Nonconvergent sums and evolution

It's often said that evolution works in such a way as to maximize the number of descendants that an individual has.

More formally, X's children share one-half of their genes with X, X's grandchildren share one-quarter of their genes with X, and more generally X's nth-generation descendants share 1/2ⁿ of their genes with X. So if you buy the whole "selfish gene" theory that genes act in such a way as to maximize the number of copies of them which are made, the quantity individuals should be attempting to maximize might be half the number of children, plus one fourth the number of grandchildren, plus one eighth the number of great-grandchildren, and so on.

It's an infinite sum.

What's more, if you assume a "total fertility rate" of k -- that is, the average female bears two children -- then this sum is k/2 + k²/4 + k³/8 + .... And if k = 2, which corresponds to population not growing or shrinking, this sum is just 1 + 1 + 1 + 1 + ... which doesn't converge. (Similarly if k > 2, but populations which grow indefinitely don't seem sustainable.)

Of course, biologically populations don't live for infinitely long. And in reality people don't at least consciously think more than a couple generations into the future. So practically speaking this is all a bit meaningless.

edited, Monday, 8:42 am: I did mean maximizing the number of copies of genes, in the sense of Dawkins' idea of the selfish gene, and this post is not meant to be taken seriously.

Presidential science debate?

the Presidential Science Debate is scheduled for April 18, at Philadelphia's Franklin Institute. (The Pennsylvania primary is April 22.)

The New York Times asks: will the candidates come? A lot of people commenting there seem to think that it would be a bad move for a candidate to go, basically because they either have to claim that global warming and evolution are real (and thus anger the right) or that they're not (and thus anger the left). Yes, I'm caricaturing. But science shouldn't be a political football.

I would clearly support candidates going to this thing, if only because we may actually get an idea to what extent they're members of the reality-based community. (I'm still not endorsing a candidate, but you can probably guess I'm not endorsing Mike Huckabee.) And as a lot of people point out, such a debate will almost certainly include questions about scientific education; as an educator I'd like to see how those get handled. More funding for the schools. And stop sending us at the college level students who can't do algebra properly. This could be tied into funding -- from what I've heard a lot of the school teachers don't know how to do it, because teaching doesn't pay well enough to hire competent people.

Also, if anyone denies evolution on the grounds that there's no way a process based on "random chance" would create an organism, they will lose the all-important probabilist vote. (Okay, that might just be me.)

Some probabilistic ramblings on evolution

The Repeater (Olivia Judson, in an NYT blog). The post begins:

Here’s an evolutionist’s dream: 10,000 planet Earths, starting from the same point at the same time, and left to their own devices for four and a half billion years. What would happen? Could you go on safari from one planet to the next seeing an endless procession of wildly different organisms? Or would many of the planets be home to life forms that are broadly similar?

This is the sort of question that's hard to answer a priori. Basically, evolution is made up of a ridiculously large numer of random decisions, each with a very small effect. There are a lot of classes of combinatorial structures for which we can generate members of the class uniformly at random (or according to some other probability distribution; the details don't matter here) and they'll all basically look the same. Why shouldn't evolution be like that? The details will be different every time; but in broad outline one can imagine that a "law of large numbers" and "central limit theorem" could apply to evolution -- if we consider some numerical measure of some evolutionary trait, then if we average that numerical measure over many independent "runs" of evolution we should approach some limit, and the deviations from that average might even be spread out according to a normal distribution.

Of course, this isn't something that has to be true -- the many events that make up a single evolutionary process aren't exactly independent, some of them can only happen if others happen, and so on. And whatever numerical measure I was talking about in the previous paragraph might only exist in some runs and not in others. That would seem to argue against my hypothesis. But on the other hand, evolution isn't just a random walk. There are selection pressures which are the more standard explanation for what's known as "convergent evolution", which is the indepedent evolution of similar traits in evolutionarily distinct populations.

By the way, on the topic of convergent evolution: the eye has evolved something like forty times. This suggests that eyes are very likely to arise via the evolutionary process; things that have only evolved once among all life, like language (although that's open for debate), are given the state of our current knowledge less likely to arise. One might be able to compute something like the "probability" that eyes, language, or some other complex trait evolves by looking at how many times it has arisen independently. But this is the sort of probability that is very hard to interpret -- what would it mean to let evolution happen more than once?

Am I a naked brain?

Guess what, folks? Probability and cosmology are weird when they interact. (Big Brain Theory: Have Cosmologists Lost Theirs?, by Dennis Overbye, January 15, 2008 NY Times.)

Basically, it appears to be more likely that we are some sort of naked brain living in an illusion of a world than that we live in the actual world we perceive. Roughly speaking, this occurs if we assume that the universe is infinite -- and thus everything that can occur does occur -- because a naked brain is supposedly much more likely to form by chance than the reality we think surrounds us does.

