Gaussian anomaly

Intelligence is highly heritable- so generally, when someone is far smarter than average, his parents are at least somewhat above average.

But Gauss’s parents sure looked ordinary: his father had menial jobs such as gardener and street butcher, while his mother was barely literate. Nonpaternity is not very common, but it is more likely if a kid is vastly different from his parents: and Gauss (arguably the greatest mathematician of all time) was vastly different. Perhaps the most different kid in history.

So if we dug him up and sequenced him, we might discover something interesting. In the immortal words of Zuckerberg: “He’s not your dad.”  Maybe.

Generalizing this case has obvious implications.






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73 Responses to Gaussian anomaly

  1. Agree is would be good to sequence Gauss. Also Ramanujan for contrast. Pity that Gauss did not, as far as I know, work out the chance of having been born to parents with menial occupations.

  2. Coagulopath says:

    So if we dug him up and sequenced him, we might discover something interesting.

    I recently had my DNA tested and got a similar surprise. My dad wasn’t my dad. A few surprises, actually: I look Caucasian, but I have about 10% Sub-Saharan ancestry. A few more drops of negritude and I’d be Jayman.

    Before my adoptive dad died, he made me custodian of his half of the family tree. Whenever there’s a birth or a death, I write it down. It all seems like theater: I’m related to none of these people.

    Nonpaternity is not very common

    Yes, but have you thought about why that is? Because it’s hard to get away with it. And hard for whom? People of average intelligence.

    A super genius could impregnate a woman, have another man raise the child, and nobody would be any the wiser. It’d be easy.

    Were any famous scientists or thinkers hobnobbing around Germany in 1777 or so? We should make a list. Clearly, we stand at the threshold of the greatest paternity scandal since Jesus of Nazareth. Maury Povich has nothing on this.

    • Tanturn says:

      “A super genius could impregnate a woman, have another man raise the child, and nobody would be any the wiser.”

      I’ve known a few people with Rick and Morty tier iqs. That certainly wasn’t the impression I got from them.

    • Space Ghost says:

      So how many von Neumann bastards do you think are running around out there? Apparently he was a real ladies man.

      • gcochran9 says:

        He was interested, far from blind, but I’ve never heard that he was a player.

        • Toddy Cat says:

          There can’t be that many Von Neumans around – things would be going a lot better in this country if there were. I mean, I know, regression to the mean and all, but that was one Hell of a mean…

          • Greying Wanderer says:

            “things would be going a lot better in this country if there were”

            the New York and London banks have been recruiting super smart people from all over the world for decades now – the chance of the world getting better as a result is zero

    • crew says:

      Nonpaternity is not very common

      Yes, but have you thought about why that is? Because it’s hard to get away with it. And hard for whom? People of average intelligence.

      Perhaps an arms race over time.

      Women would be selected for discriminating against cads who would get them pregnant and leave, while men would be selected for discriminating against females who were likely to cuckold them.

  3. catte says:

    Generalizing this case has obvious implications.

    That if we dig up enough dead geniuses, we could sequence the Kryptonian genome?

  4. Candide III says:

    Gauss’s father also worked as a treasurer for a local insurance company [German wiki], but never mind. The inference from Gauss’s parents’ occupations is invalid, because in XVIII century the ability-education-status sorting machine didn’t work anywhere near its today’s efficiency, women often received less education or none at all, and most of the population was illiterate and worked what we’d now consider menial jobs regardless of ability.

    • Jim says:

      Wasn’t one of his uncles also fairly well educated for the times?

    • Michel Rouzic says:

      Agreed, if we go back up my (French) family tree quickly enough you find nothing but farmers and seamstresses and would have difficulty explaining why children of mere peasants found success in Paris. Look at George Green, he could have easily been stuck in a windmill his whole life instead of just most of it. Had he been born 200 years later he would have gone straight for a PhD instead. Potential brilliant mathematicians died being known as nothing but modest peasants, people could have much unrealised potential until recently, mostly women, and mostly women and genetic potential for mathematics.

      If we suppose that Gauss received most of his good math genes from his mother’s side it’s very easy to see how that could have gone unnoticed in a 18th century woman. I for one also get my math skills from my mother’s side (skills which are shared with my maternal male cousins) but you wouldn’t know it from knowing anything about my mother’s life.

