Thursday, September 13, 2018

Tim Gowers' conceptual thinking sucks

Four days ago, I discussed – and many of us talked about – the retroactive silent erasure of Ted Hill's already published article from an electronic journal because the paper implies conclusions that are considered politically undesirable. The focus was on the sociology and dirty politics.

Here I want to focus on the content of Ted Hill's paper (read on the arXiv) and the quality of its arguments and counter-arguments, especially those from Tim Gowers, a famous mathematician. Another famous mathematician, Terry Tao, wrote just three paragraphs whose main purpose seems to be to express some loyalty to the powerful leftwingers in his environment.

Thomas Young who lived two centuries ago was labeled the last man who knew everything, the last true polymath. Aside from the interference pattern, he knew and did lots of other things that were linked to the musical harmony, Egyptian hieroglyphs, and others.

I think it's obvious that these days, a person who would be at the very top in all disciplines is extremely unlikely, basically impossible. On top of that, I don't really believe that even the polymath Thomas Young was one of the history's greatest minds. So I think it is also foolish for someone to try to achieve such a versatile outcome today. On top of that, I think that a proper scientist is really driven by the instinctive curiosity, not by the desire to be "great" or "versatile".

But some degree of versatility is obviously a big plus. And let me admit that despite my admiration for the whole occupation, I consider most professional 21st mathematicians to be close to Fachidiots i.e. people who largely lack versatility, extremely overspecialized people who usually know just a very limited strip of the – mathematical – thought. It's fun to be great at some patterns in arithmetic progressions. But there are hundreds of comparably important subfields of mathematics all of which may be considered idiosyncratic hobbies.

So in my understanding of the mankind, mathematicians are among the smartest people but the #1 expert in arithmetic progressions – one of hundreds of similar mathematical topics – is probably less smart than the #20 theoretical physicist because there's really one theoretical physics where the practitioners are largely commensurable. So even if there were no dramatic pressures and filters that choose politically correct mathematicians, I wouldn't have too great expectations about the number theorists' ability to resolve conceptual questions outside their narrow expertise.

For this reason, and because of the obvious selection that only keeps the politically correct people in that system, I wasn't surprised at all that Tim Gowers' take on Hill's paper failed to be intelligent.

Hill's paper

I have independently invented the same "explanation" of the greater variability of men decades ago. To keep things simple, the reason for the greater male variance is the following:
Assumptions (on selectivity): Females only want to date men from the best X percent where the quality is measured by the quantity Q. Males are OK with women from the best Y percent where Y is greater than X.

To make things intuitively clear, you should imagine that X is much less than 50 (percent) while Y is more than 50 (percent). But it's not hard to see that the only real assumption we need is that X is smaller than Y.

OK, there are genes that control the variability of the offspring. Given the assumption above, it will be a successful strategy for Nature to favor genes that increase the male variability relatively to the female one. Nature wants to generate some "really good men", as measured by Q (e.g. top 10%), and experimentation or variability is simply a good strategy to get these men.

On the other hand, women really want to avoid being in the bottom 10%, because that's the layer that the men avoid, so trying to be average is a better strategy for them.
It's common sense. It obviously works, as Hill shows in some differential equations and theorems. The logic why it works is analogous to the strategy of an ice-hockey coach who removes the goaltender when things look really bad at the end of the match.

You know, in the last 2 minutes, if your foe leads 1-to-0, the probability of another goal is already small. 1-to-1 would be good for you but another goal's probability is shrinking. 2-to-0 is worse than 1-to-0 but not so much, it doesn't change the winner. So if the probability of your victory is 1%, you would obviously prefer if it were 10%. And if you dramatically increase the probability of a goal by making the dynamics "hotter and more chaotic", the game without a goaltender could predict some 40% probability that your team without the goaltender will score the first goal. And that's high enough.

