Things equal to the same thing are equal to each other; if 4=1, and 2+2=4, then 2+2=1.
Yes. But it still equals 4. If you are trying to demonstrate a case where 2+2 isn’t 4, this isn’t it. You’ve demonstrated a difference, just not the difference you say you demonstrated. Hence, your original example was bad.
Ah. What does it mean to say that the integers (not mod 3) exist, if it is an observed fact that 2+2=1? That is, that what we think of as an integer number of objects is actually an integer mod 3? If I dispute your assertion, and say that the integers (not mod 3) do not actually exist, what experimental outcome will demonstrate that I am wrong?
There’s are a few problems with your question. Firstly, when we ordinarily say “2+2!=1”, this is implicitly a statement about integers, not integers mod 3. That is to say, “2+2!=1″ isn’t a complete statement at all; it only makes sense to the extent that we know what “2”, “+”, and “1” indicate. As a statement about integers, with the usual interpretations of thse symbols, it’s true. As a statement about integers mod 3, with the usual interpretations of these symbols, it’s false. It’s not one statement being true in one universe and false in another, it’s just two different statements.
Secondly, numbers are models. Remember, the universe does not run at the level of “objects”, it runs at the level of particle fields (or something like that). We count objects using whatever system of numbers is appropriate for counting objects (to the extent that “object” is a sensible notion). But even if whole numbers were somehow not a good model for counting objects, there are still plenty of other things that they would be a good model for, and they come up pretty naturally mathematically even without that.
So aside from a proof of the inconsistency of mathematics, not much could convince me that the whole numbers don’t exist. There are plenty of experimental outcomes, however, that could convince me that they aren’t a good model for something we’re currently using them for (e.g. counting objects, if 4 objects were indeed the same thing as 1 object, whatever that means).
Yes. But it still equals 4. If you are trying to demonstrate a case where 2+2 isn’t 4, this isn’t it. You’ve demonstrated a difference, just not the difference you say you demonstrated. Hence, your original example was bad.
There’s are a few problems with your question. Firstly, when we ordinarily say “2+2!=1”, this is implicitly a statement about integers, not integers mod 3. That is to say, “2+2!=1″ isn’t a complete statement at all; it only makes sense to the extent that we know what “2”, “+”, and “1” indicate. As a statement about integers, with the usual interpretations of thse symbols, it’s true. As a statement about integers mod 3, with the usual interpretations of these symbols, it’s false. It’s not one statement being true in one universe and false in another, it’s just two different statements.
Secondly, numbers are models. Remember, the universe does not run at the level of “objects”, it runs at the level of particle fields (or something like that). We count objects using whatever system of numbers is appropriate for counting objects (to the extent that “object” is a sensible notion). But even if whole numbers were somehow not a good model for counting objects, there are still plenty of other things that they would be a good model for, and they come up pretty naturally mathematically even without that.
So aside from a proof of the inconsistency of mathematics, not much could convince me that the whole numbers don’t exist. There are plenty of experimental outcomes, however, that could convince me that they aren’t a good model for something we’re currently using them for (e.g. counting objects, if 4 objects were indeed the same thing as 1 object, whatever that means).