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.