Mrs. Laquedem asked me, with a dangerous glint in her eye, what I thought of the January 15 remarks of Lawrence Summers, the president of Harvard College, to the NBER Conference on Diversifying the Science & Engineering Workforce.

More bravely than sagely, I told her what I thought. After the resulting discussion [omitted here purely for the sake of brevity] I read his actual remarks, available here. A few things stand out. First, Mr. Summers said that he was speaking to provoke discussion and not to report on what Harvard is doing to address gender equality in the sciences. Second is that he described his specific topic as "women's representation in tenured positions in science and engineering at top universities and research institutions," not as a general survey of women and mathematics.

Top universities don't have a lot of women holding tenured positions in science and engineering. The question is why. Mr. Summers described three competing theories: (1) the "high-powered job hypothesis," that men are more willing than women to work at 80-hour-per-week jobs, (2) innate biological differences, the theory that got him into trouble, and (3) discrimination and socialization.

If you read his remarks (and I hope that you will), you may notice something interesting about how he described theory no. 2. He said that possibly men and women have different standard deviations (not means) in their aptitude for science, and that this results in more men than women at the extremes.

For example, suppose that science ability can be measured on a IQ-like scale, with 100 as the mean, and that a science quotient (SQ) of (say) 130 is required to achieve tenure at Harvard. If men and women have a mean SQ of 100 (that is, the same average ability), but men have a standard deviation of 15 and women have a standard deviation of 10 and both are normally distributed, then about 2.5% of the men will have an SQ above 130, but only 0.25% of the women will have an SQ above 130, even though the average abilities of men and women are identical. So even though the two populations have identical average abilities, at the high end men will outnumber women by about 10 to 1. (If this is true, then men will outnumber women at the low end by 10 to 1 also.)

The controversy that he stirred up (which he said was his goal in giving the speech) brought to mind another story of women and mathematics, this one involving Marilyn vos Savant, who claims to have the highest IQ of any living person and who writes a column in Parade magazine called "Ask Marilyn." In her September 9, 1990 column, she included a brain-teaser that's now called the "Monty Hall problem," after the original host of Let's Make a Deal. You must choose among three doors. Behind one door is a car; behind the other two are goats. You choose one door. Before the door is opened, Mr. Hall opens a second door to reveal a goat, and then asks if you want to swap the door you picked for the third door. The question is whether you should switch. Ms. vos Savant said that you should switch: you have a 1/3 chance of winning the car if you stick with your original choice, and a 2/3 chance of winning the car if you switch.

Professional mathematicians (all men) were outraged. . .

. . . and they wrote in droves. Ms. vos Savant printed some of their responses on December 2, 1990, and more on February 17, 1991, when she said that her mailbag was running 9 to 1 against her answer. Many said that (as put by Paul Hoffman in his biography of Paul Erdos, *The Man Who Loved Only Numbers*) "she was the goat and that women look at mathematical problems differently from men." Mr. Hoffman cheekily preserves some of their names and institutional affiliations in his description of the incident. ["As a professional mathematician, I'm very concerned with the general public's lack of mathematical skill. Please help by confessing your error." -- Robert Sachs, George Mason University; "There is enough mathematical illiteracy in this country, and we don't need the world's highest IQ propagating more." -- Scott Smith, University of Florida; and my favorite, "You are utterly incorrect . . . how many irate mathematicians are needed to get you to change your mind?" -- E. Ray Bobo, Georgetown University]

Erdos himself, one of the finest mathematicians of the last century, said she was wrong. But she was right, and the professional mathematicians ultimately had to admit it.

To prove her answer, Ms. vos Savant listed all possible outcomes and counted how many yield the car and how many a goat. It occurred to me that Mr. Summers was simply using the same technique, or trying to, in his remarks, and not doing it very well. Here's how I break down the possibilities:

A. Women and men have different innate abilities for mathematics, and this is the main cause of women being underrepresented in tenured chairs.

B. Women and men have different innate abilities for mathematics, and this is one cause of underrepresentation, but not the main cause.

C. Women and men have different innate abilities for mathematics, but this is not a significant cause of underrepresentation.

D. Women and men do not have different innate abilities for mathematics, and other factors cause underrepresentation.

E. It is not possible to determine whether women and men have different innate abilities for mathematics.

Mr. Summers chose B, and his critics chose D, the reverse of the grades that they would probably give each other on this issue.