In the Science Wars of the 1990s, 'postmodernist' scholars claimed that the social structure of the scientific community influenced the development of scientific thought and argued that this influence somehow invalidated science. To the working scientist, it is obvious that the course of scientific development is influenced by the humanity of its practitioners and is not merely a function of some predetermined logical path. Most scientists (with the delightful exception of Alan Sokal) ignored their postmodern critics, allowing flying airplanes and nuclear power plants to argue in their stead.

Mathematics, like any science, is a human enterprise and is not driven solely by logical or scientific forces. The social structure of the mathematics community plays an important role in determining what questions get attention and funding. What determines which questions mathematicians find interesting?

For the purpose of discussion, I will divide the universe of mathematical problems into three nebulous subsets. (i) There are problems which arise directly from science and engineering questions. For example, physicists might determine that some aspect of high energy particle physics is modelled by a previously unexplored system of partial differential equations. Do these equations have solutions? Are the solutions physically reasonable? Questions like this fall directly in the laps of mathematicians. (ii) Mathematicians attempting to answer questions arising in science or engineering repeatedly observe similar types of mathematical problems arising in disparate fields. They abstract common elements from these similar problems and try to develop a theory which will provide a general framework for simultaneously answering many problems arising in diverse fields. (iii) Pure mathematics generally refers to questions and investigations whose applications to the external scientific and engineering community are neither obvious nor motivating.

Mathematics has an amazing history of developing tools and ideas decades - or even centuries - before applications of these tools to science, engineering, or business are discovered. For example, in my youth, I often thought of number theory as coffee table mathematics, meaning it was pretty but of no practical value. As so often happens in mathematics, however, techniques developed to answer abstract questions in number theory became the foundation of the modern cryptographic systems employed by the world's financial institutions. This transition from pure mathematics to very applicable mathematics happens frequently and blurs my above division of the mathematical universe.

Mathematicians working in problems of type (i) often rely on the judgments of their nonmathematical collaborators in assessing the relative importance of their work. Hence we will shunt the sociological questions associated with this section of mathematics to other fields of science. Problems of the second type are a hybrid of the first and third; hence we focus on the third species. Problems in pure mathematics gain prominence in several ways, including: longevity, centrality, and social promotion.

Problems which have been well known for many decades or centuries but have resisted the attacks of well known mathematicians become more "valuable" with each passing year, even if their potential applications to science or mathematics have not grown in the interim. Mathematicians simply do not like to be defeated. Solving a well known problem which has survived a half century of attack will usually land you a professorship at a prestigious university.

In the development of a subfield of mathematics, problems often arise whose solution will greatly extend the scope of the subfield or perhaps simply fill an obvious gap in the theory. The greater the extension, or the more important the gap, the higher the solution is valued. Of course, the importance of the extension is still partially a social construct. Mathematicians develop their own sense of the relative importance of various aspects of their field, but it is impossible to ignore the opinions of their teachers and colleagues when forming these judgments.

Some prominent mathematicians are proselytizers. They identify an area of mathematics which they think deserves more attention and push others to devote energy to attacking problems in these areas. The more students, postdocs, invited lectures, and important publications they have, the greater the impact of their proselytizing. The greater your contributions to those problems deemed important by the community, the larger is your role in defining what the community deems important.

Hence, social constructs play an important role in the community's determination of the importance of mathematical work. This determination has coercive power through the grant, referee, and promotion processes. The great strides (as defined by the community...) mathematics has made over the last century argues that mathematics has functioned quite well subject to these social influences. In fact, only a believer in some abstract utopian vision of science should find the social aspect of science surprising.

## Sunday, April 25, 2010

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