Is there a limit beyond which mathematics, if used correctly, cannot be applied to reality?

Question

And if so, why, and which?

Take this case: If I've two apples and I believe that with two more I'll have four apples, then I implicitly believe that summation applies to reality. Yet there are arguments, such as the Ugly Duckling Theorem * developed by Satosi Watanabe, which results can be extremely counterintuitive. But if the theorem is sound, shouldn't we accept its results? (In this case that without bias a duckling is just as similar to a swan as two duckling are to each other).

My opinion is that if I believe that 2+2=4 can be applied to the apples, I should also believe that operations like 2^1000 + 2^1000 would, although (let's say) there are no 2^1000 + 2^1000 apples in the universe. So, if I accept that part of the mathematics can be applied to reality, I should accept that the whole mathematics could, if used correctly.

(The same can be said about Logic)

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*: The Ugly Duckling is an argument asserting that classification is impossible without some sort of bias. More particularly, it assumes finitely many properties combinable by logical connectives, and finitely many objects; it asserts that any two different objects share the same number of (extensional) properties. The theorem is named after Hans Christian Andersen's story "The Ugly Duckling", because it shows that a duckling is just as similar to a swan as two duckling are to each other. An in-depth study can be read here, a short summary here.

Answer 1

Mathematics is a tool. While two real apples plus two real apples will give you four real apples, there are no physical numbers in reality.

Take negative numbers for example. If I have one apple, and give you one apple, then I have zero apples. If I say you can have the apple for $1, but you do not currently possess $1, then we would say you are in debt and have -$1. However, in reality you have $0 and I also have $0. Therefore, while the negative numbers associated with debt may have real world consequences, nobody actually possesses a negative amount of money (or any physical substance) in reality.

Answer 2

Once you fix the symbols of your theory, assume a certain amount of rules of induction to be valid (let's say in a first order logic), and one or more statements (axioms) to be true, you have as a byproduct a (large) number of true statements, i.e. all the ones that are provable in the theory build on those symbols, induction rules and axioms.

Among the true statements there may be some that cannot be applied to reality, but nonetheless they remain true as a byproduct of the axioms. Of course the basic axioms should be in agreement with sensible experience, if we want our theory to be applicable to reality at all.

Answer 3

This question have a profound sub-question implicity, that is, what is the relation between mathematics and reality? This comes in line with the study of the nature of mathematics itself.

The first thing i would point out is to differentiate mathematics from the symbols and/or language that we use to express this "thing" (or propriety, or accident). Is all math reducible to symbols? Godel as a Platonist probably understood that no, and his theorem is part of that mentality.

There is a predisposition today in the scientific community to think that math is everything in a certain way, basically because of physics, the way that some physicists think. In a Aristotelian perspective, for example, this would clearly not be true. The quantitative aspect is just an accident in a thing, in a being. The reality of this quantitative aspect, is outside the scope of this answer, but there is some good works in this direction like the books of professor Wolfgang Smith.

If all being have some quantitative accident, in it's existence, then the question is if math is the study of this aspect or not. What is math? (Is possible for a being not to have a quantitative aspect in its existence?).