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.
politics, psychology, sociology, statistics, philosophy, insight-porn
2018-12-152019-02-18 finished certainty: highly likely importance: 5


Online & main­stream media and social net­work­ing have become increas­ingly mis­lead­ing as to the state of the world by focus­ing on ‘sto­ries’ 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.” This must now be extended to a global scale for a hyper­-net­worked global media cov­er­ing anom­alies from 8 bil­lion peo­ple—all coin­ci­dences, hoax­es, men­tal ill­ness­es, psy­cho­log­i­cal odd­i­ties, extremes of con­tin­u­ums, mis­takes, mis­un­der­stand­ings, ter­ror­ism, unex­plained phe­nom­ena etc. Hence, there will be enough ‘mir­a­cles’ that all media cov­er­age of events can poten­tially be com­posed of noth­ing but extreme out­liers, even though it would seem like an ‘extra­or­di­nary’ claim to say that all medi­a-re­ported events may be flukes.

This cre­ates an epis­temic envi­ron­ment deeply hos­tile to under­stand­ing real­i­ty, one which is ded­i­cated to find­ing arbi­trary amounts of and ampli­fy­ing the least rep­re­sen­ta­tive dat­a­points.

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, Scott Alexan­der in March 2017 pointed out a fol­lowup to a news story which got much less press than did the orig­i­nal:

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 an Israeli Jew who may have a per­son­al­i­ty-al­ter­ing brain tumor. The Atlantic has a pretty good post­mortem of the whole affair.

(His autism/brain tumor defense did not suc­ceed, and he was ulti­mately con­victed & sen­tenced to 10 years.)

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 con­vinced his friend to mur­der him on the orders of British intel­li­gence? 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 because air­planes are so safe, 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 Air France 447, 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 did­n’t real­ize was even pos­si­ble (“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 post-WWII, 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 eye­balling the data before­hand to decide on sta­tis­ti­cal approach, each paper has the equiv­a­lent of 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, . 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 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, , but the orig­i­nal pho­to­graph edi­tor at the respected firm did­n’t intend to ‘forge’ any­thing: it was sim­ply some polit­i­cal humor, part of a set of 3 edited pho­tographs along with (sit­ting on an ele­phant) and (don­key), which was dredged up and repur­posed for Inter­net memes. (Speak­ing of ele­phants, you may have seen an ele­phant at an Irish riot c.1970—pho­to­shopped for an amus­ing series of Irish satires, but sub­se­quently taken for real.) More recently 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 mul­ti­-sea­son Net­flix mock­u­men­tary series. Which series? Well, one you’ve almost cer­tainly never heard of (much less watched), inas­much as thanks to Net­flix & other trends there are now >400 scripted TV series annu­ally 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 organ har­vest­ing video hoax­es?) 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. Snopes is chock­-a-block with these. When it comes to media, .

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, “it all adds up to nor­mal­ity”!

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 of cov­er­age as your first anec­dote—right?

  • 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 in today’s edi­tion, there might be a giant void with a tiny lit­tle 2-line wire ser­vice item for Africa, while the (much small­er) USA sec­tion requires a micro­scope to read all the mate­r­ial in it.

  • 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 1953 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 Debunked! ESP, Telekine­sis, and Other Pseu­do­science, 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 The Sci­en­tist as Rebel (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.)

Dyson’s def­i­n­i­tion of events hap­pen­ing one per sec­ond seems fairly rea­son­able, and it then fol­lows that dur­ing one’s most active hours, a mil­lion will hap­pen dur­ing a month. 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.

Where does ‘month’ keep com­ing from, any­way? I sus­pect that the appeal of month as the unit of time, rather than any other unit like minute or hour day or year or decade, reflects the essen­tially memetic aspect of Lit­tle­wood’s obser­va­tions: he is skep­ti­cally exam­in­ing those sto­ries that peo­ple retell end­less­ly. If an appro­pri­ately ‘mirac­u­lous’ story could be turned up every hour, it would quickly lose all nov­el­ty; but one good story every decade, or even year, is too rare, with pen­t-up demand, and a mag­a­zine could steal cir­cu­la­tion from more ret­i­cent rivals by report­ing exam­ples more fre­quent­ly, lead­ing to an inter­me­di­ate equi­lib­ri­um. Once a week or month sounds about right: a reg­u­lar source of enter­tain­ment by the extra­or­di­nary, but not so fre­quent as to wear out its wel­come & won­der and become ordi­nary. (Since Lit­tle­wood was writ­ing in a less glob­al­ized media envi­ron­ment, with a smaller effec­tive pop­u­la­tion size, a thresh­old of one in a mil­lion was appro­pri­ate; but these days, to main­tain an appro­pri­ate click­bait drip rate, a more strin­gent thresh­old may be required, such as one in a bil­lion.) One can see this sort of tem­po­ral limit in out­rage cycles on social medi­a—they can’t hap­pen too often, because par­ti­sans will become exhausted and top­ics will lose nov­el­ty, but since poten­tial out­rages are always hap­pen­ing which can feed the need, quiet peri­ods won’t last too long; thus, there seems to be a peri­od­ic­ity around the week range, rather than, say, hour or year. Per­haps sim­i­lar­ly, in big soci­ety-wide issues not based on sin­gle inci­dents or out­liers, there is the mul­ti­-year “issue-at­ten­tion cycle” (/).

Are there any other sources besides Dyson?

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 artillery/geometry math­e­mat­i­cal for­mula, and there is an amus­ing crit­i­cism of a math­e­mat­i­cal logic text­book by Boo­los 1986: “[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 (which was a key source for ), although the sources for are also plau­si­ble (per­haps con­flat­ing Alexan­dre Pri­vat d’An­gle­mont with his quon­dam patron “Lord Henry Sey­mour”):

    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 Paris anec­doté (1854)/Paris Inconnu (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 Jul­lien 2009:

    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 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).

    ↩︎
  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 be hyp­ocrites & expelled.↩︎

  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, Lud­wig Wittgen­stein: A Mem­oir and a Bio­graph­i­cal Sketch by G. H. Von Wright, Sec­ond ed., with Wittgen­stein’s Let­ters to 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 lep­rechauns 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 unnamed clus­ter 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 work­shop, 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.↩︎