(I hope this is not too personal. If you want to get to the point scroll down to the end, where my questions are.)
I'm a philosopher who's been -- gradually -- coming around to mathematics. I have wide interests in philosophy, although I tend towards analytic-oriented work (philosophy of language, maths, Logic, etc.) Ever since I got hooked on everything logic, I've been trying to teach myself things from allied areas: universal algebra, analysis, category theory, etc. It's a bit of a transition, but all of the mathematics I've seen is very interesting and beautiful. This has been a slow, sometimes discouraging, but often rewarding, process.
There are a few philosophers who work in pure mathematics, or at least prove major theoretical results in areas closely allied to pure or even applied mathematics / sciences, who don't necessarily have Math PhD's (or proper math training). And in fact traditionally this has happened as a somewhat common occurrence. (More noticeable is how many mathematicians ended up working in philosophy departments, but that is another story.)
Examples include: Willard Van Orman Quine, William Craig, Hillary Putnam, Noam Chomsky (context-free grammars, formal languages hierarchy, etc.), Wilfrid Hodges, Hartley Rogers, etc. I believe Paul Halmos might also belong on this list, as he had a math and philosophy background, and in fact started his PhD in Philosophy originally. I assume then that he may have felt his philosophy background was stronger, originally, than his training in mathematics.
Note that the above list mostly includes logicians.
(1) Are there known examples beyond the short list given above of philosophers who ventured into mathematics / formal areas?
(2) Are there known examples of philosophers who ventured into areas outside of logic?
(3) Any ideas of how hard they had to work to get good in these areas? (I'm looking for inspiration, I suppose. And the answer to this question is not obviously the same as asking how people with earlier training in mathematics got good at their chosen field.)
(4) Are there examples of humanists / liberal arts majors (beyond Hartley Rogers -- I believe his background was in English Literature before he went into mathematics) who made similar transitions?