(Note: I added a link in my previous comment)

First, if you accept that mathematical constructs are fictional, why do you consider it valid to define a concept in terms of them? Second, I admit I wasn't clear on this issue: The salient part of intuitionistic type theory isn't intuitionism, but rather that it is a structural theory. This means that statements of the form "exists x, P(x)" are not well defined, but rather only statements of the form "exists x in A, P(x)" can be made.