I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any mathematical insights from them.
My naïve expectation would be that philosophy might take a difficult construction or proof, and clarify it by isolating the key ideas behind it. Having isolated the key ideas, philosophy might then highlight their relevance and thus point the way forward. Beyond this, I would hope that philosophy might elucidate the `true meaning' of axioms and of definitions by examining their ontology in a wider context.
In reality, to the best of my knowledge (please prove me wrong!) both of the above tasks seem to be carried out exclusively by mathematicians, physicists, computer scientists, and other natural scientists, as far as I can see. To play the devil's advocate, philosophy seems to me like it might historically have largely played an opposite role, labeling certain objects as "unreal" and "unnatural" which in fact later turned out to be fruitful to study (negative numbers, irrational numbers, complex numbers…).
Question: Has it ever happened that philosophy has elucidated and clarified a mathematical concept, proof, or construction in a way useful to research mathematicians?
Philosophers have created much new mathematics (e.g. the work of C.S. Peirce, much of which is bona fide mathematical research), but the question is not about this, but rather about philosophy as practiced by philosophers providing elucidation, explanation, and clarification of existing mathematics.
Best Answer
Two points: one, firstly understanding mathematical processes can be of immense pedagogical value. See e.g. Polya's How to Solve It (and he wrote a more academic book with these themes), or Lakatos' Proofs and Refutations. I found this book New Directions in the Philosophy of Mathematics which had interesting essays as well.
Secondly, remember that broadly the point of philosophy is to make things not philosophy. In extremely simplistic historical terms, once natural philosophy becomes rigorous it becomes science, once philosophy of language became rigorous it became linguistics, and today we're seeing philosophy of mind turn to neuroscience.
So philosophy that elucidates mathematics is simply... mathematics. Most obviously Russell and the development of set theory. Modernly I don't know: I think the interesting stuff is happening at computer science/philosophy and physics/philosophy which trickles into mathematics. I'm posting this largely because I think the question is slightly broken because philosophy doesn't really work to clarify a field where it has already been clarified.