▻ FIRST-ORDER SUBJECT PROMPTS PHILOSOPHICAL QUESTIONS
I should say that the study of philosophy as a sole course is losing its appeal and rightly so. Historically the best work in philosophy was done very often by thinkers who were involved in other matters and turned to philosophy in order to elucidate problems in independent fields - Descartes, Leibniz and Russell all turned to philosophy to aid or clarify their work in science or mathematics, for example. They did this not just for interest' sake but because their first-order inquiries raised problems that turned out to be philosophical. One can go back further, and further afield : Aquinas was a theologian who, within theology, hit rock bottom philosophical problems. Croce in the early 20th century turned to aesthetics and the philosophy of history because in his experience of art and his practice as a historian, he encountered problems that could not be solved or even addressed within art and history but only in the philosophy of these subjects.
It seems to me that mathematicians and scientists - Polkinghorne, Penrose, Raymond Tallis, three that come instantly to mind - have genuine and deep philosophical interests that have stemmed from their professional work.
▻ FIRST-ORDER SUBJECTS AND PHILOSOPHY COMBINE
It is not possible to do the philosophy of mind any longer without a reasonable scientific knowledge of the brain and central nervous system. Philosophy detached from or empty of such knowledge is just mental chess. Equally political philosophy, a very different field, needs to be informed by theories of rational choice and the theory of games drawn from economics and strategic studies.
Philosophers can contribute some conceptual finesse and facility in argument to scientific teams working at the intersection of science and philosophy - in regard for instance to the status of 'folk psychology'.
▻ SCIENCE AND PHILOSOPHY - THE BLURRED FRONTIER
There have always been scientists, from Newton to Brian Cox, who do not feel the need for any engagement with philosophy in their work. But consider the case of Einstein, mentioned here before. His work on space and time - or space/ time - was both physics and philosophy. Physics and philosophy were related as opposite sides of a curve as he not only did physics but in the process revolutionised the concepts of space and time. From this viewpoint it's otiose to ask whether he was a physicist or a philosopher. In his work this was a distinction without a difference.