Two understand why this is so, you have to understand the distinction between "false" and "incorrect".nYour question is caused by conflation of the two. I must admit that my usage of the two terms is not universal, but the distinction is common even when people use other terms for it (confusingly enough, textbooks on formal logic tend to use "false" for what I call "incorrect" here!).
All branches of science make models, which are representations of reality. These models contain statements. A statement is correct or incorrect with regard to a model, and true or false with regard to reality. Correctness here means a good fit to the rest of the model (usually, being derivable within the model, since scientists are usually interested in models as means of deriving predictions), and trueness means being a good fit to reality.
As a first simple example, take the model of classical conditioning, demonstrated in the famous experiment of Pavlov. The model makes the prediction that "the dog will salivate when the light goes on", and this statement is both correct within its model, and true in reality.
Note that a statement can be correct without being true, and vice versa. Imagine a situation in which a person of average build is standing upright in the middle of a room. I walk up to the person and push them forcefully on the sternum. A physics model predicts that the person will topple over. A model of interpersonal psychology predicts that the person will remain upright, and will slap me. Each of these statements is correct within its own model, but at most one is true in a given situation. This is not just about the domain of the model - a second model of interpersonal psychology can predict that I will get shouted at, but not slapped. They are simply different models, producing different predictions (statements). And note that, while in most real-life situations the psychological models are more likely to be true, one could imagine situations where the physics model is true, for example performing in a theater.
Sometimes people naively assume that science is (or should be) looking for the truest models, that is, the ones which fit reality the closest. This is not true (no pun intended). Scientists value parsimony, which means that they prefer the simplest model whose prediction is true enough for their intended use case. As a famous statistician quipped, all models are wrong, but some are more useful than others. This is why we are using widely both Newtonian physics and quantum mechanics - each of them is the preferred model for a given application.
It is also entirely possible that a model which describes one situation in reality perfectly (it produces true statements in that situation) is not at all applicable to other situations. The inverse square law is a true model when you consider the intensity of light, and not a true model when you describe the speed of a ball rolling on a horizontal plane.
Now on to your mathematics example. Mathematical models are very abstract. They can be used to represent some situation present in reality, or they can be used as mere mental constructs without any claims to be representative of a real situation. Let's first use your examples as representations of reality. In situation A, you get some sheep delivered and put them in a paddock. You want to predict the number of sheep you will have at the end. In that case, the model of addition of natural numbers will provide you with a true statement, while the second model, that of modulo arithmetic, will provide you with a false statement. Note that both statements are correct within their own models - an incorrect statement would be "1 + 1 = 1" in modulo 2 arithmetic. But one is true and one is false in your use case. If your neighbour is getting sheep delivered to two paddocks and has to ensure equal number of sheep between the two paddocks, then I also have a model which will predict how many sheep she has left over - here the modulo 2 arithmetic produces the true statement, and the addition over natural numbers produces a false one.
But if there is a pure mathematician who has no aspiration of applying your mathematics to real situations, the question of "is it true or false" doesn't even arise. Since true/false is about fitting a model to reality, and she is not fitting your model to any reality, she cannot measure the statement's trueness anymore than she can measure its physical weight or its radioactivity - it just doesn't have this quality. All she can decide about the statement is whether it is correct within its model or not.
If you think that there is just "truth", it is easy to see why you would compare the two situations you described, "1 + 1 = 2" being false* in modulo 2 arithmetic, and "each electron is in one place at a time" being false in quantum mechanics. But in fact, the two statements are not false, but incorrect in their respective models. Besides, the second one is false in the physical reality of the universe we live in. The first one is not traditionally attached to any specific use case in reality, so cannot be said to be true or false before you also provide such a use case.
* OK, not strictly false, but my point still holds with examples that are strictly false.
