Certainly it is possible to argue about what the meaning of “5324 + 2326 = 7650” is, but this is not where the really interesting weaknesses of arithmetic are to be found (or at least, to be looked for). The statements that should excite the most controversy all begin with “for all numbers n...” or “there is a number n...” That is where your interpretations 1. and 2. really start to diverge.
For instance, “for all integers n, there is an integer m such that applying the Goodstein operation m times to n yields 0.” This certainly can be given a formal syntactical meaning in Peano arithmetic, and a formal proof as well (but not in PA!), so we are good as far as interpretation 2 goes.
But the only physical meaning we can try to give it is that a certain process (fighting hydras) will stop eventually. Since there’s no guarantee it will stop before the universe ends, this claim can’t be tested. Then is there a fact of the matter about this claim?
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.
Certainly it is possible to argue about what the meaning of “5324 + 2326 = 7650” is, but this is not where the really interesting weaknesses of arithmetic are to be found (or at least, to be looked for). The statements that should excite the most controversy all begin with “for all numbers n...” or “there is a number n...” That is where your interpretations 1. and 2. really start to diverge.
For instance, “for all integers n, there is an integer m such that applying the Goodstein operation m times to n yields 0.” This certainly can be given a formal syntactical meaning in Peano arithmetic, and a formal proof as well (but not in PA!), so we are good as far as interpretation 2 goes.
But the only physical meaning we can try to give it is that a certain process (fighting hydras) will stop eventually. Since there’s no guarantee it will stop before the universe ends, this claim can’t be tested. 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.