      • gcochran9 says:

        It’s awfully hard to score six standard deviations above average with an average dad and a not-noticeably-special mom. Most famous mathematicians did not have average-seeming parents – in fact, I don’t know of a single example other than Gauss.

        • Peter North says:

          What about Newton?

          His father was illiterate – he signed his name with a cross.

          • Reading and writing are different skills, and plenty of people a few centuries ago could read well because of their native intelligence, but could not write, because you usually had to be trained in that.

          • tautology5628 says:

            Newton was smart, but later Mathematicians were on another level. Newton needed 20 years to prove that point masses and speheres have the same gravitational field. Gauss found a five minute proof.

          • Patrick Boyle says:

            I now sign my name with a ‘X’. The delivery men no longer give you a piece of paper to sign with a pen. Now they give you a tablet or a smartphone and expect you to sign your name on the slippery surface with your finger. No one can actually write under these conditions so I just make a crude X.

            There is some lesson about progress here.

        • Jim says:

          Yes and there are a lot of related mathematicians – Emmy Noether’s father Max Noether was a very famous mathematician and one of her brothers was a physicist of some note. Pascal’s father Etienne was a mathematician who wrote books on geometry. The curve called Pascal’s Limacon is named after Etienne not Blaise. I believe Euler’s father was a professor of mathematics and one of Euler’s sons was a pretty good mathematician. There are the fathers and sons – Elie Cartan and Henri Cartan, Emil Artin and Michael Artin, Benoit Mandelbrot’s father was part of the original Bourbaki group. Harold Bohr’s older brother Niels was a physicist of a some note. Emmanuel Lasker had a brother Berthold who was also a mathematician whose work I have seen references to. Heinz Hopf had a physicist brother who was pretty good. Markov’s grandson was a noted mathematician. Voevodsky’s father was a physicist and director of a top Soviet physics lab. His mother had a PhD in I think physical chemistry or something.

          Then there’s the Bernoulli’s and the Novikov’s.

          • Jim says:

            I forgot to mention Alfred North Whitehead and his famous nephew J.H.C. Whitehead. Also there are the Birkhoff’s – father and son.

            By the way Alfred North Whitehead often doesn’t get sufficient recognition for the Principia compared to Russell. Russell himself said that Whitehead did most of the really technically difficult parts of the Principia.

        • Hugh Mann says:

          But his dad probably wasn’t average. I don’t know much about Germany in that period, but in the UK your class position pretty much determined your station in life, unless you were fortunate (Humphrey Davy gives you a job in his lab – Faraday’s father was IIRC a blacksmith) or connected. And your class position had little to do with your IQ.

          It’s that the “ability-education-status sorting machine didn’t work anywhere near its today’s efficiency” , or as Thomas Gray put it

          Perhaps in this neglected spot is laid
          Some heart once pregnant with celestial fire;
          Hands, that the rod of empire might have sway’d,
          Or wak’d to ecstasy the living lyre.

          But Knowledge to their eyes her ample page
          Rich with the spoils of time did ne’er unroll;
          Chill Penury repress’d their noble rage,
          And froze the genial current of the soul.

          Full many a gem of purest ray serene,
          The dark unfathom’d caves of ocean bear:
          Full many a flow’r is born to blush unseen,
          And waste its sweetness on the desert air.

          Some village-Hampden, that with dauntless breast
          The little tyrant of his fields withstood;
          Some mute inglorious Milton here may rest,
          Some Cromwell guiltless of his country’s blood.

          • gcochran9 says:

            ” your class position pretty much determined your station in life,”

            I don’t think that’s true.

            • Hugh Mann says:

              UK prime ministers were almost exclusively recruited from the upper classes (David Lloyd George being the single exception) until the 1960s, when a combination of selective state education (‘grammar schools’) and rising living standards (your bright child didn’t have to leave school at 14 to bring in some money) produced the 1964-1997 run of state-educated prime ministers. Wilson, Heath, Callaghan, Thatcher, Major.

              But in the late 60s, just as this cohort was coming to power, a privately educated Labour politician abolished the grammar schools and the pipeline stopped. Three out of four PMs since have been privately educated (Gordon Brown went to an academically selective Scottish school).

          • Rosenmops says:

            I was thinking of this poem too.

