Abstract
‘Mapping accounts’ of applied mathematics hold that the application of mathematics in physical science is best understood in terms of ‘mappings’ between mathematical structures and physical structures. In this paper, I suggest that mapping accounts rely on the assumption that the mathematics relevant to any application of mathematics in empirical science can be captured in an appropriate mathematical structure. If we are interested in assessing the plausibility of mapping accounts, we must ask ourselves: how plausible is this assumption as a working hypothesis about applied mathematics? In order to do so, we examine the role played by mathematics in the multiscalar modelling of sea ice melting behaviour and examine whether we can indeed capture the mathematics involved in the kind of mathematical structure employed by the mapping account. Along the way, we note that the cases of applied mathematics that appear in discussions of mapping accounts almost exclusively involve the employment of a single clearly circumscribed mathematical field or domain (e.g. ‘the use of arithmetic in counting physical objects’). While the core assumption of mapping accounts may appear plausible in such situations, we ultimately suggest that the mapping account is not able to handle the important added complexities involved in our sea ice case study. In particular, the notion of mathematical structure around which such accounts are framed does not seem to be able to capture the way in which some applications of mathematics require that very different pieces of mathematics be related to one another on the basis of both mathematical and empirical information.
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Notes
For ease of expression, I will occasionally speak generically of ‘the mapping account’ instead of ‘mapping accounts’, but this is merely stylistic. There are, of course, a variety of mapping accounts in the literature, united by their commitment to the claim articulated in this section. These include Bueno and Colyvan (2011), Bueno (2016), Bueno and French (2018), Pincock (2004a, b, 2007, 2011), and Nguyen and Frigg (2017).
Although the mapping account falls pretty naturally out of a broadly structuralist philosophy of mathematics, its proponents do not intend for it to depend on any particular view about the foundations of mathematics. In fact, both Bueno and Colyvan (2011) and Nguyen and Frigg (2017) suggest that it is a strength of their mapping accounts that they do not depend on any particular philosophy of mathematics.
Thank you very much to anonymous reviewer for raising this point.
This is helpful, all things considered, since once we begin to think about more complex uses of mathematics than those that have heretofore featured in discussions of mapping accounts, we might find that it is more and more difficult to clearly distinguish exactly what single purpose some particular piece of mathematics is serving at some particular point in time.
It may well be open to the proponent of the mapping account to respond to a purported counterexample by narrowing their account such that it only applies properly to some particular restricted class of uses. There are, it seems to me, two reasons that this is unlikely to represent an attractive option to proponents of the mapping account. The first is that, as mentioned in the previous footnote, complex cases of applied mathematics may involve questions of prediction, explanation and representation dovetailing in ways that might make the question of what single purpose some mathematical construction is serving at some particular time somewhat difficult to untangle. Perhaps more importantly, the case we will discuss later on at the very least includes mathematics used both for the purpose of representation and the purpose of facilitating predictions. Amongst the purposes for which mathematics is used in empirical science these seem particularly central, and it seems unlikely that proponents of mapping accounts would want to abandon the claim that their account is intended to cover them.
Of course, Shapiro is not offering a full-blown account of applied mathematics in the way that Bueno, Colyvan, French and others are.
As an example, Fourier’s Law for heat conduction can be written using relatively simple differential operators:
$$\begin{aligned} \frac{\partial ^2 T}{\partial x^2} + \frac{\partial ^2 T}{\partial y^2} + \frac{\partial ^2 T}{\partial z^2} = \frac{\rho c_p}{k}\frac{\partial T}{\partial t}. \end{aligned}$$However this PDE can only be solved directly in the most simple cases. As such, solutions are typically approximated by powerful software packages that employ numerical methods and algorithms which are not properly part of analysis itself. See Bergman et al. (2011).
Thanks to an anonymous reviewer for pushing me on this.
It is worth noting that the proponents of mapping accounts that we have considered appear to take themselves to be bound by requirements like these, although they do not consider the question as explicitly as we have done. In particular, both Pincock (2011, p. 212) and Bueno (2016, p. 2597) consider the expectations we might apply to mapping accounts of applied mathematics. The standards they put forward seem to me closely related the two requirements outlined here.
