[butterthebuddha]

When people talk about different language, they mean more than just notation. For example, I can write "2 | 4" as "4 = 0 in Z/2Z". There is more than just a notational difference here.

A more non-trivial example: Say you want to say that 2 x 4 = 8. One way is to just write 2 x 4 = 8. Another way is to say that a finite set of cardinality 8 is the product of a finite set of cardinality 2 and cardinality 4 in the category of finite sets (in the sense that 8 satisfies the universal property of 2 x 4).

[ajarmst]

How is your example different from claiming that the statements "Aaron and Elizabeth" and "My friends whose names begin with vowels" are statements in different languages? I also note that with the exception of the symbols for multiply and equality, your final statement is written entirely in (an admittedly stilted and formal dialect of) English. I'm not saying that Mathematics is anything but beautiful and expressive---but the only way to justify a declaration that it's a distinct language is to do irreparable harm to the meaning of the word "language". And, of course, you make my point in your first sentence. I agree that what we call language is more than just notation. Which is one of several reasons (among them concepts related to things like mutual intelligibility, orthography vs. grammar, native speakers, effortless learning during early childhood, communication as an evolutionarily selectable trait, ...) that I disagree that mathematical statements or reasoning are the same species of thing as British Received English, American Sign Language or even Elvish and Klingon.

[dfischer]

Precisely. Language under the context that it is a restricted set of abstractions used to communicate a set of consistencies.

Language itself is an exclusive operation on the possible coordinates of anything within a universe.

[ajarmst]

Well, if we're allowed to just redefine the terms, you can make and prove whatever claim you wish, but its unsatisfying and ultimately pointless. I'm not sure why you think the above (which, I'm sorry, parses as gibberish for me) is useful given my assertion that mathematics is not the same species of thing as human language, as opposed to whatever the thing you're describing above is.