        • JMcG says:

          I scored a 1420 on the SAT back in the early 80’s. Not stunning, but not bad. My parents grew up in rural Ireland, born in the 30’s.
          My father left school at age 14 to work. He was a mechanic without parallel as far as I can tell. My mother went to nursing school in England. Dad was drafted shortly after coming to the states. After testing, he said the army offered him flight school or OCS. He wasn’t against a little exaggeration, so I don’t know how true that might be.
          I think that in pre-modern times there were lots and lots of lights hidden under bushel baskets.

          • Rosenmops says:

            My dad ( born 1913) also left school at 13 or 14. But he didn’t take up a trade . (His father had been a blacksmith)

            He just worked at an assortment of jobs and got in trouble for trying to start a union at a mine. When ww2 came along he was selected to train as a pilot. (He used his brother’s highschool transcript). They had to take courses which used trigonometry and such. So he got a book on high school math and learned it on his own. He ended up tying with one other person for the highest mark. So he became a pilot in the RCAF. Then he we to university after the war. (Also using his brother’s transcript)

        • Return of Shawn says:

          Is Gauss a clone of Cochran who time traveled, or is Cochran a clone of Gauss? I was just looking at some photos of Gauss and became confused.

          In any case, maybe Gauss’s parents just really liked their menial jobs? Chris Langan and Richard Rosner are both super smart guys who worked many menial jobs.

      • Jim says:

        If his father worked as a treasurer for an insurance company his father was probably quite a bit above average in intelligence. There probably weren’t many such jobs available at that time.

  5. Phille says:

    I once did a back of the envelope calculation, what you would get if you could select each chromosome of the parents for maximum IQ of the kid. I got something like 50 IQ points above the parent’s mean.

    • Eponymous says:

      That’s flipping 46 heads in a row. A 1 in 2^46 improbability.

      Assuming the children of two 100 IQ individuals have the same IQ distribution as the population as a whole (not sure about that), that works out to an IQ of 224. Probably smarter than anyone who has ever lived.

      • Eponymous says:

        Okay, I think I worked it out: 100 IQ parents should have kids with mean 100 and standard deviation (1-h^2)15. The 1 in 2^46 kid would be a 7.6 sd event, so IQ = 100 + (1-h^2)15*7.6.

        If h^2 = 0.5, that’s 162, not 224. Of course, h^2 is not a constant, but varies with the society. Probably higher in 18th C Germany.

    • savantissimo says:

      Things get rather murky if you try to do a realistic model of crossover.* OTOH, if you could select the best sections of each chromosome pair for each parent, you could potentially do quite a bit better than simply choosing the better chromosome of each pair for each parent, even if you only allowed one fixed crossover point per pair.

      Another possibility that might not be so hard as engineering the desired crossovers would be taking the best (most average, lowest genetic load) chromosome of each of the 23 or 24 types from a large population.

      I think Eponymous’ calculation is simplistic because it assumes that one chromosome within each pair is substantially better than the other, which isn’t what you would usually find.

      *I’m not all that knowledgeable about genetics or even the terminology; Prof. Cochran could teach more about the genetic calculations and (im)practicality of this than he probably cares to. I did recently come across an interesting related post from the UC Davis Population and Evolutionary Genetics Lab “How many genomic blocks do you share with a cousin?” that gives a start on how to do the calculations, though one of the comments indicates a factor of 2 may have been dropped.

      • Glengarry says:

        Interesting post, thanks. Does anyone here have some rules of thumb for how many recombination events occur and the distance between events, for some typical cases of crossover? (Pref. human.)

    • gwern says:

      I did that a few months ago too: A more realistic estimate would be like +30 points, to get 50 you need to assume something like 100% heritability, but of course almost all traits are less and SNP heritability is much less, like 30%, so that’s how I got +30.

      It’s surprising (I know I was surprised) but this is just how order statistics and the CLT work – you want to push selection as far down the ‘stack’ as possible, increase the variance of each group as much as possible, and do selection at multiple levels. (Person, chromosome, sperm/egg, embryo, person?) The gains from increasing variance or doing multiple levels of selection are a lot larger than you think, because of the thin tails of the normal distribution – if you can increase the width, the density of the tail can increase multiplicatively at the average position of the maximum of your n sample, and if you can start at a higher mean, likewise, because you’re ratcheting your way up. It might not seem like a big difference whether you do a single selection from n=10 or two iterative selections from n=5 and then a new n=5, since the total sample is the same size, but it does – n=10 gets you a max of 1.53SD (for N(0,1) as usual), but n=5+n=5 gets you 3.488, which is more than twice as much.

