the implicit source is the “Amazing Coincidences” chapter of Debunked!, but upon checking, Debunked! does not mention Littlewood anywhere. (The “Amazing Coincidences” chapter is, however, in the spirit of “Coincidences and Improbabilities”, and a more pleasant read.)

It is unclear why Dyson describes Littlewood as having defined “miracles precisely” as being events with “a probability of one in a million”, since no definition of “miracle” occurs in the presumed source and the only use of the word “miracle” (in the snowball Hell example) refers to a probability astronomically rarer, unless we take Littlewood’s use of 106 as his definition of a criteria & are free with putting “miracle” in Littlewood’s mouth. But even assuming this, nowhere in A Mathematician’s Miscellany can I find anything like that analysis about 8 active hours a day or things happening one per second or a million “events” a month.

Are there any other sources?

Checking Google Scholar & Google Books for “Littlewood’s Law” prior to 2004, there are no hits for anything like “Littlewood’s Law of Miracles”. (There is one hit for an , and there is an amusing criticism of a mathematical logic textbook by : “[The book] constantly violates Littlewood’s law of exposition: Do not omit from the presentation of an argument two consecutive steps.” But no miracles or statistics.) Checking several dozen discussions of the Law in general Google hits, all date after 2004 and appear to trace back to Dyson 2004 or later sources.

The closest thing to a predecessor I found was the paper , & 1989, which discusses the same topic as Littlewood/Charpak-Broch/Dyson, and in analyzing the same phenomena of “extraordinary” events in ordinary life and making some cute analyses (like an explanation of Baader-Meinhof effects as in a ) coins a law, the :

The Law of Truly Large Numbers. Succinctly put, the law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen. The point is that truly rare events, say events that occur only once in a million [as the mathematician Littlewood (1953) required for an event to be surprising] are bound to be plentiful in a population of 250 million people. If a coincidence occurs to one person in a million each day, then we expect 250 occurrences a day and close to 100,000 such occurrences a year.

Going from a year to a lifetime and from the population of the United States to that of the world (5 billion at this writing), we can be absolutely sure that we will see incredibly remarkable events. When such events occur, they are often noted and recorded. If they happen to us or someone we know, it is hard to escape that spooky feeling.

Diaconis & Mosteller 1989 anticipate Dyson 2004 in defining “one in a million” as a criteria for “surprising” based on Littlewood’s invocations of 106, and puts it in terms of individuals & days, although they do not give any estimate involving seconds or months for individuals. Importantly, despite citing Littlewood 1953, Diaconis & Mosteller 1989 do not mention or give any sign of knowing any Law.

So, by all available evidence, “Littlewood’s Law of Miracles” did not exist in print before Dyson 2004 coined it.

This suggests that Dyson, perhaps as a student at Cambridge University as he mentions (1940–1942, Fellow 1946–1949), heard an extended or folkloric version before Littlewood 1953, and only mentioned it 62 years later in print. More likely, Dyson is extending Diaconis & Mosteller 19897 but misattributing it all to Littlewood based on a old memory of the book (in a case of ) and ‘reconstructing’ an estimate of how often one million “events” would occur in a kind of which leads to a nice time unit of a month.


  1. A passage I like from by :

    “How many people do you think there are in Nessus?”

    “I have no idea.”

    “No more do I, Torturer. No more does anyone. Every attempt to count them has failed, as has every attempt to tax them systematically. The city grows and changes every night, like writing chalked on a wall. Houses are built in the streets by clever people who take up the cobbles in the dark and claim the ground—did you know that? The exultant Talarican, whose madness manifested itself as a consuming interest in the lowest aspects of human existence, claimed that the persons who live by devouring the garbage of others number two gross thousands. That there are ten thousand begging acrobats, of whom nearly half are women. That if a pauper were to leap from the parapet of this bridge each time we draw breath, we should live forever, because the city breeds and breaks men faster than we respire.”

    I have wondered if Wolfe was alluding to ’s (which was a key source for ’s ), although the sources for are also plausible (perhaps conflating Alexandre Privat d’Anglemont with his quondam patron ):

    Every town and village was a living encyclopedia of crafts and trades. In 1886, most of the eight hundred and twenty-four inhabitants of the little town of Saint-Étienne-d’Orthe, on a low hill near the river Adour, were farmers and their dependents. Of the active population of two hundred and eleven, sixty-two had another trade: there were thirty-three seamstresses and weavers, six carpenters, five fishermen, four innkeepers, three cobblers, two shepherds, two blacksmiths, two millers, two masons, one baker, one rempailleur (upholsterer or chair-bottomer) and one witch (potentially useful in the absence of a doctor), but no butcher and no storekeeper other than two grocers. In addition to the local industries and the services provided by itinerant traders (see p. 146), most places also had snake collectors, rat catchers with trained ferrets and mole catchers who either set traps or lay in wait with a spade. There were rebilhous, who called out the hours of the night, ‘cinderellas’, who collected and sold ashes used for laundering clothes, men called tétaïres, who performed the function of a breast-pump by sucking mothers’ breasts to start the flow of milk, and all the other specialists that the census listed under ‘trades unknown’ and ‘without trade’, which usually meant gypsies, prostitutes and beggars…

