So I was delighted by recent news reports that beautiful women do indeed have more daughters. But I was stunned by the reported magnitude of the effect: According to one report, beautiful people are 36 percent more likely to have a daughter than a son!

If that’s true, it throws a whole new light on certain divorce statistics. On average, parents of daughters are more likely to divorce, and I’ve argued (here, here and here, citing research by Gordon Dahl and Enrico Morretti) that daughters probably cause divorce. But now we have a rival theory: Perhaps beauty causes both divorce and daughters.

It’s not hard to see how beauty might (on average) cause divorce; for example, it seems likely that the proverbial “trophy wives” are both disproportionately beautiful and disproportionately likely to be dumped when their beauty fades. So if the beautiful really have 36 percent more daughters, that’s plenty big enough to explain the daughter/divorce correlation.

But I don’t believe it. First, the “36 percent” number is just sloppy reporting. The actual number—which I found in the original paper by Satoshi Kanazawa of the London School of Economics—is 26 percent. (He’s also written a book on this and related research, which I have not read.)

Twenty-six percent is still huge. But it’s also probably spurious. For starters, the sample size is pretty small. Professor Kanazawa started with a sample of about 3000 women, rated from 1 to 5 on attractiveness. (I’ll tell you later where the ratings came from.) Of these women, about 350 were rated 5, and it’s true that these women had a great preponderance of daughters. But that’s a pretty small sample—small enough that there’s about a 5% chance the result is just a statistical fluke. (By contrast the
daughters/divorce study by Dahl and Moretti was based on a sample—drawn from census data—of 3 million.)

Worse yet, those women rated 4 (“attractive” as opposed to “very attractive”) had a great preponderance of sons. Sons were in fact much more common among both the “unattractive” 2’s and the “attractive” 4’s, while daughters were much more common among the 5’s. Among the 1’s and 3’s (“very unattractive” and “average”), the sex ratio is 50/50. Not much of a pattern there.

In fact, Professor Kanazawa’s strong result relies crucially on the fact that he lumped the 1’s, 2’s, 3’s and 4’s into a single category and compared them to the 5’s. If he had lumped the 4’s and 5’s together into a single category called “above average”, he’d have gotten a very different result.

In fact, there are a lot of ways to lump these data, and each way of lumping them gives you another shot at finding a statistical fluke. Given that the sample size makes flukes pretty likely to begin with, it’s not at all unlikely that the professor managed to stumble across one.

And there’s another problem. The sample consists of mothers who were interviewed about their families and then rated for attractiveness by the interviewer. It is not at all implausible to me that a mother talking about her daughter–or holding a little girl on her lap—might present, on average, a different appearance than a mother talking about her son. So the attractiveness ratings are contaminated from the get-go. I would have found this study a lot more convincing if the attractiveness ratings had been based on wedding photos, or other photos from before the children were born.

In fact, all of the mothers in the sample were between the ages of 18 and 28, which means that all the children in question were very young. I’ve watched enough toddlers in my life to believe that, on average, the mother of a two year old boy is going to look a little more haggard than the mother of a two year old girl—possibly enough to keep her out of that “very attractive” category. If Professor Kanazawa has discovered anything at all, I suspect it’s not that beauty causes girls but that boys can run you ragged.

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It seems to be far less well known that Jenkin’s model also predicts that all life on earth dies out after a few generations, which would seem to cast doubt on its assumptions. Jenkin was apparently unaware of this, and so, presumably, was Darwin, who gave considerable credence to the Jenkin model in the final edition of The Origin of Species. This was in 1872.

It seems to be even less well known that the inadequacy of Jenkin’s model was identified in a little-noticed letter to the editor of Nature by the mathematician Arthur Sladen Davis. In that letter, Davis corrected Jenkin’s error and supplied an alternative model that he believed was favorable to Darwin. This was in 1871, but apparently Darwin never heard about it.

And it seems to be known only to me (and now, to the readers of this blog!) that Davis’s model is also flawed, in the opposite direction from Jenkin’s, in that it predicts that any species population must grow without bound following the appearance of a beneficial mutation. And as a result of this, the Davis model undercuts Darwin more than it supports him.

