A colleague of mine, Mark Zelcer, and I just posted a philosophy of
mathematics paper on the arxiv.
http://arxiv.org/abs/1306.4235
While the paper is not about category theory per se, the ideas in it were
inspired by category theory. There is an appendix on categorical algebra.
“Mathematics via Symmetry”
We state the defining characteristic of mathematics as a type of symmetry
where one can change the connotation of a mathematical statement in a
certain way when the statement’s truth value remains the same. This view of
mathematics as satisfying such symmetry places mathematics as comparable
with modern views of physics and science where, over the past century,
symmetry also plays a defining role. We explore the very nature of
mathematics and its relationship with natural science from this perspective.
This point of view helps clarify some standard problems in the philosophy of
mathematics.
We would be very interested in any comments or criticisms.
All the best,
Noson Yanofsky