This is the topic that I want to understand better, too. It seems to me that some books on philosophy of mathematics are too concerned with good English and style, and are too vague.
When I was choosing a book on the subject, "The Oxford Handbook of Philosophy of Mathematics and Logic" edited by Shapiro seemed OK, but I only started reading it. I also encountered a quite recent paper "Foundations of Mathematics from the Perspective of Computer Mathematics" by a prominent logician Henk Barendregt, which can be found on his home page. However, it is probably not intended to be a first introduction.