Thursday, September 18, 2008

What psychologists think they know about math

This is appallingly dumb and uninformed:

“When mathematicians and physicists are left alone in a room, one of the games they’ll play is called a Fermi problem, in which they try to figure out the approximate answer to an arbitrary problem,” said Rebecca Saxe, a cognitive neuroscientist at the Massachusetts Institute of Technology who is married to a physicist. “They’ll ask, how many piano tuners are there in Chicago, or what contribution to the ocean’s
temperature do fish make, and they’ll try to come up with a plausible answer.”

“What this suggests to me,” she added, “is that the people whom we think of as being the most involved in the symbolic part of math intuitively know that they have to practice those other, nonsymbolic, approximating skills.”

Fermi problems are entirely symbolic. You work them out something like this -- you estimate the number density of fish in the ocean, then you look up how big the ocean is, then you make a crude model of how often fish fart and how much energy that releases, and then you multiply everything -- all your estimates are powers of ten, so this step is just addition.

The point is, your estimates are generally based on either knowledge or a certain kind of abstract deductive cleverness. You virtually never know enough about the problem to have a "gut instinct."

As for the general thesis of the article:

  1. I imagine there are strong correlations between basic math instinct and intelligence, and between intelligence and formal math. One really ought to control for IQ in these studies.
  2. They seem to have stopped testing at 14. At this point, most people other than math prodigies are still basically multiplying and dividing numbers; translating this into, say, serious group theoretic ability is a stretch. Who knows, some people might use an innate approximating sense to figure out if their results make sense. But it seems like you could also use general intelligence for this purpose.
  3. Their "philosophical implications" are bunk. (Surprising isn't it.) Incidentally -- if, as they claim, Fermi problems are related to gut instinct -- neither Einstein nor Heisenberg (and almost certainly not Hawking, though I don't know for a fact) was much good at Fermi problems. In fact, Heisenberg's incompetence at this sort of reasoning had a lot to do with the failure of the German nuclear program.
  4. To reason from positive correlations in the general populace to high prevalence in the tails is to forget about Bayes' rule.
  5. I've never heard of a mathematician with an interest in Fermi problems. Math is generally one of the least quantitative disciplines.

I generally do not read the Times's neuroscience/psychology articles. Whenever I do, I promptly regret it.


Grobstein said...

The study controls for IQ, according to WaPo. Of course, I don't know whether they did it a smart way or a dumb way.

Zed said...

Thanks. The WaPo story is clearer and less annoying. Have to look at the original story; it seems a little incredible that they actively controlled for all the stuff they claim -- it'd take a lot of tests -- but I guess they had a lot of time and money.

Grobstein said...

Yeah, I hadn't read the NYT article, and still haven't, but looking over it it seems pretty bad.