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.