Figure 2.—The distribution of fitness effects of deleterious

mutations represented as either (a) a continuous or (b) a discrete

function. The dashed lines in a and the solid lines in b

represent the 95% credibility intervals. (a) A transformation

of the gamma distribution to a log-scale. Note also the difference

in the minimum values for a and b.

These graphs are taken from a paper written by Adam Eyre-Walker, Megan Woolfit, and Ted Phelps. ‘Nes’ means the effective population size (assumed to be about 10,000 in humans) multiplied by the percentage decrease in fitness associated with a particular kind of mutation. So a Nes of 100 correspond to a 1% decrease in fitness. The point is that mildly deleterious mutations, ones that reduce fitness by something like 1%, are considerably more common than ones that drastically reduce fitness. This makes sense, because most non-synonymous mutations, ones that change an amino acid in a protein, don’t cause a big decrease in fitness. A few do, as when a mutation turns an amino acid into a stop codon, truncating the protein.

Note that this describes the spectrum of new mutations. The distribution of existing deleterious mutations in a population is quite different. Dominant lethal mutations are not passed on, hence do not build up with time. The dominant lethals you see are all new, freshly generated by mutation. On the other hand, a mutation that reduces fitness by 1% is only slowly eliminated by purifying selection, so its frequency builds up with time. Its equilibrium frequency is 100 times higher than that of a dominant lethal that occurs equally often.

So… most genetic load in humans is made up of many, many mutations that each have fairly small effects. A smaller fraction of the genetic load consists of mutations with big effects on fitness.

I think that this picture has some interesting implications. Consider paternal-age effects: several problems seem to be more common in children whose fathers were older than typical. Sometimes this effect is dramatic, as with Apert syndrome, but that is a special case: premeiotic cells with this mutation, the precursors of sperm cells, seem to have a growth advantage. Something similar probably happens in achondroplasia, classic dwarfism. More generally, one expects that the mutation rate rises with paternal age due to an increasing number of cell divisions in the male germ line. There is evidence, not utterly conclusive but fairly strong, of increased rates of autism and schizophrenia with paternal age.

We can then conclude that increased rates of disease with paternal age are driven by highly deleterious mutations, rather than an increase in the number of slightly deleterious mutations. If the mutation rate in men of a certain age is 1.5 times higher than average, dominant lethals would go up by 50% in their children, while the average number of of 1%-deleterious mutations would increase by only 0.5% – a totally insignificant change. This because those slightly-deleterious mutations have accumulated over many generations: one generation can’t make much difference. Still, the number of slightly deleterious mutations does vary between individuals: the distribution should be Poisson, although with N large enough to closely resemble a Gaussian distribution. And this distribution might be modified by selection: people on the high end may suffer materially reduced fitness. Theory suggests that they should.

One important point is that a single highly deleterious mutation has a good chance of pushing the whole organism in some odd direction in phenotype space. In other words, the same mutation that drops your IQ, or damages your heart, may also make you look funny. At lower IQs, more and more kids are considered to suffer from ‘organic’ retardation. On the other hand, a higher-than-average number of small-effect mutations should also interfere with really complex systems such as the brain (and reduce IQ), but because of the law of large numbers, wouldn’t tend to have any particular direction in phenotype space. As far as I can tell, an extra-large dose of small-effect mutations, which we will henceforth call genetic noise, would not make you funny-looking.

Individuals can vary in the amount of genetic noise they carry, and populations can as well, depending on the relative intensity of selection and on the mutation rate, which might also differ. For example, although having an unusually old father does not much affect the amount of genetic noise an individual carries, a culture in which fathers were typically 55 would undoubtedly accumulate an unusually high amount of genetic noise, over a couple of millennia.

If a kid’s parents have a higher-than-average amount of genetic noise, on average the kid will as well. This sure looks like what we usually call non-organic or familial retardation.

Most of the within-population variation in IQ looks to be familial rather than organic. If I’m right, this means that most IQ variation – what we might call the normal range – is caused by differences in the number of slightly deleterious mutations. None of them would show up in a QTL search, because all are rare. And that is where we stand thus far: no intelligence QTLs have been found – although you never know what you’ll see in the next population. On the other hand, shared chromosomal segments would mostly contain the same slightly deleterious mutations, and so IQ should correlate with genetic similarity, which is what Visscher has found.

