A few things here. First, would someone like Kurt Gödel be considered a philosopher of math, a logician, or a mathematician? Maybe all three (or something else not listed)? And what are the differences between the three? Thanks.

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A philosopher of mathematics is interested in questions like: what are numbers? what kind of necessity to arithmetical truths have? how do we know the basic laws of arithmetic are true? what about sets -- do they really exist over an abover their members? is there a universe of sets? there are various set theories, how can we decide which is true of the universe of sets? And so on. You don't have to be a working mathematian to think about these matters (though obviously you have know a little about the relevant mathematics you want to philosophize about). Nor do you have to be a logic-expert.

Most mathematicians aren't interested in the philosophy of mathematics (just as most scientists aren't interested in the philosophy of science, and most lawyers aren't interested in the philosophy of law). They just go about doing their maths. And among mathematicians, a serious interest in logic is a rarity: you can certainly be a mathematician without being a logician (e.g. by being a fluid-dynamicist or an algebraic topologist). Of course, mathematicians use logical reasoning. But you can be a mathematician without being concerned with the articulation of a formal theory about good reasoning.

And I suppose you can at some level be a logician of sorts without being a serious mathematian. People write whole books on `informal logic', without being mathematicians (or being interested in the philosophy of mathematics).

Kurt Gödel though was indeed not just a great mathematician, who was concerned with formal theories of logic (so was a mathematical logician in particular), but he was also profoundly interested in issues the philosophy of logic and mathematics.

Indeed, Gödel thought that his philosophical ideas were in part responsible for some of his major logical discoveries. For example, there was a tendency in the 1920s to run together the idea of a proposition's being a mathematical truth with the idea of its being provable. Gödel thought this was a philosophical confusion. He was a platonist, in the sense of holding mathematical truth to be one thing, and our being able to get at the truth via a proof is something else. And clearly distinguishing truth from proof enabled him to discover his First Incompleteness Theorem which shows that, for any 'nice enough' theory containing a bit of arithmetic, there will be truths of arithmetic it can't prove (for a bit more explanation of that, the reasonably accessible first chapter of my book about Gödel's incompleteness theorems is freely available here!)

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