One night several years ago while watching TV, I was surprised to see a mathematical equation make an appearance on the "Tonight Show." The occasion was an interview with Paul Ehrlich, author of The Population Bomb and popularizer of population control as a solution to the world's problems. At that time the ecology movement had just started to capture the attention of the public, and Mr. Ehrlich was arguing that the solution, as always, was in population control.
Johnny Carson was in top form, but the show could have bogged down if his guest had delved into subtleties or overly serious discussion. However, Ehrlich had the perfect solution. He took a piece of posterboard and wrote in large letters for the TV audience:
D = N * I
"In this equation," he explained, "D stands for damage to the environment, N stands for the number of people, and I stands for the impact of each person on the environment. This equation shows that the more people, the more pollution. We cannot control pollution without controlling the number of people."
Johnny Carson looked at the equation, scratched his head, made a remark about never having been good at math, and commented that it all looked quite impressive.
Who can argue with an equation? An equation is always exact, indisputable. Challenging someone who can support his claims with an equation is as pointless as arguing with your high school math teacher. How many of Johnny Carson's viewers had the sophistication necessary to question Ehrlich's equation? Is Ehrlich saying that the "I" for the president of Hooker Chemicals is the same as the "I" for you and me? Preposterous, isn't it? But what if the viewer is too intimidated by a mathematical equation to apply some common sense? Ehrlich knew how to use his time on the show well.
Political Theory
Of course, it will surprise no one to find low standards of intellectual honesty on the "Tonight Show."
But we find a less trivial example if we enter the hallowed halls of Harvard University, where Professor Samuel Huntington lectures on the problems of developing countries. His definitive book on the subject is Political Order in Changing Societies (1968), in which he suggests various relationships between certain political and sociological concepts: (a) "social mobilization," (b) "economic development," (c) "social frustration," (d) "mobility opportunities," (e) "political participation," (f) "political institutionalization," (g) "political instability." He expresses these relationships in a series of equations (p. 55):
social mobilization / economic development = social frustration (a / b = c);
social frustration / mobility opportunities = political participation (c / d = e);
political participation / political institutionalization = political instability (e / f = g).
When he is called upon to summarize his book (e.g., in Theories of Social Change, Daniel Bell, ed.), he emphasizes these equations.
Huntington never bothers to inform the reader in what sense these are equations. It is doubtful that any of the terms (a) - (g) can be measured and assigned a single numerical value. What are the units of measurement? Will Huntington allow us to operate with these equations using the well-known techniques of ninth grad algebra? If so, we could infer, for instance, that
a = b * c = b * d * e = b * d * f * g
i.e., that "social mobilization is equal to economic development times mobility opportunities times political institutionalization times political instability!"
A woman I know was assigned an article by Huntington for her graduate seminar on historial methodology. The article summarized his work on modernization and cited these equations. When she criticized the use of the equations, pointing out the absurdities that follow if one takes them seriously, both the professors and the other graduate students demurred. For one, they had some difficulty following her application of ninth grade algebra. Moreover, they were not used to questioning an eminent authority figure who could argue using equations.
Huntington's use of equations produced effects - mystification, intimidation, an impression of precision and profundity - which were similar to those produced by Paul Ehrlich's use of an equation on the "Tonight Show." But Huntington operates on a more serious level. He is no mere talk-show social scientist. When he is not teaching at Harvard, he is likely to be advising the National Security Council or writing reports for the Trilateral Commission or the Council on Foreign Relations.
Slavery
Before leaving Harvard, let us look in on another professor, this time in the Department of Economics. Robert W. Fogel's specialty is applying quantitative methods to economic history. He and a collaborator, Stanley Engerman, produced a sensation in 1974 with a book called Time on the Cross. Using statistical arguments with voluminous computer-processed data, they purported to show that the slave system in the South was both more humane and economically more efficient than the free labor system that existed at that time in the North.