Many great scientists and mathematicians have likely had relatively low levels of genetic noise combined with some fairly deleterious de novo mutations; with the net effect of a powerful mental engine strangely focused on some particular topic not directly related to fitness. Low noise, high weirdness. Math, not sheilas. One might look for advanced paternal age in such cases.

We know that IQ is associated with a number of good outcomes – greater longevity, for example. Some of that, maybe most, may be people practicing good health habits, but having less genetic noise could help in a more direct way.

I am interested in why you think less genetic noise could be tied to longevity other than correlation, what would the causal mechanism be?

With less noise, just about any organ would work better.

Mr Cochran,

You always have fascinating posts. I check this blog often. Could you recommend one or two textbook style books that would good introduction to the genetic topics discussed here?

Seconded. More specifically: What tools do you need to understand a reasonable chunk of Fisher’s The Genetical Theory of Natural Selection? Where should you start?

Might I suggest you check out another outstanding blog, Gene Expressions/ Discover Magazine. Click on Razib on Books and then under the topic Genetics you will get reviews and recomendations of exactly the kinds of books you are looking for.

Regarding the supposed weirdness of mathematicians. I think this is a bit overdone. The part of it that is true it that mathematics happens to be a field in which one can be successful with an almost complete lack of social skills. Someone like Gregor Perelman would presumably have no chance of being successful in business say because of his strange personality. So you rarely see someone like him successful in most other fields.

But I don’t think it is true that most outstanding mathematicians are like him. When I was an undergraduate Atle Selberg visited my university and the math department had a dinner honoring him. I was one of the undergraduates who were invited to sit at the same table with him. He was there with his wife and I recall that they were both extremally pleasant and totally normal in appearance. They talked mostly about cross-country skiing of which they were both avid devotees. No sign of any deleterious mutations

Casual conversation isn’t a very good test of anything. I can manage that much and I am extremely autistic. What I can’t do, because dealing with others takes too much conscious attention for me, is to deal with others when I also need to think about any kind of problem. Or to deal with the same people for relatively long periods, as in working in a team, I don’t have the automatic responses normal people possess for interacting with others and it is very stressful, I can only manage a week or two before I have a meltdown.

There are at least two ways to do work that is seen as creative or innovative. One is to systematically explore some possibility space until you find something nobody’s done before. Great for generating bland computer art and crossover cuisine.

The other is to be genuinely different – to have a mind that generates unusual thoughts and intuitions, often with an unusual set of priorities. Steve Mann is a great example: http://upload.wikimedia.org/wikipedia/commons/3/3f/Wearcompevolution.jpg (he wore an earlier prototype to prom).

The rewards are different in different fields. If your perspective is too far from the ordinary in art or music, you simply produce incomprehensible outsider art that will only get you nervous laughs.

It’s a bit better in tech: Steve Mann’s contraptions actually works, and were enough to get him tenure, despite being the results of what could only be described as an unhealthy obsession.

So what field rewards unusual thoughts and priorities the most? Well, correct me if I’m wrong, but there is no such thing as a crank mathematician – as long as you prove your results, you’re as much of a mathematician as anybody else. If you don’t, you’re not a mathematician at all.

You only need to step down to physics for the situation to be radically different. Kurt Gödel was no coincidence.

Sorry, I misspelled “extemely”

You think Selberg was autistic?

To Olof – I agree that in mathematics you can suceed even if you have a very strange personality. Someone like Goedel could be very successful in mathematics despite his very introverted and non-social personality. But I don’t agree that you need to be autistic to be successful in mathematics. You just see some exotic personality types in that field because social skills are completely irrelevant there. But that doesn’t mean that you have to be autistic to be successful in mathematics. When I was in grad school Samuel Eilenberg was my algebra teacher. He was highly extroverted, cracking jokes at every opportunity,

the life of the party at social functions (at least to the extent that math department social functions had a life). I don’t think he was faking anything. That was just his natural personality.

I never said that you had to be austistic (a word I am beginning to hate) – or neurologically weird in some way – to be a world-class performer in subjects like math or theoretical physics. Assume for the moment that social agility played no role in top-level success in those fields: then you would assume that the neurologically weird might show up at their base level in the population, which would be higher than what you see among politicians or car salesmen. But that base level isn’t very high. It certainly looks to me as if they show up far more often than that. It isn’t just Borcherds, or Godel, or Perelman, or Dirac, or Newton. So, although it may not be necessary, cerrtain kinds of weirdness seem to help.

