Sunday, October 12, 2008

McGivern on Multiscale Structure

Back in July I made a brief post on multiscale modeling from the perspective of recent debates on modeling and representation. So I was very happy to come across a recent excellent article by McGivern on “Reductive levels and multi-scale structure”. McGivern gives a very accessible summary of a successful representation of a system involving two time scales, and then goes on to use this to question some of the central steps in Kim’s influential argument against nonreductive physicalism.

To appreciate the central worry, we need the basics of his example. McGivern discusses the case of a damped harmonic oscillator, like a spring suspended in a fluid, where the damping is given as a constant factor of the velocity. So, instead of the simple linear harmonic oscillator
my’’ + ky = 0
we have
my’’ + cy’ + ky = 0
Now this sort of system can be solved exactly, so a multiscale analysis is not required. Still, it is required in other cases, and McGivern shows how it can lead to not only accurate representations of the evolution of the system but also genuine explanatory insight into its features. In this case, we think of the spring evolving according to two time scales, t_s and t_f, where t_f = t and t_s = εt and ε is small. Mathematical operations on the original equation then lead to
y(t) ~ exp(-t_s/2)cos(t_f)
where ~ indicates that this representation of y is an approximation (essentially because we have dropped terms that are higher-order in ε). McGivern then plots the results of this multiscale analysis against the exact analysis and shows how closely they agree.

McGivern’s argument, then, is that the t_s and the t_f components represent distinct multiscale structural properties of the oscillator, but that they are not readily identified with the “micro-based properties” championed by Kim. McGivern goes on to consider the reply that these are not genuine properties of the system, but merely products of mathematical manipulation. This seems to me to be the most serious challenge to his argument, but the important point is that we need to work through the details to see how to interpret the mathematics here. I would expect that different applications of multiscale methods would result in different implications for our metaphysics. I hope that this paper will be studied not only by the philosophy of mind community, but also by people working on modeling. If we can move both debates closer to actual scientific practice, then surely that will be a good thing!

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