Tuesday, October 09, 2007

Bounded cognition

Many people lack standard cognitive tools useful for understanding the world around them. Perhaps the most egregious case: probability and statistics, which are central to understanding health, economics, risk, crime, society, evolution, global warming, etc. Very few people have any facility for calculating risk, visualizing a distribution, understanding the difference between the average, the median, variance, etc.

A remnant of the cold war era curriculum still in place in the US: if students learn advanced math it tends to be calculus, whereas a course on probability, statistics and thinking distributionally would be more useful. (I say this reluctantly, since I am a physical scientist and calculus is in the curriculum largely for its utility in fields related to mine.)

In the post below, blogger Mark Liberman (a linguist at Penn) notes that our situation parallels the absence of concepts for specific numbers (i.e., "ten") among primitive cultures like the Piraha of the Amazon. We may find their condition amusing, or even sad. Personally, I find it tragic that leading public intellectuals around the world are mostly innumerate and don't understand basic physics.

Language Log

The Pirahã language and culture seem to lack not only the words but also the concepts for numbers, using instead less precise terms like "small size", "large size" and "collection". And the Pirahã people themselves seem to be suprisingly uninterested in learning about numbers, and even actively resistant to doing so, despite the fact that in their frequent dealings with traders they have a practical need to evaluate and compare numerical expressions. A similar situation seems to obtain among some other groups in Amazonia, and a lack of indigenous words for numbers has been reported elsewhere in the world.

Many people find this hard to believe. These are simple and natural concepts, of great practical importance: how could rational people resist learning to understand and use them? I don't know the answer. But I do know that we can investigate a strictly comparable case, equally puzzling to me, right here in the U.S. of A.

Until about a hundred years ago, our language and culture lacked the words and ideas needed to deal with the evaluation and comparison of sampled properties of groups. Even today, only a minuscule proportion of the U.S. population understands even the simplest form of these concepts and terms. Out of the roughly 300 million Americans, I doubt that as many as 500 thousand grasp these ideas to any practical extent, and 50,000 might be a better estimate. The rest of the population is surprisingly uninterested in learning, and even actively resists the intermittent attempts to teach them, despite the fact that in their frequent dealings with social and biomedical scientists they have a practical need to evaluate and compare the numerical properties of representative samples.

[OK, perhaps 500k is an underestimate... Surely >1% of the population has been exposed to these ideas and remembers the main points?]

...Before 1900 or so, only a few mathematical geniuses like Gauss (1777-1855) had any real ability to deal with these issues. But even today, most of the population still relies on crude modes of expression like the attribution of numerical properties to prototypes ("A woman uses about 20,000 words per day while a man uses about 7,000") or the comparison of bare-plural nouns ("men are happier than women").

Sometimes, people are just avoiding more cumbersome modes of expression -- "Xs are P-er than Ys" instead of (say) "The mean P measurement in a sample of Xs was greater than the mean P measurement in a sample of Ys, by an amount that would arise by chance fewer than once in 20 trials, assuming that the two samples were drawn from a single population in which P is normally distributed". But I submit that even most intellectuals don't really know how to think about the evaluation and comparison of distributions -- not even simple univariate gaussian distributions, much less more complex situations. And many people who do sort of understand this, at some level, generally fall back on thinking (as well as talking) about properties of group prototypes rather than properties of distributions of individual characteristics.

If you're one of the people who find distribution-talk mystifying, and don't really see why you should have to learn it, or perhaps think that you're just not the kind of person who learns things like this -- congratulations, you now know exactly how (I imagine) the Pirahã feel about number-talk.

Does this matter? Well, in the newspapers every week, there are dozens of stories about risks and rewards, epidemiology and politics, social trends and psychological differences, with serious public-policy implications, which you can't understand without understanding distribution-talk. And usually you won't just feel baffled -- instead, you'll think you understand, and draw the wrong conclusions.

In fact, the people who write these stories mostly don't understand distribution-talk themselves, and in any case they believe that they need to write for an audience that doesn't understand it. As a result, news stories on these topics are usually impossible to understand correctly unless you go back to the primary sources in order to recover the information that's been distorted or omitted. I imagine that something similar must happen when one Pirahã tells another about the deal that this month's river trader is offering on knives.

