Friday, December 19, 2008

Meyer on Field-style Reformulations of Statistical Mechanics

Glen Meyer offers an in-depth discussion of Field's program to nominalize science with special emphasis on the challenges encountered with classical equilibrium statistical mechanics (CESM). He makes a number of excellent points along the way, but what I like most is his focus on the prevalence of an appeal to what some call "surplus" mathematical structure, i.e. mathematics that has no natural physical interpretation. As he argues, Field could reconstruct configuration spaces for point particles using physical points in space-time, but would face difficulties extending this approach to phase spaces and probability distributions on phase spaces.

One novelty of the paper is a distinction between interpretation and representation. Mathematical theories have some mathematical terms with a semantic reference and a representational role, but other mathematical terms may have a semantic reference with no representational role. When idealizations involve this latter kind of term, Field-style reformulations are in trouble. For example, Meyer discusses the need to treat certain discrete quantities as continuous in the derivation of the Maxwell-Boltzmann distribution law:
The intended ('intrinsic') interpretations of axioms describing a certain structure forces that structure to represent, as it were, in its entirety, i.e., that this structure be exemplified in the subject matter of the theory. Any introduction of the idealization above at the nominalistic level will therefore force us to adopt assumptions about the physical world that the platonistic theory, despite its use of this idealization, does not make. Unlike the case of point particles, this idealization is not part of the nominalistic content of the platonistic theory and therefore does not belong in any nominalistic reformulation. Without it, however, we have no way of recovering this part of CESM (p. 37).
Here we have a derivation that ordinary theories can ground, but that Field-style nominalistic theories cannot. I agree with Meyer here, but it raises the further issue: why is it so useful to make these sorts of non-physical idealizations? It may just be a pragmatic issue of convenience, or perhaps there is something deeper to say about how the mathematics contributes without representing? (See Batterman's recent paper for one answer.)

No comments: