Reread my post. I didn't use them in reference to that mathematical operation, except in the end, where the problem domain would be different (and hence the operators could conceivably mean something different). I in fact said that "Which is not to say that one plus one does not equal two. It is, however, to say that one plus one may not be meaningful as a concept outside a very limited domain."
I -did- do this in my response to you, because the confusion was in a sense important; you can't outright deny the existence of sheep interactions, you can only point out that this isn't addition. Which allowed me to make this point: "It's very close to addition... and may reflect reality better than addition."
I'm not attempting to define this operation, only present its conceivable existence. There are two points to this post: First, that any defined subset of mathematics is not universal. (That is, mathematics is not in fact a universal language, any more than "Language" is a universal language.) Second, that any defined subset of mathematics is a nonideal representation of reality, and that it would frankly be surprising if an advanced intelligence chose to use the same mathematics we chose through our biased processes.
Eliezer's post How To Convince Me That 2 + 2 = 3 has an interesting consideration - if putting two sheep in a field, and putting two more sheep in a field, resulted in three sheep being in the field, would arithmetic hold that two plus two equals three?
I want to introduce another question. What exactly are you counting?
Imagine one sheep in one field, and another sheep in another. Now put them together. Do you now have two sheep?
"Of course!"
Ah, but is that -all- you have?
"What?"
Two sheep are more than twice as complex as a single sheep. It takes more than twice as many bits to describe two sheep than it takes to describe a single sheep, because, in addition to those two sheep, you now also have to describe their relationship to one another.
Or, to phrase it slightly differently, does 1+1=2?
Well, the answer is, it depends on what you're counting.
If you're counting the number of discrete sheep, 1+1=2. However, why is the number of discrete sheep meaningful?
If you're a hunter counting, not herded sheep, but prey - two sheep is, roughly, twice as much meat as one sheep. 1+1=2. If you're a herder, however, two sheep could be a lot more valuable than one - two sheep can turn into three sheep, if one is female and one is male. The value of two sheep can be more than twice the value of a single sheep. And if you're a hypercomputer running Solomonoff Induction to try to describe sheep positional vectors, two sheep will have a different complexity than twice the complexity of a single sheep.
Which is not to say that one plus one does not equal two. It is, however, to say that one plus one may not be meaningful as a concept outside a very limited domain.
Would an alien intelligence have arrived at arithmetic? Depends on what it counts. Is arithmetic correct?
Well, does a set of two sheep contain only two sheep, or does it also contain their interactions? Depends on your problem domain; 1+1 might just equal 2+i.