> Just because automatic differentiation is a different technique than numerical differentiation doesn't mean it isn't a numerical method.

It does mean that. Pls see the overview paper http://jmlr.org/papers/volume18/17-468/17-468.pdf in particular section "2.1 AD Is Not Numerical Differentiation"

> substantial arguments to people that try to temper the breathlessness

Which arguments in particular?

> our comments regarding math and science always give me the distinct impression that you know the name of a lot of things without knowing much about the thing.

Ok be more specific, which ones do you care about? Maybe I don't always have time to answer everyone.

> some universal concept of duality

It's the concept of adjointness. And yes, it's very universal.