How should we characterize the relationship between mathematics and philosophy of mathematics?
Specifically, in what ways might the study of philosophy of mathematics make a mathematician better at his work, and which contributions from philosophy of mathematics might be considered the most critical or urgent to mathematicians?
My view as a mathematician:
One can do logically correct but horrible mathematics. By that I mean one can invent a totally arbitrary system, with rules of deduction and axioms chosen so as to be bizarre as possible. Take this system (say formalized in first order logic), and start discovering theorems, verifying along the way with a computer proof verification system. This would be mathematics, technically, but the study of such a system would be a horrible waste of time in the practical mathematical sense.
So, given the finite capacities of any mathematician, one must focus, and what one focuses on must be guided. This guidance must come from an extra mathematical source... from anything the mathematician has access to: visualization, intuition, cultural conventions, etc... Mathematicians get work done somehow, using informal meta-mathematical cognitive processes which are somehow reduced to a demonstrable result (in the strict sense, a formal theorem).
One can stop there, and leave these processes unexamined. But going up another level, POM can provide a framework for thinking about how mathematicians act as cognitive beings and how this process may be modified for the desired results.
So I would assert POM is important, at least enough to learn the basics.
My motivation is that I like to look at things from their roots and go up. So, is POM where I should start, or is POM irrelevant to the progress of Mathematics?
I'd argue for the latter position; I think that the Philosophy of Mathematics is pretty much orthogonal to mathematical progress, and is not likely to be of much practical value if your project is to actually do mathematics.
And, in fact, one prominent view in the philosophy of mathematics-- Fictionalism is predicated on precisely that orthogonality: that one is fully able to do whatever one wants to do within mathematics without having to make any ontological commitments whatsoever regarding the status of mathematical objects.
The OP's question has been discussed here. I agree with the claim in the (linked) paper that the philosophy of mathematics can contribute to mathematics to the degree that it affects the practice of mathematics. Mathematicians are pragmatists in the sense that they see the foundational questions classically considered in the philosophy of mathematics as unfruitful and are therefore abandon these questions. In fact, the roots of pragmatism stretch back to the investigation of C.S. Peirce of the clarity of ideas. (His investigation was perhaps spurred by the non-formal quality of clarity in mathematics.)
One can find, in the work of Hadamard and Poincare, a description of the mental processes of mathematicians, as well as a claim that mathematics is discovered by thinking really hard until one is stuck and then letting the subconscious take over. Even if this is true, it doesn't really settle anything and sounds borderline superstitious.
Along another line, Polya has presented heuristics of the way mathematicians solve problems. Particularly, in his work on induction and analogy in mathematics, points out the important role of mathematical analogy, where such analogies are characterized by carefully respecting the parts of the two things compared. Analogy between theories is viewed by some leading mathematicians as a driving force for the development of mathematics.
Still, there are many questions that are not addressed by the above lines. I'll give one below:
One working definition of mathematics I have heard is "The manipulation of stable mental objects" (Manin). This suggests that we need to consider more carefully what these "mental models" are. In some areas of philosophy, the existence of mental images like this have been disregarded because of a belief that language is identical with thought. Please see the question here for a strong indication of where philosophy could add to mathematical practice.
It seems that the mathematical literature is a recording of a network of formal systems (or perhaps language games) that is (at least locally) consistent, but what it seems mathematicians do is form their own mental models of patches of this logical landscape and work very intuitively with the models formed. The mathematicians who do this the most effectively can predict and prove results that continue the development of the literature.
(I could only include two hyperlinks...if my reputation increases enough I can come back and add more.)
This questions reminds me to an answer of the physicist Richard Feynman when asked about the use of philosophy of science to scientists. Feynmans answer: Philosophy of science is as useful to scientists as ornithology is to birds. (Se non è vero, è ben trovato)
My opinion: You become a better mathematician only by doing mathematics yourself, e.g., study the results of your colleagues and apply the current methods of research in your own work.
It cannot. People who are good at the philosophy of mathematics are mathematicians and not philosophers. This is because mathematics is philosophy. One can argue that Deligne, Grothendieck, Weil, Serre, Tate, etc. are all philosophers in a very deep sense. However if the ideas are not good enough for the very high bar that mathematical research has put, it is dubbed philosophy of mathematics.
