Rethinking the connections between ‘mathematical sciences’ and application areas

The National Research Council has made available a pre-print of the forthcoming report of its Committee on The Mathematical Sciences in 2025, chaired by Caltech EE/applied physics professor Thomas Everhart. Like all NRC publications, it’s a long and dense document, but the summary remains fully accessible to the general reader. Though it makes all the usual pleas for funding of basic research without undue hope for immediate practical application, the report also starkly underlines what should now be obvious connections between mathematical knowledge and rapidly accumulating advances in a wide array of other disciplines and real-world applications. Even within mathematics itself, the report argues, boundaries between sub-disciplines are breaking down, and mathematicians who would formerly have seemed past their prime years of creativity can now still make important discoveries because it pays increasingly to have long experience of these interconnections.

What I found remarkable was how hard this committee came down on the core discipline itself, calling mathematicians generally “incognizant” (fighting word!) of the expanding role that the mathematical sciences now play in other realms of theory and practice. “It is easy,” the authors write, “to point to work in theoretical physics or theoretical computer science that is indistinguishable from research done by mathematicians, and similar overlap occurs with theoretical ecology, mathematical biology, bioinformatics, and an increasing number of fields.” By implication, the authors are calling their colleagues insufficiently appreciative of these connections. And in practical fields, it seems that everyone — biotechnologists, communication-system engineers, and financial-market “quants,” to take just a few examples — has proved more aware of these interdependencies than mathematicians themselves.

Consequently, the committee calls for fundamental rethinking of the “mathematical sciences” broadly, and for new curricula that produce graduates not only knowledgeable across the core discipline, but also sympathetic to computational sciences and prepared to bring their expertise into those other areas where advances are increasingly dependent on sophisticated understanding rather than rote-learned technique. None of this is necessarily new: it’s essentially the philosophy underlying dramatic cross-disciplinary projects like Eric Lander’s redirection of his mathematical training to difficult problems in molecular biology at the Broad Institute. What did strike me as new was emphasis on rethinking education for students (undergraduate, graduate and continuing/professional) whose core interest lies in other disciplines but “who need expert guidance and mentoring from successful mathematical sciences.”

The authors suggest that current budgetary pressures and changes in the structure of higher education threaten the departmental “service tradition” of supporting special courses for non-majors, and recommend instead that the departments rethink the use of advanced calculus as a “gateway” to the major (if taught in theorem-proof mode, really a “sieve” to filter out those not primarily motivated by “analysis,” in the mathematical sense of the word). That brought to mind my own experience at Princeton, where the math department offered dual tracks (non-major and major-bound) at the level of freshman and sophomore calculus and linear algebra, but if you were interested in differential equations as a non-major, you were really better served taking the subject in an applications-oriented course offered by the engineering school (my mistake for not so doing). Now, this report suggests, math and statistics departments themselves need to embrace “deep rethinking of the different types of students they are attracting and wish to attract.”

Of course, to do that, faculty in math departments will have to give up cherished traditions, like hazing non-majors through “proof by intimidation” (or even funnier, “proof by erasure” at the blackboard). As reported long ago by my freshman-year roommate, who was well enough prepared from high school to take the junior-level course on real analysis, the instructor — a still-living scholar of truly global reputation and distinction who will remain unidentified here — once replied to a query from a confused student, “I’m not here to answer questions! I’m here to teach!”

This report is well worth a look. I particularly recommend it for anyone interested in “big data” issues.

1 comment
  • Matt Mitchell January 21, 2013, 1:09 am

    Hi David … thanks for the thoughtful post. I’ve been taking calculus classes at my local junior college toward passing into a higher secondary level math credential, and can tell you that the “sieve” is alive and well at much lower levels in the educational hierarchy than Princeton.

    Nonetheless fun to spend some time on derivatives and integrals. Matt

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