What makes you so sure they do exceed their base rate? Many of the greatest were superbly normal and capable (think of Gauss); do you have any sort of systematic survey?

For example, http://blog.computationalcomplexity.org/2011/07/disproofing-myth-that-many-early.html looks at ~50 of the most eminent logicians recently – and logicians are supposed to be mad as hatters, with, yes, Godel leading the list – but only finds 4 mad ones.

I start with a useful operational definition of insanity as a Darwinian disease: any behavior that would have drastically deceased fitness in the past. Surprisingly, dipping an apple in cyanide and eating it would fall into that category. Yet your source does not consider that Alan Turing was crazy. Your source is mistaken.

Anyhow, it’s easy to find super-prominent mathematicians who are strange at a level that is rare in the general population. If the incidence of such strangeness was the same in mathematicians as in the general population, it would not be easy.

A idiosyncratic definition which omits trivial issues like being forcibly injected with 1950s state of the art government hormones…

With that sort of definition, all modern intelligent people are insane because they have such low fertilities, and you still do not provide any evidence that eminent mathematicians or scientists are unusual in this respect.

It is a sensible definition, much more sensible than DSM, for example. Turing was weird in all kinds of ways: not many people at Bletchley Park spent an hour each night discussing a children’s radio show with their mother. Anyhow, anyone who thinks that professional mathematicians, especially at the highest level, don’t have a significantly higher fraction of truly weird people than the population as a whole is ignorant of mathematicians, or of the general population, or both. I’m not just talking about Turing: I’m thinking about Borcherds, and Perelman, and Grothendieck, and Erdos. And others. Borcherds once said that every math department he had ever visited had at least one person who who clearly more strange than he was… Maybe you are thinking of ‘normal’ as some sort of moral judgment. That’s not what I mean. When I read about a prominent mathematics professor, who, having dinner with friends, spends most of the time under the table reading the Britannica, I do not condemn. But I know it is unusual, and likely not conducive to fitness.

Now if you think that ‘more than expected’ means ‘all’ or even ‘most’, you’re hopeless.

There are more ways to be weird than just lacking social skills. You can have excellent social skills and still be ostracized – even if you know when to keep your mouth shut, every second you spend thinking

about recent human evolution is a second you don’t spend thinking about golf. Your boss can tell.

Of course you can make great contributions to mathematics while being completely normal. You could be smarter than the competition, work harder, luck out in a systematic evaluation of possibilities, be familiar

with two fields that usually aren’t studied by the same person and make a connection, and so on.

However: There is no reason to think that Gödel was autistic, but he had an unusual mind, to put it mildly. As it happens, Gödel’s incompleteness theorem is an unusual piece of mathematics: A

significant fraction of the greatest minds in mathematics got their life’s work reclassified as a fools errand overnight. I’m no expert, but as far as i know none of them saw it coming, and nobody but Gödel was looking in that direction.

What should the null hypothesis be?

To Olof – As for proving your results in mathematics that is desirable but not necessary. Ramanujan never proved a damn thing in his entire life. My favorite Ramanujan story is that when investigating the Ramanujan numbers he formed a hypothesis that they were all divisible by some number, 692 or something. So he checked his hypothesis by testing the first 10,000 Ramanujan numbers. Not a single blasted one of them was divisible by 692. So he drew the obvious conclusion. He published a paper in which he stated that all but a finite number of the Ramanujan numbers were divisible by 692. His empirical evidence agreed exactly with this hypothesis.

By 1993, Fermat’s Last Theorem had been proven for all primes less than four million, so I suppose Andrew Wiles proof was a complete non-event.

“Low noise, high weirdness. Math, not sheilas. One might look for advanced paternal age in such cases.”

Wouldn’t this be the extreme version of what is likely to be often the case with late-in-life paternity? Under most social conditions one can imagine, men fathering children at an advanced age would tend to be higher status, and their higher-than-average social status would tend to indicate lower-than-average genetic noise. So children with lower noise but moderately higher weirdness.

It’s worth checking out.

