requesting a book recommendation

Would anyone here be willing to recommend me a decent book on different philosophies of mathematics (eg formalism, intuitionism, etc)?  I've been searching amazon and I'm having a difficult time distinguishing what's worth the purchase.  I'm looking for something relatively rigorous in its details.

I have some background in predicate calculus (mathematical logic - I've read through the proofs of Gödel's theorems and briefly touched on second-order logic), set theory (I'm familiar with ZFC as well as basic ordinal and cardinal arithmetic), and a tiny bit of category theory (products, equalizers, duality, etc).  I wouldn't mind reading something that reviews those things, as long as the main subject is the philosophy and it's aimed at someone with an undergraduate background in math.

Thanks to anyone who bothered to read this.

It sounds like you are looking for physics (tangible proofs) and metaphysics (intangible proofs) -intuition- mathematics.

Though, I wonder if predicate calculus, logic, Gödel's theorems, second-order logic, and set theory are enough. You may want to look into long-division. A math book that reviews a number of contemporary maths that I do have is called, Topics in contemporary mathematics, though I'm not sure that is practical for what you are talking about. You also may want to check out Chaos mathematics (the study of math) to get an understanding that our math system is fundamentally imperfect which is the reason for theorems and other equations to equalize or get close to as real an answer as possible. Like the use of Pi.

Other than that, I would look into a list of mathematics philosophers and see where they left off and research what maths we've come up with since then with regards to the subjects and topics they were proving. I believe there is a list on Wikipedia to get started with.

Good luck!