The obvious rebuttal, if one is wedded to this particular model of cosmology, is an evolutionary one -- maybe naked brains aren't so likely after all, because brains are produced (or so we think) by evolutionary processes, so is one really so likely to find a brain just sitting there without the biology in which it evolved? Overbye's article only mentions physicists; I wonder what (if anything) the biologists have to say. And I don't think our probabilistic understanding of evolution is quite to the point where the first sentence of this paragraph can be made rigorous. (On this point, I'd love to be told I'm wrong!)

edit: Sean at Cosmic Variance has written about this much more insightfully than I, and with links to a lot of the relevant research.

links for 10 August

Why Stuff Is Hard, at the Everything Seminar. I asked this in physics class in high school and never got a satisfactory answer. I suspect I'm not alone here. I was told it was some sort of electromagnetic repulsion between the outermost electrons, but apparently the Pauli exclusion principle is really doing most of the heavy lifting.

Mark Chu-Carroll at Good Math, Bad Math comments on the way math is taught at certain religious schools. If you don't want to bother reading the post, check out the course descriptions at one such school. They all start out "Students will examine the nature of God as they progress in their understanding of mathematics". The descriptions of the non-mathematics courses begin similarly. Today I was reading parts of Laplace's A Philosophical Essay on Probabilities (available in Hawking's anthology God Created the Integers: The Mathematical Breakthroughs That Changed History); among other things, Laplace mocks the idea of Pascal's wager. I couldn't help but thinking of what Laplace is said to have said to Napoleon when asked why he didn't mention God in his work on celestial mechanics: "I had no need of that hypothesis."

Compound interest isn't intuitive, from Adventures of BruteForce; if you invest a little money now that's like investing a lot of money later. People just aren't set up to understand exponential growth, which isn't surprising; unrestrained exponential growth isn't common in the situations for which we evolved. A population can't keep doubling every ten years without pretty quickly running out of space; a sum of money can. There's a persistent rumor that Ashkenazi Jews are actually better equipped for understanding this particular sort of abstraction than other classes of people, because of certain unique historical circumstances -- for quite some time they lived among Christians, who were forbidden to lend money for religious reasons, but these same Christians wanted to borrow money, and therefore turned to the Jews, who were not subject to those same religious laws. The Jews who were better at understanding this fact ended up with more money themselves, their kids didn't starve, and supposedly this explains why about a quarter of Nobel laureates are Jewish. I can't find exact numbers overall. One thing I can find is in this Wikipedia article which says that "Of American Nobel Prize winners, 37% have been Jewish Americans (19 times the percentage of Jews in the population) [...]" But I'm not sure who they count as "American". (Wikipedia used to have a list of Jewish Nobel laureates, but it's been deleted.)

It would be interesting if this were true, because it seems to imply that evolution can work ono the scale of a few hundred years. As humanity heads more and more towards working with its brains instead of its hands, will we get smarter? (On the other hand, at least at the present time in the United States, intelligence and number of children seem to be inversely correlated; it seems difficult for Darwinian evolution to work in a population when almost everyone survives long enough to have children.)

turning our backs on Bourbaki

From yesterday's New York Times (July 31): In Games, an Insight Into The Rules of Evolution, a profile of Martin Nowak, a mathematical biologist. His interests lie in trying to understand cooperation, which is important in evolution; he has been coming up with mathematical models for it; one example that's given is descendants of the Prisoner's dilemma where various members of a population interact preferentially with certain other members, instead of randomly. (The members that interact with each other more frequently are the ones that are "near" each other, either geographically or in some more abstract sense.) Cooperation turns out to emerge under conditions with are rather simple to identify.

I think that it's important for mathematicians to collaborate with people outside of mathematics, and to be exposed to ideas that at first glance don't necessarily appear mathematical. We're always going on and on to our students about how mathematics can be applied to large parts of everyday life, and I believe that's true. But at the same time, the mathematical community seems to look down on those who actually try to do so, instead lionizing people like Wiles and Perelman who have solved problems that have apparently no relevance to the real world.

The linguist Steven Pinker says that “Martin has a passion for taking informal ideas that people like me find theoretically important and framing them as mathematical models... He allows our intuitions about what leads to what to be put to a test.” I believe that this is one of the most important services we can render to the world. What mathematics is good at is stripping away the parts of a problem that are irrelevant and reducing it to its essence; seeing that that essence has something in common with many other apparently different essences; and finally using that knowledge to solve the problem. (It's like the "applications" exercises in most calculus textbooks, except not stupid. In such textbooks the translation from a real-world problem to mathematics is usually so simple as to just be an annoyance.)

Now, I believe that mathematical research that apparently has no use in the "real world" should be allowed to continue, and be funded with tax dollars, because we don't know what bits of mathematics that we're coming up with now will turn out to be "useful" a couple hundred years from now. (The canonical example here, I think, is that number theory has turned out to be very important for cryptography.) But at the same time, it seems to me that the mathematical community turns its backs on those who dare to actually think about those practical applications.

I hope I'm wrong.