That would mean that the high profiled mathematicians were individuals with low genetic noise, like their parents, but, unlike their parents, with one or more highly deleterious mutation. This, if true, rises two questions.

1.Why the children became geniuses and not their parents, which presumably lacked deleterious mutations, hence, had lower level of noise than their offspring?

2. Strictly speaking, these children had higher degree of genetic noise than the parents (their weirdness); despite that, they clearly had higher intellectual achievements. Do some of the deleterious mutations in fact boost creativity / or the IQ of its carriers? Are they beneficiary in a (non-darwinian) way, only for a single particular trait?

To Olof – Whose life’s work was reclassified as a fool’s errand overnight by Goedel’s Incompleteness Theorem?

http://en.wikipedia.org/wiki/Formalism_%28mathematics%29#Principia_Mathematica , to name two of the most prominent. Both prolific gentlemen, so “life’s work” might be a stretch, but certainly something they spent a significant part of their lives on (and had great hopes for).

To Olof – There is some doubt about the computer work on Fermat’s Last Theorem. Most of that computer work was checking divisibily conditions on Bernoulli numbers. The relevance of this to Fermat’s Last Theorem depends on a theorem of Vandiver on cyclotomic class numbers. Iwasawa expressed reservations about Vandiver’s proof of his theorem. Maybe the experts have cleared this up.

At any rate the Modularity Theorem, a special case of which was proved by Wiles, has an importance which goes far behind the use of it to establish Fermat’s Last Theorem.

I’m somewhat puzzled as to exactly what point you are trying to make.

I’m simply saying that an unusual perspective increases the probability of doing original work, all else being equal. Neurological weirdness is a reliable way of obtaining an unusual perspective.

This is a concrete example: http://en.wikipedia.org/wiki/Synesthesia#People_with_synesthesia . An unusual perspective has different costs in different fields: If you find rotten meat delicious due to some neurological

quirk, your highly original work in cuisine will not bring you a lot of glory. As far as I can see, mathematics is the field where an unusual perspective has the lowest costs (and possibly the highest utility).

Therefore, we should expect to see more neurologically weird people among prominent mathematicians than among prominent people in other fields.

But people who like rotten milk (cheese) have done quite well in cuisine.

Where in the Principia Mathematica is there any attempt to prove the consistency of arithmetic? As far as the Principia goes Type Theory as formulated there as part of logic has been generally abandoned as too cumbersome. In some sense Type Theory has been shifted out of logic into set theory. The assumption that the class of inaccessible ordinals is unbounded when added to Zermelo-Fraenkel set theory gives a stratification of the universe that is both far more powerful and far less cumbersome than Type Theory. As for the rest of the Principia to the extent that it is useful it has been absorbed into standard notation. Probably like Grassmann’s work people will mine it for centuries for overlooked material.

Everything in the Principia that is mathematical survives although to be sure it is of varying importance.

Although the stricly logical part of the Principia – Type Theory – has been abandoned as it stands many of the basic ideas of Type Theory live on in somewhat disguised form. Only nearly everybody now agrees that it was a mistake to put it into logic. It belongs instead in set theory. That way you don’t complicate the basic logic and you can get a vast extension of type theory in set theory if you assume the existence of arbitrarily large inaccessible ordinals.

Also you refer to Goedel’s Incompleteness Theorem. This name is usually given to the theorem that the true statements of elementary first order arithmetic are not recursively enumerable. The statement that the consistency of arithmetic cannot be proven by finitistic methods is a different theorem. Hilbert had broached the problem of establishing the consistency of arithmetic but it was hardly the life’s work of that remarkable mathematician.

Type Theory was mostly Russell’s baby. Whitehead’s contributions were mostly on the mathematical side . All of it remains valid.

I’m unfortunately not qualified to respond to that, but I’m pretty sure http://en.wikipedia.org/wiki/Formalism_%28mathematics%29#Principia_Mathematica is the mainstream view. It’s certainly all I’ve ever heard from any mathematician. You could always edit it, but be prepared to defend your unusual perspective.

Give me a reference in the Principia where any mention is made of proving the consistency of arithmetic.

I’m sure this is an interesting controversy to some, but probably not to most readers of this blog, including me. If you want to continue, there must be endless hours fun to be had debating the editors of every single wikipedia article that mentions Gödel, Hilbert, the Principia, Formalism or Logicism.

