I agree there's much value in lawful, powerful building blocks. I would
formalize that with composition law (i.e. exists F. forall X,*,Y. P(X*Y) =
F(P(X),'*',P(Y)), for some useful set of properties P). We can find many
lawful, powerful building blocks in category theory or algebraic topology.
I agree, even with your depiction of the tension between law/power and
approachability. My monoid example was just that, an example. The
notion of structure (as in group, ring or monoid) is certainly
much more important, and perhaps there's more fundamental concepts.
Maybe category theory should
Yes. A concrete way to teach an abstraction is to provide multiple concrete
examples each of which are different excepting their abstract commonality.
For example, addition is an abstract concept when compared to the concept of
identity. It is known that one way of teaching it is by grouping
