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)
*: 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.