Now, I only used Gödel as an example of an unusual, highly atypical mind making an enormous and unexpected breakthrough. Is this controversial?

“Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Godel’s achievement.” —John von Neumann

Seems legit to me. If there is a coverup and Gödel was in fact completely normal and/or a mundane mathematician, you could replace him with Borcherds, Perelman, Dirac, Newton or quite a few others and address my main point.

Also what statements in the Principia when disentangled from the Type Theory logic are not generaaly acceped as valid?

I’m older than average to have kids – 49 yo – and had a daughter a year ago, and NYU asked me if I’d be interested in enrolling in a 5-year study of older fathers, which I willingly did. They said that they expected that I would harbor a number of mutational defects, some maybe serious. They were testing for 130 micro-insertions/micro-deletions and 90 other conditions, using a micro-array. The results for my wife, me and daughter were NEGATIVE for everyone on everything.

My father’s father had 7 children between the age of 55 and 70 yo. All very bright and higher than average IQ. His third child, a daughter, born when he was about 60 yo, was studying Calculus at primary school age. My grandfather was an autodidact with no formal education, and had taught himself Calculus and other subjects after he retired at 53 yo, and largely home schooled his 7 kids. She would go on to get a PhD in Chemistry, and Lever Brothers sent a plane from London to pick her up for an interview there. She was the first girl from Irish Midlands to go to college (the first boy was an uncle on my mother’s side). She was hired immediately and worked there for years. Lever Brothers would later be renamed Unilever.

Right, so perhaps your father got lucky. It is well established that older men (and high birth order children) have more genetic problems.

On an related point, if I am 26, and don’t plan to have children before 40, should I have some sperm frozen? That seems like an implication.

So, regarding IQ, are you suggesting that between population differences are the result of different average mutational loads…?

…by which I mean that the differences we see are due some groups having accumulated more deleterious mutations? Sorry if I’m not making myself clear, it’s been a long day and much of this would be over my head on a good one.

Good question.

If different populations suffered different effects from the same mildly deleterious mutation, we’d expect the two different populations to have different distributions of mutations. To use a simple example, if there are dozens of mutations that slightly increase your susceptibilty to malaria, those only have a fitness cost in malaria-prone areas. In populations where intelligence was being strongly selected for (ashkenazi Jews in Eastern Europe, say), those mutations would have had a bigger effect on fitness and been more quickly removed. I’m not sure how big or relevant this effect would be, though.

What would cause different rates of germline mutation in different populations? Viral infections? High local level of carcinogens in the environment?

African girls become fertile about five years younger than North Europeans and the age they reproduce (females and males too) is probably lower. The theory says they should have less “noise” and be fitter while late-maturing North Europeans should carry a heavier mutational load. This is a subject I know little, so if the comment is “stupid” please just ignore it. No need to insult me.

In the US, where we have some halfway decent numbers, about 3 months earlier. Not much different.

Thanks.

I think we would have to be looking at length of time after sexual maturity, not exactly total age, since # of divisions of the sperm seems to be causing the buildup, and as I understand it, males don’t produce sperm before puberty?

Just because genetic load affects intelligence, it doesn’t follow that all diffences in intelligence are due to genetic load. Racial differences in intelligence are probably due to different traits being selected for in different environments. In addition to having children at younger ages, Africans populations–living under more primitive conditions with higher rates of infant mortality–are probaby be more effective at shedding the genetic load which comes from mutations with only a small fitness cost. That doesn’t mean that Greg is wrong, but if you are looking for explanations for racial differences in intelligence, you should look in a different direction.

Wasn’t late marriage fairly common in Northern Europe?

First marriages came at a relatively late age in northern and western Europe, for hundreds of years. I don’t know what the fraction of 55-year old fathers was there, or how it differed from the fraction in other parts of the world.

Jim:

Isn’t the consistency of arithmetic *assumed* throughout the Principia? After all, Hilbert’s program, and logicism in general, make no sense unless any underlying logic (strong enough to establish the Peano axioms) is assumed to be consistent. What you call the ‘broaching’ of it by Hilbert seems more than a bit understated: Logicism was the dominant paradigm until Godel scuttled it, with roots going back at least as far as Leibniz, if not Aristotle himself.