    As the Breton peasant discovered to other people’s cost, begging was a profession in its own right. Beggar women sold their silence to respectable people by making lewd and compromising remarks about them in the street. They borrowed children who were diseased or deformed. They manufactured realistic sores from egg yolk and dried blood, working the yolk into a scratch to produce the full crusty effect. A judge at Rennes in 1787 reported ‘a bogus old man with a fake hump and a club foot, another man who succeeded in blacking out one eye to give a terrible, dramatic impression of blindness, and yet another who could mimic all the symptoms of epilepsy. ’Idle beggar’ was a contradiction in terms. As Déguignet insisted in his memoirs, it was no simple task to hide behind a hedgerow and to fabricate a stump or ‘a hideously swollen leg covered with rotten flesh’.

    These rustic trades were also found in cities. In the 1850s, one of the first amateur anthropologists of Paris, the Caribbean writer [Alexandre] Privat d’Anglemont, set out to explain [in (1854)/ (1861); no English translations available] how seventy thousand Parisians began the day without knowing how they would survive ‘and yet somehow end up managing to eat, more or less’. The result was a valuable compendium of little-known trades. He found a man who bred maggots for anglers by collecting dead cats and dogs in his attic, women who worked as (a speedy woman in a densely populated quartier could serve up to twenty clients), ‘guardian angels’ who were paid by restaurants to guide their drunken clients home, a former bear-hunter from the Pyrenees who exterminated cats, and a goatherd from the Limousin who kept a herd of goats on the fifth floor of a tenement in the Latin Quarter.

    To expand a little more from :

    His books are filled with tales of quaint encounters, and describe the bizarre trades of old Paris. The reader is introduced to a killer of cats, who sells the skins as sable and the flesh as rabbit (113), a painter of turkey feet, expert at giving them the glossy look of freshly killed fowl (50), a breeder of maggots for the many fishermen of Paris (23), a retailer of used bread crusts to feed rabbits (52), a guardian angel who escorts drunks back home safely (66), a maker of artificial rooster crests (116), a renter of leeches to patients who cannot afford to buy them (121), and—strangest of all—even a lyric poet who makes a living with his poetry (139). The list goes on.

    Milord l’Arsouille, a.k.a Lord Henry Seymour (1801–1859), the eccentric English millionaire who held court in the Paris slums, haunts the final pages of the book (228–240). Although Privat never met him in person, but only heard of him, he is the benign ghost who provides the author with a kind of aristocratic patronage. Milord l’Arsouille, often emulated (but never surpassed) by young and wealthy Parisians, became a legend for the poor people, a real-life replica of ’s Rodolphe Gerolstein, the hero of his fantastically popular serial novel (1843)…Like Prince Rudolph, Milord L’Arsouille is a protector of the weak and punisher of the evil, and outrageous anecdotes proliferate around him (239–240).

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  2. The difficulty of doing veganism safely, or its compatibility with athleticism is apparently a sore point for vegans, given the online popularity of vegan gurus claiming to do both but who then eventually are discovered to .↩︎

  3. Describing the news or media as having a “selection bias problem” is a bit odd, and like describing bombs as having a mortality problem; arguably, the sole function of the news is to be a giant global selection bias.↩︎

  4. Norman Malcolm, 1984, pg 30, pg35 (emphasis in original).↩︎

  5. Not that any source is 100% reliable, but at least tracing it back eliminates the many serious distortions which happen along the way. I can’t count how many times I’ve found when I traced back a claim or story to its original source or paper, and discovered that a major caveat had been left out, the original was fake or otherwise worthless, or the original actually said the opposite of what had finally been relayed to me. (And often the best & most interesting version is the original, anyway.)↩︎

  6. And the of these involving social contagion.↩︎

  7. Would Dyson have read Diaconis & Mosteller 1989? Entirely possible. Aside from being an interesting paper Dyson might read anyway, while Dyson & Diaconis do not seem to overlap at any institutions, they both have worked on theory and eg both were speakers at a 2002 , so that is one way they might be acquainted with each other’s work. Diaconis’s advisor Frederick Mosteller has no connection with Dyson that I noticed although as a major statistician, founder of Harvard’s statistics department, and president of multiple major academic organizations, he needs no particular connection to have potentially interacted with Dyson many times.↩︎