As a novice amateur, the similarities between mathematics and analytic philosophy seem striking to me. At least in a caricature view of analytic philosophy, it is the project of establishing the necessary and sufficient conditions for the validity of different claims given a framework of well-defined concepts. And, on the other hand, mathematics is also about exploring the implications of (i.e., exploring as to which claims are true or false given) a set of axioms and definitions. The similarity between the two projects seems difficult to ignore. I realize that one can make a distinction in that the well-defined concepts of analytic philosophy usually correspond to (or come out of) ideas arising out of the natural experience whereas the mathematical objects can be purely abstract. However, given the potentially infinite range of natural phenomena, in principle, one can imagine a variety of exotic concepts which can be subjected to the usual process of conceptual analysis/analytic philosophy. In this vein, it appears that mathematics is nothing but precisely this project--it is the conceptual analysis of all logically conceivable frameworks of concepts, the ultimate culmination of analytic philosophy.
I would like to know if this is a valid argument or a result of my misunderstanding of what analytic philosophy is. I would also appreciate if one can direct me to references relevant to the question of the relation between mathematics and analytic philosophy.
Another way of stating the claimed similarity is that both mathematics and analytic philosophy seem to be in the business of exploring a priori truths given a framework of concepts or definitions and axioms. They do not, in and themselves, make statements whose truth might be subject to empirical evidence. However, when concepts from philosophy (or mathematical objects) are identified with aspects of natural phenomena (as an axiomatic step in science), if the identification is empirically correct, the necessary truths of philosophy (or mathematics) automatically translate into empirical truths that one can verify.