I would disagree with that. First of all, it’s really just a matter of definition. I think we all know (intuitively, at least) what the individual symbols “2”, “+”,… and so on mean. Certainly we can define those symbols in an alternate manner so that 2 + 2 is no longer 4, but I don’t think that’s a valid objection to your original claim. When we say, “2+2=4 is an immutable fact”, we’re not speaking of any intrinsic meaning contained in the symbols themselves, we’re speaking directly of the *concept* related by the sentence “2+2=4”. That concept is an immutable fact, independent of the symbols used to express it; if we then change the meaning of the symbols, we haven’t altered the truth value of the concept originally expressed by them.

That’s partly what’s going on here, if you’re speaking of the integers mod n. To elaborate on some of what’s been said before, the integers mod 3 (for example) are constructed by grouping the integers into three classes:

{…,-6,-3,0,3,6,…}

{…,-5,-2,1,4,7,…}

{…,-4,-1,2,5,8,…}

These **sets** I just listed are the elements of Z3. However, a shorthand is used. In Z3, when we write 2 + 2 = 1, what we *really* mean is:

{…,-4,-1,2,5,8,…} + {…,-4,-1,2,5,8,…} = {…,-5,-2,1,4,7,…}

In other words, “2+2=1” means: If you take the class containing 2 and add it to the class containing 2, the result is the class containing 1. “2+2=1” is just a shorthand way of writing this, by picking an element from each class and using it as a symbol to represent that entire class. To illustrate a bit further:

-4 + 8 = 7

8 + 2 = -5

2 + (-4) = 1

8 + 8 = 7

Each of these expresses the same equation represented by the original 2+2=1; I’ve only picked a different representative from each class, but you should see that each equation really expresses:

{…,-4,-1,2,5,8,…} + {…,-4,-1,2,5,8,…} = {…,-5,-2,1,4,7,…}

To get to the point, in the integers mod 3 (or mod any n), we’ve *changed* the meaning of “2”. Previously, “2” referred to the integer two; in Z3, “2” refers to the class {…,-4,-1,2,5,8,…}. As I argued originally, this doesn’t alter the fact that the integer two plus the integer two equals the integer four; we’ve only changed the meanings of the symbols “2”, “4” and “1”.

Furthermore, you may notice, in fact, that even in the integers mod 3, we still have 2+2=4 (i.e., {…,-4,-1,2,5,8,…} + {…,-4,-1,2,5,8,…} = {…,-5,-2,1,4,7,…}, and I picked 2 from each of the first two classes, and 4 from the last). In the integers mod n (any n), we still have 2+2=4, even though, as I mentioned before, we changed the meanings of those symbols. In the integers mod 3, it is true to say any of the following:

2+2=1

2+2=7

2+2=10

2+2=13

2+2=-2

2+2=-5

2+2=-8

and, of course,

2+2=4.