Batterman starts from the claim that there are cases of mathematical entities doing explanatory work. The question he is asking himself is "How does this work?" He argues that accounts of this sort of phenomenon by Pincock and Bueno and Colyvan (so-called "mapping accounts") cannot work. He offers a sketch of his own approach.
The idea behind the mapping account can be seen in an example like the seven bridges of Königsberg. It is impossible to walk across each bridge only once and get back to where you started because the bridges instantiate a particular mathematical structure which has the property of being "non-Eulerian". It is in virtue of this property that it is impossible to cross each bridge once and get back to where you started. Generalising the idea, a mathematical explanation works because the mathematical structure maps onto the structure (or a crucial part of the structure) of the physical situation.
Batterman makes two criticisms of this account of mathematical explanation which, to my mind, pull in different directions. The first criticism is that these mapping accounts don't have global measures of "representativeness". This amounts to saying that there is no general way in these accounts to determine which of two models of some phenomenon is "closer to the truth". This means that one standard ("Galilean") approach for explaining the idealisations in models is not available. The Galilean approach has it that one can justify the use of an idealisation by showing how replacing this idealisation with a "more representative" component would make the model more accurate. Without a rank-ordering of models by representativeness, this sort of thing isn't an option for mapping accounts.
Batterman's second criticism is that sometimes the idealisation plays an important (essential, ineliminable) role in the explanation. He has several examples. One is the criticality behaviour in fluids by reference to the thermodynamic limit, the mathematical construction whereby the number of particles in the substance is taken to approach infinity. Another is the explanation of rainbows in terms of geometrical optics. In each of these cases, there is nothing in the physical system to which the crucial bit of mathematical apparatus can be straighforwardly mapped, hence at least a prima facie difficulty for a mapping account.
These two criticisms don't seem to work together. Suppose, as Batterman seems to think, there are two kinds of mathematical explanations: ones that involve (only) Galilean idealisations, and ones where the limit is essential. Then it seems to be a good thing that the mapping account people don't commit themselves to only dealing with Galilean idealisations and the relatively rigid system of mapping from model to world that goes along with them. Pincock and Bueno and Colyvan presumably don't think that the distinction between Galilean and non-Galilean explanations is so important, and would rather have a looser system of mapping that lacks a general measure of representativeness.
The behaviour of a system as one approaches some mathematical limit, and the behaviour at the limit are qualitatively different in some important cases. This seems to be an important point for Batterman. But why can't the mapping-proponent argue that the (mathematical) behaviour at the limit can map onto some physical phenomena, and thus explain Batterman's problem cases?
In the Galilean cases, the "ways to drop the idealisations" offers insight into how the mapping from mathematical structure to world is supposed to go. In cases where idealisation plays some essential role, this insight isn't available. But this isn't irreparably damaging for the proponents of mapping accounts.
The sketch Batterman offers of his own account of mathematical explanation is too short to judge fairly. It rests on a distinction between mathematical structures (which are "static") and mathematical operations (which are "dynamic"). I haven't fully understood what is being hinted at with this distinction, so I don't feel qualified to assess whether his sketch of an account shows promise. [Pincock, in a reply to Batterman, says that he doesn't think there is much distance between his own and Batterman's account...]
In the Galilean cases, the "ways to drop the idealisations" offers insight into how the mapping from mathematical structure to world is supposed to go. In cases where idealisation plays some essential role, this insight isn't available. But this isn't irreparably damaging for the proponents of mapping accounts.
The sketch Batterman offers of his own account of mathematical explanation is too short to judge fairly. It rests on a distinction between mathematical structures (which are "static") and mathematical operations (which are "dynamic"). I haven't fully understood what is being hinted at with this distinction, so I don't feel qualified to assess whether his sketch of an account shows promise. [Pincock, in a reply to Batterman, says that he doesn't think there is much distance between his own and Batterman's account...]
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