With some numbers, taking the goaltender off brings the chances of a tie closer to 50-to-50, and that's a good thing for the team in trouble whose chances are much worse than 50-to-50 at that point. I think that whether this strategy is actually good in ice-hockey does depend on the numbers because by removing the goaltender, you probably increase the other team's ability to score a goal much more than the probability of your team (otherwise it would be a great idea to play without goaltenders). But this problematic asymmetry is assumed not to affect our analysis of male and female variances.

The main spirit of Hill's paper could be said to be equivalent to:

If you need some really exceptionally good men, e.g. because women insist on them (or because you want your nation to discover a breakthrough of relativity's caliber), then your clan of men should better do more experimentation. The experimentation and therefore higher variability should become (and actually has become) a part of the men's nature in the long term.

It's really common sense and even some extreme leftists – e.g. Lee Smolin – have verbally understood this simple argument. Variability, experimentation, and risk is good whenever the average is not good enough. The point is that being an average member of her sex is pretty much good enough for women but not for men, so Nature does more experiments with men because the experiments increase the probability of being sufficiently highly above the average.

It's common sense, it obviously works, and this simple logic is clearly relevant for the discussion of the real-world male-to-female variance ratios, too. Your main complaint against this "explanation" presented so far could be that it could be said to be circular. It says that the more selective demands by women are equivalent to the greater variance of men. But it doesn't really say why women are more selective. It just relates two quantities but the existence of neither is really demonstrated for our real world situation by this argument.

However, Hill does have an explanation – a not really new one – why women are more selective. (I don't claim any co-authorship in this observation.) Plain English maybe enough for that: It's because they spend more time (like 9 months) with the pregnancy, not to mention some expected extra years with nursing the baby etc. The choice of the right sperm will affect their life for a long time, and it will affect the value of a "product" that results from a year or years of work. (Well, in the past when biology mattered, we talked about the survival chances of the offspring.) The woman wants such "products" to be good, most of it is decided by the choice of the partner, and that's why the selection is more picky. On the other hand, men produce some millions of sperms every day. If they initiate a child that will have some disadvantages inherited from the mother, they may have sex with someone else soon, too. (Some differences in the promiscuous behavior of men and women may therefore be derived from the different timescales of egg and sperm cycles.)

Let's discuss what Gowers did write.
You can see, or probably already know, where this is going: some people like to claim that the reason that women are underrepresented at the top of many fields is simply that the top (and bottom) people, for biological reasons, tend to be male. There is a whole narrative, much loved by many on the political right, ...
In reality, the much lower percentage of women in STEM, especially near the top, isn't any narrative. It's a simple fact. Another fact is that this gap hasn't shrunk, at least not in a significant way, after 50 years of affirmative action. Many particular numbers describing the gap have grown. So there is quite some evidence that the gap has very little to do with any "discrimination of female STEM workers" by the society that may have been widespread up to the 1950s or so.

If you or your movement prefers to operate with narratives, it's too bad because every sane and impartial person knows that facts trump narratives (yes, the Donald is the main reason why my usage of the verb "trump" has skyrocketed). On top of that, Gowers says that "the narrative" is "much loved by many on the political right". Why is it true? Because this question is politically flavored. Why? Because the differences between groups contradict some ideas about equality and egalitarianism that some leftwingers, pretty much by the definition of the Left, believe. So the people who don't believe and who don't need to believe in such assumptions ("the many people on the political right") don't have a problem with the aforementioned facts.

The existence of this correlation with politics has obvious reasons. But we may ask a different question: Why did Gowers think it was a good idea for him to mention the correlation between the belief and one's allegiance to political camps? Why was it a good idea for the main points he wanted to convey? Let me tell you why. Because when he says that "ideas XY are favored by the political right", it will automatically make XY less popular in the environment where he operates – because that environment is partially a political one and wants to hurt the political right.

But the sensible and impartial people – on the right and on the left, assuming that being on the left isn't in contradiction with being sensible and impartial – evaluate the causal relationship between the statements very differently. If "ideas XY are favored by the political right" and "ideas XY are facts", then it means that "the political right is closer to the truth about XY and similar things than the political left". The fact that Gowers or most of his environment can't get this correct conclusion indicates that the environment is not only predominantly left-wing. It also fails to be sensible or impartial or both.