      (Did you know plant breeders deliberately increase the meiotic crossover rate for “reverse breeding”? Can be both chemically and genetically increased. Makes one think.)

  6. Maciano says:

    If his father was IQ100 and his mother IQ90, how big is the chance for a kid to have an IQ160 or IQ180?

    Compare to the total German population of the time of birth, we’d get a reasonable estimation of this happening by chance.

    • dearieme says:

      But if you think he was a strong candidate for the best mathematician ever, why base your calculation on one time, one place? Why not use the total human population since, say, Archimedes?

      • gcochran9 says:

        In the same sense, surely it wouldn’t have been too surprising if a kid was a full two feet taller than the parental average – without a pituitary tumor.

  7. Phille says:

    The German wikipedia says his mother was described as “smart, cheerful and of steadfast character”. Together with his father’s job as a treasurer I don’t see why his parent’s shouldn’t average >130 IQ.

  8. jb says:

    Would it really be that unusual in 18th century Germany for a man with an IQ of, let’s say, 140, to work in random menial jobs? Even today there are highly intelligent people with little drive or ambition who just sort of drift through life. How much more likely back then when there were so few opportunities for those born poor? Maybe Gauss picked up his drive from his mother, who for all we know may have been a remarkable woman who just never learned to read — again, how unusual would that be for the time? Given how little we know about Gauss’s parents I don’t see how we can draw any strong conclusions about them.

    • Wency says:

      Hear, hear.

      Last I knew, the smartest guy in my high school class, who aced every calc exam without effort, still lived at home and drifted between unemployment and retail jobs.

  9. megabar says:

    I was just having a somewhat related discussion on another site: I hear the term “regress to the mean” used often. Sometimes it’s used to argue that average people who are of “good stock” are still likely to produce smarter kids then a smart kid who came out of an average family. Sometimes it’s used to argue that two smart people of different races are less likely to produce a smart kid as compared to equally smart parents of the same race.

    It seems to me that these arguments require that the trait has at least some genes that require two copies of the allele to confer advantage. In this world, more closely related parents that are smart are more likely to have the advantages alleles at the same loci, increasing the chances of the kid to ‘double up’, and thus express the advantage. It’s also possible in this world for a kid to be average, but contain many unpaired beneficial alleles, and so is a “carrier” for high IQ (particular if that kid goes on to marry someone that is from the same stock).

    One the other hand, if only a single copy is needed at a locus, than having less related parents that are smart, and who thus may have beneficial alleles are different loci, are actually more likely to have a very smart kid, because their beneficial genes are possibly additive (if the kid rolls the dice well). There might be more variance in this scenario altogether.

    In reality, I could imagine that both are true, and some genes need 1 copy, and some need 2. In which case it is harder to say which breeding scenario is more likely to produce very smart kids.

    Is there evidence that suggests which interpretation is more accurate in real life? Or am I missing something more fundamental?

  10. David says:

    “Nonpaternity is not very common”

    A questionable assertion if ever I saw one. Although, I suppose it does depend on one’s definition of “common”. If 5-15% can be considered “not very common”, I suppose we could gloss over the error.

  11. Bob says:

    I posit a changeling now and then is part of how the alien overlords manage the herd. It explains a lot.

  12. Peter North says:


    In the Germany of those days, the working class, generally, was barely educated.
    There was no such thing as ‘compulsory state education’, and generally, working class parents saw education as more of a hindrance than a help – it had nothing to offer them in their struggle for life.

  13. epoch2013 says:

    I don’t know.

    My wife is fairly intelligent as I am. She comes from a family of non-educated people. My family was a lower class family during most of the industrial revolution yet most of my nephews and nieces are highly educated.

    So I reckon there was a non-tapped amount of intelligence hiding in lower classes that have recently been given the opportunity to blossom. When I look at my genealogical records I see religious outcasts, Mennonites, where higher class women would rather marry lower class Mennonites. Considering the religious ideas of Mennonites I’d say out of ideological reasons.

