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?](http://research.microsoft.com/pubs/155608/209-3.pdf) and [A. Blass, Y. Gurevich, Algorithms: a quest for absolute definitions](http://www.math.lsa.umich.edu/~ablass/abs.pdf)).
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](http://www.andrew.cmu.edu/user/avigad/Talks/baltimore.pdf#page=21),
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](https://mathoverflow.net/questions/23829/solutions-to-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](http://www-bcf.usc.edu/~kleinsch/NYU_Schedule.html),
the [Workshop on set theory and the philosophy of mathematics at U Penn 2010](http://philosophy.sas.upenn.edu/Workshop%20on%20Set%20Theory%20and%20the%20Philosophy%20of%20Mathematics/description),
the conference on [Set theory and higher-order logic: foundational issues and mathematical developments](http://www.bbk.ac.uk/philosophy/our-research/ppp/summer-school) in London, the [Workshop on infinity and truth, NUS 2011](http://www2.ims.nus.edu.sg/Programs/011aiic/wk2.php) and the
[EFI series at Harvard 2012](http://logic.harvard.edu/efi.php). 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](http://data.stackexchange.com/mathoverflow/query/5153/who-does-the-most-typing-on-stack-overflow), 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.