[hazov]

You realize there are tomes and tomes defining languages standards just that

def foo (Real): Real

do not have any ambiguity, right?

I really don't see how this is much more clear than

r:R^m -> R^n

Anyway, functions in mathematics are not the same thing as functions in programming languages, one generally can simply can be called subroutine, the other is commonly called mapping, they are not even same stuff. They're not in the same class.

There's really no "mathematical notation" as you think there is, people just use conventions for what we have, you can invent your own and advocate that's it's better, physicists and engineers do that all the time, and people will decide about it, maybe it is and you invented the new decimal numbering system for all of mathematics but probably you're just too naive.