Philosophy of mathematics for the masses : extending the scope of the philosophy of mathematics

How should we understand mathematics? Are we discovering truths about reality when doing mathematics? Or are we merely constructing an elaborate story, which happens to be very useful? These questions have long been the subject of philosophical debate, a debate that has most recently been structured around Benacerraf’s Dilemma. Theories have been put forward on both sides, some arguing that we discover truths about reality, and some arguing that mathematics fails to attach to anything in the external world. I argue that while these theories might be able to answer the question I started with in the case of professional mathematicians, they cannot answer it for ordinary people. So, I argue for an extension of the scope of the philosophy of mathematics. It should not only look at professional mathematicians, but it should also attempt to understand the mathematical practices of ordinary people. In order to establish these claims, I provide an extensive survey of the current theories in the philosophy of mathematics. For each of these, I point to shortcomings that prevent them from describing the practices of ordinary people. I appeal to relevant empirical research, both from psychology and from pedagogy, and provide some new interpretations of this research. Since each of these theories fails to describe the practices of ordinary people, the conclusion that current research needs to start looking at these practices follows.