Also, that there are true statements of arithmetic that can’t be proved by any ‘finitistic methods’ is a consequence of the fact that “the true statements of elementary first order arithmetic are not recursively enumerable”. The two proofs by Godel are not logically independent. That’s the whole point of Godel numbering, right? At least that’s what Leon Henkin thought.

To Olof – I don’t know what you’re talking about. Of course Goedel was a great mathematician but his work

did not render the Principia ” a fool’s errand”.

To Deckin – Hilbert was not a logicist but a formalist. Read the Principia or Russell’s Principles of Mathematics. I can recall nothing said there concerning the consistency of arithmetic. I see no reason to

believe that the question even occurred to Russell or Whitehead. You are confusing a lot of different people together.

I’m not sure what you mean by “two proofs by Goedel”. The Goedel Incompleteness Theorem is the statement that “the true statements of elementary first order arithmetic are not recursively enumerable”.

Goedel proved many theorems. One of the many other theorems he proved is that there can be no finitistic proof of the consistency of arithmetic.

It might help if you learned something about mathematical logic before talking about it.

From http://plato.stanford.edu/entries/principia-mathematica/#HOPM :

“…Written as a defense of logicism (the view that mathematics is in some significant sense reducible to logic)”

“In Bertrand Russell’s words, it is the logicist’s goal “to show that all pure mathematics follows from purely logical premises and uses only concepts definable in logical terms” (1959, 74).”

Here is a list of people to write angry letters to: http://plato.stanford.edu/board.html . Good luck in your quest.

http://fair-use.org/bertrand-russell/the-principles-of-mathematics/part-ii

“We have now briefly reviewed the apparatus of general logical notions with which Mathematics operates. In the present Part, it is to be shown how this apparatus suffices, without new indefinables or new postulates, to establish the whole theory of cardinal integers as a special branch of Logic.”

Isn’t that a rather strong statement about the consistency of arithmetic? Those eight chapters aren’t enough to suggest that the question might have occurred to him?

To Olof – Yes Goedel was very introverted and became very strange in his last years. But you focus on some strange people in mathematics and take them as typical of outstanding mathematicians. There are not.

This is what I said: “we should expect to see more neurologically weird people among prominent mathematicians than among prominent people in other fields.”

Does this mean “outstanding mathematicians are typically strange people” in your language?

Sorry – I meant to say “They are not”

Jim:

That there is no axiomatic system whose theorems can be *effectively* produced by an algorithm or procedure is often called (by almost everyone) *one* of Godel’s Incompleteness Theorems. Leon Henkin (heard of him??) used to call it ‘Godel’s First Incompleteness Theorem’.The corollary of that, that any such system will contain true statements whose truth cannot be demonstrated from within that system. This is often called (by people like William Craig, Benson Mates, Charles Chihara, Howard Kiesler–heard of them?) Godel’s Second Incompleteness Theorem.

By the way, the old joke is that only three people actually read the Principia, are you claiming to be the fourth? If the whole project of the Principia isn’t to give a logical foundation sufficient to ground arithmetic (which Godel’s result showed to be impossible), then what in God’s name was the point? Another logic text?

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Genetic load sounds like what a lot of writers (Lombroso, Morel) were trying to talk about a century ago with the concept “degeneration”. But I can’t remember most of the factors they thought were at play. Alcohol consumption, I think, and living in an urban environment. Even though I suppose they were pretty Lamarckian about it, is it possible that they were at least in some minor way on to something?

According to wiki the average age of puberty around northern europe in the 1840s was 17 which was when it started to drop sharply. If the average age of first reproduction remained at 24-26 then the increasing gap between age of puberty and age of first reproduction may have led to a lot more load.

Low noise, high weirdness. Math, not sheilas. One might look for advanced paternal age in such cases.Hmm, is it reasonable to suspect that some men’s weirdness leads to an interest in whatever-the-Australian-for-dudes-is instead of Sheilas, with math falling by the wayside? Do you think genetic load could underlie various fitness-reducing sexual preferences? I like my gay cad hypothesis more.

If most within pop variation is caused by genetic load, then common SNP genotyping like 23&Me does/did won’t ever be useful. Deeper sequencing and matching against consensus sequences to find genetic tendencies.

