Observations

The traditional division of university faculty into separate academic departments has been an educational necessity. But it has had the effect of creating separate cultures, each with its own dialect, behavioral norms, and professional valuation and reward standards. These distinguishing characteristics in turn have been reinforced by the professional societies that serve the respective disciplines. And in some degree the editorial policies of professional journals have acted similarly.

Language differences between disciplines have become a serious problem. Each group has generated its own technical vocabulary (i.e., jargon), and in many instances a given word or phrase carries different meanings in different disciplines. This situation is troublesome enough between, say, materials science and chemistry, or between condensed matter physics and molecular biology. But many researchers report that the situation is particularly acute between these fields and mathematics, especially since the last seems to celebrate the style of expression that has aptly been called "abstract minimalism."

Deeper, methodological differences also should be recognized, beyond mere vocabulary and mode of expression. The physical and biological sciences often concentrate on inductive processes, while mathematics is primarily deductive, and this distinction can cloud opportunities for fruitful communication. By way of extreme illustration, mathematics appears to have no analogs to the purely reconstructive fields of paleontology and archeology that operate on a preponderance of physical evidence but are not amenable to direct demonstration (proof) or refutation (disproof). Put another way, the standards of "proof" can often be very different in the physical and biological sciences as compared to mathematics.

To the extent that these features have created a sense of isolation of mathematics from the rest of the scientific intellectual community, we all suffer. One unfortunate scenario that can result has recently unfolded at my undergraduate institution, the University of Rochester: the administration decided that the mathematics doctoral program could be scuttled as a cost-saving measure, citing as one reason the disconnection between the mathematics department and the rest of the university. Luckily the original draconian proposal seems to have been softened by the effect of widespread outrage from the external mathematical and scientific communities, but it is unfortunate that such a display of solidarity required the presence of a threat.

We need each other. The physical (and biological) sciences would be a dreadful morass of confusing empiricism and blind experimentation if it weren't for well-posed mathematical models to organize data and to generate predictions of novel phenomena. Mathematics would likely drift into aimless pedantry without the real-world focus and drive imposed by science and technology. In the best of circumstances mathematics rises constructively to the challenge of rigorously characterizing concepts and models, and in that process enforces precision of thought among physical scientists who often slide into lazy habits of fuzzy thinking.

We ought to be able to invent relatively painless remedies to reduce the barriers to cross-discipline communication and collaboration. Furthermore, these remedies should not, and indeed need not, erode the distinctive character of mathematics as an estimable profession. The goal instead is to improve "diplomatic relations" between mathematics and the other sciences in order to stimulate research at the interfaces with inevitable benefits for society.



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