Comment by bubblyworld

Mmm, the problem with computable foundations (in my opinion anyway) is that it takes a lot of stuff that is trivial in standard foundations (equivalence relations, basic operations of arithmetic and their laws, quotients, etc) and fills them with subtle logical footguns.

As you say, some view this as a feature, not a flaw. But to my mind mathematics is a tool for dissecting problems with hard formal properties, and for that I'd like the sharpest scalpel I can find.

For me classical foundations strikes a good balance between ease of use and subtlety of reasoning required to get results. I'm not sure the non-constructive and self-referential bits bother me, they don't really get in the way unless you're studying logic (in which case you're interested in computability and other foundations anyway).