Load shedding will mostly happen in gametes and in the womb, right? The miscarriage rate is high enough that tons of selection pressure can be hiding there.

Advanced paternal age *might* be a signal of a society in which male competition is a matter of cognitive specialization – succeeding in specialized niches. Such that men do not achieve high status until after many years of learning. In other words, a signal of relatively complex, sedentary, centralized societies – the same kind of societies which seem to select for higher general intelligence.

As far as I know, such societies are ‘always’ patriarchal, and women are allocated to men in arranged marriages – by their parents. So the fact that such ultra-specialized ability is not sexy is irrelevant.

Also, in such societies high (above replacement) fertility was pretty much universal (except for sickness) – and differential reproductive success was mostly a matter of differential mortality.

So successful men were perhaps relatively old cognitive specialists (manye similar to what gets called Asperger’s nowadays) and they were allocated (i.e. didn’t have to ‘attract’) young healthy wives because they were good providers; and because they were good providers, more than two of their offspring survived on average.

Meanwhile, nearly everybody elses kids all (or nearly all) died (of disease and starvation, mostly) – no matter how many kids they had.

Maybe.

Interesting theory, what I am wondering is how soon it can be tested. Perhaps the BGI study which is using super computers to compare the entire genomes of 100 extremely bright people and 100 average folk will be the start of our answering the question why there is such large variability in inherited human intelligence. We live in interesting times.

To Olof and Deckin – this is getting very tiresome.

First to Olof – The quote you give from Russell’s Principles says absolutely nothing about a consistency proof for arithmetic. Giving a demostratration of some theorems in an axiomatic system is completely different from giving a consistency proof for the system. If I write down proofs of a bunch of properties of the integers in say Zermelo-Fraenkel set theory that does not mean that I have established the consistency of ZF or even tried to.

Hilbert had a program that attempted to show that classical mathematics could be given a finitistic interpretation that he hoped would make it acceptable to non-realists. It is from this viewpoint that he desired a finitistic consistency proof for arithmetic. Hilbert was well aware of the deductions that could be made in the system of the Principia. But making these deductions is not giving a consistency proof for arithmetic. Neither Russell nor Whitehead or any else until you came along ever conflated giving the deductions in the Principia with a finitistic proof of the consistency of arithmetic. The Principia is not what anyone would call a finitistic system.

To Deckin -

Your silly parade of names of eminent mathematicians is highly amusing. Almost as amusing as your discovery of logicism in Aristotle. You are probably both the first and last person to make this remarkable discovery.

Also amusing is your hilarious statement

“any such system will contain true statements whose truth cannot be demonstarated from within that system”

What do you mean by “contained within the system”? The only sense I can make of this is that “contained within the system” means “provable in the system”. But on that interpretation your statement quoted above says that there are provable statements which cannot be proven. You would have driven Goedel nuts long before he got there on his own.

Look if a statement of elementary first order arithmetic is not provable in ZF then in what sense is it “contained within the system”? Obviously it isn’t contained within the system.

To Olof –

You stated that Goedel’s work rendered the life work of Russell and Whitehead a “fool’s errand”. At the time that Goedel published his famous results both Russell and Whitehead had long moved on to other areas and neither regarded the Principia as their life’s work. but in any case nothing in Goedel’s work rendered anything in the Principia invalid. Goedel’s work does show that Hilbert’s hope of a finitistic prood of the consistency of arithmetic cannot be realized. However this does not render Hilbert’s fabulous career in mathematics a fool’s errand. So whose life’s work was rendered a “fool’s errand”?

Zermelo? Skolem? Von Neumann? Your statement is complete, total and utter nonsense.

Back to Deckin -

I have just notices another idiocy in your comment. You state that Goedel’s Incompleteness Theorem

says that there is no axiomatic system whose theorems can be produced by an algorithm. What a howler! The definition of an axiomatic system is that it’s theorems can be produced by an algorithm. The theorems of EVERY axiomatic system can be produced by an algorithm!! Otherwise it wouldn’t be an axiomatic system by definition.

This is pathetic.

To Deckin -

Any idiot can write a computer program which will produce all theorems of say the Principia or all theorems of Zermelo-Fraenkel set theory. If you cannot do so I would advise you to abandon mathematical logic and take up some other career such as bee-keeping.

Jim:

Cute.

