The Marriage of Rationalism and Empiricism: A Naturalistic Solution to the Access Problem for Realist Mathematics
Mathematical facts are abstract in a way that makes our knowledge of them seem hard to explain. I argue that we can explain knowledge of mathematics by drawing on knowledge of combinatorial possibility: possibility with regard to the most general principles about how any objects can be related by any relations.
On the one hand, having accurate general principles of reasoning about combinatorial possibility would suffice to give us access to facts about what mathematical objects there are. For, I claim, what it takes for a mathematical object to exist is (just) for certain things to be combinatorially possible. I show how we can uniquely characterize the intended structure of the numbers and the sets purely in terms of nested claims about combinatorial possibility. Given good general principles of reasoning about combinatorial possibility, we can work out that various further claim must be true of the numbers and the sets given that they satisfy these characterizations.
On the other hand, I argue that there is ultimately no access problem with regard to combinatorial possibility. As inquirers we try to predict and explain the behavior of concrete objects. There are more and less economical ways of doing so. When we are dealing with sufficiently diverse and plentiful collections of concrete objects, the most economical explanations will often appeal to a combination of general principles which are expected to constrain the behavior of all objects and relations, and specific physical or metaphysical laws whose application is restricted to certain particular kinds of objects or relations. This push to predict and explain the behavior of concrete objects by appeal to facts about combinatorial possibility can be leveraged to explain the accuracy of our mathematical intuitions with respect to the more powerful claims about combinatorial possibility which are needed to make sense of standard mathematics.