Although this thesis contradicted the conclusions of all major conventional historians, the book was received enthusiastically. Harvard historian Stephan Thernstrom called it "quite simply the most exciting and provocative book I've read in years," and Columbia economist Peter Passel wrote in his New York Times book review that it has "with one stroke turned around a whole field of interpretation and exposed the frailty of history done without science."
The initial acclaim lasted long enough to produce an effect outside academia. Fogel appeared on the "Today Show"; the book was reviewed in the Wall Street Journal, Time magazine, Newsweek, and over three dozen other major publications. The public was told that a sentimental and subjective view of slavery had given way to a "scientific" view based on computer analysis of hard quantitative facts.
But then historians of the slave period and specialists in the use of quantitative methods in history ("cliometricians") undertook careful studies of the book, and the honeymoon ended. They found such an accumulation of outright errors, fallacious inferences, dubious assumptions, and disingenuous use of statistics that the entire project lost any validity. Here is a typical example, as explained by Thomas L. Haskell in the New York Review of Books:
. . . readers of Time on the Cross are inclined toward a benign view of slavery when they read that the average slave on the Barrow plantation received only 0.7 whippings per year. In the first place the figure is too low because it is based on an erroneous count both of the number of slaves Barrow owned and the number of times he whipped them. But more important, the figure is not the most relevant measure of the importance of whippings. A whipping, like a lynching, is an instrument of social discipline intended to impress not only the immediate victim but all who see or hear about the event. The relevant question is "How often did Barrow's slaves see one of their number whipped?" - to which the answer is every four and a half days. Again, the form in which the figures are expressed controls their meaning. If one expressed the rate of lynchings in the same form Fogel and Engerman chose for whippings, it would turn out that in 1893 there were only about 0.00002 lynchings per black per year. But obviously this way of expressing the data would cause the reader utterly to misunderstand the historical significance of the 155 Negro lynchings that occurred in 1893.
Other examples would take too long to go into here; the interested reader is referred to Haskell's excellent article (NYRB, Oct. 2, 1975) and to the three volumes critiquing Time on the Cross which Haskell reviews.
Haskell regards Time on the Cross as an aberration, and refrains from indicting the entire "cliometric" approach because of one unfortunate case. However, he makes some insightful comments on the dangers inherent in any application of mathematics to the social sciences:
On the surface, cliometrics is an austere and rigorous discipline that minimizes the significance of any statement that cannot be reduced to a clear empirical test ("operationalized"). But beneath the surface one often finds startling flights of conjecture, so daring that even the most woolly-minded humanist might gasp with envy.
The soft, licentious side of cliometrics derives, paradoxically, from its reliance on mathematical equations. Before the cliometrician can use his equation to explain the past, he must assign an empirical value to each of its terms, even if the relevant empirical data have not been preserved or were never recorded. When an incomplete historical record fails - as it often does - to supply the figures that the cliometrician's equation require him to have, it is considered fair play to resort to estimation, just so long as he specifies the assumptions underlying his estimates. And although cliometrics requires that these and all other assumptions be made explicit, it sets no limit at all on the number of assumptions one may make, or how high contingent assumptions may be piled on top of each other - just so they are explicit.
Fogel, like Huntington, understands the propaganda value of mathematics. In some quarters, invoking an equation or statistic can be even more persuasive than citing a well-known authority. An argument which would be quickly disputed if stated in plain English will often acquire some momentum if accompanied by numbers and formulas, regardless of whether or not they are relevant or accurate. The threshold of expertise and self-confidence needed to challenge an argument becomes much higher if it is enshrouded in science. It is no wonder that quantitative methods have become a bit of a fad in the social sciences.
The impact of Time on the Cross reached outside the academic world. Slavery is perhaps the most profound and emotional issue in American history. How one regards slavery has clear implications for attitudes toward present-day grievances of black people and methods proposed to address those grievances, such as busing, affirmative action, compensatory education, etc. It was because of these implications that the book received so much attention outside scholarly circles.
Another example of pseudo-quantitatve argument injected into an emotional issue with wide repercussions is the IQ controversy.
-snarfed from Volume III of Mathematics: People, Problems, Results - 1984