Now, Gowers also writes:
There is also a counter-narrative that says that people on the far right keep on trying to push discredited claims about the genetic basis for intelligence, differences amongst various groups, and so on, in order to claim that disadvantaged groups are innately disadvantaged rather than disadvantaged by external circumstances.
OK, we are told that there is a "counter-narrative" – these people are all about narratives and brainwashing, not about the truth or intelligent debates. What is the content of this counter-narrative? It's a mixture of a tautology with a would-be insulting description of the people who believe in that tautology and an insulting, untrue description of some facts.

The insulting part directed against the people is their identification of "far right". That's very interesting because people like Steven Pinker who is clearly left-leaning must obviously be included in this "far right". Needless to say, this label by Gowers and thousands of others that we encounter every day show that "far right" is the label that dishonest leftists use for the advocate of any inconvenient truth. Years ago, the label "far right" may have been damning and people may have tried to avoid it but it's no longer true. The label has lost its power because it's been used for most of the good people on Earth. Certain people who keep on using it have overlooked how childish and inconsequential their slurs have become.

The insulting word directed against facts is the word "discredited". There is obviously nothing "discredited" about the role of genes for intelligence. Our higher intelligence relatively to chimps surely results from our different genes. People who deny that and who say that chimps only look dumber because of some reactionary political system of the chimps are just plain lunatics.

The tautological part is that some people claim that if there are biological reasons for some group differences, then the disadvantaged status of some groups may be explained by genetics. What can be controversial about this reasoning? Nothing. Why does Gowers repeat the same thing twice? Let me tell you why. Because his text has nothing to do with an intelligent take on the topic. It's another collection of slogans that are written so that they maximize the emotional reactions by brainwashed readers who lack intelligence. If he repeats the same thing about the genetic causes of group differences twice, using slightly different words, he can make a larger number of the readers upset.

I myself, as will be obvious, incline towards the liberal side, but I also care about scientific integrity, so I felt I couldn’t just assume that the paper in question had been rightly suppressed.
How would a sensible people even talk about the possibility that the paper has been "rightfully suppressed"? It was a published paper so its erasure obviously couldn't have been "rightful", could it? Maybe Gowers didn't "assume" that the paper has been "rightfully suppressed". But that's pretty much the conclusion that we heard from him at the very end, anyway, and evidence is strong that he was pre-programmed from the beginning to end up with the only imaginable – politically correct – conclusion.

The very fact that he needed to discuss the possibility that he would immediately declare the erasure of the published paper "rightful" proves that his scientific integrity isn't in perfect shape. And it looks very likely, although not rigorously provable, that this whole game about his hesitation was just a theater.

Gowers does everything he can to delegitimize Hill's proof as mathematics:
...bear in mind that since the article is written by a disgruntled author, there is almost certainly another side to the story. In particular, he is at pains to stress that the paper is simply a mathematical theory to explain why one sex might evolve to become more variable than another, and not a claim that the theory applies to any given species or trait.
He's disgruntled as much as most Jews who were in a concentration camp – rightfully disgruntled. Incidentally, concerning the other side of the story, Amie Wilkinson claimed she was innocent and she completely avoided the discussion of Hill's assertion that she hired her father and husband as attack dogs against Hill's paper. Well, main conclusion is that Hill's side of the story is almost certainly the only correct one.

If you read Hill's paper, you may easily check that this is a mathematical paper about some theorems. In fact, even at the ad hominem level, Hill has no record of writing politically flavored papers. The fact that these theorems are useful to think about the real-world situation is a corollary. The existence of such real-world corollaries cannot possibly decrease the legitimacy of a mathematical paper. Instead, intelligent people who think about the real world try to take all relevant mathematical facts into account at all times.

After several sentences trying to question whether Hill's paper is mathematics at all, we read:
I was worried that I would find [Hill's arXiv preprint] convincing, ...
Oh, I see. So Gowers was worried that he would find the paper convincing. That says quite something about Gowers. How could an honest, let alone curious mathematician be worried that the paper he reads is convincing? Let me tell you the answer. He couldn't. A proper mathematician or scientist is always worried that the paper he reads turns out to be wrong or a waste of time – simply because it's a bad thing to waste one's time!

