The importance of analogies in math and science

“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, …).

UV & skin color

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)….

Observed Expected Different between expected & observed
Cambodia – Khmers 54 38.99 -15.01
Saudi Arabia – Saudi 52.5 38.65 -13.85
Peru (Nunoa – Az) 47.7 34.89 -12.81
Philippines (Manila – Filipino) 54.1 41.53 -12.57
China (Tibet – India Mussoorie – Tibetans) 54.17 41.78 -12.39
Vietnam 55.9 43.59 -12.31
Afghanistan/Iran 55.7 44.55 -11.15
Algeria (Aures – Chaouias from Bouzina) 58.05 47.91 -10.14
India (Rajasthan – Rajputs) 52 42.19 -9.81
Iraq/Syria (Kurds) 61.12 51.5 -9.62
Israel 58.2 48.67 -9.53
Libya (Cyrenaica) 53.5 44.19 -9.31
India (Southern) 46.7 37.6 -9.1
India (Northern – Baniya, Jat Sikhs, Haryana Jats, Khatris, Brahmans, Aroras) 53.26 44.23 -9.03
China (Southern, Hong Knog – Han) 59.17 50.49 -8.68
Pakistan 52.3 44.15 -8.15
Jordan (Non-village Arabs, All Arabs) 53 45.36 -7.64
India (Goa) 46.5 38.93 -7.57
Lebanon 58.2 50.74 -7.46
India (Punjab, England, Dehli – Sikhs, Punjabi) 54.24 47.89 -6.35
Morocco 54.85 49.09 -5.76
Libya (Tripoli) 54.5 48.83 -5.67
PNG (Port Moresby – Hanuabada) 41 35.45 -5.55
India (Bengal – Low Caste, Kayastha, Brahman, Vaidya, Rarhi Brahman) 49.73 44.33 -5.4
Tunisia 56.3 52.03 -4.27
Nepal (Eastern – Jirel, Sunwar, Sherpa, Tamang, Brahman, Chetri) 50.42 46.31 -4.11
Spain (Leon – Meseta, Cabrera, Bierzo, Montana, Maragateria) 64.66 60.8 -3.86
PNG (Mt. Hagan – Western Highlands) 35.35 31.56 -3.79
Turkey 59.15 55.56 -3.59
Spain (Basque – Basque and non-Basques) 65.7 62.38 -3.32
Botswana (Kalahari – Central Bushmen, Yellow Bushmen at Lone Tree, Central San, Yellow Bushmen at Takashwani, Central San, Yellow Bushmen at Ghanzi, Central San) 42.4 39.45 -2.95
South Africa (Namaqualand, Hottentot) 46.8 43.91 -2.89
Libya (Fezzan) 44 41.31 -2.69
South Africa (Warmbath – Hottentot) 43.75 41.14 -2.61
Ethiopia (Highland – Residents of Debarech (3000 m altitude)) 33.55 31.35 -2.2
Sudan 35.5 33.45 -2.05
Brazil (Parana – Guarani) 47.2 45.29 -1.91
Germany (Mainz – German and American Whites) 66.9 65.21 -1.69
Netherlands (Dutch (mainly resident in Utrecht)) 67.37 65.94 -1.43
Brazil (Caingang Indians) 49.4 48.53 -0.87
Peru (Maranon – Aguarana Indians) 43 42.28 -0.72
South Africa (Cape – Cape Coloureds) 50.96 50.71 -0.25
India (Nagpur – Mahar) 41.3 41.53 0.23
UK (Cumberland) 66.75 66.99 0.24
Average 46.18 46.52 0.34
Mali (Dogon) 34.1 34.54 0.44
PNG (Goroka) 33.3 34.2 0.9
Ethiopia (Residents of Adi-Arkai (1500 m altitude), 31.7 32.7 1
UK (Wales – Isle of Man, Merthyr Tydfil, North Pembrokeshire) 65 66.15 1.15
Ireland (Carnew) 64.5 65.84 1.34
UK (Northern) 66.1 67.49 1.39
Kenya 32.4 34.21 1.81
Ireland (Ballinlough) 65.2 67.11 1.91
PNG 35.3 37.26 1.96
Ireland (Rossmore) 64.75 66.73 1.98
Ireland (Longford) 65 66.99 1.99
Belgium 63.14 65.66 2.52
Japan (Central) 55.42 58.51 3.09
Japan (Southwest) 53.55 56.68 3.13
South African (S. A. Negroes (73% Tswana and Xhosa), Bantu (96% Xhosa)) 42.5 45.67 3.17
Tanzania (Sandawe) 28.9 32.13 3.23
UK (London- Europeans) 62.3 65.84 3.54
Namibia (Rehoboth Baster – Black Bushmen at Bagani) 32.9 36.49 3.59
India (Angami Nagas) 44.6 48.85 4.25
Zaire (Congolese except 3 Cameroon females) 33.2 37.46 4.26
Japan (Hidakka – Ainu) 59.1 63.58 4.48
PNG (Karker – Karker Islanders) 32 37.25 5.25
Russia (Chechen) 53.45 59.04 5.59
Burkina Faso (Kurumba from Roanga) 28.6 34.23 5.63
PNG (Lufa – Lufa villagers) 31 36.88 5.88
Japan (Northern) 54.9 61.34 6.44
Tanzania (Nyatura) 25.8 34.12 8.32
Swaziland 35.6 44.62 9.02
India (Orissa, Koraput Town – Bareng Paroja, Bado Gadaba) 32.05 41.52 9.47
Zaire 29.4 39.43 10.03
Chad (Ndila Sara – Madjingay) 24.6 34.77 10.17
Liberia (Mainz – Africans from Ghana and Liberia) 29.4 40.52 11.12
Malawi (Mainly Cewa) 27 38.67 11.67
Nigeria (Yoruba) 27.4 39.62 12.22
Namibia (Kurungkuru Kraal, Tondoro) 25.55 38.29 12.74
Cameroon (Fali Tinguelin) 21.5 34.37 12.87
Nigeria (Ibo) 28.2 41.86 13.66
Greenland (Southern – Eskimo Ammassalimiut) 55.7 70.31 14.61
Namibia (Okavango Bantu, M’bukushu at Bagani, Kuangali) 22.92 38.63 15.71
Australia (Darwin – Aborigines) 19.3 36.24 16.94
Mozambique (Chopi) 19.45 43.84 24.39

