> Perhaps a better way of saying is is that hidden data is a characteristic of abstraction. Which is certainly fair, and fits the analogy to algebra quite well.

I'd disagree with even that much; I'd say abstraction is much more about parameterisation - factoring out the common structure from different constructions - than outright hiding anything.

> But in any case, parent's comment that "OOP has no special claim" to hidden state/data (or abstraction) is sound; after all, ADTs already do that.

ADTs don't hide mutable state. That much is something OO has a special claim to.