Well sure, there is a constructive subset of the topic we call "mathematics", just like there is a subset that admits only numbers less than or equal to 5. But if you want the whole topic of mathematics, it's going to include nonconstructive theorems like the Banach-Tarski paradox. One could take a philosophical view for or against the idea that the non-measurable set in the paradox platonically "exists", but either way, those theorems are a legitimate part of mathematics. At best you can say the theorems are about mythological entities rather than "real" ones.