“mathematical Social Sciences;” an Oxymoron?

Be honest. Deep down, most mathematicians probably accept “mathematical social sciences” as an oxymoron. I did — until after examining this area more seriously. Comfortable in my own research emphasizing the Newtonian N -body problem, I probably would respond to news that someone was working in, say, mathematical economics with a polite, “That’s nice,” where this noncommittal comment carried the tacit, private, yet universal translation among mathematicians, “What a ‘wishy-washy’ topic! Guess he can’t do real mathematics.” But approaches and events have significantly transformed fields. The social sciences have changed to the point that it now is appropriate to re-examine our beliefs. Of course, it always is possible to find disturbing examples. But when conducting a serious search of some of the social sciences, expect to discover that our thoughts are so dated that many (maybe not all) of them qualify as outright prejudices. Times truly are changing. We just have to review what has been happening in our field of mathematics over the last thirty to forty years to recognize that this probably is the most exciting period in history to be a mathematician. New areas and topics are constantly developing. New approaches now permit mathematical advances which would have been met with skepticism only a decade or so earlier. We are experiencing a fascinating time when even major named problems keep falling. Much more is happening. Mathematical excitement extends beyond the doors of a mathematics department to include a significant increase in the mathematical sophistication and advances of other disciplines. We all have a sense of this; all of us can point to another discipline — computer science, physics, biology, engineering — and recognize important, challenging mathematical concerns. But, the social sciences? The point of these lectures is to indicate that the social sciences can be the source of a considerable mathematical interest. A major reason is that while the social sciences always have addressed deep and serious societal concerns, often the heaviest mathematical tool outside of statistics has been elementary calculus. This is changing; more and