Mathematics, Proper Function and Concepts

Continuing the previous post, I think what bothers me about the proper function approach is the picture of cognitive faculties as innate. This forces the naturalist to account for their origins in evolutionary terms, and this seems hard to do for the faculties that would be responsible for justified mathematical beliefs. A different approach, which should be compatible with the spirit of the proper function approach, would be to posit only a few minimally useful innate faculties along with the ability to acquire concepts. If we think of concepts as Peacocke and others do, then acquiring a mathematical concept requires tacitly recognizing certain principles involving that concept (an "implicit conception"). Then, if a thinker generated a mathematical belief involving that concept and followed the relevant principles, we could insist that the belief was justified.

This would lead to something like:

Jpfc: S’s belief B is justified iff (i) S does not take B to be defeated, (ii) the cognitive faculties producing B are (a) functioning properly, (b) truth-aimed and (c) reliable in the environments for which they were ‘designed’, and (iii) the production of B accords with the implicit conceptions of the concepts making B up.

In certain cases, perhaps involving simple logical inferences, the innate cognitive faculties would themselves encode the relevant implicit conceptions, and so clause (iii) would be redundant. But in more advanced situations, where we think about electrons or groups, clause (iii) would come into play. As far as I can tell, this proposal would allow for justified beliefs in demon worlds. For example, if an agent was in a world where groups had been destroyed (if we can make sense of that metaphysical possibility), her group beliefs could still be justified. In fact, the main objection that I foresee to this proposal is that it makes justification too easy, but presumably that is also an objection that the proper function proposal faces for analogous cases.