Functions map members of a set A to members of a set B. These can simply be Cartesian products whose members are tuples. In my dream PL syntax a function call would be a function name followed by a tuple, and that tuple would be no different than the tuples you would use in any other part of the program (and so you could use all the tuple manipulation library goodies). If the function preserves any other structure of the type, like an identity element, that could be stated so you could have morphisms. And that identity element or other properties could be declared just as other stuff like 'const' are declared, and since the compiler can't verify all these stated properties, it's on the user to provide correct info, just like it's on the user to write a correct program, so nothing lost here, and anything more, like verification by the compiler, would be a bonus.

Mathematicians have been packing all this stuff nicely for a couple of centuries now, maybe we could use more of their work on mainstream computing, and it could also be a nice opportunity to get more people to appreciate math and structure.

Something that has side effects all over the place should just not be called a function, but something else, maybe "procedure" would be an appropriate, clear term.