I can't imagine mathematics without sets, but the question "what was mathematics like before there were sets" is not answerable I think. Instead, a good answer to the question (see title) what the point was Cantor felt that there should be the theory he invented should cover a certain aspect of the more general question.

I can't imagine mathematics without sets, but the question "what was mathematics like before there were sets" is not answerable, I think. Instead, a good answer to the title question should cover a certain aspect of the more general question.