[cwzwarich]

Mathematics existed long before logic came to explain it, and most practicing mathematicians don't care that much about formal logic or think about it in their day-to-day work. The incompleteness theorems add a further disconnect.

[hackinthebochs]

I don't think it makes sense to say one can do mathematics without logic. Even if early math users didn't have a concept of formal logic, their mathematical reasoning was still dependent on it.