Philosophers who became mathematicians -- how did they do it? And who were they? (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?
Thanks again!    
 A: I definitely believe that Abraham Robinson, although not prominently a philosopher in the sense of working in purely philosophical concepts, contributed to the philosophy of mathematics and invented nonstandard analysis. He was familiar with the works of Leibniz and Plato and wrote a influential article on the foundations of mathematics entitled "Formalism 64" which included a great deal of his mature philosophy concerning mathematics in general. He was something of an algebraist in his younger years before working under Abraham Fraenkel in set theory and publishing his dissertation.
Another one is Edmund Husserl. He was foremost a philosopher, which I am sure you are familiar with his introduction of phenomenology. He began as a research assistant under Weierstrass (developed the theory of analysis and gave the rigorous $\epsilon-\delta$ definition for continuity, integration, etc). He wrote his dissertation on the calculus of variations, and definitely had a strong background in mathematics, taking astronomy and working under Weierstass in his prime as mentioned. He supposedly was friendly with Hilbert and, in one way or another, worked on the Russell paradox in set theory and critiqued Cantor and Frege on the (philosophical) concept of number. I am not sure the exact details of this final part though. 
These are probably the first of a long list, but I see no reason for philosophers to not work in mathematics, as I believe it is fruitful to be well rounded in many areas of thought. I imagine Husserl, Cantor, Frege, and Robsinson would all agree with me on this small point.
Good Luck!
A: (1) Quine, Davidson, Suppes, Dummett, Lewis all instantly come to mind, amid legion others. 
(2) All of the above
(3) This isn’t a helpful or necessarily meaningful question for you to ask.
(4) See #2
A: Check out Alain Badiou. Many people in analytic phil. don't love him because he doesn't really git into the analytic or continental trends, but that's sort of part of his thing.
He's known for saying "mathematics is ontology" but that's very misleading, as he's admitted. You might be interested.
A: Maybe not exactly fitting your question, but Felix Hausdorff published two philosophical books, and several articles, under the pseudonym Paul Mongré, before starting the mathematical work in set theory and topology which he is mostly known for today. The books were Sant'Ilario. Gedanken aus der Landschaft Zarathustras (1897) and Das Chaos in kosmischer Auslese — Ein erkenntniskritischer Versuch (1898).
He was not a "professional" philosopher though even at that time, but rather already had a doctorate in mathematics. The two books are heavily influenced by Nietzsche (not universally accepted as philosopher in the academic sense either, and even less so at that time). In volume VII of Hausdorff's collected works, the editors do argue however that some of the issues considered in the second book in particular (where, if I recall correctly, he e.g. tried to refute Nietzsche's concept of "Eternal Recurrence") might have led him to considerations that were fruitful in his approach to set theory later.
