Mark Povich (Washington University, St. Louis)
An account of distinctively mathematical explanation (DME) should satisfy three desiderata: it should account for the modal import of DMEs; it should distinguish uses of mathematics in explanation that are distinctively mathematical from those that are not (Baron 2016); and it should also account for the directionality of DMEs (Craver and Povich 2017). Baron’s (forthcoming) deductive-mathematical account, because it is modeled on the deductive-nomological account, is unlikely to satisfy these desiderata. I provide a counterfactual account of distinctively mathematical explanation, the Narrow Counterfactual Account (NCA), that can satisfy all three desiderata.
NCA satisfies the three desiderata by following Lange (2013; but not Lange 2017, apparently) in taking the explananda of DMEs to be of a special, narrow sort. Baron (2016) argues that a counterfactual account cannot satisfy the second desideratum, because such an account, according to Baron, holds that an explanation is a DME when it shows a natural fact to depend counterfactually on a mathematical fact. However, this does not distinguish DMEs from non-DMEs that employ mathematical premises. NCA satisfies the second desideratum by narrowing the explanandum so that it depends counterfactually *only* on mathematical fact. Such an explanandum is subject to a DME. This narrowing maneuver also allows NCA to satisfy the first desideratum. Since the narrowed explanandum depends counterfactually only on a mathematical fact, changes in any empirical fact have no "effect" on the explanandum.
Narrowing the explanandum satisfies the third desideratum, because Craver and Povich's (2017) "reversals" are not DMEs according to NCA. To see this, consider the case of Terry's Trefoil Knot (Lange 2013). The explanandum is the fact that Terry failed to untie his shoelace. The explanantia are the empirical fact that Terry's shoelace contains a trefoil knot and the mathematical fact that the trefoil knot is distinct from the unknot. Craver and Povich (2017) point out that it is also the case that the fact that Terry’s shoelace does not contain a trefoil knot follows from the empirical fact that Terry untied his shoelace and the mathematical fact that the trefoil knot is distinct from the unknot. (One can stipulate an artificial context where the empirical fact partly constitutes the explanandum.) However, if we narrow the explananda, NCA counts Terry’s Trefoil Knot as a DME and not Craver and Povich’s reversal of it. This is because the first of the following counterfactuals is arguably true, but the second is arguably false: 1) Were the trefoil knot isotopic to the unknot, Terry would have untied his shoelace that contains a trefoil knot. 2) Were the trefoil knot isotopic to the unknot, Terry would have had a trefoil knot in the shoelace that he untied. (I use Baron, Colyvan, and Ripley’s [2017] framework for evaluating counterfactuals with mathematically impossible antecedents, so that these two counterfactuals get the right truth-values.) The same is shown for all of Lange’s paradigm examples of DME and Craver and Povich's "reversals".
