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 underMore generally, when it is unclear how to think of the relevant feature of the explanadum as a variable, Woodward rejects the explanation as causal.interventionson the dimensionality of space-time (p. 220).

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.

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