Sociology of mathematicians?

Has anybody seriously looked at mathematicians from an anthropological or sociological perspective? I was talking to some sociologists last night, and apparently various people coming from the disciplines that study people have looked at biologists in this way; this got me wondering.

Correlation coefficients and the popularity gap

The Popularity Gap (Sarah Kliff, Newsweek, May 15 issue).

Apparently, the people who end up being successful later in life are the ones who think people like them in middle school, not necessarily the ones who are actually well-liked in middle school. This reports on a study by Kathleen Boykin McElhaney, who is not particularly important to what I'm going to say, because I'm going to comment on something that I assume was introduced by the folks at Newsweek.

The Newsweek article continues:

One of McElhaney's most interesting findings is that self-perceived and peer-perceived popularity don't line up too well; most of the well-liked kids do not perceive themselves as well liked and visa versa. The correlation between self-perceived and peer-ranked popularity was .25, meaning about a quarter of the kids who were popular according to their classmates also thought they were popular. For the other three quarters, there was a disconnect between how the teen saw themselves and what their peers thought.

I can't read the original journal article (the electronic version doesn't become available for a year after publication, and I'm not going to campus in the rain and looking around an unfamiliar library just to track this down!) but the Newsweek article says enough to make it clear that the study wasn't using a two-point "popular/unpopular" scale. I'm inclined to think that the "correlation" here is what's usually referred to as the "correlation coefficient" -- and this is usually explained in popular media by saying that "one-fourth of the variation in how popular students believed they were was due to how popular they actually were" or some such similar phrase. I'm not a statistician, so I won't try to explain why that phrase might be wrong; if you are, please feel free to weigh in!

But let's assume that half of students are actually popular, and half of students think they're popular. (This might be a big assumption; recall the apocryphal claim that 75 percent of students at [insert elite college here] come in thinking they'll be in the top 25 percent of their class.) Then if only 25 percent of the students who are actually popular think they're popular, there's actually a negative correlation between actual popularity and perceived popularity! More formally, let X be a random variable which is 0 if someone's not (objectively) popular and 1 if they are; let Y play the same role for their self-assessed popularity. Then E(XY) is the probability that a randomly chosen student both is popular and thinks they are, which is 1/8 in this case; E(X) E(Y) = 1/4, which is larger.

Then again, if there actually were a negative correlation -- if people were so bad at self-assessment as to be worse than useless at it -- then that would be quite interesting. As it is, there seems to be in general a weak positive correlation between how P someone is (where P is some desirable trait, say popularity in this case) and how P they think they are.

And the fact that I bothered to write this post probably will lead you to guess -- correctly -- that I wasn't all that popular in high school.

hard science, and calculating the area of a circle

The Really Hard Science, by Michael Shermer, from the October 2007 issue of Scientific American. The author makes the claim that the traditional distinction between "hard" and "soft" science is backwards; the social sciences might more reasonably be considered harder. I'm not sure how much I buy this linguistic claim, because "hard" has two antonyms in English, "soft" and "easy", and I suspect there's a reason we call the social sciences "soft sciences", not "easy sciences". The point that all these disciplines are valuable, I support. Shermer points out that modeling a biological or social system is often more difficult than modeling a physical system. We tend to think of the mathematics used by physicists as more complicated than that used by social scientists, but how much of that is a reflection of the subject matter and how much of that is due to the historical alliance between mathematicians and physicists, which means that physicists have tended to know lots of mathematics and, say, sociologists don't know as much? (I struggle with this in teaching calculus; our textbook, like many calculus textbooks, draws a lot of its examples from physics -- centers of mass, moments of inertia, and such -- and the students aren't as likely to be conversant with physics as the authors of the text seem to assume.)

He writes the following:

Between technical and popular science writing is what I call “integrative science,” a process that blends data, theory and narrative. Without all three of these metaphorical legs, the seat on which the enterprise of science rests would collapse. Attempts to determine which of the three legs has the greatest value is on par with debating whether π or r² is the most important factor in computing the area of a circle.

Which is more important in calculating the area of a circle? I say r². It's interesting that there is a constant, but the way in which the area of the circle grows when you increase the radius (namely, twice as fast) is more important to me. I may only be saying this, though, because I don't have to actually calculate the area of a circle. The people who made the circular table I'm sitting at certainly care about π to tell them how much raw material they will need.

The "twice as fast" statement there is a bit vague, but I mean it in the sense that if one increases the radius of a circle by some small fraction ε (as usual, ε² = 0), one increases its area by the fraction 2ε. That is,

π(r(1+ε))² = π(r² + 2εr²+ ε²r²) = (πr²)(1+2ε)

where the last equality assumes ε² = 0. For the most part this is how I think of derivatives -- at least simple ones like this -- in my head. The ε² = 0 idea is the sort of thing one might see in the proposed "wiki of mathematical tricks" of Tao and Gowers (the real meat of the discussion is actually in the comments on those posts), although perhaps at a slightly lower level than they're aiming at.