Maddy on Applied Mathematics

In her recent Review of Symbolic Logic article “How Applied Mathematics Became Pure” Maddy offers a rich discussion of the various changes that have occurred in mathematics, science and their relationship. While I am generally sympathetic to her main conclusion that mathematics as it is practiced today has its own science-independent standards of success, I am surprised by her pessimistic conclusions concerning the sort of project that I am engaged in for applications.

Maddy first summarizes the history of scientists who took a modest perspective on the degree to which their mathematical representations were capturing ultimate causal mechanisms:

we have seen how our best mathematical accounts of physical phenomena are not the literal truths Newton took them for but freestanding abstract models that resemble the world in ways that are complex and sometimes not fully understood (p. 33).

She continues that

One clear moral for our understanding of mathematics in application is that we are not in fact uncovering the underlying mathematical structures realized in the world; rather, we are constructing abstract mathematical models and trying our best to make true assertions about the ways in which they do and do not correspond to the physical facts (p. 33).

After surveying some successful accounts of particular cases where we can make these distinctions, she concludes

Given the diversity of the considerations raised to delimit and defend these various mathematizations, anything other than a patient case-by-case approach would appear singularly unpromising (p. 35).

But nothing in the article precludes a useful classification of these sorts of successes into kinds. Of course, such a classification must start with individual cases. This would be just the beginning, especially if we could find patterns across cases. Indeed, it seems like this is just what applied mathematicians are trained to do, as a review of any applied mathematics textbook would reveal.

I grant that this is just a promissory note at this stage, but the attempt to understand and classify successful cases of mathematical modeling is really just another instance of the naturalistic methods that Maddy has applied to set theory.