Dangerous Abstractions first appeared in the April 7, 1996 NY Times OpEd pages. Copyright John Allen Paulos
At first glance, a Ph.D. in mathematics, the subject often termed the handmaiden of science and engineering, seems an odd credential for a Luddite, Yet Theodore John Kaczynski, the alleged suspected Unabomber, received such a degree from the University of Michigan in 1967. What are we to make of this?

A disclaimer first. The sample size of such terrorists is happily quite small, so I make no claim to conclusiveness in the speculative remarks to follow.

Certainly the arrest of Mr. Kaczynski (coming, coincidentally, just before Mathematics Awareness Week) does not enhance the public image of mathematics and mathematicians. Indeed it furthers some of the worst stereotypes. Believe it or not, mathematicians are generally humorous, not asocial loners. The only time most use the phrase "blow up" is when they consider division by zero. Still, it's fair to ask if the background of Mr. Kaczynski is as anomalous as it appears on the surface?

Several aspects of mathematics suggest that it isn't. One is that the various subdisciplines of mathematics as well as the subject as a whole are axiomatic - that is, they are based on a few fundamental assumptions from which all else follows logically. Mathematicians often view themselves as radical thinkers in the literal sense of the word: they get to the root of the matter.

In a similar sense, the Unabomber's manifesto, which shows a mathematician's meticulous attention to detail, has the feel of an extended proof based on a few presuppositions about what constitutes the good life - personal control, self-reliance, minimal environmental impact. From these elements he attempts to derive a radical alternative theory of society - a non-Euclidean approach, if you will, to our social and psychological problems.

Another obvious characteristic of the subject is its abstractness; and the subdiscipline in which Mr. Kaczynski got his degree, complex analysis, is certainly abstract. (In fact, complex analysis has little to do with the common meanings of either word. Analysis is the branch of mathematics that grew out of calculus, and complex refers to complex numbers, a generalization of our number system that accomodates so-called imaginary numbers.)

The ability to think abstractly is a precious one too often absent from public debate. (England's recent panic over mad cow disease, for example, is fueled by its citizens' failure to calmly consider the slim chances of contracting the disease.) Nevertheless, abstract thinking has been associated with various pathologies, and it is easy to see how one trained in such reasoning and in thrall to an ideal could come to justify murderous acts as a nebulous "good."

Mathematics is also beautiful, but its aesthetic - minimalist, austere - can blind one to the messiness and contingencies of the real world. To use mathematical principles to devise a grand socio- economic theory, one must simplify. In doing so, mathematicians can forget that their mathematical or economic model is not the real world.

Reality, like the perfectly ordinary woman in Virginia Woolf's essay "Mr. Bennett and Mrs. Brown," is indefinitely complex and impossible to capture completely in any model. Thus, it is hardly surprising that, for all the Government's high-tech surveillance and analysis, the real break in the case came out of the murky realm of family dynamics.

Of course, Mr. Kaczynski may not be the Unabomber. If so, my speculations are but another example of a mathematician's failure to produce a model that works in the real world.

Professor of Mathematics at Temple University, John Allen Paulos is the author of Innumeracy and A Mathematician Reads the Newspaper.

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