A typical abstract axiomatic exposition in a subject like mathematics is extremely terse, functioning partly as a hierarchical filing system for the subject’s materiel, ensuring that nothing goes missing and content can readily be found by members of the professional community, but offering few clues to the original motivating insights and meaning of the subject.
It is the job of the philosophers (I believe), to help interpret the meaning of the contents of the abstract axiomatic filing system of mathematics, and also to help preserve the praxis and understanding of the subject between generations and also between mathematicians and non-mathematicians. The philosophy of mathematics is, then, an ongoing reinterpretive, partly pedagogical, partly popularising narrative in the margins, rather than as usually thought, a foundational monolith.
A successful philosophy of mathematics (or of art, science, ethics …) will offer an intuitive visceral approach to the subject, with suggestive concepts, metaphors, similes and viewpoints, simulating the praxis of a creative professional. A philosophy-of paraphrases, circumlocutes and metaphorises, to infuse its subject with near-praxic life.
• A successful philosophy of mathematics (or of art, science, ethics …) will never directly add new mathematical truths (or works of art, scientific facts, direct moral guidance…) – though it should improve the teaching of these subjects, and may assist talented creative minds to more quickly adopt a viewpoint and mantle from which praxis and discovery of new truths in them is possible.
• Philosophers-of are probably often would-be or retired practitioners-of, but who are now for whatever reason apraxic in their discipline – which is as it should be, since who better motivated to develop tools to allow non and future professionals to glimpse, and in some (rare) cases attain for themselves, the praxic viewpoint and capability of creative practitioners ?
• Practitioners-of will often view philosophers-of with suspicion, and the feeling that they add no new content to their subject (which is true) , and that they are nothing but second or third rate practitioners (which is (one guesses) often true, but is not necessarily relevant to their rating as philosophers-of).
• Contrary to intuition, while mathematics provides eternal truths, the philosophy of mathematics does not. This is because the philosophy of mathematics is concerned with the human binding and vivification of its subject – and this is a contingent activity, dependent on the intuitive ways of human thought, in each generation and in each human audience. Mathematics and science are not culturally relative, but their philosophies-of are.
• The philosophy of a subject is never finished, even should the subject it deals with eventually become completed. This is because the task of preserving and transmitting the meaning of a subject across deep time to future generations and across human space to contemporary non-specialists and students - is never finished.
• The philosophical narrative in the margins itself has marginal narrative and so on, recursively extending the subject and its philosophy as it were breadth-wise, rather than as is usually thought foundationally depth-wise. This accounts for the failure of philosophy ever to find bedrock and terminate.
Without an ongoing philosophy of mathematics (or of art, science, ethics ...) , updated and reinterpreted in each generation or two, we cannot be certain that our current terse axiomatic filing-system-like presentations of (say) abstract group theory will escape the fate of ancient Egyptian hieroglyphics - by which the language’s tokens were preserved, but meaning and worldview became completely lost. Luckily the ancient hieroglyphics had been semantically replicated into two other scripts on the Rosetta stone so their meaning could be recovered. The job of the philosophy of a subject, even of one as purely logical as mathematics, is as a marginal but vital narrative using non-specialist language, to help communicate and preserve the real meaning and insights of the subject, and not just the contents of its axiomatic filing system, across deep time and human space.