Øystein Linnebo

“..First, we need to get clear on the subject matter of mathematics. What is mathematics about? Is it really concerned with abstract objects..? We obviously need to listen to what mathematics itself has to say.. As Frege emphasized, however, the questions are also in part concerned with language. How should the language of mathematics be analyzed? Should apparent talk about numbers and sets be taken at face value? This concern with language means that we shall also need assistance from linguistics and perhaps also psychology. Second, we need to understand how mathematicians.. settle on their first principles (or axioms), and how do they use these to prove mathematical results (or theorems)? .. The challenge is to make our answers to these two sets of questions mesh. How is it that our ways of forming mathematical beliefs are responsive to what mathematics is about? How are the practices and mechanisms by which we arrive at our mathematical beliefs conducive to finding out about whatever reality mathematics describes? In short, why is it not just a happy accident that our mathematical beliefs tend to be true? There must be something about what we do that keeps us on the right track., Since the challenge is to integrate the metaphysics of mathematics (namely, what mathematics is about) with its epistemology (namely, how we form our mathematical beliefs), we shall call this the integration challenge.”
― Øystein Linnebo