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Sunday, September 30, 2007
"Good mathematicians see analogies. Great mathematicians see analogies between analogies."
--Stefan Banach A recent Cognitive Daily post called "Why aren't more women in science" (part 1) reviews some of the lit on sex differences in cognitive abilities. Dave Munger notes: In the verbal portion of the [SAT] test, the male advantage is eliminated if the analogy portion of the test is eliminated; arguably this is more a test of mapping relationships than literacy. The analogy portion was, of course, scrapped as of the spring 2005 SAT. [1] The boldfaced clause above shows why it matters more than the other Verbal portions: figuring out relationships between ideas matters, and reporting what some author said does not. Analogies are highly g-loaded, reading comprehension much less so. But aside from better detecting who the smarties are, analogies are more reflective of real-world math, science, and engineering. (And they matter in the humanities too [2].) If A got one more math question than B, but B got three more analogy questions than A, I'd bet on B doing better in math, even if an IQ test showed they had the same IQ. What follows is mostly a diversion to show the importance of analogies in math, starting with high school material and moving to some college material. I hope you learn something new, but mostly the goal is to put it on the record, with examples, how important a person's verbal analogy score is in predicting their success in math and science. Example 1. A bouncy-ball is dropped from 2 feet, and after hitting the ground, bounces up only 1/2 as high as its previous maximum height. Pretend that it bounces forever like this. In the long run, how much distance does the ball travel? We can make a table that shows how much distance the ball travels in a particular trip, either up or down, like so: Trip 1, 2, 3, 4, 5, 6, 7, ... Dist. 2, 1, 1, 1/2, 1/2, 1/4, 1/4, ... This problem is introduced in a pre-calculus class during the unit on the sum of an infinite geometric series -- infinite because it starts but never ends, and "geometric" meaning you multiply by the same number to get from one term to the next. The formula for such a sum is t1 / (1 - r), where t1 is the first term, and r is the constant that multiplies one term to get to the next. So if we only had these values, we'd be all set! Unfortunately, if we guess that r is 1/2, when we try to go from 1 to 1 -- we don't multiply by 1/2 anymore (or from 1/2 to 1/2). Damn. Plainly, the above series is not geometric, and at that point most students will opt to make better use of their time by yakking with friends on their cell phone. Ah, but the students in the class who are good analogical thinkers will notice a geometric series hiding behind the series above -- in fact, they'll discover two of them. The terms of one are interlocking with the terms of the other, like two rows of teeth that complete a zipper. That analogy suggests a strategy: unzip the above series. Then we have two series that go: 2, 1, 1/2, 1/4, ... and 1, 1/2, 1/4, ... Bingo! In each of these, you multiply by a constant (1/2) to get from one term to the next. And we know the first term of each, so we can plug in values for t1 and r in the sum formula. We get 2 / (1 - 1/2) = 4, and 1 / (1 - 1/2) = 2. So all together, the ball traveled 6 feet. That's a neat analogy, but it only makes sense when there are two series meshed into one. We'd like to generalize to any number of series that dovetail into one -- and no one makes zippers with more than two rows of teeth. So a better analogy might be the following:  Here there are two strands woven one around the other infinitely, with beads bearing numbers that face us, and there is a knot at the start where the strands fuse. Could we think up series with three or more geometric series hiding inside them? Sure, just as we could make a rope with three or more strands. And to make that series easy to solve, we would just unbraid the strands and work with the beads of each one separately. See note [3] for more uses of this braid analogy. Example 2. Here are some (x,y) pairs associated with a function. What is the degree of this function? That is, does it look like x, x^2, x^3, etc.? x = 1, 2, 3, 4, 5, 6... y = 2, 14, 34, 62, 98, 142... This problem also comes from high school math -- or middle school, if you took algebra then. There, you were taught to look for the difference between consecutive terms, and maybe repeat this process, until you got a sequence of the same number. The number of runs you have to make is the degree of the function. So for the above, the differences are: 12, 20, 28, 36, 44 OK, not the same number, but take the difference again: 8, 8, 8, 8 Ta-da. We had to go through 2 runs, so it must be some function like x^2 (in fact, it is 4x^2 - 2). I guarantee you never knew why this worked when you learned it -- and even after calculus or more advanced math, you may still have treated it as a mysterious trick. But there are analogies between discrete and continuous areas of math, and they are pervasive. If you took at least a semester of calculus, you know that if you take the 1st derivative of a function like 4x^2 - 2, you get something with the independent variable still in it -- 8x. And sure enough, in our discrete case, the first differences are 8x plus a constant 4. But if you then take the derivative of the derivative, you get a constant -- 8, the same 8 that appeared in our constant sequence after the 2nd run. A constant second difference in the discrete case is analogous