To find out that a paper that we just read is good and actually teaches us something that is convincing is always a good thing for every honest researcher who reads a paper. So we can't possibly be worried about such an outcome. We are hoping that this is the outcome.

Gowers' "worries" that a paper he reads is convincing proves that he is not a curious yet impartial researcher into this part of mathematics. But we can be a little bit more quantitative in what it means. The obvious reason why he was "worried" is that if such a paper were found "convincing", it would undermine Gowers' preferred political ideology and/or it would make him harder for him to defend that ideology or the behavior of his soulmates. To summarize, the reasons why he could be "worried" are all about the importance he assigns to the well-being of (political) "narratives".

A scholar could have mixed feelings. On one hand, he wants to read papers that teach him something convincing, new, and true. On the other hand, he may have some political biases. "Hopes" are fighting against "worries". We have learned that "worries" have won which proves the following: The political narratives are more important for Gowers than the convincing nature of the mathematical content!

So whether you like it or not, it's still true that even folks with awards such as Gowers are primarily left-wing political activists and only secondarily, they are researchers in mathematics who search for the truth wherever it is. That's the only possible reason why the "worries" could have trumped the "hopes".
...but in fact I found it so unconvincing that I think it was a bad mistake by Mathematical Intelligencer and the New York Journal of Mathematics to accept it, but for reasons of mathematical quality rather than for any controversy that might arise from it.
That's a standard fraudulent obfuscation of the reasons why people and their work are harassed by authoritarian regimes. At some point (but not always), such regimes try to nurture their image of enlightened systems, so all the terror against the politically inconvenient people has to be described as meritocratically, not politically, justified. Except that this justification – which is always manifested as an attack on the prosecuted people's professional qualities – is a plain lie.

Even if there were something seriously wrong with the paper, it would still be objectively unethical to silently erase Hill's paper after it was published.

At any rate, the main theorems described in the paper are clearly correct and Gowers' inability to see it shows that Gowers is simply no good as a thinker about these general mathematical issues.

In the subsequent paragraphs, Gowers tries to follow Hill's paper. He has a hard time to find something wrong with it. He ends up saying that he's OK with the definitions of the variability. However, he's dissatisfied with the "selectivity" because it's too crude.
The definition of selectivity in the paper is extremely crude. The model is that individuals of one sex will mate with individuals of the other sex if and only if they are above a certain percentile in the desirability scale, a percentile that is the same for everybody. For instance, they might only be prepared to choose a mate who is in the top quarter, or the top two thirds. The higher the percentile they insist on, the more selective that sex is.

When applied to humans, this model is ludicrously implausible. [...]
A model with a sharp cutoff for a well-defined variable Q is obviously not an exact rigorous description of anything in the real world. The real world doesn't have such sharp cutoffs. But what's enough for the validity of the qualitative conclusions is that the assumptions at least qualitatively or approximately resemble what is happening in the real world.

The idea that the woman can identify the variable Q of the men and "measure it" precisely is exaggerated, of course. On the other hand, this comment is just a description of the very general and obvious problem that social sciences aren't exact sciences and it is always difficult to translate complex systems such as this one into the language of mathematics. This non-exact character of the social sciences isn't Ted Hill's fault – on the contrary, he is trying to reduce the gap between social sciences and sharp mathematical thinking. On the other hand, it is equally obvious that there is "something true" about it, that the mathematical model resembles the real world behavior.

In particular, there is no sharp cutoff below a certain particular percentile. But even if the cutoff were fuzzy, if the probabilities of picking were continuously dropping, we could define an "effective cutoff" which, if substituted to the discontinuous model of the selection, would yield the same results for a quantity as the "fuzzy cutoff". The mathematical model is clearly "close" to the real world situation in some sense so it tells us about the real world situation and the predictions might be, in fact, rather accurate. To say the least, having an approximate model is arguably much better than drawing conclusions from pure prejudices with no model at all.

