> I do not believe that mathematical notation can be improved (a poor analogy would be trying to improve, say, English language itself)
I strongly hope you mean that heavy-handed top-down approaches aren't likely to work. Because this reads as though you're saying we've somehow reached the optimal point on every axis for both mathematics and general-purpose communication.
The (original) purpose of mathematical notation is to facilitate (to some degree, automate) reasoning and calculations (mathematicians tend to conflate them) on paper. For the existing frameworks it’s already as good (efficient) as it gets, given the 2-dimensional nature of said paper. Would adding new characters beyond the existing set (which is already quite large), introducing new alphabets in addition to the Latin and Greek (OK, we have the aleph), adding more font styles and sizes - would any of that constitute an improvement worthy of note? We have already exhausted the freedom given to us by the 2-dimensional paper-space as a computational medium - consider, for example, machinery that heavily relies on graphs (diagram chasing; Dynkin diagrams, etc.) or tables (matrices, tensors, character tables of groups, etc.).
I strongly hope you mean that heavy-handed top-down approaches aren't likely to work. Because this reads as though you're saying we've somehow reached the optimal point on every axis for both mathematics and general-purpose communication.