Your Sunday Brunch Special #13. Miracles: Littlewood’s Fourth Principle.

John Edensor Littlewood (1885-1977) is still a bit of a legend in the (mathematics) profession — he was best known for important results with his frequent collaborator Godfrey Hardy and others. But the things that come to mind most readily when I think of Littlewood are his four principles.[1] The first three pertain to objects relevant to mathematical analysis, and while I will state them, I’m not going to say anything about them.

J. E. Littlewood, when he wasn’t so stuck up.

It’s the fourth principle, usually called Littlewood’s Law, that I wish to consider. Anyway, here they are, as found in my memory, which may differ some from what Littlewood actually claimed:

1. Every set is nearly open.
2. Every function is nearly continuous.
3. Every convergent sequence is nearly uniformly convergent.
4. Every person may expect to experience a miracle about once a month.

It’s number 4 that I find most interesting (properly contextualized, the others are obvious). For Littlewood, a “miracle” is an exceptional event, something that follows the old saw, “one in a million.” Given this quantified version of events we might call miracles, Littlewood offers that (assuming you’re asleep for eight hours, vegetative for eight hours and alert for the other eight hours in a day, and assuming that during those alert hours you experience one event per second) “on average,” a miracle will happen to you about every 35 days. Therefore, miraculous events (probably!) happen all the time.

There are several things about this I see as rather fishy. First, when I think of a miracle, I think not of winning the lottery, but someone being raised from the dead or, nearly as wild, a four-year-old landing a 747, sans autopilot.

Evans, trying to land. “No lady, I don’t have a license. But I finished P90X today.”

But perhaps this merely pushes the probabilities another couple of decimal places. Is this the old “Paleyesque” problem that appears now and again from well-meaning folk? You know, wishing to promote faith, they tender statements about things like the human eye — such an intricate thing arising spontaneously is so improbable as to make belief in such ridiculous? Yes, says I. The confusion here stems from several sources, but partly it’s a matter of spelling. Improbable ≠ Impossible.

But then again, Littlewood’s probabilistic argument leaves one cold. It can’t measure feelings or the distribution of the Holy Ghost, can it? Does a miracle depend on perception? When you catalog the miracles in your life, truly “supernatural” out of the ordinary events as you see them (no, I’m not counting those poetic things like experiencing the sunrise from the top of Kilimanjaro or the birth of your child), how often would you say they occur in your life?[2] Once a month, once a year, once in a lifetime, never? What is a miracle?[3]

There is another aspect to Littlewood’s idea though and it depends on the nature of the universe. If the universe is infinite,[4] then it’s possible, even reasonably likely, that there is another (likely, infinitely many) “you(s)” walking around on some world(s) somewhere. Someone exactly like you in every detail with the same life history. Would that be a miracle, or does it cheapen every miracle? Indeed, assuming that the process of populating worlds with embodied spirits has been going on “forever,” then it is quite likely that you are not very unique at all. Given the nature of baryonic matter, there’s a pretty high probability that you can count ω StevePs out there somewhere. Does this mean a long stay in hell?

[1] Littlewood is lesser known perhaps for a wonderfully large shaped-charge ego.

[2] There is a thread in Mormon thinking that stems from its materialistic background, suggesting that “supernatural” doesn’t even make sense as a category. I think I disagree for a number of reasons, but now is not the time.

[3] Naturally we should keep in mind D&C 84:73.

[4] Actually this is rather likely, given that the universe appears to be flat (so far).


  1. I think miracles do depend on perception — one person sees something, and sees the hand of our God — someone else sees the same thing, and doesn’t/can’t/won’t see God in it — there is some choice here.

  2. Chris Kimball says:

    With one million events per month and the very human tendency to find patterns, we will all see “one in a million” type miracles if we look for them. (That just restates Littlewood’s 4th principle, I think.) A corollary might be that if we are looking for miracles in the sense of an active influence, the “God’s hand” kind of miracle, then there has to be something more than one per month. What’s “more”? More frequent? Probably not. After all, if the average is once a month, then the likelihood of two in some months is fairly high.

