At the beginning of the twenty-first century, mathematicians sought tangible experiences with mathematical theories and objects; in turn, they advocated an experiential, embodied, and sensorial mediated approach to mathematical training.
Intellectual Merit
Through archival research, in-depth interviews, and participant observation in conferences, classrooms, and labs, this project aims to locate the shift characterized above in its historical and institutional context. Specifically, this project asks what prompted and enabled mathematicians to seek a more embodied and procession understanding of mathematical ideas, and then tracks what this change might entail for the way mathematics is now communicated and taught.
To answer the above questions, PhD candidate will examine five case studies starting in the 1960s and leading up to the present: (1) mathematical videos produced in the late 1960s to illustrate otherwise abstract and difficult to imagine mathematical problems; (2) a 1987 drawing manual that called on topologists to draw pictures and illustrate their work; (3) An NSF-funded mathematical research center at the University of Minnesota, established in 1991, that was dedicated to developing visualization programs for mathematicians; (4) the appropriation of fiber arts (such as knitting and crocheting) to investigate the properties of mathematical surfaces; and (5) the emergence over the past two decades of computational origami, which brings the ancient art of origami under the purview of professional mathematics.
All five cases represent examples of "mathematical manifestation," which denotes the concrete, demonstrable, exemplary, displayable, and presentable aspects of mathematical research. By focusing on the experiential aspects of mathematical work, this project seeks to bring to the history of mathematics practice-based investigations and theoretical tools developed by science studies scholars, as well as offer a unique theoretical intervention into theories of mathematical practice.
Potential Broader Impacts
This project will contribute to scientific pedagogy by examining a under examined approach to mathematical education, which supplements the axiomatic abstract method by cultivating the experiential, hands-on, and embodied nature of mathematical work. By focusing on the application of computer graphics in mathematics, the project will provide a new perspective from which to understand debates within the Mathematical community regarding the application of computer technology to mathematical work.
Mathematics is often presented as the most historically stable discipline whose methodology of proof and logical deduction dates back to Ancient Greece. Yet mathematics is also incredibly malleable: the precipitous growth of the field and its institutional remaking in the aftermath of World War II triggered worries over its intellectual boundaries. I am here interested in the relationship between ideas and institutions, showing how the two are mutually inclusive in the case of twentieth century mathematics. The growth of applied mathematics as an independent field of study, and the expansion of computing, operations research, and game theory were greatly aided by war-related projects and an increase in federal funding in the postwar period. Yet the growth of pure mathematics during the same time follows a different path. The development of mathematics as an abstract field with no direct relation to the external world actually increased during this period. Mathematicians enjoyed the fiscal benefit of the Cold War while maintaining the autonomy of their research by continuously redefining the nature of their field. I identify two distinct yet overlapping periods in the development of mathematics in the postwar period. In the first three decades following World War II, the influx of federal money into the field caused mathematics, like the rest of the sciences, to increase in both size and scope. Yet, in an attempt to maintain autonomy over the growth of the field, conservative mathematics departments continued to emphasize and promote mathematics as an independent field of study. Consequently, they discouraged experiential and experimental approaches to mathematical discovery and presentation, maintaining a separation between pure and applied, abstract and concrete. Starting in the 1970s, the boundaries between theoretical mathematical research and its applications were less clearly delineated. As a result, mathematics admitted a diverse set of approaches to research and training. Undergraduate and graduate enrollment, having been supported by federal funding, had steadily grown for two decades. The field was much bigger than it once had been, and encompassed a diverse range of approaches and interests. This trend, paired with the advent of computer technology into mathematics, prompted numerous mathematicians to challenge prevailing dogma. They began promoting an expansive view of mathematical practice, including physical and digital modeling and visualization techniques, and began designing new educational initiatives to underwrite these interests. Though mathematics is considered the field of the sciences least impacted by the vagaries of international conflict, domestic economy, and cultural trends, this research shows how the Cold War and repercussions spurred arguments over the boundaries of pure and applied mathematics, disputes over funding, and trends toward concrete, visualizable, and experimental inquiry.
