2.1. Asymptotic Explanation

Let me begin my response by emphasizing one point about explanation that, I believe, has been overlooked in these criticisms. I intended the arguments in the book, in part, to demonstrate how very different the kind of explanation required for an understanding of various universal features of, say, the rainbow, is from the types of explanations talked about in the extant philosophical literature. That is to say, most contemporary philosophical accounts of explanation simply do not have the resources to incorporate the types of asymptotic analysis that Belot presents in his explication of the explanation of the rainbow. As I said, and as Belot quotes, “… [t]hese emergent phenomena are not derivable in any straightforward sense from the underlying wave theory. They are not, as it were, from-first-principle solutions to the wave equation. … They are deeply encoded in that equation but are apparent only through its asymptotic analysis” (Batterman Reference Batterman2002, 118).

The nature of this asymptotic explanation is really quite different (as Belot's extended technical discussion ably demonstrates) from the ‘standard’ (D-N type) accounts, whereby one plugs the appropriate initial conditions and boundary conditions into the fundamental (wave) equation, and then deterministically/deductively grinds out the appropriate wave function for the system's state at some later time. The asymptotic investigation, involving the principle of stationary phase, the projection of the bicharacteristic strips from the appropriate Lagrangian manifold onto configuration space, etc., really does not much resemble standard accounts of ‘deriving the explanandum from the explanans’.

Yet, Belot's footnote number 34 suggests that there is no genuine fundamental distinction between the type of asymptotic explanation I highlight and the D-N type accounts. He takes me to be providing a ‘friendly amendment to the D-N account’. He cites Hempel, regarding the D-N model: “given this notion of explaining a particular occurrence of a solar eclipse or of a rainbow, etc., one can speak derivatively of a theoretical explanation of solar eclipses or rainbows in general: Such an explanation is then one that accounts for any instance of an eclipse or rainbow” (Hempel Reference Hempel1965, 423). Belot continues:

In the book and in an earlier article (Batterman Reference Batterman1992), I emphasized a distinction between kinds of explanatory why-questions. On the one hand, there are what I called type (i) why-questions. These ask for the explanation of why a given instance of a pattern obtained. Type (ii) why-questions, on the other hand, ask why patterns of a given type can be expected to obtain. They are, in effect, question about the existence of universal behavior. I argued that it is far from obvious how D-N based models are capable of answering type (ii) why-questions. In the present context, this distinction is just the distinction between explaining each occurrence of the rainbow (type (i)) and explaining the rainbow in general (type (ii)).

The question now concerns how to understand Hempel's idea that we may “speak derivatively of a theoretical explanation of solar eclipses or rainbows in general.” The D-N model provides an account whereby we can explain the particular occurrence of a solar eclipse or of a rainbow. It is unclear how such an account (or even a collection of such accounts) can allow us to speak ‘derivatively’ of a general account that would answer the corresponding type (ii) question. The point here, as I have emphasized in the book, is that each such D-N account of a particular occurrence will be remarkably different from all of the others. As Belot himself notes, each account will involve different initial and boundary conditions—different shapes of the raindrops, for example. Had we an explanation that answers the type (ii) question, then it seems that, yes, we may very well speak ‘derivatively’ of explaining any given instance. We will have an account that tells us why many/most of those individual details can be ignored. That is just what asymptotic explanation, as I have presented it, provides. Hempel and Belot, it appears, get things backwards. The individual accounts—answers to type (i) why-questions—are (in part anyway) dependent upon there being an answer to the type (ii) question.Footnote ¹

I think that is fair to say that asymptotic explanation as I have presented it is, at the very least, quite different from standard philosophical accounts. Furthermore, it seems fair to say that Belot endorses the idea that explanation often does involve asymptotic analysis and that, in particular, such analysis is often involved in the demonstration of important stability results required to understand universal behavior. As a consequence, I believe he ought to be convinced that we really are discussing and employing a philosophically new (or at least widely ignored) approach to explanation. The question now concerns how radically different this approach really is. How are we to understand the kind of asymptotic decoding of information contained in the fundamental theory? Are we in any way forced to say that despite this ‘containment’ the fundamental theory is explanatorily inadequate?

