There is a general pattern of inquiry in mathematics and the sciences by which an investigation begins in philosophy, using philosophical ideas that may be initially quite vague, but which become increasingly clear upon further philosophical analysis, in such a way that the ideas eventually mature and the investigation finds a home in its natural discipline, unmoored from the philosophical origin. The history of science is replete with instances of this philosophy-into-science phenomenon.

And so is mathematics. Consider, for example, the work of Alan Turing, much of which is essentially philosophical in nature. Before Turing, Gödel had despaired that we could achieve an acceptable answer to the question, What does it mean to say that a function is computable? Philosophers have speculated that Gödel had had in mind a diagonal argument, whereby if we had an effective means of enumerating the computable functions, then we could diagonalize against them (and this diagonalization succeeds with Gödel's primitive recursive functions, showing that they do not capture the notion of computability). So Gödel had expected a hierarchy of computability. Meanwhile, Turing undertook a philosophical inquiry into what it means for a human to undertake a rote computational procedure, arriving in this way at his Turing machine concept, an idea so robust that it gave birth to the entire fields of computability theory and complexity theory, if not also helping us into the modern computer age. (Meanwhile, Gödel's hierarchy expectation is surely realized in complexity theory and many other parts of the subject.) So this is a clear case where philosophical ideas, which vexed even our greatest thinkers, matured into purely mathematical developments, and extremely important ones at that.

Other prominent examples would be (1) the resolution of the truth/proof distinction, from Frege and Russell through to Hilbert and then Gödel's refutation of Hilbert, and (2) Cantor's ideas on cardinality and the transfinite. These were cases where purely philosophical ideas eventually transformed into our current purely mathematical investigations.

But the phenomenon is not at all restricted to such high-profile historical cases like this; rather, it is a pervasive and on-going phenomenon, by which philosophical developments, even small ones, often proceed into mathematics, and one can sometimes witness the process in philosophy department seminars. A contemporary analogue of Turing's investigation, for example, would be the current work on the question, What is an algorithm? (for example, see Y. Gurevich, What is an algorithm? and A. Blass, Y. Gurevich, Algorithms: a quest for absolute definitions). In his plenary talk at the recent JMM in Baltimore, Jeremy Avigad challenged mathematical logicians to develop better philosophical ideas concerning some fundamental concepts, such as what it means to verify mathematics at an appropriate level of abstraction, and to develop formal methods for everyday mathematical language and formal methods of everyday proof, among others. This kind of analysis begins as philosophy and ends up as mathematics.

With respect to your suggestion that connections between philosophical logic and mathematical logic have weakened, I disagree. In the case of set theory, these connections appear if anything to be strengthening, and set theoretic research is increasingly preoccupied with philosophical concerns. The fact of the matter is that set theory is currently grappling with several extremely difficult and troubling philosophical issues, concerning for example the criteria by which we adopt new axioms in mathematics and set theory (such as large cardinal and determinacy axioms) and the nature of mathematical truth (such as the raging debate on pluralism, and the question of definiteness of truth) in a context of a pervasive independence phenomenon. We still don't have agreement on the status of the continuum hypothesis, and the obstacles are philosophical rather than mathematical.

The situation is complicated by the fact that many of the most interesting philosophical issues in set theory concern highly technical parts of the subject, especially forcing and large cardinals. For progress, therefore, we need philosophically minded set theorists who can operate in both realms. Several set theorists are now undertaking explicitly philosophical work, including Woodin, who has just taken up a joint appointment in philosophy and mathematics at Harvard. (And my own work has become in part explicitly philosophical.) There is an increasing interaction between set theorists and the philosophers of set theory. In recent years, for example, we've had conferences devoted specifically to this interaction, with participation both from mathematicians and philosophers, such as the NYU Conference on philosophy of mathematics, 2009, the Workshop on set theory and the philosophy of mathematics at U Penn 2010, the conference on Set theory and higher-order logic: foundational issues and mathematical developments in London, the Workshop on infinity and truth, NUS 2011 and the EFI series at Harvard 2012. Several of those meetings have published proceedings volumes.

Postscript. Lastly, let me mention that this appears to be my one-thousandth answer on MathOverflow. (I have apparently typed over three million characters, for which I should apologize for my lack of greater brevity.) I have learned enormously from all the great mathematical posts here, and I am grateful to be a part of this remarkable community. Thank you, MathOverflow; it's been great.