Here's a great comment:

For many years I attempted to teach Biology and Genetics students the rudiments of statistics, with, alas, only limited success. The notions of population, sample, variance, hypothesis testing, etc. require more time and practice than can be devoted to them in such courses. Most students in the life sciences are math-phobic and few take statistics courses until they reach graduate school. Even among professional biologists publishing in journals like Science and Nature you can find examples of statistical ignorance. Is it any wonder that the average man on the street doesn't understand them either? Practical statistics needs to be incorporated into high school math courses and, possibly, earlier. But I'd remain doubtful that even then the average person would understand enough to be critical of what they read in the papers.

Posted by: Dale Hoyt | October 7, 2007 8:21 PM

For a real life example, see Gary Taubes' book on nutrition and public health research, reviewed here. Even the medical establishment adopted hypotheses that were not in any way supported by good statistical data.


RA said...

That's an interesting perspective on replacing calculus w. Prob & Stat. On the one hand, Prob would be much more useful for those going into the social sciences (Sociology, Psychology, etc.) On the other hand, calculus is a prerequisite for a rigorous course of probability, so a number of students would need to retake Prob after taking calculus.

Anonymous said...

You can learn a lot of useful statistics without calculus. Some of the most useful for research -- Design of Experiments -- (which shows up in Google Analytics) can be done without calculus.
A friend who is working on his PhD in Physics knows calculus well. But he is unfamiliar with design of experiments methods. Yet that is what he needs to use to be able to do more efficient experiments for his thesis.


Anonymous said...

I think Lieberman is displaying the usual smug and politically correct attitude so dear to "linguists" (a bunch of monoglot universal polyfools,like their guru: UschBeurkIgitt Chomsky ).
His argument boils down to "you are just as ignorant as a recursiveless Piraha.But the Piraha is not ignorant".
Why, there might even be
a few million Americans out there who don't know that the hypercohomology of the De Rham complex of a smooth complex affine scheme calculates its singular cohomology...

Go study Hungarian, Chinese and some harder math, Liebermann, then we'll listen to you.

Hildebrand Spencer Poynt de Burgh John Hannasyde Coombe-Crombie

Steve Hsu said...

The point isn't that there is some body of knowledge that average people lack. It's that they lack something of immediate practical value to them -- e.g., counting for Piraha traders or an understanding of statistics for Americans trying to make sense of health or economic data in the newspaper. It's hard to argue that the average person can find concrete benefits from understanding advanced modern mathematics.

Anonymous said...

Dear Steve,
of course you are perfectly right.
I was in a bad mood and wrote my comment in total bad faith.
I am a great admirer of your blog and of yourself and it was wrong to bring my quarrel with some linguists to this fine place.

Sincere apologies and best wishes,

Hildebrand Spencer Poynt de Burgh John Hannasyde Coombe-Crombie

mhnin said...

Do you have any reading recommendations for remedying a P&S deficit?

steve hsu said...

Thinking in terms of distributions is step 1. But another leap is to understand that there are uncertainties in your probability estimates, and different kinds of uncertainties. See, e.g., http://www.ellsberg.net/documents/Risk_Ambiguity.pdf

You might also find this essay of interest (are you a Bayesian? must two rational Bayesians always converge to the same conclusions?): http://infoproc.blogspot.com/2008/12/frequentists-vs-bayesians.html

LaurentMelchiorTellier said...

1/5 of Americans believe that the sun orbits the Earth. Seriously. 
A majority of Americans believe in astrology.
http://www.gallup.com/poll/3742/new-poll-gauges-americans-general-knowledge-levels.aspxI've checked the polls in question. No "tricky wording" or confounding factors that I can detect.*I've come to believe that scientists frequently fail to grasp how profoundly uninterested the majority population is in verity and verification, as understood by science. Even when understanding that it is profoundly to their personal advantage. I've reminded myself of this many times, yet I constantly regress to making incorrect assumptions to the contrary. Unhappily, the 1% estimate on language log might be accurate, :-( however contrary it may be to our anecdotal experience in academia/business. 

Inverse said...

Where I go to school, biology majors must take statistics and calculus. Statistics is obviously of great utility -- calculus has far fewer applications in biomedical research.

Many people on the street intuitively understand statistical notions like variance and effect size. Maybe they even understand P values and the importance of large samples. They merely use different language.

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