But the degree of proximity between the model and reality isn't a subject of Hill's paper or any paper in a pure mathematical journal. Mathematical journals are supposed to describe the pure and rigorous heart of some ideas that people may need, even if they deal with all the messy situations of the real world.

On one hand, Gowers wrote that he was going to debunk the paper because it's bad mathematics. On the other hand, it is very obvious that all the things that Gowers finds "wrong" about the paper are related to the possible applications of the mathematical paper in the real-world situation. That's what Gowers finds primary in his whole critique and his claim that he is finding something wrong about the paper as a mathematician does in another mathematician's paper is simply a lie.
...the idea that some huge percentage of males are simply not desirable enough (as we shall see, the paper requires this percentage to be over 50) to have a chance of reproducing bears no relation to the world as we know it.
The percentage was assumed to be over 50 to make the point and to prove the theorem. But it's very obvious, as I said, that the relationship is completely general. The sex that is more selective will have a lower variability and the sex that is less selective will have a higher variability. Hill's choice of stronger assumptions for his theorem was useful for the situation to be intuitively understandable and for his theorems to be proven more directly, too.

So while the paper may require "over 50" somewhere, it's clearly not the case that the general argument requires the number to be over 50. Gowers is obfuscating the real target of his criticism. But without obfuscation, there is nothing to criticize because
  1. when it comes to the pure mathematicians' perspective, there is nothing wrong about a paper that simply makes a somewhat stronger assumption (than a random other assumption one could imagine) and then proves some theorems
  2. when it comes to the implications for the real-world situation, there is nothing to criticize because the generalized theorems surely do work even with weaker assumptions
Then, another talking point by Gowers:
And even if we were to accept that something like that had been the case, it would be a huge further leap to assume that what made somebody desirable hundreds of thousands of years ago was significantly related to what makes somebody good at, say, mathematical research today.
The relation is obviously significant. Doing mathematics isn't too different from cleverly behaving in the wild nature. Already hundreds of thousands of years ago, humans were trumping other animals when it came to clever solutions. This was their comparative advantage, this was what their survival increasingly depended upon. Those talks weren't identical to a mathematics exam but not to understand the significant relation between these tasks means to be stupid.

Cavemen didn't need a Fields-Medal-level mathematical intelligence to fool a mammoth but they still needed much more than the average chimp – that survived differently. The same mechanisms that allowed the cavemen to become much smarter than the chimp also leads (or greatly contributes) to a "side effect", the existence of Fields-Medal-level mathematicians.

Here is the first well-defined complaint by Gowers that could make you say "maybe he has at least one point, after all":
But there is something very odd about this. Those poor individuals at the bottom of population P aren’t going to reproduce, so won’t they die out and potentially cause population P to become less variable? Here’s what the paper has to say.
If you think for a minute, however, you will realize that Gowers doesn't understand what is going on at all. "The higher male variability" is a gene shared by the whole subpopulation that competed against other subpopulations – and because the higher-male-variance subpopulation won, it's a gene shared by basically the whole mankind today. Another subpopulation used to have a different gene that adds smaller random terms to Q to the newborn boys. These two subpopulations' genes aren't "destroyed" when individual members of the subpopulations die. So the subpopulation with the larger male variance will still add larger random terms to Q of the newborn boys even if some boys in that subpopulation die out.
If females have a higher cutoff than males, wouldn’t that suggest that males would have a much higher selection pressure to become more desirable than females?
Right, as long as you focus on the percentage of "very desirable ones". That's indeed the conclusion and what is observed, too.
And wouldn’t the loss of all those undesirable males mean that there wasn’t much one could say about variability?
As I said, no, it wouldn't. The variability of men and female – the magnitude of the random terms added to Q – is encoded in genes. You can't change the genes by individual deaths.
Imagine for example if the individuals in P were all either extremely fit or extremely unfit. Surely the variability would go right down if only the fit individuals got to reproduce.
No, it's not unavoidable at all. As I said, the higher variability for every newborn boy is a property of the gene. It is this gene that guarantees this behavior – higher male variability – permanently that is giving the subpopulation of men an advantage.