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.

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Neandertal mtDNA in Siberia & Central Asia?

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

Authors: Johannes Krause, Ludovic Orlando, David Serre, Bence Viola, Kay Prufer, Michael P. Richards, Jean-Jacques Hublin, Catherine Hanni, Anatoly P. Derevianko & Svante Paabo

Morphological traits typical of Neanderthals began to appear in European hominids at least 400,000 years ago and about 150,000 years ago in western Asia. After their initial appearance, such traits increased in frequency and the extent to which they are expressed until they disappeared shortly after 30,000 years ago. However, because most fossil hominid remains are fragmentary, it can be difficult or impossible to determine unambiguously whether a fossil is of Neanderthal origin. This limits the ability to determine when and where Neanderthals lived. To determine how far to the east Neanderthals ranged, we determined mitochondrial DNA (mtDNA) sequences from hominid remains found in Uzbekistan and in the Altai region of southern Siberia. Here we show that the DNA sequences from these fossils fall within the European Neanderthal mtDNA variation. Thus, the geographic range of Neanderthals is likely to have extended at least 2,000 km further to the east than commonly assumed.

Good in the name of God

Thich_Quang_Duc_-_Self_Immolation.jpgI have to say, this Ian Buruma op-ed, Religion as a force for good, read my mind in relation to the events of the past few days. Another rebellion civil society against an autocracy coalescing around the predominant religion of a society. What’s surprising? The Iranian revolution against the Shah, the Christian led protests against the dictatorship of Syngman Rhee (himself a Christian) in South Korea, Buddhist protests against the persecution of the government of Ngo Dinh Diem in South Vietnam, the role of liberation theology across Latin America. The list goes on. Of course, religion has also been implicated in horror, and given the imprimatur of godliness to abomination, whether the accusation was infidel, apostate, takfir or heathen.

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Lap dancing for science

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 site

This 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.

D. S. Falconer Obit

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.

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The rate of cultural evolution, jerky or smooth?

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?

Atheists still more hated than Mormons & Muslims

Pew has a new survey out, Public Expresses Mixed Views of Islam, Mormonism. The table to the left summarizes the most important points in the survey: Americans dislike Islam somewhat more than Mormonism, and they think Mormonism is a pretty weird religion (and on the whole, barely Christian). And of course atheists are the gold-standard in terms of being disliked. There are a few points which I think are important to keep in mind. First, many Americans have vague understandings of their own religion, asking them about Mormonism or Islam is pretty humorous. What you’re gauging here are vague impressions and reflections of the Zeitgeist; not well thought out opinions. Second, as I’ve been saying for a while one of Mitt Romney’s major problems is that he is running as a Republican whose target constituency is the segment which has the most inbuilt hostility toward his religion. In other words, Mitt probably has more general election appeal than he does within the primaries. Third, asking about Pope Benedict seems really a stretch, I don’t think most Americans are keeping track of the goings on in the Vatican. I do keep track of these surveys; I’m a data fiend, what can I say? Yet despite that sometimes I look at the questions and imagine the average respondent, and wonder if the Memos at Pew are titled “So what are we going to ask stupid people this month?”