2.2. Explanatory Inadequacy of Fundamental Theory

As Belot notes, the cases I consider in the book are cases in which a certain partial differential equation, depending upon some parameter, needs to be investigated in an asymptotic limit in which that parameter approaches a limiting value (0 or $\infty $ ). This investigation proceeds via an auxiliary problem relating a set of curves in the cotangent bundle $T^{*}X$ of the manifold X upon which the original equation is defined. These curves are the integral curves of a vector field generated by the so-called principal symbol of the original partial differential equation. One is interested in the projection of these curves (bicharacteristic strips) in $T^{*}X$ down onto the manifold X. Such projections yield so-called bicharacteristic curves, and we primarily care about various lines of singularities that appear in the projection—intersections of bicharacteristic curves in X projected down from the bicharacteristic strips inhabiting the appropriate Lagrangian manifold in $T^{*}X$ .

Belot's view is that this is, so far, a purely mathematical exercise. In fact, it is the precise description of the apparatus involved in the asymptotic decoding of a partial differential equation depending upon some parameter.Footnote ² So if we set a mathematician, who has no knowledge of the physical interpretation of the partial differential equation, the task of performing this asymptotic investigation, she will be able to construct a function characterizing the asymptotic approximate solutions to the original equation. She will even be able to show how such a solution is relatively stable under perturbations of certain kinds—that is, she will be able to show that the structure of the corresponding bicharacteristic curves on X will be diffeomorphically related to each other under such perturbations.

But so what? Well, Belot argues that if we now tell the mathematician that she has been investigating the wave equation and that she has ‘derived’ Airy's integral, she will see that she has explained the fringe spacings and intensities of rainbows. This requires that we

Furthermore, “none of this would appear to require reference to the concepts of geometrical optics [or ray theory]” (Belot Reference Belot2005, 151).

I believe that (ii) and (iii) are intertwined. That is, I believe that the very characterization of the appropriate boundary conditions is related to the explanation of why the perturbations are to be interpreted in terms of changes in the drop shapes. Furthermore, contrary to Belot's claim, I believe that fulfilling both (ii) and (iii) will essentially involve reference to concepts from geometrical optics.

Let me begin by asking about the relationship between the perturbation in the shape of a macroscopic object like a raindrop, say of radius r, and the stability of patterns of intensity of light in the asymptotic limit of the wave equation as $(\lambda / r) \rightarrow 0$ ?Footnote ³ One sees that when $(\lambda / r) $ approaches zero, the contributions to the integral describing the intensity of light become concentrated near certain lines (these are the bicharacteristic curves) that exit the raindrop at certain angles. (This is what the principle of stationary phase tells us.) Now some families of these lines of concentrated intensity intersect to form an envelope (or curve) outside the raindrop. How do we fulfill (iii)? We must assert that changing the shape of the raindrop (in effect, letting r vary in certain ways) doesn't effect the general ‘shape’ of the envelope of the family of lines.

In addition, all of this involves considering the nature of the boundaries appearing in the physical problem. These characterize how the light reflects off the back of the raindrop and refracts as it enters and exits the drop. In order to see what boundary conditions to impose on the partial differential equation in the first place, we must conceptualize the problem as one in which (to a first approximation) we are considering specular reflection off the back of the raindrop. It involves, that is, thinking about light behaving as rays on the physical boundaries. Without the physical interpretation to begin with, we would not know what boundary conditions to join to the differential equation. Neither, would we know how to join those boundary conditions to the equation. Put another way, we must examine the physical details of the boundaries (the shape, reflective and refractive details of the drops, etc.) in order to set up the boundary conditions required for the mathematical solution to the equation.

