"pure mental activity" as opposed to "mental activity related to a reality outside of the mental activity".

I don't see how you could distinguish those. Even the purest form mathematics involves things like rules and symbols and those have to exist in reality (certainly they are at least as real as any emotion).

counterpoint:

remove all blackboards and chalk, and paper and pencils.

can you still do math?

Sure, it just takes a lot of remembering. But the point is that still that isn't "pure" mental activity. I am still imagining symbols and rules, those exist in reality at least as much as emotions do.

hmm, but imagining rules and symbols IS a pure mental activity, isn't it???

They are very similar to language, which cettainly is part of reality.

My first foray into math involved counting with my fingers.