Discussion Note of Batterman on Mathematics and Explanation

I have written a short discussion note of Batterman's recent article on mathematical explanation in science. If you have looked at the article, you may recall that he criticizes my "mapping account" as an account of how mathematics helps in explanation, especially the sorts of explanations using asymptotic reasoning which Batterman himself has spent so much time on. The basic point I make in my reply is that I was trying to provide an account of descriptive or representational content in terms of mappings and that I agree that this approach to description is not sufficient to ground explanatory power in all cases. Still, I argue that a theory of explanatory power can build on what I offer for descriptions, and that any account of explanation must say something about how explanations differ from descriptions.

Comments are, of course, appreciated!

PSA Symposium: Applied Mathematics and the Philosophy of Science

As the final version of the PSA program is finally online, it is about time for me to promote the symposium that I will be in. Here are the details:

Applied Mathematics and the Philosophy of Science
PSA 2008 Symposium
Parallel Session 6: Saturday, November 8, 9-11:45 am
Room CCA (Conference Center A)
Chair: Paul Teller

Proposed schedule:
9:00-9:30 Christopher Pincock, “The Value of Mathematics for Scientific Confirmation”
9:30-10:00 Stathis Psillos, “What If There Are No Mathematical Entities? Lessons for Scientific Realism”
10:00-10:20 discussion
10:20-10:25 break
10:25-10:55 Mark Wilson, “Leibniz’ ‘Possibilities’ and Our Own”
10:55-11:25 Robert Batterman, “Essential Models and Explanatory Mathematics”
11:25-11:45 discussion

Abstract: This symposium will explore the relevance of philosophical reflection on the details of applied mathematics for current debates in the philosophy of science along four dimensions: (i) scientific representation, (ii) confirmation of scientific theories, (iii) idealization and scientific explanation, (iv) scientific realism. In all four cases the participants aim to show that a clear focus on the contribution that mathematics makes to science sheds new light on traditional positions in the philosophy of science. In some cases the viability of a philosophical view is called into question, while in others a standard thesis receives new support. The symposium is motivated by the realization that the philosophy of mathematics has changed considerably in the last twenty years and the hope that philosophers of science can benefit from this transformation.

For those of you who can't be there, here is a link to a rough draft of my paper. Constructive comments appreciated! Update (April 11, 2009): I have removed this old draft and hope to repost a final version sometime this spring.

Batterman on "The Explanatory Role of Mathematics in Empirical Science"

Batterman has posted a draft tackling the problem of how mathematical explanations can provide insight into physical situations. Building on his earlier work, he emphasizes cases of asymptotic explanation where a mathematical equation is transformed by taking limits of one or more quantities, e.g. to 0 or to infinity. A case that has received much discussion (see the comment by Callender in SHPMP) is the use of the “thermodynamic limit” of infinitely many particles in accounting for phase transitions. In this paper Batterman argues that “mapping” accounts of how mathematics is applied, presented by me as well as (in a different way) Bueno & Colyvan, are unable to account for the explanatory contributions that mathematics makes in this sort of case.

I would like to draw attention to two claims. First, “most idealizations in applied mathematics can and should be understood as the result of taking mathematical limits” (p. 9). Second, the explanatory power of these idealizations is not amenable to treatment by mapping accounts because the limits involve singularities: “Nontraditional idealizations [i.e. those ignored by traditional accounts] cannot provide such a promissory background because the limits involved are singular” (p. 20). Batterman has made a good start in this paper arguing for the first claim. The argument starts from the idea that we want to explain regular and recurring phenomena. But if this is our goal, then we need to represent these phenomena in terms of what their various instantiations have in common. And it is a short step from this to the conclusion that what we are doing is representing the phenomena so that it is stable under a wide variety of perturbations of irrelevant detail. We can understand the technique of taking mathematical limits, then, as a fancy way of arriving at a representation of what we are interested in.

Still, I have yet to see any account of why we should expect the limits to involve singularities. Of course, Batterman’s examples do involve singularities, but why think that this is the normal situation? As Batterman himself explains, “A singular limit is one in which the behavior as one approaches the limit is qualitatively different from the behavior one would have at the limit”. For example, with the parameter “e”, the equation ex^2 – 2x – 2 = 0 has two roots for e ≠ 0, and one root for e = 0. So, the limit as e goes to 0 is singular. But the equation x^2 – e2x – 2 = 0 has a regular limit as e goes to 0 as the number of roots remains the same. So, the question remains: why would we expect the equations that appear in our explanations to result from singular, and not regular, limits?

Batterman makes a start on an answer to this as well, but as he (I think) recognizes, it remains incomplete. His idea seems to be that singular limits lead to changes in the qualitative behavior of the system and that in many/most cases our explanation is geared at this qualitative change. Still, just because singular limits are sufficient for qualitative change it does not follow that all or even most explanations of qualitative change will involve singular limits. Nevertheless, here is an important perspective on stability analysis that I hope he will continue to work out.