to a constant second derivative in the continuous case. That also shows why you knew, back in high school, that you didn't have a polynomial function like x or x^2 or x^3 when you saw something like this: x = 1, 2, 3, 4, 5, 6... y = 2, 4, 8, 16, 32, 64... You can take differences of differences of differences of... and you'll never get a constnant sequence for this function, which is 2^x. In first-semester calculus, you learned that e^x is its own derivative, so that if you keep taking the derivative over and over, you always get back e^x -- the independent variable never goes away, so you never get a constant. This resilience to your effort to tease a constant derivative out of it is true of all exponential functions, which by analogy tells us that we'd never come up with a constant difference in the discrete case above. Since there are a billion other discrete-continuous analogies, I'll leave it there. I don't think they're that neat since it's only like switching between a British and American accent, not like translating between Farsi and Chinese. On a closing note, the entire domain of represenation theory in algebra is based on finding good analogies: they attempt to better understand how some group works by casting the problem in terms of matrices and linear algebra, which are better understood. All of this shows how indispensable this way of thinking is to fields that many assume are primarily about visuospatial skills (though those are key too). Analogies are to all types of thinkers what SONAR and nets are to deep-sea fishermen regardless of which species they hunt. [1] According to CollegeBoard's 2007 national report of college-bound seniors, it does appear that within the past couple of years, the male mean for Verbal is only about two points above the female mean, shrinking from a difference of about 11 to 12 points that had persisted since about 1980. And at the high end, in 2007, 1.98 % of males and 1.84 % of females scored 750 - 800. Data from other years on the elite scorers are not contained in the 2007 report, and I'm not interested enough in this topic to pursue them. The point is that gutting the analogy portion seems to have served its purpose. [2] When the retiring of the analogy questions was announced, an educator named Ted Sutton got an op-ed into the very liberal Boston Globe and made a guest appearance on the very liberal radio show On Point (which airs on NPR). He lamented the change, focusing on the centrality of analogies to the great philosophical and humanistic traditions. Older-style liberals like Sutton appear unaware that their social engineering cousins are the ones responsible for flushing great ideas down the drain, so that the gap between the sexes on a test might close. At least there are still analogies on the GRE -- despite a plan to re-vamp the test with the same gap-narrowing agenda in mind. And thank God for the Miller Analogies Test -- not a single "how does the author most likely feel about X" question at all! [3] The braid idea can also guide your intuition when you have a homework problem in a college-level course that says: "Prove that a countable union of countable sets is countable." I provided a visual proof here (with a more detailed proof at the end), but I didn't think of the braid analogy, which makes it even easier to picture. The argument is as I wrote before, but when you're introducing yet another countable set into the union, it's like adding a new strand to a rope. You look at the place where the n strands have shown themselves once -- and before the first strand winds around the second time, you push it over and braid in your new strand. When they n strands have shown up twice, you push the first strand over before it winds around the third time, and there's the second place where the new strand goes. And so on to infinity. The union of these strands is a rope whose beads are countable and, more importantly, ordered in a straightforward way. More explicitly, we can think of the strands as equivalence classes and the rope as the space they fill out. We can imagine a rope that extends infinitely in either direction, like the even and odd integers woven together. We've already seen a rope with a knot but which continues to weave itself forever in one direction. A rope with knots at both ends is pretty boring -- unless they were the same point, i.e. the rope circled back so that each strand fed back into itself, as with a sequence that's cyclic (for instance: x, y, x^2, y^2, x^3, y^3, x, y, ...). Labels: general intelligence, mathematics  Update: I've added some geographic and ethnic notations to the ones that are relevant. For example, the Indian groups which are the darkest for their latitude turn out to be a Dalit and Tribal sample. In contrast, the other groups are more socially diverse. In South Afica the Capetown sample consists of mixed-race Coloureds. I've also added geographic data for places like Ireland, since I know there are readers who might be able to confirm with local knowledge (or disconfirm).End Update From The Evolution of Skin Coloration by Nina G. Jablonski Figure 1: "The potential for synthesis of previtamin D3 in lightly pigmented human skin computed from annual average UVMED. The highest annual values for UVMED are shown in light violet, with incrementally lower values in dark violet, then in light to dark shades of blue, orange, green and gray...In the tropics, the zone of adequate UV radiation throughout the year (Zone 1) is delimited by bold black lines. Light stippling indicates Zone 2, in which there is not sufficient UV radiation during