Lyon & Colyvan on Phase Spaces

In their recent article “The Explanatory Power of Phase Spaces” Aidan Lyon and Mark Colyvan develop one of Malament’s early criticisms of Field’s program to provide nominalistic versions of our best scientific theories. Malament had pointed out that it was hard to see how Field’s appeal to space-time regions would help to nominalize applications of mathematics involving phase spaces. As only one point in a given phase space could be identified with the actual state of the system, some sort of modal element enters into phase space representations such as Hamiltonian dynamics where we consider non-actual paths. Lyon and Colyvan extend this point further by showing how the phase space representation allows explanations that are otherwise unavailable. They focus on the twin claims that

All galactic systems that can be modeled by the Henon-Heiles system with low energies tend to exhibit regular and predictable motion;
All galactic systems that can be modeled by the Henon-Heiles system with high energies tend to exhibit chaotic and unpredictable motion.

The mathematical explanation of these claims involves an analysis of the structure of the phase spaces of a Henon-Heiles system via Poincare maps. As the energy of such a system is increased, the structure changes and the system can be seen to become more chaotic.

For me, the central philosophical innovation of the paper is the focus on explanatory power, and the claim that even if a nominalistic theory can represent the phenomena in question, the nominalistic theory lacks the explanatory power of the mathematical theory. This is an intriguing claim which seems to me to be largely correct. Still, one would want to know what the source of the explanatory power really is. Lyon and Colyvan focus on the modal aspects of the representation, and claim that this is what would be missing from a nominalistic theory. But it seems that a Field-style theory would have similar problems handling cases of stability analysis where phase spaces are absent. For example, I have used the example of the Konigsberg bridges where the topology of the bridges renders certain sorts of paths impossible. There is of course a modal element in talking of impossible paths, but the non-actual paths are not part of the representation in the way that they appear in phase spaces. What the bridges have in common with this case is that a mathematical concept groups together an otherwise disparate collection of physical phenomena. While all these phenomena may be represented nominalistically, there is something missing from this highly disjunctive representation. I am not sure if what is lost is best characterized as explanatory power, but something is surely worse.

Three different elements come together, then, in Lyon and Colyvan’s case, and it is not clear which contribute to explanatory power: (i) non-actual trajectories in a phase space, (ii) a mathematical concept that groups together a variety of physical systems (“the Henon-Heiles system”) and (iii) stability analysis. Maybe they all make a contribution, but more examples are needed to see this.