Batterman (2021) contains several helpful discussions and illustrations of the way that homogenizations techniques allow us to extract information about the behaviour of heterogeneous composites by connecting continuum and mesoscale level descriptions. In this respect, our sea ice example falls into a broader category of material scientific problems concerned with heterogeneous materials, including that of understanding the behaviour of a mixed conductor, for instance.
A periodic medium is one that, although not homogeneous and thus composed of at least two different kinds of material, is arranged in such a way it exhibits some periodic internal structure (e.g. some characteristic or property of the materials involved varies periodically as you traverse some path through the medium).
A helpful overview of these can be found in McPhedran (2015).
An integer lattice in d dimensions is one in which the lattice sites are labelled according to d-tuples of integers (for example, (1, 2, 3) is a site in a 3-dimensional integer lattice).
Such lattice models were originaly developed in order to investigate the effective electrical conductivity of composite materials, but may be repurposed to help us understand the fluid permeability of our ice-brine composite. In fact, there are deep analogies between the way that problems of electrical transport and fluid permeability are tackled by way of homogenization techniques (McPhedran, 2015, p. 502).
To clarify, by ‘mathematical’ here I mean related to the formal relational properties of the elements of some purported mathematical structure, whereas by ‘empirical’ I mean related to some physical feature of the system in question.
At this point one still might wonder: what kind of application of mathematics have we just met? The mathematics in our sea ice case seems to serve a variety of purposes at once, such a prediction, representation, and so on. In particular, is the mathematics in our case study used for the purpose of working with the model involved or simply framing that model by way of a set of equations? I am skeptical that there will be a definitive and clear way to assign a single ‘purpose’ to applications of mathematics that exhibit the kind of complexity we see in the case just covered. Nonetheless, the point made in Sect. 2 bears reiterating: since proponents of mapping accounts take their account to cover any application of mathematics, we need not worry about whether the purpose for which the mathematics is employed in our case study is primarily related to framing a model or working with it. The point is that whatever the various purposes involved might be, the application that features in our case requires that we both frame a model using some mathematical tools and then investigate the behaviour of that model in certain tractable cases. If the mapping account is to handle such a case then we require a ‘mathematical structure’ that can capture the mathematics relevant to both of these related tasks.
The point we develop here suggests a way of responding to the way that Bueno and French (2018) dismiss the criticism of mapping accounts found in Batterman (2010). Batterman suggests that scientific explanations often rely on asymptotic features of the mathematical description of a system for which there is no physical analog. He further suggests that the mathematical operations that allow us to understand these asymptotic features are not the sort of thing that can be captured in a ‘mathematical structure’. Bueno and French simply reply that insofar as these operations are run-of-the-mill mathematical operations, they “can be characterized set-theoretically and hence represented within our framework” (Bueno & French, 2018, p. 187). One way of paraphrasing Batterman’s point in the terms of this paper is that the limiting relationships on which such asymptotic relationships rely are not purely mathematical relationships. It is not because they are operations that Batterman suggests they cannot be captured in the kind of structure required by the mapping acount, but rather because we require both mathematical and empirical input to establish that the limiting relationship exists.
References
Batterman, R. W. (2010). On the explanatory role of mathematics in empirical science. British Journal for the Philosophy of Science, 61(1), 1–25. https://doi.org/10.1093/bjps/axp018.
Batterman, R. W. (2013). The tyranny of scales. In R. W. Batterman (Ed.), The Oxford handbook of philosophy of physics (pp. 255–286). Oxford University Press.
Batterman, R. W. (2021). A middle way: A non-fundamental approach to many-body physics. Oxford University Press.
Bergman, T., Incropera, F., DeWitt, D., & Lavine, A. (2011). Fundamentals of heat and mass transfer. Wiley.
Bueno, O. (2016). An anti-realist account of the application of mathematics. Philosophical Studies, 173(10), 2591–2604. https://doi.org/10.1007/s11098-016-0670-y.
Bueno, O., & Colyvan, M. (2011). An inferential conception of the application of mathematics. Noûs, 45(2), 345–374.
Bueno, O., & French, S. (2018). Applying mathematics: Immersion, inference, interpretation. Oxford University Press.
Bursten, J. (2018). Conceptual strategies and inter-theory relations: The case of nanoscale cracks. Studies in the History and Philosophy of Modern Physics, 62, 158–165.
