One interesting observation that popped into my mind when I read the OP: Mathematicians don't write mathematical notation in papers/books by hand anymore, they use a far more verbose language called LATEX.
Wouldn't it be great if every time you saw a mathematical formula there was a little widget to push that would show you the "source code" in LATEX++, and LATEX++ was like LATEX but made up of stringently defined mathematical operations (like '\element_wise_multiplication' instead of '\plus_sign_with_circle_around_it')? :D
Now we're getting close to the heart of the matter: In LaTeX it's "good practice" to define the semantics of operators, in programming languages it's an absolute requirement.
Yeah, it's a bummer that LaTeX math notation is really just markup and has no indication of what belongs together and what the purpose of various symbols is.
Well, when I write TeX (and I stick to the "good practice" of defining "semantic" macros for common symbols), I often notice myself finding the TeX source code more readable than the PDF it produces.
Wouldn't it be great if every time you saw a mathematical formula there was a little widget to push that would show you the "source code" in LATEX++, and LATEX++ was like LATEX but made up of stringently defined mathematical operations (like '\element_wise_multiplication' instead of '\plus_sign_with_circle_around_it')? :D