    I think the European religious wars and feuds caused a lot of potential achievers to remain among the underclass.

  14. Warren Notes says:

    Horizontal gene transfer from a crafty fox.

  15. Chris B says:

    Do we have portraits of either parent?

  16. Lior says:

    If his kids intelligence regressed a lot that would be evidence in favor of him being the child of his ordinary father.
    wikipedia states:’Gauss eventually had conflicts with his sons. He did not want any of his sons to enter mathematics or science for “fear of lowering the family name”, as he believed none of them would surpass his own achievements’

  17. wontgetthtough says:

    Click to access heideman-johnson-etal1985.pdf

    “In a recently published history of numerical analysis [9], H. H. GOLDSTINE
    attributes to CARL FRIEDRICH GAUSS, the eminent German mathematician, an
    algorithm similar to the FFT for the computation of the coefficients of a finite
    FOURIER series. GAUSS’ treatise describing the algorithm was not published in his
    lifetime; it appeared only in his collected works [10] as an unpublished manuscript.
    The presumed year of the composition of this treatise is 1805, thereby suggesting
    that efficient algorithms for evaluating coefficients of FOURIER series were developed
    at least a century earlier than had been previously known. “

  18. JRM says:

    Gauss’s mother lived to be 95 years old.

    Gauss’s paternal half brother lived to be 85 years old

    Longevity and intelligence have a correlation. This is some evidence that Gauss’ parents had good genetics.

  19. Paul Conroy says:

    I reckon your suggestion of an NPE in Gauss’s case is very likely true – based on my experiences with genetic genealogy.

    In my wife’s family, her father’s father was tall lanky (6′ 6″ and about 190 Lbs) Polish immigrant steelworker who never read a book in his life, and came from a peasant background South of Poznan. Family describe him as genial but kinda dumb and he looked like Dolph Lundgren. Her father’s mother was a short, smart bossy Sicilian woman (5′ 2″) who came from a business family, and who totally dominated her husband. Yet their son, my father-in-law, in his prime was a muscle bound 6′ 1″ and around 270 Lbs, and looked like Brock Lesnar. He was always an A student in his sleepy upstate New York, steel town. He was also an excellent athlete. He won full academic scholarship to Princeton, Yale and Harvard, and 2 full athletic scholarship for basketball and 2 full athletic scholarships for football (linebacker). He graduated valedictorian of his class at Princeton, was recruited as an early programmer at IBM, then recruited from there by the CIA, which he left to return to law school. By his early 40’s he had started a law firm that employed over a hundred attorneys and had run for major of Philly. I would estimate an IQ of about 150.

    My wife’s family maintained that he got all his smarts from his mother. But DNA testing paints a totally different picture. It would seem that his actual father was a successful MD from Buffalo, that his half brother is a retired Astrophysicist who worked for NASA, and there are 2 other cousins with PhD’s in Physics, and multiple other professionals of note, a total of 12. Plus he has 2 other close relatives who also match the same Doctor and are likely also NPE’s.

  20. Sean says:

    Guass’s father was a gardner when he started work, but went on to being a merchant’s assistant, “master of waterworks,” [ for the city] and the treasurer of a small insurance fund. Well above average and a hell of a rise from a standing start back then. He was also a domineering a-hole and not the sort to be oblivious to, or put up with, being cuckolded. Mother was kind and intelligent, daughter of a skilled stonemason who died young.

  21. tommy says:

    Greg, I’d love to know if you have any interesting hypotheses or even just vague suspicions on the nature of subfactors of intelligence. We’ve got the fluid vs. crystallized distinction and verbal vs. visuospatial distinctions (or verbal vs. nonverbal vs. visuospatial). I suspect there may be a lot more to be learned here, however.

  22. Whyvert says:

    I know of two geniuses that were illegitimate: Leonardo da Vinci and Erasmus. Both were born out of wedlock. In both cases the father had diddled the servant girl. So the mothers were of very humble position, the fathers in the educated class, though hardly stellar.

  23. SlushFundPuppie says:

    From wikipedia: Gauss was born … in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony)

    A resident of Saxony was smart. Such an anomaly. You can’t have WASPs without Saxons.

  24. Kenny says:

    What’s a “street butcher”?

  25. NumberOneCustomer says:

    Wikipedia says Fourier was the son of a tailor

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