Philosophy of mathematics seems to focus primarily on set theory, which is probably a historical accident (just like it is a historical accident that set theory became the prevalent "foundation" in the 20th century). If you want to see things from other perspectives you could read things like:

* S. Awodey: [Structure in mathematics and logic: a categorical perspective.][1] Philosophia Mathematica (3), vol. 4 (1996), pp. 209--237. (See also the [subsequent discussion][2].)

* F. W. Lawvere's "[Down with "Foundations"! Up with algebra!][3]" and "[Why are we concerned? II][4]" posting on the categories mailing list, which include further references.

Perhaps other, more knowledgable readers, can suggest additional references in this direction.


  [1]: http://philmat.oxfordjournals.org/content/4/3/209.full.pdf
  [2]: http://www.andrew.cmu.edu/user/awodey/preprints/awodeyVhellman.pdf
  [3]: http://permalink.gmane.org/gmane.science.mathematics.categories/3167
  [4]: http://permalink.gmane.org/gmane.science.mathematics.categories/3175