Good books on Philosophy of Mathematics
I think there are probably some good introduction to "classical" philosophy of math that I'm not aware of, but what I find most interesting are modern treatments of philosophy of math.
Lakoff and Nunez's book called Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being uses modern cognitive science to discuss philosophy of math.
David Corfield's Toward a Philosophy of Real Mathematics is a fascinating look at what mathematicians actually do (as opposed to most philosophy of math done by philosophers that don't know much math).
I have lots of other suggestions, but I'll let other people have their say as well (and moreover this wasn't quite what you were asking for).
Philosophy of Mathematics: Selected Readings, edited by Paul Benacerraf and Hilary Putnam, is one of the standard essay collections, and introduces the classical schools: formalism, intuitionism and logicism.
But there are newer points of view. Personally, I liked The Nature of Mathematical Knowledge by Philip Kitcher because of its sophisticated historical point of view. Kitcher's approach is empiricist.
Another interesting approach is Charles Chihara's structuralism, as presented in his book Constructibility and Mathematical Existence.
This next one is not a book but it is famous enough that I think it should be mentioned here. With respect to axiomatic set theory and philosophy, there is the two-part essay entitled "Believing the Axioms" by Penelope Maddy:
Maddy, Penelope (Jun. 1988). "Believing the Axioms, I". Journal of Symbolic Logic 53 (2): 481–511.
Maddy, Penelope (Sept. 1988). "Believing the Axioms, II". Journal of Symbolic Logic 53 (3): 736–764.
If you are interested in the foundations of set theory in particular, there is the classic book Foundations of Set Theory by Abraham Fraenkel, Y. Bar-Hillel and A. Levy. The standard system of axiomatic set theory ZF is named after Ernst Zermelo and Abraham Fraenkel.
Russell's book is probably not what you're looking for. Firstly, it is mostly his opinions on the subject, and some of his arguments are surprisingly weak (as I see them anyways). Here is the online version of Russels book. Looking at the content list, it does not seem what you are looking for.
Personally I can recommend "The Mathematical Experience", by Davis and Hersh. http://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395929687 It presents some of the main philosophical views on mathematics. (among other things)
When I took a course on the philosophy of mathematics, our textbooks were Shapiro's Thinking about mathematics and Parsons' Mathematical Thought and its Objects. I recommend them both.