I presented a co-authored paper with Ian O’Loughlin at the 9th conference on Embodied and Situated Language Processing in Pucon, Chile. We’re currently working on submitting for publication, which we hope to do in the near future.
DOING MATHEMATICS: CONTENT AND PRACTICES IN SITUATED ABSTRACTION
Ian O’Loughlin and Katie McCallum
Understanding the meaning of, expression of, and learning of abstract, symbolic subject matter such as mathematics and logic in embodied or situated terms has long proven difficult. Whereas the language of everyday pursuits, culture, arts, or natural science can be—and to some extent each of these has been—grounded in the practices of embodied agents situated in the human world, the resilience with which mathematics and logic resist this grounding in situated practice, the apparent inexorability with which these “abstract” subjects lend themselves to a priori, internalist, rationalist, even Platonist characterizations has been often noted. In recent decades, researchers from an array of traditions—philosophy, cognitive science, linguistics, education, sociology, anthropology—have begun to consider and advocate alternatives to these characterizations, but these alternatives have suffered from at least two challenges inherited from the collection of disciplines addressing the issue. First, although there has been a significant amount of work which examines the actual practices of mathematics, especially in math education and the sociology of science, devoted to showing how easily overlooked the situated aspects of the activities of mathematics are, and how important these situated, embodied elements are to learning and doing mathematics, these research programs have not always made a clear distinction between the claim that these situated elements are enabling conditions for learning and doing mathematics and the claim that these situated elements are constitutive of mathematics itself. Even the most doctrinaire subscriber to rationalism and Platonism can unproblematically acquiesce to the former, but the latter, that the very meaning of mathematical expressions is grounded in situated practices, is a bolder thesis, deserving of careful attention. The second challenge is inherited from neighboring and complementary disciplines: studies in the philosophy of mind—as well as cognitive science and linguistics—have also examined the possibility of an alternative and situated characterization of abstract cognition and discourse, taking their lead from antecedents in the history of philosophy. Philosophers attend more carefully to distinctions between conditions and constitution, but philosophy, as a discipline, less frequently and less materially engages with the actual practices of mathematics in the human world—and this kind of engagement is made particularly difficult by the fact that, unlike anthropology or sociology, philosophy does not offer schemas for how or what to observe when attending to human practices. This lacuna leaves philosophers who would situate abstract cognition and language with a difficult task. This project is an attempt to elucidate, and to begin to respond to, both of these challenges: through observation and perspicuous demonstration of the actual practices of mathematics, we investigate whether the doing of mathematics, situated among embodied, human agents, can be a suitable candidate for the bolder claim that the very concepts of mathematics are grounded in situated practice. We identify a number of difficulties for treating mathematical practices as such, but argue that these difficulties are not insurmountable.