I am interested in a special case of the general question about whether the philosophy of X has an effect on the research or practice of X.
My special interest is in the area of mathematics. I am a research mathematician working in applicable, if not applied, mathematics and I also have some academic background in philosophy, though no research experience.
I guess there are some connections or interdisciplinary areas between philosophy and mathematics - presumably in logic and maybe elsewhere. I am not interested in these as such. Rather, I am looking for work in philosophy of mathematics (as defined in e.g. both answers at What is the Philosophy of Mathematics?) that is used by practicing mathematicians when doing mathematics, or used to argue the importance of this or that mathematical theory, or that is formalized in the language of mathematics.
A concrete sign of this would be a work of philosophy cited by mathematics papers or textbooks. Mathematics papers and textbooks can be defined, for the purposes of this question, as those where proving theorems is a significant preoccupation.
I am interested in recent developments; say, philosophy within during this millenium. If there is a thriving research community in this direction, than some overview of that or a major research question or two would be nice to know. If not, examples, or a persuasive argument that this does not happen, would be good answers.