Thus, although Belot's pure mathematician can be given initial conditions and boundary conditions and be told to investigate the asymptotics of the relevant partial differential equation, those initial and boundary conditions are not devoid of physical content. They are ‘theory laden’. And, the theory required to characterize them as appropriate for the rainbow problem in the first place is the theory of geometrical optics. The so-called ‘pure’ mathematical theory of partial differential equations is not only motivated by physical interpretation, but even more, one cannot begin to suggest the appropriate boundary conditions in a given problem without appeal to a physical interpretation. In this case, and in others, such suggestions come from an idealized limiting older (or emeritus) theory.Footnote ⁴

The lines of concentrated intensity leaving the raindrop—the bicharacteristic curves—are the rays (normals to the wave fronts) and the envelope of a family of such lines is a caustic. If one is going to say how the perturbation of the shape of a macroscopic object, characterized by variations in r, figures in the stability of the fringe spacings and intensities, then it seems one needs to refer to the relationship between that macroscopic object and the lines of concentrated intensity exiting the object as its shape is altered. In other words, the very investigation of the stability under perturbation of the equations involves considering perturbations to the boundaries understood in terms of structures foreign to the fundamental theory. In this way, the boundary conditions referred to in Belot's (ii) are indeed crucial for the stability problem in his (iii).

Note further that the investigation of the shortwave limit of the wave equation presents an interesting kind of problem. When $(\lambda / r) =0$ , the wave equation blows up. That is to say that the limiting behavior as $(\lambda / r) \rightarrow 0$ is qualitatively distinct from the behavior at the limit, when $(\lambda / r) =0$ . What is the motivation for studying this singular limit? One way to think about this is to realize that we are asymptotically investigating a boundary problem—we are looking at a place where the fundamental theory breaks down completely. One attitude toward this might be to say that the fact that the law breaks down at this boundary (in this limit) just shows that the boundary is unimportant to the physics. After all, the mathematics of the fundamental theory is incapable of saying what goes on at the limit—the wave equation makes no sense when there are no waves!

But boundary problems such as this crop up all over the place in physics. One example, discussed by Mark Wilson (Reference Wilson2003) concerns the development of shocks. Suppose we are interested in understanding the behavior of a gas as it moves through a tube. If a collection of the molecules are given a push (say by blowing into the tube at one end), then they will begin to catch up to those in front resulting in a more densely populated region separating two regions of relatively lower molecular density. Across this region, molecules will exchange momentum with one another as if “some kind of permeable membrane were present.” The region occupied by this ‘membrane’ is a shock. Of course it is far too complicated to track the behavior of the individual molecules as they move through the tube and undergo the collisions in the shock region. And, once again, since we are interested in recurring phenomena, the individual (molecular) details of the development of any one shock will be quite different from the details of the development of another. Those details will, in other words, be largely irrelevant to the pattern being investigated. As a consequence (familiar story!) one approaches the problem asymptotically by taking a continuum limit. We model the collection of molecules in the tube as a continuous fluid. Such a limit will shrink the shock region onto a two dimensional boundary. Sufficiently far from either side of the boundary, the behavior of the fluid will be governed by the relevant partial differential equations of fluid mechanics. However, the behavior across the boundary is not governed by any differential equation at all. The fluid mechanical equations blow up.

The shock boundary represents a situation in which the theory fails to describe what is going on. Nevertheless, the boundary is a dominant feature of the phenomenon studied that, far from being unimportant, is crucial for scientific understanding. As Wilson (Reference Wilson2003) puts it, “the allegedly ‘suppressed details’ have become crushed into a singular (hence not law-governed) factor that still dominates the overall behavior through the way in which it constrains the manner in which the ‘law governed regions’ piece together.” It constrains the manner in which the law governed regions fit together by informing us about the nature of the boundary conditions we must impose upon those equations that are operative in those law governed regions.

The mathematical problem here is analogous to the problem of investigating the rainbow from the point of view of the asymptotics of the wave theory. The blow up at zero wavelength represents a boundary upon which the fundamental theory itself fails. Yet that boundary dominates the phenomenon—the behavior of the rainbow. In saying that the fundamental theory is explanatorily inadequate, I mean to be saying that we must take this boundary upon, which the fundamental theory fails, into consideration in our explanation of what is going on. And, in fact, that boundary or non-law-governed region (from the point of view of the fundamental theory) is characterizable only by reference to structures that are present in the older superseded theory.