> Mathematical notation is a programming language.
It actually isn't - the notation has lots of ambiguities that don't confuse mathematicians because they are aware of the context.
On Proof and Progress in Mathematics[1][2] is a great essay by a Field's medalist. Part of the essay discusses this very topic, and why mathematicians choose not to use a formal notation for every day work. Some quotes from it:
> The standard of correctness and completeness necessary to get a computer program to work at all is a couple of orders of magnitude higher than the mathematical community’s standard of valid proofs. Nonetheless, large computer programs, even when they have been very carefully written and very carefully tested, always seem to have bugs.
> When one considers how hard it is to write a computer program even approaching the intellectual scope of a good mathematical paper, and how much greater time and effort have to be put into it to make it “almost” formally correct, it is preposterous to claim that mathematics as we practice it is anywhere near formally correct.
Yes, but I think this presents the core of the problem in modern pedagogical methods when it comes to mathematics. The Bourbaki attempted to reduce math to a highly axiomatic foundation, while disregarding the intuition and visualization that used to be a part of mathematics. The issue is that this sort of "code only" or "language only" approach really works when mathematics is a true "perfect language", the likes of which philosophers were attempting to construct, but is likely in fact impossible to create. Unfortunately, not only did the ideas of the Bourbaki fail, as modern research advances mostly still work with intuition instead of their ideas, but their approach polluted and ruined education. Many "textbooks" are terribly written reference books that have gaps and ambiguities that only people already knowledgeable in the field know about. Rudin's Analysis textbooks are probably the classical example of this. I would argue that any notation or abuse of notation is fine within insular fields and private practice, but there does need to be a leaning towards universal notation within all pedagogical works, at least up through all the core Algebra, Analysis, and Geometry and Topology work that you would see within a PHD qualifying exam.
The notation section is type definitions.