Stephen Jay Gould, the eminent evolutionary biologist and stylish essayist who died last month, used mathematics to elucidate many ideas, both in and out of science.
Gould's was a wide-ranging and impassioned voice of reason that will be keenly missed. One aspect of his work that particularly appealed to me was his use of simple mathematical observations and analogies to help clarify his multifarious arguments.
Baseball, Bacteria, and the Complexity of Life
Gould was, for example, famously interested in baseball, bacteria, and the complexity of life and characteristically managed to connect them in an enlightening and non-trivial way. Consider his explanation for the disappearance of the .400 hitter in baseball in his book, Full House, The Spread of Excellence.
He argued that the absence of such hitters in recent decades was not due to any decline in baseball ability but rather to a gradual decrease in the disparity between the worst and best players, both pitchers and hitters. When almost all players are as athletically gifted as they are now, the distribution of batting averages shows less variability. The result is that .400 averages are now very scarce. Players' athletic prowess is close to the "right wall" of ultimate human excellence in baseball.
Leading to bacteria and complexity, Gould next asked his readers to consider an imaginary country in which initially every adult receives an annual salary of $100. Assume that every few years each person's salary is either adjusted upward by $100 (say 45 percent of the time) or downward by the same amount (say 55 percent of the time) with the proviso that no one's salary ever declines below $100, the minimum wage.
After a number of generations the largest salary in the country will likely be quite a bit larger than $100 and the average will rise somewhat as well. This is because there is, at first, only one direction for salaries to grow; there is a "left wall" below which salaries can't decline. Although the $100 salary becomes less common over time, it nonetheless remains the most common salary.
And bacteria and complexity? Gould was interested in reconciling the obvious growth in the complexity of organisms over time with the Darwinian contention that the complexity of highly developed species is as at least as likely to decrease as it is to increase. His resolution is the following. Bacteria, the first life form on Earth, have minimal complexity, roughly analogous to the $100 minimum salary of the imaginary country above.
The apparent trend in overall complexity growth is a consequence of the fact that no life forms are simpler than bacteria, just as a rise in average salary follows from the fact that no one can make less than the $100 minimum wage in the imaginary country. Random changes in biological complexity can initially be in only one direction even though, in more developed species, decreases in complexity are at least as likely to occur as increases. Evolution provides no inherent drive for species to become more complex; global biological progress is an illusion. (Gould's argument is, of course, much more nuanced than a brief description can capture.)
Medians, Cancer, and I.Q.s
A more heartening use of a mathematical notion appeared in his well-known article, "The Median Isn't the Message." There Gould discussed being diagnosed in 1982 with abdominal mesothelioma, a cancer whose median survival time is (or at least was) eight months; that is, half the people with it live less than eight months, half longer. He took comfort from the fact that the mean or average survival time might nevertheless be considerably longer than eight months. This could occur in the same way that the average home price in a neighborhood might be $1,000,000 even if the median price were just $50,000 (if, say, there were a few palatial mansions surrounded by many quite modest homes).
Gould's book, The Mismeasure of Man, is also full of mathematical observations, most devoted to undermining the foundations of I.Q. testing. He criticized disguised confusions of correlation and causation in the misapplication of statistical techniques such as factor analysis and regression analysis. In a related New Yorker piece on The Bell Curve he illuminated the I.Q. debate with a discussion on the heritability of height in impoverished villages in India. Tall fathers there, say over 5 feet, 7 inches, generally have tall sons, while short fathers there, say under 5 feet, 2 inches, generally have short sons.
Despite this undisputed heritability of height (much more so than for I.Q.), most would conclude that improved nutrition and sanitation would raise the average height of the villagers to that in the West. The conclusion: the heritability of characteristics (such as height or I.Q.) within populations is not an explanation for average differences between populations.
Being an evolutionary biologist, Gould was always conscious not merely of averages, but of variations, exceptions, and diversity. Appropriately, his writings across a wide range of disciplines were themselves various, exceptional, and diverse.
He was a rare throwback to a less myopically specialized era.
Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on ABCNEWS.com appears the first weekend of every month.