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Origin of ‘Littlewood’s Law of Miracles’

Leprechaun hunting the origins of the famous skeptical observation that because millions of events are constantly happening, ‘miracles’ happen once a month; it was actually coined by Freeman Dyson.

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, making it a leprechaun, 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 & Mosteller1989, in a case of Stigler’s law.

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

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 A Mathematician’s Miscellany/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 101046.1.) 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] Oxford Group. But one could not say the adverse chance was 10^6:1. 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 2, 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 importance 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 8 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 parapsychology cards 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 20 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 3 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 3 obstacles. To reach any scientifically credible conclusions, we would have needed to overcome all 3.

The history of the card-guessing experiments, carried out initially by Joseph Rhine 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 3 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.

Dyson2004 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 Boolos1986: “[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 Dyson2004 or later sources.

The closest thing to a predecessor I found was the paper “Methods for Studying Coincidences”, Diaconis & Mosteller 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 effect as regression to the mean in a Poisson process) coins a law, 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.

Diaconis & Mosteller1989 anticipate Dyson2004 in defining “one in a million” as the 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 & Mosteller1989 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 Dyson2004 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 & Mosteller19891 but misattributing it all to Littlewood based on an old memory of the book (in a case of Stigler’s law of eponymy) and ‘reconstructing’ an estimate of how often one million “events” would occur in a kind of Fermi estimate which leads to a nice time unit of a month.

  1. Would Dyson have read Diaconis & Mosteller1989? 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 random matrix theory and eg. both were speakers at a 2002 Mathematical Sciences Research Institute 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.↩︎

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