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.)

Tuesday, December 16, 2008

The Limits of Causal Explanation

Woodward's interventionist conception of causal explanation is perhaps the most expansive and well-worked out view on the market. He conceives of a causal explanation as providing information about how the explanandum would vary under appropriate possible manipulations. Among other things, this allows an explanatory role to phenomenological laws or other generalizations that support the right kind of counterfactuals, even if they do not invoke any kind of fundamental or continuous causal process.

Given the recent debates on mathematical explanation of physical phenomena, it's worth wondering if Woodward's account extends to these cases as well. In a short section in the middle of the book, he concedes that not all explanations are causal in his sense:
it has been argued that the stability of planetary orbits depends (mathematically) on the dimensionality of the space-time in which they are situated: such orbits are stable in four-dimensional space-time but would be unstable in a five-dimensional space-time ... it seems implausible to interpret such derivations as telling us what will happen under interventions on the dimensionality of space-time (p. 220).
More generally, when it is unclear how to think of the relevant feature of the explanadum as a variable, Woodward rejects the explanation as causal.

Still, some mathematical explanations will qualify as causal. This seems to be the case for Lyon and Colyvan's phase space example, but perhaps not for the Konigsberg bridges case I have sometimes appealed to. To see the problem for the bridge case, recall that the crucial theorem is
A connected graph G is Eulerian iff every vertex of G has even valence.
As the bridges form a graph like the figure, they are non-Eulerian, i.e. no circuit crosses each edge exactly once.

I would argue, though, that as with the space-time example, there is no sense in which a possible intervention would alter the bridges so that they were Eulerian. We could of course destroy some bridges, but this would be a change from one bridge system to another bridge system. It seems that to support this position, there must be clear set of essential properties of the bridge system that are not rightly conceived as variable.

Monday, December 15, 2008

Globe and Mail Drops the Ball

If you needed any more evidence that the general public, including Toronto's Globe and Mail newspaper, doesn't know what "philosophy" means, check out this profile of a self-described "cyberphilosopher" and his campaign to expose the dark side of the parent-child relationship:
One might have hoped for at least one sentence in the article explaining that this has nothing to do with philosophy.