at least one month of the year to produce previtamin D3 in human skin. Zone 3, in which there is not sufficient UV radiation for previtamin D3 synthesis on average for the whole year, is indicated by heavy stippling." Below the fold I've reproduced a table that compares expected skin color and observed skin color for indigenous people. The expected is derived from a prediction equation which uses the observed values and combines them with the values from the UV map above: Predicted skin color = (annual average UVMED) X (-0.1088) + 72.7483 I also added a column which measures the difference between expected and observed and ordered it from populations which were lighter than expected to those which were darker than expected. Many of the values seem explicable via historical information (go to the paper and in the appendix you see what populations they used, that's important information); nevertheless, I am wondering about possibilities of different diet and its affect on skin color (more later)....
Notes: I'm skeptical of the accuracy of some of the reflectance measures. The authors report which ethnic groups they used for sampling in the appendices, so I would ask readers to look in there if they think some of these measures are questionable (I'll have a follow up post on this). They also assume that these "indigenous" peoples (which is, admittedly, a flexible definition) are well adapted to their local UV regime, and that other factors are controlled. Jablonski's thesis is that skin color is driven by two opposing forces: adaptation to high levels of UV which break down folate and increase birth defects, and, the need to synthesize vitamin D through the interaction of UV and biochemicals in the skin. Variation in diet and other possible selective forces aren't of much concern to her, and so she generated her expected skin color values assume that UV is the primary independent variable. My own hunch is that the far lighter than expected skin color across much of Asia is due to Vitamin D deficiency induced by the extreme carbohydrate biased diets of these populations. At this point this is just a tentative hypothesis, but, there has been selection for alleles known to be implicated in generating lighter skin in both South and East Asia within the last 10,000 years. Labels: Genetics
I don't know if we should believe Svante Paabo anymore, but his lab has some new findings re: Neandertal mtDNA:
Neanderthals in central Asia and Siberia Nature advance online publication 30 September 2007. doi:10.1038/nature06193 Labels: Evolution  The role of biology in constraining/enabling human culture is largely underappreciated outside of, well, the small group of people who study biology and culture. But that role is clearly enormous. Consider, for example, what is sometimes referred to as "cryptic ovulation"-- the fact that human females do not conspicuously display the fact that they are ovulating. In many other primates, the females have a patch of hairless skin that, as ovulation approaches, swells up bright red (see the picture of this in a baboon), signaling that she is fertile and driving the males a little crazy. Humans clearly do not do this, and I don't think it's an exaggeration to claim this was a biological prerequisite (or as close as you can get to one) for today's mixed-gender offices and the large-scale incorporation of women into the workforce. But how cryptic is the human cryptic ovulation? Women are generally aware of where they are in their cycle, and men with long-term girlfriends/wives have probably noted subtle physiological changes (in breast size, for example) that correspond to their partner's hormonal fluctuations. Is this a subtler version of the sexual swelling in other primates? There is some evidence that this is the case, but Geoff Miller and colleagues take a rather novel approach to the question:  To see whether estrus was really "lost" during human evolution (as researchers often claim), we examined ovulatory cycle effects on tip earnings by professional lap dancers working in gentlemen's clubs. Eighteen dancers recorded their menstrual periods, work shifts, and tip earnings for 60 days on a study web siteThis is a nice way at getting around subjective measures of "attractiveness" in studies like this-- the amount of money made by a stripper probably corresponds pretty well to how physically attractive the males in the audience find her. And as seen in the graph on the right, there's a noticeable peak in earnings among normally-cycling women at around 10 days (ovulation). The sample size is small, of course, but the effect is consistent with other evidence than human females modulate their physical appearance and behavior according to the menstrual cycle, so I'm inclined to believe it. And needless to say, if this is the case, it suggests a rather simple profit-maximizing strategy for the professional lap dancer. Labels: Evolutionary Psychology
Anyone trying to understand heritability, or other aspects of quantitative genetics, is likely to rely heavily on D. S. Falconer's Introduction to Quantitative Genetics. I find that Falconer died a few years ago, and there is a fine Obituary by W. G. Hill available here. I love this anecdote from Falconer about D'Arcy Thompson:
I asked him at the beginning for recommendations as to what to read and he said 'Just browse, my boy, just browse.' So I worked away on my own ... and at the end of the year he came along to me and said 'Well, Douglas, my boy, you're a very good lad and I don't think we need give you an examination this year.' They don't make them like that any more.
Saturday, September 29, 2007
There has long been a tiresome debate in evolutionary biology (or at least in pop science books about evolutionary biology) whether evolution generally proceeds gradually or in bursts alternating with stasis. But I wonder: what about cultural evolution? With evolutionary biology we can look at fossils and the molecular substrate to determine the nature of change; with culture it is a little different because of its amorphous character. Some aspects are pretty easy to quantify, for example baby names for example drift like genes subject to purely random forces. On the other hand, my perception is that attitudes toward homosexuality have changed very fast over the last 15 years, so that some of the positions staked out by "social conservatives" in 2007 would be out of the mainstream for being too pro-gay in the late 1980s (here are polls). Has anyone out there plotted changes of attitudes from sources like Gallup and noticed whether the changes were gradual or subject to sharp increases or decreased in frequency?
Labels: culture
The Lancet has the case report.
Friday, September 28, 2007
What is contingent across the arc of human cultural development? What is inevitable? Interesting, if difficult to answer, questions. Last year I posted No fear of Patrick Henry College - the Borg shall assimilate. My argument was simple: an explicitly Christian institution which attempts to take over "secular" culture will be assimilated. There are long, and tiresome, historical debates about whether this in fact happened to the Christian churches when the Roman state adopted them and turned them into the Universal Church. But more recently, and specifically in the context of universities, there has been a long track record in the United States of Christian institutions being founded to stem the tide, only themselves to be swallowed up by the rising waters.
Harvard was originally a training ground for Calvinist ministers. Over its first century it became progressively more heterodox. Princeton was founded explicitly to serve as a second Harvard, a bastion of Calvinist orthodoxy. It too was suborned. Wheaton college is in many ways the Harvard of contemporary evangelical America; and it reaffirmed its Protestant credentials when it fired a professor who converted to Catholicism. Nevertheless, the act itself was not without controversy on the campus, suggesting that the commitment toward ideological purity has wavered. Additionally, it seems clear to me that Wheaton's loyalty to one American subculture has resulted in constraining its influence. Patrick Henry College reached out, its aim was to conquer the public space. But last spring while I was busy at something I like to call "life" a shakeup occurred at Patrick Henry, half a dozen faculty members left (there are fewer than two dozen told faculty members). Why? Ideological conformity and theological purity were being compromised. Patrick Henry aimed for the stars, recruited bright students and challenged the faculty. But such an environment naturally leads to intellectual hubris and the pushing of boundaries. Mental meekness and dullness often go together. Like an invasive species unleashed to control a pest any attempt to conquer the mainstream by mastering its toolkit may inevitably be self-defeating. This is not just true of the evangelical Christian subculture. Books like Bobos in Paradise document the paradoxical stances of the bohemian bourgeois; 60s radicals turned "socially conscious" entrepreneurs & mercenary professionals. American culture is a massive and uncontrollable river. On occasion it changes course or jumps its bed, but it has its own will and logic and can process anything thrown into its maw. The extruded cultural material is often totally transformed, but the the human tendency to self-delude is great enough that those who have been reprogrammed by the river truly believe that they have won. There's no point in standing athwart history if it will only drown you; 'tis far more productive to make use of the power of the current and outfit your ship appropriately so that your journey is as smooth and pleasant as possible. Related: The New York Times has an interesting article about a new Christian college, New St. Andrews. I obviously don't share their presuppositions, but I do respect their passion for learning. As long as books & faith are their focus they will persevere on their island surrounded by the river. If they challenge it then I suspect their fate is predestined. Labels: Religion
Thursday, September 27, 2007
I don't follow the non-science news very closely. I'm curious about what's going on in Myanmar/Burma, if you have an interesting link, drop it in the comment box. Thanks.
