For a while I've imagined that higher orders of logic "quantify over sentences" of lower orders of logic.
In Dr. Eugenia Cheng's lecture in Sydney today, she showed a slide saying that category theory is the mathematics of mathematics:
Category theory is the logical study of
the logical study of
- how logical things work.
This reminded me of that notion. Is there a connection here? Is this what Category theory is doing? Taking maths to a level of abstraction where the operators are quantifying over, and acting on, themselves, in such a way that no higher level of abstraction is possible? Does category theory effectively quantify over its own sentences?