Then is there a fact of the matter about this claim?
There is no guarantee that any statement of arithmetics would have a sensible direct physical interpretation. However many propositions of form “for all n, P(n)” have such an interpretation. For instance, starting from the original example we can replace it by “for all n, putting one apple to a group of n apples produces a group of n+1 apples” which may be physically checked for different values of n.
In some sense the two interpretations get closer to each other concerning quantified propositions. Having “for all n, P(n)” not only one can check the outcome of the corresponding physical process (having in mind the first interpretation) but it also seems reasonable to check “P(n)” formally for each n, using a subsystem of arithmetics which doesn’t include quantifiers. If we found a single n allowing to prove “not P(n)”, arithmetics would be inconsistent. Since we don’t know for sure that arithmetics is consistent, to test it in such a way is not completely useless.
There is no guarantee that any statement of arithmetics would have a sensible direct physical interpretation. However many propositions of form “for all n, P(n)” have such an interpretation. For instance, starting from the original example we can replace it by “for all n, putting one apple to a group of n apples produces a group of n+1 apples” which may be physically checked for different values of n.
In some sense the two interpretations get closer to each other concerning quantified propositions. Having “for all n, P(n)” not only one can check the outcome of the corresponding physical process (having in mind the first interpretation) but it also seems reasonable to check “P(n)” formally for each n, using a subsystem of arithmetics which doesn’t include quantifiers. If we found a single n allowing to prove “not P(n)”, arithmetics would be inconsistent. Since we don’t know for sure that arithmetics is consistent, to test it in such a way is not completely useless.