The Aboutness Problem for Mathematical Realism

According to mathematical realism, mathematics is the study of a necessary and unchanging reality that exists independently of human activities, beliefs, and capacities. It seems to follow that gaining mathematical knowledge consists merely in acquiring knowledge that accurately describes this mind-independent mathematical reality. Against this, I will argue that, even if there is a mind-independent mathematical reality, gaining mathematical knowledge requires more than merely acquiring knowledge that accurately describes it: it requires knowledge that employs specifically mathematical ways of thinking about mathematical entities – so-called modes of presentation . Hence, even if mathematical realism is true, a full understanding of the nature of mathematics calls for an understanding not only of the ontological status of mathematical objects – as has been the realists’ focus of research so far – but also of the specifically mathematical ways of thinking about these objects. We will see that, for realists, providing such an understanding is by no means trivial.