Applicability of mathematics: special functions
...it seems reasonable to maintain that many special functions are special for more than simple pragmatic reasons. They are not special simply because they appear in the physicist’s, applied mathematician’s, and engineer’s toolboxes. Furthermore, special functions are not special simply because they share some deep mathematical properties. Recall this is the point of view of Truesdell and of Talman/Wigner. On their proposals, what makes some functions special is that despite “surface” differences, they are each solutions to the “F-equation” (for Truesdell), or they possess similar group representations (for Talman and Wigner). While these classificatory schemes suffice to bring some order to the effusions of the Divine Mathematician, they do not fully capture the special nature of the special functions.
From the point of view presented here, the shared mathematical features that serve to unify the special functions—the universal form of their asymptotic expansions—depend upon certain features of the world. What Truesdell, Talman and many others miss is how the world informs and determines the relevant mathematical properties that unify the diverse special functions.
As I noted, in many investigations of physical phenomena we find dominant physical features—those features that constrain or shape the phenomena. These are things like shocks and the highly intense light appearing in the neighborhood of ray theoretic caustics. They are features that are most effectively modeled by taking limits.
Limiting idealizations are most effective for examining what goes on at places where the “laws” break down—that is, at places of singularities in the governing equations of the phenomena. These “physical” singularities and their “effects”—how they dominate the observed phenomena—are themselves best investigated through asymptotic representations of the solutions to the relevant governing equations. The example of the Airy integral is a case in point. By using Stokes’ asymptotic representation we get superb representations of the nature of the diffraction relatively far (large z) from the dominating “physical” singularity—the caustic.
There's a curious tendency when it comes to discussing the applicability of mathematics to polarise one's response to the question of which of mathematics and science owes the other the most. It seems each side is only too happy to exaggerate the role of their favoured discipline, finding the benefits mathematics bestows as miraculous, or deflating mathematics to a bunch of tautologies which physics is gracious enough to give an interpretation to. In the philosophical literature of the past century, with one or two notable exceptions, the latter attitude has been prevalent. Of the two, it is the role of mathematical understanding which tends to get passed over or taken for granted.
Batterman is more sensitive to the narratives various mathematicians and physicists tell about special functions, but I wonder if there might not be a richer mathematical story which would make the still sharp dichotomy he maintains between physical and mathematical considerations less sharp. Is it not possible that the richer mathematical understanding of special functions gained since Talman (1968) could intersect with the physical considerations treated by Batterman? For instance, might the asympotic results he discusses have something to do with current ideas from the group representation understanding of special functions? And how do q-deformed special functions fit in? It's surely common to find mathematics and physics narratives creatively intertwined.
I hope that the discussion here leads us to question the anthropocentric role of the mathematician’s appreciation for beauty (or formal analogy) as an important criterion for what arguably should be paradigm examples of mathematics’ applicability to the world; namely, the special functions.
I'd like to hear some contemporary mathematicians' aesthetic narratives about special functions.