Thursday, 14 December 2006

Devil in the Details - Robert Batterman's odd notion

Bob Batterman gave a talk a week ago, and I've been meaning to say something about it. Here's the abstract:
This paper discusses the nature and role of idealizations in mathematical models and simulations. In particular, it argues that sometimes idealizations are explanatorily essential--that without them, a full understanding of the phenomenon of interest cannot be achieved. Several examples are considered in some detail.
Bob says that the traditional philosophy behind models is that an idealization is justified when the behavior of a "complete" model (say, molecular dynamical model) converges on the idealized model. This is a natural idea; essentially, the idealization captures some pattern that does exist in the complete model but is perhaps too complicated or too subtle to notice.

As usual, things turn out to be more complicated than we philosophers would like. Some idealizations do not reflect convergence behavior in the complete model, but Bob argues that they are nevertheless genuinely explanatory. An example (one of Bob's) may help.

If we compare two models of a pole breaking under strain (one from molecular dynamics, the other with a continuum idealization), we find that breakage in the continuum model comes from a singularity. On the other hand, in the molecular models, breaks arise from the contingent details of the system's evolution. There is no convergence on singularity. In either model, the pole breaks under the increasing strain, but the continuum model wipes out precisely the detailed initial conditions on which the molecular model depends. Batterman says that the imperfections in the molecular system are explanatory of single events, but not of the class of breaking behavior. Singularities in continuum models fill that explanatory role. How can two fundamentally different kinds of explanations still count as genuine explanations?

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