    For myself, I feel . . . I believe . . without being able to explain, that I live in a miraculous universe where God is active. “Active” is not a clock winder, and “active” is not the operation of random events in an infinite universe where everything will happen sometime. To see that, the “more” I look for is usually some kind of correlation. An ask and answer kind of prayer. A petition and a reply. A need and a fulfillment. There’s still a lot of room for chance and pattern matching, but the probabilities shift.

    To answer the question, in my life the ask-and-answer kind of miracle is also on the order of once a month. In other words, not daily, not annual, more like two-hands countable per year. .

  3. I firmly believe in the Great Video Replay In The Sky, a time somewhere in the future where our time on Earth will be reviewed, in the context of everybody’s else’s journey. I deeply suspect we will be flabbergasted at all the time the hand of God intervened in a miraculous way in our lives. That accident we did not have, that hot dog we ‘should’ have choked to death on, but swallowed instead, that plane we missed, the stop sign we decided not to run that avoided that catastrophe…on and on. I think we miss most of the miracles in our lives, but I am certain they are there, and numerous. Once a month sounds rather sparse, now that I think about it.

    I do hope there’s popcorn. I love popcorn, and the heavenly kind must be even better.

  4. At the opposite end of the spectrum as ji, I think there are also people who look too much to the hand of God in absolutely everything. As in God helped them find their favorite brand of Peanut Butter on sale this week–A MIRACLE!!!

    I know the old adage is to count your blessings, but applying a mathematical formula to them, or the law of averages makes them feel cheap in my opinion, as well as creates a false expectation. How does one explain this to a person who sees a miracle once every 5 years? IMO it is best to receive blessings graciously and humbly and be thankful for every one no matter how many or how few.

  5. Intervention is a really interesting question. Absolutely fascinating is a better way to say it.

  6. Indeed I recently attended a ω SteveP conference and we all stayed at Hilbert’s Grand Hotel, but let me tell you, we all got pretty tired of changing rooms to the successor room every time a new one of us showed up.

    I think Littleton may have underestimated. If you calculate the probability that the two cars parked in front of the church today would be there when they were manufactured (one in Detroit one in Japan), (which would include the probability of winding up on the same country, the same state, the same city, being purchased by members of the same ward, finding an empty parking space etc) it’s astronomical, if you include the probability of the third car, it’s inconceivably high, add a forth for some really wild improbabilities. But if you go for the probability of the configuration off all the cars in the church parking lot today, at the time of individual manufacture of all those cars, you might as well call it so improbable as to be impossible. Yet there it was. A miracle every Sunday.

    Bill I love these. They always leave me thinking more than I should.

  7. Since you’re all Steves, you probably tried to crowd into the same room. Knowing yous though, you all slept in the lobby. The lobby is a set. So there was just one of you after all!

  8. themormonbrit says:

    Just a thought about supernatural miracles.

    To be honest, I don’t find miracles all that…miraculous. iI am very grateful for them (most of the time) and I am often pleased by them, but I am rarely amazed at them. To me, the entire universe is a miracle. If you hold up a pen and let it go 100 million times, and 99,999,999 times it falls to the floor, and one time it just stays hovering in the air, to me that is not a miracle. The miracle is that it did exactly the same thing over and over again 99,999,999 times. Of course, to me the only way that I can explain why gravity works infallibly (so far) is because I believe in an all-powerful God. Scientists, and all human beings, always start everything with an assumption of a constant, ordered universe with constant, unbreakable laws which bind all matter. That’s fine, until they start arguing that everything can be explained in terms of these laws, and hence there is no need for a God. Then I get really irritated, because they have no way of explaining where these constant, unbreakable laws come from. They begin with an assumption of an orderly universe, but have no way of explaining why the universe is that way. That’s why I really don’t marvel at miracles anymore. Instead I marvel at the wondrous, beautiful orderliness of the laws that (with the exception of ‘miracles’) constantly govern the physical world.

  9. You know, the most amazing thing happened to me today. On my way to church I came in through the parking lot and you wouldn’t believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance I would see that particular one today? Amazing!

    The miracle, of course, is that I went to church.

  10. Mathematicians have an uncommon sense of humor. I suspect Littlewood was poking fun at the believers around him by trivializing an idea they held as sacred and mysterious (perhaps extra sacred because it was mysterious – sacredly mysterious?). There are at least two dangers to this type of joke: 1) The believers respond seriously (inevitably poorly, because mathematicians are masters of precision and religionists, by and large, are not). Then, those who are in on the joke naturally see the believers as thickheaded zealots, something clearly to be avoided. 2) The joker strikes a nerve with those whose faith is wavering because the joke seems to be on to something. (I say seems to, because I find that our religious intuition is very underdeveloped.) I think Joseph Smith, who likely had well-developed religious intuition (at least after a while) would laugh with Littlewood and then proceed to give a more divinely insightful definition for miracle. Therefore, in most cases the joke hurts the believers and the nonbelievers, which is not all that surprising as it is born of pride and scorn and feeds on willful ignorance and false piety. A better course of action for Littlewood, I think (pride?), would have been to consider the intellect of Hardy’s great “discovery” Ramanujan and think it a miracle that such penetrating insight was within the grasp of mere mortals. (Related query: Does God view His miracles as miraculous or, since He understands the mechanics underlying them, are they mere events to Him?)

  11. Brian you’re on the money about one thing: Littlewood was trying to deflate unexamined belief. Or at least, that’s what he thought it was. I cannot speak to God’s experience, but there must be something interesting up there, otherwise the monotony must get to you.

  12. Latter-day Guy says:

    …one person sees something, and sees the hand of our God — someone else sees the same thing, and doesn’t/can’t/won’t see God in it — there is some choice here.

    This reminds me a little of the tendency among some––I’ve noted this in a few different wards––to ever-so-patiently correct my use of the word ‘lucky’ with the gentle remonstrance: “Don’t you mean ‘blessed’?”

    No, as a matter of fact I didn’t. If I’d meant it, I would’ve damn well said it. It’s not a huge issue, but it is a bit irksome. However, turnabout is fair play, so the next time one such individual mentions being unlucky or unfortunate, I’ll chip in with, “Don’t you mean ‘CURSED BY GOD’!?”

  13. 12 haha that was funny. I’m gonna use that on some of the old biddies and stiff collars in my ward.

  14. Miracles are hard to define but we know one when we see one.

    Action adventure movies are exercises in miracles, and we love them. We would not like them if they were exercises in high order improbability.

  15. Sarah W. says:

    I think miracles frankly are bullsh:t. I am an PICU nurse at one of the nations leading children’s hospitals. Every day I see parents praying for miracles. Sometimes a child gets better – sometimes they don’t. There is literally no rhyme or reason outside of what we can expect from medicine.

  16. So, I presume that the great parsimony is at play with regard to miracles.

    The great parsimony:

    God is a monopolist. He has the monopoly on the spiritual interaction in the world. Nothing goes in or out without his consent. He, like all good monopolists, uses that monopoly to maximize his return. I assume his return is the number of souls which return to him in a state of progression. As in all monopolies, too much of the good cheapens it, too little and it is ignored and replacements are found. The point of operation of any monopoly is the point of maximum pain to the system because that is where the monopolist is extracting the maximum return from the system. In the case of God, it is the point of greatest spiritual pain. Thus all those parents anguishing in the PICU.

    Jesus commented on this condition. Remember the Syrian general who came to Elijah to be healed? Jesus commented that there were plenty of Hebrew lepers who were not healed, the Children of the Promise, but God cured a gentile. My observation is that after that miracle, every Jewish leper prayed for a miracle, and has to this very day. This focuses our minds mightily on spiritual things.

    I longed for a miracle when my wife died of cancer. I gave her a blessing up front and I knew that no miracle was forthcoming. What has happened after she died could be considered a miracle. And, I hope and pray, she is waiting and still cares. What has transpired is according to her wishes.

    So, while my heart goes out to those parents, I know that God, the Monopolist, lets people be tortured to death. This must be the great curse of being God, who, with perfect empathy, works to maximize the return from this planet. How valuable it must be to bring enlightenment to even a few of his children that this pain must be endured. Bow down, for these sufferers do it for you. Christ is not alone.

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