One of the popular approaches to the role of mathematics in scientific modeling and explanation has been what is called the "mapping account," which supposes that there is some isomorphism or homomorphism between a mathematical representation and the physical phenomenon it is representing. A notable recent formulation of the mapping account, given by Christopher Pincock (Pincock, 2007b, 2012), introduces the notion of a matching model in order to accommodate infinite idealizations (Pincock, 2007a). While this account is an improvement on the "naive" mapping model, I will argue that Pincock's account and all mapping accounts rely on a correspondence between a mathematical structure and some structure in the physical world. Pincock's matching models address certain concerns about this correspondence critique, but I will argue that they still rely on a correspondence between the physical magnitudes represented in mathematical equations, and the world. 

To challenge the mapping accounts' correspondence requirements on physical magnitudes, I will investigate the concept of viscosity in the Navier-Stokes Equations and Prandtl's Boundary Layer Theory, particularly on the mapping account interpretation of these equations. Building on Morrison's (Morgan & Morrison, 1999) analysis of Boundary Layer Theory, I will argue that the implementation of viscosity in Navier-Stokes and Boundary Layer Theory involves a balancing between macro- and microlevel phenomena. The derivation of the Navier-Stokes equation involves both molecular assumptions about viscosity and macro-level intuitions, particularly in Navier's choice to move from a molecular to continuum-mechanical treatment, which is an example of infinite idealization. I argue that the mapping account interpretation of Navier-Stokes fails to adequately represent these mechanics of how the viscosity concept is functioning in the Navier-Stokes equation, which leads to the correspondence problems the mapping account runs into when trying to reconcile Navier- Stokes and Boundary Layer Theory. In particular, I will use Wilson's (Wilson, 2006) and Chang's (Chang, 2007)accounts of how scientific concepts evolve in physical theory. Using a case study on the concept of force, Wilson discusses how the historical development of a concept affects its current use. Similarly, Chang discusses the problems in coordinating theoretical conceptions of temperature and measurements of temperature. While these accounts differ, they both recognize the fine grain resulting form the historical development of these concepts. These details do not enter into the mapping account, and this is what results in the correspondence problem that I identify.

Chang, H. (2007). Inventing temperature : measurement and scientific progress. Oxford ; New York: Oxford University Press.

Morgan, M. S., & Morrison, M. (1999). Models as Mediators: Perspectives on Natural and Social Science (M. S. Morgan & M. Morrison Eds.). Cambridge: Cambridge University Press.

Pincock, C. (2007a). Mathematical Idealization. Philosophy of Science, 74(5), 957-967. doi: 10.1086/525636

Pincock, C. (2007b). A Role for Mathematics in the Physical Sciences. Noûs, 41(2), 253-275. doi:10.1111/j.1468-0068.2007.00646.x

Pincock, C. (2012). Mathematics and Scientific Representation: Oxford University Press.

Wilson, M. (2006). Wandering significance : an essay on conceptual behavior. Oxford : New York: Oxford : Clarendon Press New York : Oxford University Press.