  In this essay, I argue that mathematics is a natural science---just like
physics, chemistry, or biology---and that this can explain the alleged
"unreasonable" effectiveness of mathematics in the physical sciences. The main
challenge for this view is to explain how mathematical theories can become
increasingly abstract and develop their own internal structure, whilst still
maintaining an appropriate empirical tether that can explain their later use in
physics. In order to address this, I offer a theory of mathematical
theory-building based on the idea that human knowledge has the structure of a
scale-free network and that abstract mathematical theories arise from a
repeated process of replacing strong analogies with new hubs in this network.
This allows mathematics to be seen as the study of regularities, within
regularities, within ..., within regularities of the natural world. Since
mathematical theories are derived from the natural world, albeit at a much
higher level of abstraction than most other scientific theories, it should come
as no surprise that they so often show up in physics.
  This version of the essay contains an addendum responding to Slyvia
Wenmackers' essay and comments that were made on the FQXi website.
