The central positive claim of the book is that the justification of agent’s beliefs should be characterized in terms of proper function rather than reliability:
Jpf: S’s belief B is justified iff (i) S does not take B to be defeated and (ii) the cognitive faculties producing B are (a) functioning properly, (b) truth-aimed and (c) reliable in the environments for which they were ‘designed’ (133).The main advantage of this proper function approach over a simple focus on reliability is that a person in a demon world can still have justified perceptual beliefs. This is because their faculties were ‘designed’ for a normal world and they can still function properly in the demon world where they are not reliable.
Still, it is not clear how the proper function approach marks an improvement when it comes to the justification of mathematical beliefs. This is not Bergmann’s focus, but I take it that a problem here would call into question the account. To see the problem, consider a naturalized approach to proper function that aims to account for the design of the cognitive faculties of humans in terms of evolutionary processes. I can see how this might work for some simple mathematical beliefs, e.g. addition and subtraction of small numbers. But when it comes to the mathematics that we actually take ourselves to know, it is hard to see how a faculty could have evolved whose proper function would be responsible for these mathematical beliefs. The justification of our beliefs in axioms of advanced mathematics does not seem to work the same way as the justification of the mathematical beliefs that might confer some evolutionary advantage. If that’s right, then the proper function account might posit a new faculty for advanced mathematics. But it’s hard to see how such a faculty could have evolved. Another approach would be to side with Quine and insist that our justified mathematical beliefs are justified in virtue of their contribution to our justified scientific theories. The standard objection to this approach is that it does not accord with how mathematicians actually justify their beliefs. More to the point, it is hard to see how the cognitive faculties necessary for the evaluation of these whole theories could have evolved either.
A theist sympathetic to the proper function approach might take the failure of naturalistic proper functions as further support for their theism. But if that’s the case, then the claim that proper function approaches are consistent with some versions of naturalism, defended in section 5.3.1, needs to be further qualified.