Turing has already been mentioned in previous excellent answers as someone whose ideas sit at the boundary of philosophy and mathematics, conventionally understood. I want to mention Ludwig Wittgenstein in this context as an example of someone grappling with a host of similar ideas but who arguably took a more "philosophical" approach to them, one that explicitly resists formalization.
The book "Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939" is a transcript of a seminar given by Wittgenstein which was attended by Alan Turing, among a handful of other notable young Cambridge scholars of the day. It contains an interesting exchange between Turing and Wittgenstein about what happens when people disagree as to the result of a calculation. Turing insists that some are right and some are wrong, and Wittgenstein wonders what that might mean. Turing says that if you build a bridge that depends on a wrong calculation, it will fall down. My recollection is that Wittgenstein argues that this doesn't cause the bridge to fall down, but that the bridge falling down might serve as a definition of what it means for the calculation to be wrong.
I gather that many people find Wittgenstein to be obtuse and/or just plain confused, but a more charitable reading suggests that he was wrestling with ideas related to undecidability, albeit from a much broader sociological vantage point. See, for example, "A note on Wittgenstein's notorious paragraph on the Gödel theorem". If a bridge standing or falling down depended on a proposition that was undecidable, what then? I believe, but am not sure, that the issue of "in which formal system" would not necessarily have been in the air at the time.
So, on the one hand we have a nice example of philosophical questions stimulating what went on to become a much more formal (and elaborate) theory in the work of Turing. On the other hand, I feel that the hypothetical scenario about the bridge is a "purely" philosophical question that is both interesting and challenging and that is not addressed by the subsequent formal developments. Naively put, which formal system does mother nature obey? Moreover, how can we make better sense of this question? Philosophy gropes at such questions; once they have been sufficiently sharpened, mathematics constitutes the work of refining and extending our understanding.
