It is now easy to understand what one means by generalization in mathematics.
We have demonstrated a proposition A, which states a certain property of a mathematical notion a. We compose a mathematical notion b, which includes the notion a as a particular case. Finally, with respect to this notion b, we demonstrate a proposition B that restores proposition A when we substitute for notion b its particular determination a. Theorem B is a generalization of theorem A.
[Pierre Duhem, "The Nature of Mathematical Reasoning," in Essays in the History and Philosophy of Science, Ariew and Barker, eds. Hackett (Indianapolis: 1996) p. 231.]