I guess strictly, it is a contradiction because indeed "all of existing mathematics" cannot be derived purely constructively. The way I choose to see it, howev

I guess strictly, it is a contradiction because indeed "all of existing mathematics" cannot be derived purely constructively. The way I choose to see it, however, is constructivism is the activity of seeing how far you can go with the least possible axiomatic foundation. If, at some point, you need something like the Law of Excluded Middle, you just assert the specific instance that you need as an axiom (and not as a structural rule). In this sense, then all of mathematics is valid from a constructivist viewpoint, what changes is that now you care about where you assert certain axioms.