I’ve adjusted Davis’s model much as Davis adjusted Jenkin’s, and gotten a result that could be considered favorable to Darwin. In fact, it’s more or less the result that Davis thought he’d gotten, but hadn’t.

Here comes the more technical part. If this is not your cup of tea, stop reading now. Do come back tomorrow though. I’m not always like this.

Both Jenkin and Davis start with a “blending” theory of inheritance that it is thoroughly at odds with the modern understanding of heredity: They assume that the traits of the offspring are some kind of average of the traits of their parents. These models, then, are therefore thoroughly irrelevant to modern evolutionary theory, with its discrete units of inheritance (i.e. genes). But they remain relevant to the question: What ought Darwin’s contemporaries have thought of Darwin’s theory?

Jenkin begins with the assumption that each individual has a “fitness level”. When two individuals mate, their offspring inherit the average of their fitness levels. An individual with fitness level, say, 1.2, produces 1.2 times as many surviving offspring as an individual with fitness level 1. Let’s make this concrete by saying that your fitness level is defined to be the length of your beak, and the number of your surviving offspring is proportional to your beak length.

Initially, everyone’s beak has length 1. Then along comes a mutant (or, in 19th century terminology, a “sport”), with beak length 2. The sport necessarily mates with an individual of beak length 1, producing offspring with beak length 1.5; they in turn mate (almost surely) with individuals of beak length 1 and produce offspring of beak length 1.25. It’s not hard to show that no matter how long this process runs, the sport can never have more than about 2.38 descendants, whose beak lengths approach 1 over time. The mutation has essentially no lasting effect.

Unfortunately, Jenkin made the phenomenal blunder of assuming that to maintain a fixed population size, each individual must have just one surviving offspring. But because each offspring has two parents, this a recipe for rapid extinction.

Davis corrected the error by assuming that each pair of non-mutant parents produce two offspring, which is indeed a recipe for stable population size as long as there are no mutants. From this assumption he finds (though he doesn’t put it quite this way) that over time the population-wide excess beak length approaches 2.38. In other words, if several generations down the line you’ve got 1000 creatures, their total beak length will be 1002.38. This looks like a small but lasting effect, which presumably can be magnified by further mutations. On this basis, Davis seems to think he’s resurrected Darwin.

The problem, though, is that in Davis’s model, all those sports cause the population to grow without bound, while the excess beak length stays constant. Starting with a population of 100, in generation 10, we have 2439 sports, each with a beak length of 1.0098, for a total excess beak length of about 2.38. In generation 15, we have 78,124 sports, each with a beak length of 1.00003, again for a total excess beak length of about 2.38. Those sports are going to be pretty hard to distinguish from a creature that never mutated in the first place. So Davis’s model predicts that mutations can lead to population growth, but not that they can alter a species.

By way of repairing this, I’ve modified Davis’s model by culling the population every generation to keep it constant (with sports culled in proportion to their numbers). The result: Once again the total excess beak length goes to 2.38, but this time it’s shared among individuals of a fixed population size. This seems to be the pro-Darwin result that Davis thought he’d gotten but really hadn’t.

My calculations, for anyone who wants to see them, are here. Let me add several caveats:

First, I am not an evolutionary biologist, nor anything that could be considered a reasonable approximation thereto. But I think that this might give me a slight advantage here, since the point of the exercise is to view evolution through the 19th century eyes of someone who knows nothing about 20th century evolutionary theory.

Second, neither am I a historian of science. Maybe there was other relevant literature, contemporary with Jenkin and Davis, that I don’t know about. I’ve searched pretty diligently and convinced myself otherwise but I could be badly mistaken.

