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Highlighting the role of applying mathematics within the life and social sciences through research produced by undergraduate participants of Arizona State University's (ASU) Mathematical and Theoretical Biology Institute (MTBI), we offer a brief account of what this community has learned from a research model that deliberately relinquishes the scientific agenda to its undergraduate participants. Over the past 21 years MTBI's summer research program has focused on training a diverse population of student researchers in the art of identifying and investigating questions primarily at the interface of the mathematical, life, and social sciences. At the heart of this paper are insights gained from the research leadership of twenty-one cohorts of undergraduate students. We highlight three selected projects that capture MTBI's philosophy of student-driven research, a model that has motivated hundreds of students to enroll in quantitative graduate programs across the nation. The first models collaborative active learning and its role in building robust communities of learners; the second studies the effects of oil spills on the spatial dynamics of loggerhead sea turtles; and the third develops a new, student-created version of pair-approximation modeling which has led to new research and a wide variety of applications. [ABSTRACT FROM AUTHOR]

Applying mathematics to solve any problem inevitably grants an aura of objectivity and truth to the results generated through such methodology. However, mathematics remains a means of elucidation that functions within strict constraints of tractability and stylization. Despite the aesthetics of models and the precision of computation, mathematical representations are but a shadow of the reality they seek to characterize. Nonetheless, because mathematics' pure form hides its own limitations, it becomes a privileged site of scholastic reason and, at the same time, of symbolic violence, two notions that Bourdieu theorized in his work. Despite this handicap, Bourdieu recognized the importance of formalizing tools such as those derived from mathematics. He thus devised many strategies to take advantage of their power of elucidation while avoiding the distortions and dogmatic stance they unavoidably bring about. This paper examines these strategies in detail to further inspire the identification of a number of reflexive approaches that can deflect the potential symbolic violence of mathematical reasoning. In this analysis, particular attention is given to the specific needs of practitioners engaged in mathematical formulation such as economists, statisticians, operations researchers and other quantitative scientists. This process of reflexive discovery is then used to enrich the notion of 'specific intellectual' that Foucault introduced and that Bourdieu further specified and 'lived'. The paper concludes by foregrounding the perhaps unexpected moral dimensions of applied mathematics, a discipline that otherwise often appear distant and esoteric. [ABSTRACT FROM AUTHOR]

This paper argues for a heuristic rather than a fundamental role for mathematics in the social sciences. The a priori argument for insisting on a natural-scientific and mathematical approach to the study of social phenomena is found to rest on an untenable assumption about the powers of standard logic. Further, the requirements of description in the natural sciences are found to conflict irreconcilably with the nature of social interaction and social organization. As an alternative to the natural-science model, it is proposed that the regularities described by mathematical models depend upon and reflect underlying institutional arrangements. Consequently, mathematical models represent the outcomes of basic social processes rather than those processes directly. Nevertheless, it is argued, mathematical models are extremely useful and, indeed, indispensable in attempting to unravel the complexities of social phenomena. [ABSTRACT FROM AUTHOR]