Mathematicians use logic to talk about the mathematical world. But logic is not the world.

Not even the most dogmatic of the set theorists ever argued mathematics was possible without reason, however. For mathematics, logic is the world, as the copula makes no distinction between substance and existence. In the same sense that the earth is not matter itself, but it is a material thing.

Putting that aside, to make things more clear: computer science is mathematics. Computer scientists are mathematicians. That was a categorization decided long before you and I ever lived. In the sense that you mean, you're only referring to a very small fraction of what "mathematics" refers to In the true sense of the word. It is just as irreconcilably disjointed as Logic is, not unified and fundamentally non-unifiable.

I too think it would be better if "mathematics" was reserved for the gated suburb of ZFC. But that's not the world we live in, courtesy of the same people who pushed ZFC as a foundation to begin with.