What divides these rigorous branches of philosophy form mathematics then? If it is because it is applied in some sense then they could fall under applied mathematics.
There are similarities, but in analytic philosophy you find a much greater focus on foundational issues. So Frege tries (and fails) to derive all mathematics from logic. Later Bertrand Russell tries to derive mathematics from set theory. (Leading to the most unintentionally funny line in all philosophy. Somewhere around page 300, he gets to the line: "From this proposition it will follow...that 1+1=2")
If I could restructure our naming system is there a good reason for me to label those who focus on foundational issues in mathematics as philosophers rather then mathematicians? Right now I would be inclined to label them the latter.
The reason is largely historical [1]. Before about 100 or so years ago, almost every serious scientist or mathematician was effectively a philosopher as well (hence the Ph in PhD). Aristotle was (and still is, albeit augmented with more modern interpretations) foundational to almost every scholastic curriculum. "Philosophy" literally means "the study of knowledge", and for much of human history, it has been where big ideas came from.
Philosophy gave us many of the things we consider science: the scientific method is the direct result of many of the enlightenment philosophers' quest to determine "what is real" versus "what is not real". Once you have an agreed upon, repeatable definition of a process to get to "real", you can move on from there.
Once the total knowledge of our species began to expand toward the middle of the 19th century, it was no longer sufficient for a scholar to be a "philosopher". You had to specialize in something. So philosophy became dominated by the left-over "soft" topics like ethics and personal philosophy, leaving many people working in "hard" disciplines to shun the label of "philosopher".
Interestingly enough, the classification problem is still one of the most hotly debated problems in philosophy. How much of our classification comes from a truly singular concept versus linguistic commonality? I don't have the answer, but we've been asking the question for a long time.
Maybe, I'm not that well versed on the mathematics side (and only moderately well read on the philosophy side) It is also my understanding that most of the work on logic occurs on the philosophy side. See Frege, Kripke, Hintikka, and others. My particular interest (never finished the degree) was Bayesian Epistemology--basically looking at knowledge as uncertain, and trying to model degrees of certainty using Bayesian methods.
Very little; to be honest. But the convergence has happened from both sides over the last 150 years or so. Much of theoretical mathematics could probably land in the philosophy camp, but there's no clear delineation even from university to university. In some schools, symbolic logic is considered philosophy, in others it's mathematics, and in others it's computer science.
