Intellectual Diversity in MathematicsJanuary 12, 2007
Letters to the Editor
To the Editor:
To the outside world mathematicians are much alike---cerebral, spacey, or nerdy, but uniformly bright. Factions within the community, however, divide the field into disparate neighborhoods. This division is largely the result of a tendency toward narrowly focused research, and a lack of awareness of other areas. The narrowness can lead to people in one area disparaging the intrinsic worth of work of others: "He/she is not a mathematician" or "he/she proves things which have been known for decades." The purpose of this letter is to initiate a discussion of these tendencies, which are symptoms of a dysfunctional "family," with the eventual aim of strengthening the mathematical sciences.
A few definitions regarding the breadth of the mathematical sciences: At one end of the spectrum is pure mathematics, regarded by its practitioners as the proving of theorems using the maximal degree of rigor. At the other end is applied mathematics, whose practitioners work to impact the natural sciences and engineering by developing new methods and deepening one's understanding of new physical phenomena. Another way to characterize the two extremes is "problem-driven" versus "techniques-driven" mathematics. The dividing line is diffuse, and rare individuals have feet in both camps.
My thesis is that good work at every point along the spectrum is intrinsically valuable to mathematics and the scientific endeavor at large. This is true in part because there should be a two-way transmission of insights gained between the pure and applied sides, which will lead to the creation of new mathematics. Good work in all areas deserves respect from all. To be sure, the key word here is "good."
What is good, and who decides what is good? Clearly, others working in an area must see the work as valuable. For pure mathematics, good work might take the form of an insight needed in the specification of a weak hypothesis or the ingenuity required for the proof of a theorem. On the applied side, valuable research might involve the development of a new approximation method, the derivation of a new model equation, or the elucidation of a new physical mechanism. With the dialogue I hope to spark, ideas and techniques in one area would lead to new insights in another.
Being respectful involves some degree of humility. Those at the pure end of the spectrum need to be aware that standards for rigor can change with time---what was rigorous a hundred years ago may not be considered so now. In any case, a rigorously proved result may not overlap with what is needed to understand a phenomenon being modeled. Given the short lifetimes of many models in the newest research areas, moreover, the demand for rigor can be unwarranted. Finally, it is important to remember that many interesting new areas of mathematics, e.g., dynamical systems, singular perturbation theory, evolutionary PDEs, and, indeed, geometry, originated in applied mathematics.
Applied mathematicians, for their part, need to keep in mind that formal solutions, asymptotic or numerical, can be very instructive but must be used with care: There are examples of numerical "solutions" in the literature, for example, that use algorithms that do not converge, and depend on assumptions that make them incapable of converging. Even with validation by "numerical convergence," the choice of algorithms can be guided by rigorous theory. A priori estimates of PDEs can impact convergence criteria in numerics. Finally, as most readers will know, developments in pure mathematics---e.g., in number theory, graph theory, knot theory, coding, complexity, and turbulence---are often directly applicable to scientific problems.
In terms of policy, funding, and long-range planning, representation by a unitary voice, in tune with the entire spectrum, would benefit the mathematical sciences. Institutions like the American Physical Society and the American Chemical Society are persuasive in the "corridors of power" in Washington because each has found a voice that represents a balanced view of its membership. In my opinion, the American Mathematical Society and SIAM do not provide similarly powerful representation for mathematical scientists.
Unity in mathematics is exceedingly desirable in academia as well. Too many science and engineering departments decry the mathematical training offered to their students by mathematics departments; the other departments end up offering their own courses, thereby straining their resources and delivering inadequate perspectives. Given a unified mathematical sciences community, these objections might vanish; students in all sciences could learn mathematics from mathematicians. Finally, the evaluation for promotion and tenure of pure and applied mathematicians in mathematics departments could be made transparent by the formulation of criteria for evaluating each candidate by standards appropriate to his/her scientific aims.
My hope is that readers will be motivated to engage in discussions of these ideas about the discipline and the need for all in the community to respect all parts of it. Clearly, I come from the analysis side of mathematics and have emphasized this perspective in my comments. I hope that those in other areas will respond, providing better balance to discussions. Constructive proofs only, please!---Stephen H. Davis, Northwestern University; firstname.lastname@example.org.
This letter benefited greatly from conversations with S.J. Watson.