Indeed, you could imagine different genes that determine the variability differently, e.g. in a way that Gowers suggests. A gene that would reduce the variability of the boys in the next generation. But a subpopulation with such a gene would soon behave just like any other subpopulation with a lower male variability and it would die out!

So it's only the gene guaranteeing a permanent enhanced male variability, even after generations, that trumps everyone else! That's the point and Gowers clearly failed to get it even after he read the paper. Gowers' alternative genes affecting the distributions differently (e.g. only temporarily) could exist but the subpopulations with these genes and algorithms would have died away – they are no good!

Incidentally, it's easy to empirically see that some variability in the human race is a permanent feature of our species. If the IQ (or another quantity) of the baby were simply the arithmetic average of the IQ of the parents, without any extra random terms added, the people would have already converged to IQ=100 for everybody. All the variability and diversity of humans would have already disappeared, surely after 100,000 generations. It didn't. So there is demonstrably an extra mechanism that adds random terms to IQ of the babies.

The size of variations or the speed of random mutations are largely encoded by genes as well, even though they look as "meta-parameters". But within a Darwinian picture, they are clearly parameters of genes just like any other parameters. In some epochs of the history of species, it was a better strategy to quickly (or slowly) mutate, so those subpopulations or species that did it right survived. (A trivial general example: When the external conditions were changing quickly due to an asteroid impact, species or subpopulations that were quickly mutating were able to adapt more quickly and that was an advantage. In stagnant external conditions, however, these quickly mutating species looked like a bunch of organisms suffering from too much cancer etc., and they were beaten by a more stable or predictable species.)

Incidentally, this was a key point by which I nuked an anti-evolution critique by a creationist at our 2010 brainstorming session funded by Peter Thiel. They assumed the speed to be constant and with that assumption, evolution would have had some trouble with time scales. (I unsurprisingly disagreed with their argument and conclusion but from the technical nature of that debate, you may figure out that I believe that those creationists understood evolution better than Tim Gowers does.) But these parameters dictating the variations or rates of mutations are ultimately dynamical as well, although usually less quickly changing, and that's how Nature may have achieved better outcomes with the species than any fixed-parameter model. The evolution of those parameters is surely an important part of the evolution of species.
What is the purpose of the strange idea of splitting into two subpopulations and then ignoring the fact that the distributions may evolve (and why just “may” — surely “will” would be more appropriate)?
Men are split into "subpopulations" in the paper because these subpopulations differ in some genes (which arose from some mutations, but such mutations aren't occurring too often) and these genes – which are the basis of group differences between these subpopulations – play a crucial role for the different survival chances of each subpopulation at every moment, and therefore for the evolution of the relative numbers in time.

The splitting into the subpopulations is completely analogous to the splitting to species – different subpopulations compete just like different species do. It's the genes that fight at the end!

Hill isn't assuming that the distributions are always constant. He assumes that they're constant for a long time (long enough to decide about the winning gene, i.e. many generations) because such stable distributions with large enough, non-decreasing variability (especially for men) are the strategy (e.g. the gene) that wins over other strategies (e.g. genes).
I admit that I have not spent as long thinking about the paper as I would need to in order to be 100% confident of my criticisms. I am also far from expert in evolutionary biology and may therefore have committed some rookie errors in what I have written above.
Well, Mr Gowers, you have written a long enough text to determine that you have spent at least a nontrivial amount of time with these ideas, but you still didn't get the basics of Hill's paper – and perhaps basics of evolutionary biology. That's enough to see that your thinking about general issues outside your very narrow expertise isn't good. The basic assumptions of evolutionary biology are obvious and your text makes it clear that you understand them. The rest really is mathematics, arguments similar to Hill's, and you just showed that you aren't good at this mathematics.
The only explanation I can think of for that is that it was written by somebody who worked in evolutionary biology, didn’t really understand mathematics, and was simply pleased to have what looked like a rigorous mathematical backing for their theories. But that is pure speculation on my part and could be wrong.
Yes, it is wrong and it is driven by your wishful thinking and the desire to spread politically convenient "narratives". The reviewers of that mathematical journal were mathematicians.
[...] The theory might appear to fit the facts quite well: [...]