Labels: politics
Cosma Shalizi has put up a gigantic post on IQ & heritability; he originally titled it "Duet for Leo and Razib," implying that I, and the audience here @ Gene Expression, are the targets of his eloquence (at least in part). Now, I have to admit something, I'm not really interested in psychometrics that much anymore. It has been a while since I have been, stupid people are obviously stupid and I am not interesting in debating that fact. I take my own opinions in this area as background assumptions, so I'm not going to respond to Cosma. In fact, I won't read the post right now, there's some interesting stuff on HLA & heterozygosity that I want to check out! But, I do invite readers to digest what Cosma is saying, because I guarantee you that you'll see it replicated by lesser minds elsewhere.
Labels: human biodiversity
Sadly, but unsurprisingly, the little blonde girl photographed in Morocco turns out not to be missing British girl Madeleine McCann, but the daughter of a Berber farming family, who are said to have three other blonde children.
Most of us will have been vaguely aware that blonde hair and fair skin are not uncommon among the Berbers, but it has evidently come as a surprise to the general public. It is usually explained by a hypothetical element of European ancestry, whether from Roman slaves (as in this Daily Mail article), the Vandals, or more prosaically from the colonial occupation by French soldiers and government officials (who presumably didn't just twiddle their thumbs). I wonder if there is any hard genetic evidence? Y chromosomes might at least show whether the paternal ancestry is recently European. An alternative, and more interesting, explanation would be that the Berbers are the remnants of an older, more 'Caucasian', North African population.
Wednesday, September 26, 2007
Native Americans get custom sneaker:
Nike researchers and developers spent two years designing the shoe, traveling to seven locations to look at the feet of 224 Native Americans from 70 different tribes. They created a shoe to fit the average Native American foot, which is wider than the foot the Nike Air Pegasus running shoe is designed to fit. About 164 members of the Confederated Tribes of Warm Springs tested prototypes of the shoe before its release, the company said. This is fascinating. The reason these shoes were developed was to encourage physical activity, something that comes more naturally when your feet aren't aching. Assman has noted before that South Asians might be more flexible than the typical human, which likely results in flatter feet, so this foot's-eye viewpoint might be pretty practical in tailoring shoes toward populations. Labels: human biodiversity
RPM's slamming of some silly coverage of the C-value enigma got me thinking about the problem of why we see the sorts of variation we do in the amount of non-coding DNA between species. People are right to heckle the questionable assumption that these differences in ncDNA have anything to do with the evolution of phenotypic complexity (though probably a small fraction do), but I think it might still have an interesting functional tale to tell. I'm probably not the first to think of this, but the idea is that variations in quantity of ncDNA are not functional for the organism in themselves but rather the waste product of a particular kind of functional change: gene duplication.
Recall that eukaryotic genomes are regularly bedevilled by selfish tranposons. These are rogue genetic elements with a vested interest in creating duplication events, and the basic idea is that every once in a while one of them will succeed wildly at it and in the process end up dragging a whole gene along for the ride (maybe several times). Most of the time this will be bad, but occasionally it'll be good, and sometimes it'll be nearly-neutral and you'll see functional divergence on the copied locus after the initial duplication event. In the cases where a duplicated gene confers a selective benefit, the newly formed transpositional elements hitchhike along on the newly selected gene's coattails. The upshot of this is that we should expect cases of adaptive evolution via gene duplication to be frequently be accompanied by increases in the amount of transpositional cruft in the genome of the species. This also would neatly account for much of the ncDNA variation between species, since gene duplication seems to play an important role in the emergence of species-specific traits. If this idea is correct, the amount of ncDNA should correlate more highly with how much adaptive gene duplication a lineage has undergone rather than phenotypic complexity per se. This theory should be pretty easy to test: Look at cases of adaptive gene duplication that have happened relatively recently (geologically speaking) and compare the LINEs and such around these loci with those close to the presumed "parent" locus. The further back in time you go the harder it will be to do this comparison due to drift wiping out the traces, but in the cases that are comparable they should have a very similar pattern of nonfunctional repeats. If I have this right. (EDIT: Duh. This isn't a good test, since you'd probably see the same thing under any sort of duplication. Need to think of something else. Maybe compare lineages of