Classic problem of trying to draw definitive lines between these things. I would guess this is more motivated by the social aspects of academics today than any real issue.

In any event reading the work of philosophers has certainly made me a better thinker and mathematician. Without a broad base of thought one can easily get sucked into many intellectual traps and tar pits. In my case, I found Penelope Maddy and Wittgenstein to be helpful when I was figuring out what math "is".