What do you mean when you use the term 'counting' or for that matter '+'? You seem to be using these ideas in different ways at different times, which is a language problem not a real problem: like trying to use one word that means alberzle and bargulum when you may need to use different words.
I have yet to find a concept that does not fit the description 'may not be meaningful outside a limited domain.' It seems like there is a generalizable mistake among some very intelligent people that makes them want something to be more useful than it is. Yes, addition is only useful for putting together "individual" (however you want to define that, often pragmatically) objects into "groups." You can't add x+y and get a delicious pie as a result. But any piece of knowledge is only useful within the domain of its application: why would you think otherwise?
I am using counting to refer to any process by which a number is assigned as a symbol for a property.
I use + in both its concrete - this field now contains two sheep - and its abstract - this set now contains two sheep - meanings. I hope the context makes it clear when I am using each meaning, and why; because the lack of clarity is, in fact, important. See my response to the first comment, in which I deliberately used its concrete meaning in response to somebody using the abstract meaning. Both of us are in fact correct, and the confusion itself is mea...
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.