Perry, Stephen (2021) What is "Applied Mathematics" Anyway? How the History of Fluid Mechanics Demonstrates the Role of Concepts in Applied Mathematics. [Preprint]
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Abstract
One of the more popular approaches to articulating the role of mathematics in
scientific modeling and explanation has been what is called the “mapping account.” The mapping account supposes that there is something like an isomorphism or homomorphism between a mathematical representation and the physical phenomenon it is representing. A notable recent formulation of the mapping account is given by Christopher Pincock. I use the case of representing viscosity in the Navier-Stokes Theorems and Prandtl’s Boundary Layer Theory to challenge this notion of direct correspondence between a piece of mathematics and the world.
I argue that physical concepts play a crucial role in mediating between mathematics and world, and I further argue that the way in which concepts play this role is complex, leading me to develop the notion of the "conceptual infrastructure" of a given physical concept, that is, how that concept may be used by a modeler. I draw on the work of Mark Wilson and Hasok Chang in generalizing the results about physical concepts I find in the case study, pointing the way to a different, more nuanced kind of account of not just applied mathematics, but mathematics in general.
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