(a) if you haven't read it, Chiang, Story of Your Life (1998) might have interesting aliens.
(b) the special symbols is a good point. Reading older maths papers is cool because you get to see all sorts of things people tried before we settled on what we use now. Two works that come immediately to mind: Principia Mathematica (1910) uses various numbers of dots instead of parentheses to mark precedence, while Peano, Arithmetices principia: nova methodo exposita (1889) uses very modern-looking notation, including parens as we would use them, but its expository text is all in latin!
(c) I don't think your aliens would have any more trouble with non-commutative than we have with commutative. Have you heard of the Boom Hierarchy? It starts with trees; when we add an associative law, so (AB)C == A(BC), then we only have flat lists; when we add a commutative law, so AB == BA, then we only have unordered bags; and finally when we add an idempotent law, so AA == A, then we have unduplicated sets. It turns out (exercise!) that if we have information encoded in any of these representations, we always have at least one way to represent the same information in all the other representations, such that we can "round trip" between any two levels of this hierarchy without losing any information.
So for programming, where we care about time and space, picking ordered or unordered representations can be very important, but for maths, where all that matters is the existence of invertible functions between all these representations, that decision is unimportant. Does that make sense?
(b) the special symbols is a good point. Reading older maths papers is cool because you get to see all sorts of things people tried before we settled on what we use now. Two works that come immediately to mind: Principia Mathematica (1910) uses various numbers of dots instead of parentheses to mark precedence, while Peano, Arithmetices principia: nova methodo exposita (1889) uses very modern-looking notation, including parens as we would use them, but its expository text is all in latin!
(c) I don't think your aliens would have any more trouble with non-commutative than we have with commutative. Have you heard of the Boom Hierarchy? It starts with trees; when we add an associative law, so (AB)C == A(BC), then we only have flat lists; when we add a commutative law, so AB == BA, then we only have unordered bags; and finally when we add an idempotent law, so AA == A, then we have unduplicated sets. It turns out (exercise!) that if we have information encoded in any of these representations, we always have at least one way to represent the same information in all the other representations, such that we can "round trip" between any two levels of this hierarchy without losing any information.
So for programming, where we care about time and space, picking ordered or unordered representations can be very important, but for maths, where all that matters is the existence of invertible functions between all these representations, that decision is unimportant. Does that make sense?