Mathematicians make sure their theories are logically consistent but not necessarily that they are somehow true of anything in the real world.
This may be compared with scientists whose research requires that they develop a specific mathematical theory, theory which presumably they will want to be both logically consistent and applicable to the real world.
Theories will be applied whenever they are seen as applicable by those who choose to apply them. Applying a theory may be considered as the expression of the belief that the theory is somehow true of something in the real world.
Logic seems to be a special case. Most human beings seem to have some personal logical capacity, whether they use it or not. Mathematicians are perhaps the people who make the most intensive use of their own logical capacity. So, presumably, they will be acutely aware of it and therefore of the reality of it. Boole and Frege explicitly meant to develop of logical calculus somehow true of the "laws of thought", as Boole put it. In that, they were acting more like typical scientists than like typical mathematicians.
Boole seems to have been successful in his effort. Material implication aside, I don't think anybody ever claimed Boole's algebra would somehow be unrepresentative or untrue of human logic.
However, after Boole and Frege, mathematicians working in mathematical logic seem to have moved away from the idea of a logical calculus as true somehow of the real world, and specifically true of human logic, focusing instead on the study of the formal properties of their theories.
Are there today any mathematicians, or even group of mathematicians, whose work on logic should be seen as not only mathematical but also avowedly scientific in the sense that they would try to produce a theory of logic explicitly presented as somehow true of human logic?
EDIT "For example, in two widely-read monographs both entitled Philosophy of logic (Quine  and Putnam ), the question of what logical theory is all about is barely asked." - Stewart Shapiro 1998 (Shapiro is O'Donnell Professor of Philosophy at the Ohio State University and a leading figure in the philosophy of mathematics)