Both of us:
- want to give a naturalistic account of mathematical knowledge
- in particular, want to explain how humans can have managed get "good" combination of inference patterns that count as thinking true things about some domain of mathematical objects/having a coherent conception of what those objects must be like, rather than "bad", 'tonk' like patterns of reasoning.
-appeal to causal interactions with the world, to explain how we wind up with such combinations of inference dispositions.
BUT there are some important differences. Here's why (I claim) my view is better.
Jenkins winds up positing a whole bunch of controversial, and perhaps under-explained philosophical notions to account for how experience gives us good inference dispositions. She proposes that:
Experience has non-conceptual content which grounds our acquisition of concepts so as to help us form coherent ones. Then when we have a coherent concept of something like the numbers, we inspect it to see what what must be true of the numbers and reason correctly about them.
-The idea that there's non-conceptual content is a controversial point in philosophy of perception.
-The idea that experience can "ground" concept acquisition without playing a justificatory role in the conclusions drawn is not at all clear. What is this not-justificatory, but presumably not just causal relationship of grounding supposed to be? (Kant's notion of a posteriori concepts seems relevant, but that's none-too clear either).
-Finally, what is concept inspection, (presumably you don't literally visit the 3rd realm and see the concepts) and how is it supposed to work? Jenkins admits that this is an open question for further research.
In contrast, my view gives a naturalistic account of mathematical knowledge that doesn't need any of this controversial philosophical machinery. I propose that:
People are disposed to go from seeing things, to saying things, to being surprised if we then see other things, in certain ways. When these inference dispositions lead us to be surprised, we tend to modify them.
Thus, it's not surprising that we should have wound up with the kind of combination of arithmetical inference dispositions + observational practices + ways of applying arithmetic to the actual world, which makes our expected applications of arithmetic work out.
For example: insofar as we had a conceptions of the numbers which included the expectation that facts about sums should mirror logical facts in a certain way, it's not surprising that we would up also believing the kinds of other claims about sums, which make the intended applications to logic work out (e.g. believing 2+2=4 not 2+2=5).
Note that we don't need to posit any mysterious faculty of concept-inspection, or any controversial non-conceptual experience. All I appeal to is perfectly ordinary processes. People go from one sentence to another in a way that feels natural them (whether or not they are so fortunate as to be working with coherent concepts like +, rather than doing reasoning like Frege did about extensions) And when this natural-feeling reasoning leads to a surprise, they revise.
[Well, perhaps I'm also committed to the view that innate stuff about the brain makes some ways of revising more likely than others, and certain initial inference-dispositions more likely than others, in a way that doesn't make us always prefer theories that are totally hopeless at matching future experience. But you already need something like this even to explain how rats can learn that pushing a lever releases food, so I don't think this is very controversial.]