Bursten, J. R. S. (2021). Classifying and characterizing active materials. Synthese, 199(1), 2007–2026. https://doi.org/10.1007/s11229-020-02870-2.
Fillion, N. (2019). Conceptual and computational mathematics. Philosophia Mathematica, 27(2), 199–218. https://doi.org/10.1093/philmat/nkz005.
Fillion, N., & Corless, R. M. (2014). On the epistemological analysis of modeling and computational error in the mathematical sciences. Synthese, 191(7), 1451–1467. https://doi.org/10.1007/s11229-013-0339-4.
Golden, K. M. (1997). Percolation models for porous media. In U. Hornung (Ed.), Homogenization and porous media (pp. 27–43). Springer.
Golden, K. M. (2009). Climate change and the mathematics of transport in sea ice. Notices of the American Mathematical Society, 56(5), 562–584.
Golden, K. M. (2015). The mathematics of sea ice. In N. J. Higham (Ed.), The Princeton companion to applied mathematics (pp. 694–705). Princeton University Press.
Golden, K. M., Heaton, A. L., Eicken, H., & Lytle, V. I. (2006). Void bounds for fluid transport in sea ice. Mechanics of Materials, 38(8), 801–817.
Green, S., & Batterman, R. W. (2017). Biology meets physics: Reductionism and multi-scale modeling of morphogenesis. Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences, 7161, 20–34. https://doi.org/10.1016/j.shpsc.2016.12.003.
Kasirzadeh, A. (2021). A new role for mathematics in empirical sciences. Philosophy of Science, 88(4), 686–706. https://doi.org/10.1086/713906.
McPhedran, R. C. (2015). Effective medium theories. In N. J. Higham (Ed.), The Princeton companion to applied mathematics (pp. 500–505). Princeton University Press.
Nguyen, J., & Frigg, R. (2017). Mathematics is not the only language in the book of nature. Synthese. https://doi.org/10.1007/s11229-017-1526-5.
Pincock, C. (2004a). A new perspective on the problem of applying mathematics. Philosophia Mathematica, 12(2), 135–161.
Pincock, C. (2004b). A revealing flaw in Colyvan’s indispensability argument. Philosophy of Science, 71(1), 61–79.
Pincock, C. (2007). A role for mathematics in the physical sciences. Noûs, 41(2), 253–275.
Pincock, C. (2011). On Batterman’s on the explanatory role of mathematics in empirical science. British Journal for the Philosophy of Science, 62(1), 211–217. https://doi.org/10.1093/bjps/axq025.
Rizza, D. (2013). The applicability of mathematics: Beyond mapping accounts. Philosophy of Science, 80(3), 398–412. https://doi.org/10.1086/670295.
Santosa, F. (2015). Homogenization. In N. J. Higham (Ed.), The Princeton companion to applied mathematics (pp. 103–105). Princeton University Press.
Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. Oxford University Press.
Stemeroff, N. (2021). Structuralism and the conformity of mathematics and nature. Studies in History and Philosophy of Science Part A, 86, 84–92. https://doi.org/10.1016/j.shpsa.2021.01.004.
Süli, E. (2015). Numerical solution of partial differential equations. In N. J. Higham (Ed.), The Princeton companion to applied mathematics (pp. 306–319). Princeton University Press.
Torquato, S. (2002). Random heterogeneous materials: Microstructure and macroscopic properties. Springer.
Wilson, M. (2017). Physics avoidance: And other essays in conceptual strategy. Oxford University Press.
Winsberg, E. (2006). Handshaking your way to the top: Simulation at the nanoscale. Philosophy of Science, 73(5), 582–594. https://doi.org/10.1086/518328.
Winsberg, E. B. (2010). Science in the age of computer simulation. University of Chicago Press.
Acknowledgements
I would like to thank Bob Batterman, Mark Wilson, Erica Shumener, Marco Maggiani, Stephen Mackereth and James Shaw for their helpful comments on various iterations of this paper. I would also especially like to thank two anonymous reviewers for their patient and insightful feedback.
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McKenna, T. Structure and applied mathematics. Synthese 200, 373 (2022). https://doi.org/10.1007/s11229-022-03873-x
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DOI: https://doi.org/10.1007/s11229-022-03873-x