A lot of questions on Math.SE ask for coordinate-free proofs. As an example of what this means, many standard theorems about matrices in linear algebra are actually the corollaries of theorems about linear operators that can be proven without choosing a basis and representing such operators as matrices (consider, e.g., associativity of matrix multiplication).
I recently asked for such a proof of something in analysis, and a commenter says he doubts such a proof exists. If this is true, can we prove it? I'm not really asking about the question in that post in particular, but rather, whether we can prove such non-existence claims in general. Of course, we would first have to be able to make the claim precise. I think foundational subjects might have some tools for this, since they sometimes study what can and can't be proven, so I'm tagging a few—but I don't really know how to approach this question.
