Very contentiously: Turing's original paper is about the computability properties of something "non-Platonic" (factual, etc. rather than "in math world") - a human performing calculations. (See W. Sieg's and R. Gandy's historical work.) Consequently it isn't mathematics Turing is doing, but how humans do mathematics. The generalization of this work, beyond what a human can actually do (in say, degree theory or recursion theory taken very generally) is then a branch of mathematics spawned by something "on the outside".