A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is nothing mathematics can (...) do to make an explanation mathematical ; realists think something can be done, but they are not clear about what that something is. I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's [1990] notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism. I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist. (shrink)

In his Introduction to Contemporary Metaethics, Alex Miller considered a defence that might be made on behalf of the moral non-reductive naturalist in response to Gilbert Harman’s explanatory objection, a response that uses Frank Jackson’s and Philip Pettit’s account of ‘program explanation.’ However, Miller went on to argue that program explanation fails to successfully defend the moral non-reductive naturalist against Harman’s objection. Recently Paul Bloomfield and Mark Nelson have argued against Miller that program explanation does in fact successfully defend the (...) moral non-reductive naturalist, because the only full explanation of why the relevant counterfactual, discussed in this thesis, is true requires the use of program explanation. Following Miller, I argue that the fact that counterfactuals are context sensitive undermines the argument developed by Nelson, and I also attempt to undermine Bloomfield’s recent defence of Nelson. Contrary to Bloomfield and Nelson, program explanation is not required in order to explain why the relevant counterfactual is true, so that Harman’s explanatory objection is left intact. (shrink)