recently duplicated genes: If gene B is a "recent" duplication of gene A, and gene Y is a "recent" duplication of gene X, but genes A and X diverged an extremely long time ago, then the two duplications were probably caused by different retrotransposons and so the LINEs around A and B should tend to be highly similar to each other but very different than those around X and Y, and vice-versa. You'd probably have to compare a bunch of different gene lineages to get a statistically significant result, though, and I don't know how easy it would be to find enough good candidates.) Has anyone actually looked at anything like this? Does this idea hang together? How else could we test it? Update: Looks like another beautiful hypotheses slain by an ugly fact. I'll just copy-paste what I said in the comments: Having looked into it, this doesn't work the way I thought it would. I knew that LINEs sometimes end up dragging some of the host's genetic material along in their replications, but now I know that the way this happens is that sometimes the reverse-transcription machinery grabs onto host mRNA that's floating around and splices it in. So what's being inserted is automatically a pseudogene since the mRNA has already been processed (i.e. there's no promoter attached to it). For this idea to work it would need to be an active gene. Rats. Labels: non-coding dna, selfish dna, speciation
Tuesday, September 25, 2007
In my post below I respond to Bryan Caplan's critique of Greg Clark's claim that disease can increase per capita income because it reduces population (i.e., same population has a bigger resource base to work with).1 I go the route of the two handed economist by suggesting that whether Clark or Caplan is right depends on the details.2 Herrick adds in the comments:
Caplan's big claim is that almost anything that persistently raises death rates is likely to persistently reduce output per living worker. It that true? As I suggest below I think that Caplan is wrong if he wants to claim that productivity is always decreased in direct proportion to the increased disease load (ergo, death rate) of a population. This would prevent the rise in incomes which Clark predicts as the lower productivity of each individual means that the same amount of land can support fewer people at or above subsistence. In A Farewell to Alms Clark reports a rise in incomes after the Black Death, and, amongst native peoples in the New World after Old World diseases ravaged them. Obviously this is one extreme cause: a highly lethal infectious disease which cuts down a large proportion of the population very quickly, and then recedes. The other scenario is a case where there is an endemic infection which reduces physiological fitness across the whole population, reducing lifespan and increasing death rates, but also dampening economic productivity. Then there are cases where there is a wide variance within the population in regards to susceptibility toward infectious agents. This might be more like the first scenario, a large number of people die very quickly, while many others are spared because of some immunity. And so on. From Darwinian first principles it seems that there should be a large number of pathogens which are infectious but not fatal. Though reducing physiological fitness, they don't knock out their host because to do so would result in their own reduced evolutionary fitness. But hey, Herrick asked for expert opinion. I was actually hoping that someone with medical expertise (e.g., tropical diseases?) would weigh in on that thread, but that didn't happen. So I come to you with open hands and ask you to enlighten.... Update: Greg Clark responds directly to the Caplan critique. As a non-economist I'm more interested in what the empirical historical data says, and what little I know seems to agree with the general thrust of Clark's point. 1 - That sentence should filter out chimpanzee readers since it should be totally incomprehensible to them. 2 - No shit it depends on the details! Labels: Medicine
Apropos of a previous post on race, PLoS Medicine has just published two (opinion) articles on the use of racial categories in medicine. There's only a cursory treatment of genetics (and the treatment that's there is pretty bad), but it's sometimes useful to see another take on the issue. The message I get is that, well, doctors aren't trained in genetics, so any "race-based" medicine (which is necessarily based on probabilites) is likely to become a sort of "black = medicine X, white = medicine Y" dogma.
A note for readers, there's a new book aimed at the popular audience, Justinian's Flea: Plague, Empire, and the Birth of Europe. You can find reviews here and here. I'm going to pass on it probably because it is a general interest book which doesn't introduce any original material, but it looks like some readers of this weblog get something out of it (though do read Plagues and Peoples if this genre is new to you).
Labels: History
I'm not going to spend too much time on this, but Larry Moran has responded to my post. He, of course, makes it sound as if he's being perfectly reasonable. But consider what he wrote in July: [E]volutionary biologists like Dawkins and the other adaptationists should have known about random genetic drift. Isn't it amazing that they don't?A |
