Within deductive logic, I have certain views that are quite standard and usual, and some that are probably quite odd. It may turn out that the calculus for describing intensional logic properly, a logic of thoughts and concepts, is not markedly different from a logic of things. While convenient, and following linguistic custom, this is ultimately unclear and unjustified. There is ultimately only structure. Mathematics just is the theory of all forms of well-defined structure.
All ideal understanding is ultimately mathematical.
Philosophers' understanding of the relationship of mathematics to logic is quite deluded, I think, as many mathematicians have dismissively presumed. We do not grasp or ideally understand mathematical structure through logical structure. I try to teach logic as a live philosophical discipline with philosophical issues lurking all about, and of our current views in logic as the product of an often capricious and intrinsically interesting history that is interwoven with the history of philosophy and mathematics.