But it is nothing like enough reason to declare the theory correct. For one thing, it is just as easy to come up with an environmental theory...
The last sentence is a typical example of a sentence from a person who isn't capable of thinking scientifically at all. You can come up with an environmental theory and things are complicated. But if an insight or an effect, a pressure by which "A affects B", has been found, this effect is here to stay and it doesn't get erased if you study different effects. Instead, they get added.

The real situation is surely a mixture of various mechanisms and influences. It just happens that the biological effects make sense and may be described by mechanisms that may be translated to mathematically rigorous theorems that may be published in peer-reviewed mathematics journals; while the competing environmental theory remain at the level of "narratives" and a wishful thinking that is mostly spread by political activists because they have pre-decided at the very beginning to favor certain conclusions. Here is what the "competing theory" looks like:
Let us suppose that the way society is organized makes it harder for women to become successful mathematicians than for men.
Right. What Gowers is doing is a textbook example of circular reasoning. Let's prove that all the differences are due to discrimination. How? Let's suppose that they're all due to discrimination. Then QED. We won.

Great. But if you think that this self-evidently circular comment is on par with Hill's paper, then you are a moron, Mr Gowers. All your arguments defending your wrong statements boil down to other wrong comments as assumptions. Everything is circular. But that's not the case of Hill's mechanism because Hill's mechanism ultimately boils down to facts such as the fact that a greater number of human sperms is produced per unit time than the number of human eggs. This is really the assumption and the rest – including the difference in male and female variabilities – is derived from that fact as a purely mathematical corollary.

You never assume any facts, you only assume other "narratives" picked from your ideology, i.e. other "desired conclusions". So your claims are in no way on par with Hill's and your apparent inability to see this simple point proves that you are either a very weak thinker or a brutally dishonest person pretending to think something completely different than what you believe – or both. At any rate, you suck, Mr Gowers.

And that's the memo.

P.S.: Here he has a "second argument":
A second reason to be sceptical of the theory is that it depends on the idea that how good one is at mathematics is a question of raw brainpower. But that is a damaging myth...
This claim is just plain wrong. First, there is a clear influence of the raw brainpower on the chances to do mathematics well. But even more importantly: Hill's mechanism doesn't really need the quantity Q that determines the desirability to be raw brainpower. Because it's evolutionary biology, his Q is actually rather close to chances of survival for the offspring (or some convenient life of the female for the rest of her life may be added). But assuming that these chances used to have something to do with the mathematics-like thinking, you may also say that Q is some combination of the raw brainpower, memory, speed of thinking, but even creativity, ability to imagine things, passion, patience, excitement, sensitivity, playfulness, caution, attractiveness of the adult men as a "teacher" for kids, or anything else. Whatever virtue or a mixture of virtues Q represents, Hill's argument will still work and men will have a greater distribution in either of them as long as it (this single quantity) is relevant for the selection. Are you really incapable of seeing this elementary point, Mr Gowers? Or did you associate Hill's argument with "raw brainpower" just because you figured out that "raw brainpower" is unpopular and you can hurt Hill's argument by this demagogy? In either case, it is very disappointing, indeed.

When this blog post was written, I was told about the second part of Gowers' comments on the Hill's paper. Not sure I will have time to react. At some moment, it may become a good idea to stop and I am rather busy now. A detailed response to every sentence – like above – is unlikely. But a short response is here:

In the second text, Gowers says it's generally wrong to unpublish, and he basically takes back his criticisms concerning the selectivity. He still repeats some criticisms that a "toy model was used". It seems that he doesn't understand what models are good for at all; and it seems that he still doesn't understand that the different degrees of variability are governed by different genes.

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