Third, these calculations are extremely quick-and-dirty. I assumed population sizes large enough so that sports essentially never encounter other sports. This is what Jenkin and Davis assumed, and I’ve maintained the assumption—although in fact it’s quite unwarranted, in Jenkin’s case because the overall populations shrinks so rapidly and in Davis’s because the sport population grows so rapidly. Surely if you corrected for this, some results would change. It’s also possible that the results would change if instead of assuming that the number of your surviving offspring is proportional to your fitness level, we assumed only that the expected number of your surviving offspring is proportional to your fitness level.

Finally, the really interesting question here is: How should all this have affected Darwin’s thought? Was he in any sense remiss in not having done these calculations himself? I am brimming with thoughts on this matter but I think it will probably take me a little while to get them organized and written down. Stay tuned.

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We know, as certainly as we know anything in science, that [evolution] is the process that has generated life on our own planet.

Now, I would be thunderstruck if the theory of evolution turned out to be fundamentally wrong, but not nearly so thunderstruck as if arithmetic turned out to be inconsistent. In fact, I can think of quite a few things I’m more sure about than evolution. For example:

1. The consistency of arithmetic. (This amounts to saying that a single arithmetic problem can’t have two different correct answers.)

2. The existence of conscious beings other than myself.

3. The fact that the North won the American Civil War. (That is, historians are not universally mistaken about this. I am not interested in quibbling about what constitutes a “win”; I mean to assert that the North won in the everyday sense of the word, as reported in all the history texts.)

4. The consistency of higher mathematics. (The math geeks in the audience can take this to mean the consistency of Zermelo-Frankel set theory.)

5. The special theory of relativity. (The science geeks in the audience can take this to mean that the laws of physics are locally Lorentz invariant.)

6.The efficiency of the price system. (The econ geeks in the audience can interpret this as the truth and appropriate applicability of the first and second fundamental theorems of welfare economics.)

And then somewhere down the list—though still way above anything I significantly doubt—we have:

7. The theory of evolution. (That is, all—or nearly all—living things evolved from simpler things, largely through some process involving reproduction, mutation and selection.)

I’m not at all sure I’m right about this ordering, and I’d probably have chosen a different ordering five minutes ago or five minutes from now.

What would your ordering be?

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Where does [Darwinian evolution] leave God? The kindest thing to say is that it leaves him with nothing to do, and no achievements that might attract our praise, our worship or our fear. Evolution is God’s redundancy notice, his pink slip. But we have to go further. A complex creative intelligence with nothing to do is not just redundant. A divine designer is all but ruled out by the consideration that he must be at least as complex as the entities he was wheeled out to explain. God is not dead. He was never alive in the first place.

But Darwinian evolution can’t replace God, because Darwinian evolution (at best) explains life, and explaining life was never the hard part. The Big Question is not: Why is there life? The Big Question is: Why is there anything? Explaining life does not count as explaining the Universe.

Ah, says, Dawkins, but there’s no role for God there either:

Making the universe is the one thing no intelligence, however superhuman, could do, because an intelligence is complex—statistically improbable —and therefore had to emerge, by gradual degrees, from simpler beginnings

That, however, is just wrong. It is not true that all complex things emerge by gradual degrees from simpler beginnings. In fact, the most complex thing I’m aware of is the system of natural numbers (0,1,2,3, and all the rest of them) together with the laws of arithmetic. That system did not emerge, by gradual degrees, from simpler beginnings.

If you doubt the complexity of the natural numbers, take note that you can use just a small part of them to encode the entire human genome. That makes the natural numbers more complex than human life. Unless, of course, human beings contain an uncodable essence, like an immortal soul—but I’m guessing that’s not the road Dawkins wants to take.

Now I happen to agree with Professor Dawkins that God is unnecessary, but I think he’s got the reason precisely backward. God is unnecessary not because complex things require simple antecedents but because they don’t. That allows the natural numbers to exist with no antecedents at all—and once they exist, all hell (or more precisely all existence) breaks loose: In The Big Questions I’ve explained why I believe the entire Universe is, in a sense, made of mathematics.

So while Dawkins believes that complexity can arise only from simplicity, I believe that complexity arises from even greater complexity. I’m not sure I’m right, but I’m sure he’s wrong.

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