You say: “What do you mean by “contained within the system”? The only sense I can make of this is that “contained within the system” means “provable in the system”. But on that interpretation your statement quoted above says that there are provable statements which cannot be proven. You would have driven Goedel nuts long before he got there on his own.”

No. It means that there must be sentences which, if assumed to be true, cannot be provable within that system. Truth is one thing, provability is the other. It’s Godel’s whole point that the two are different.

Have you given up on the claim that logicism is irrelevant to the Principia? How about the fatuous claim that “the true statements of elementary first order arithmetic are not recursively enumerable” and “that there can be no finitistic proof of the consistency of arithmetic” are logically independent?

As for this being very tiresome, I can’t speak for Olaf, but I for one am grateful to just get any snippet of time I can from a mind of such awesome powers as yourself. It must be a rare burden to be shouldered with such cognitive machinery. However do you do it?

Deckin –

Can you write a computer program which will produce all theorems of Zermelo-Fraenkel set theory? If you can’t there is no point in any further discussion. If you can then you can do what your statement which you confusedly call Goedel’s Frist Incompleteness Theorem says cannot be done.

The principal burden imposed by my cognitive abilities is having to deal with idiots like you.

Of course logicism was Russell & Whitehead’s philosophy when they wrote the Principia. Hilbert though was a formalist not a logicist. Neither was Aristotle.

I never said anything about “logical independence”. In any system with classical logic if “A” and “B” are both theorems then “A iff B” is also a theorem.

Jim:

You said: “Also you refer to Goedel’s Incompleteness Theorem. This name is usually given to the theorem that the true statements of elementary first order arithmetic are not recursively enumerable. The statement that the consistency of arithmetic cannot be proven by finitistic methods is a different (sic) theorem.”

Then you said: “I’m not sure what you mean by “two proofs by Goedel”. The Goedel Incompleteness Theorem is the statement that “the true statements of elementary first order arithmetic are not recursively enumerable”. Goedel proved many theorems. One of the many other (sic) theorems he proved is that there can be no finitistic proof of the consistency of arithmetic.”

I said: “Also, that there are true statements of arithmetic that can’t be proved by any ‘finitistic methods’ is a consequence of the fact that “the true statements of elementary first order arithmetic are not recursively enumerable”. The two proofs by Godel are not logically independent.”

Now you say: “I never said anything about “logical independence”. In any system with classical logic if “A” and “B” are both theorems then “A iff B” is also a theorem.”

In my world, a non-captious and generous mathematician such as yourself would, when using terms such as ‘different theorem’ and ‘other theorem’, point out, at the least, whether they took these ‘different’ and ‘other’ theorems to be logically related to the one in question or not. Since you didn’t, I made the assumption that you took them to be logically distinct.

Deckin – You are going all over the place.

Do you seriously believe that a finitistic proof of the consistency of arithmetic was something that Russell & Whitehead ever thought about at the time they wrote the Principia? Or afterwards for that matter since I doubt that either one ever gave a rat’s ass about it. Since neither cared about it how could Goedel’s demonstration that it was not possible render the Principia a fool’s errand?

Deckin maybe I have been mean to you. I apologize for hurting your feelings. I really do believe that you could write a computer program producing all theorems of Zermelo-Fraenkel set theory. So you might want to reconsider you formulation of Goedel’s Incompleteness Theorem.

I am not going to respond any further.

Good.

We all know that the more IQ someone has the better. So instead of pointing out the obvious, why not come up with ways to increase cognitive ability? There’s too much emphasis on IQ’s role in measuring the g factor. Studies have shown the benefits of having a higher IQ, so it certainly is a good measure. Logically, from there scientists would come up with ideas on how to raise it. Instead

we just inform people with a lower intelligence with redundant studies.

So Greg, any ideas?

To clarify above, I mean inform people of lower intelligence their bleak future.

@greg –

“There is evidence, not utterly conclusive but fairly strong, of increased rates of autism and schizophrenia with paternal age.”and maternal age, too, wrt autism. at least in jamaica.

Could it just be that those on the spectrum simply tend to have children later?

I agree – made the exact same observation a few years ago, based on my own family observations.

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Off-topic:

This man claims a neurological insult turned him gay. In his case it’s a stroke rather than a pathogen, but if true I’d say it lends plausibility to the pathogen theory.

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