Since I have posted at some length on the connections between mathematics and logic, I should draw your attention to a topic little known outside the profession: Generalized Logic.
This is not a new subject, but a masterful exposition of it was given by P. Dunmore in the New Zealand Mathematics Magazine in 1969. Marston Conder sent a copy to Gordon Royle, who has posted it on the SymOmega blog. As Dunmore says, it “eliminates a tiresome (and now superfluous) restriction on the growth of mathematics”. Everyone who teaches or does research in mathematics should have this subject at her/his fingertips!