So, let me just say why I think the following idea is a total nonstarter:
`You can't understand a claim (e.g., "there are infinitely many twin primes") unless there's something you'd accept as verifying it, or refuting it. Thus, if (as probably true for godel+anti-penrose reasons) there are arithmetical statements which are independent of all the mathematical reasoning we accept (i.e. that we would accept something as a proof or refutation), we don't understand them. Hence, we don't know that "C or ~C"(pardon the abuse of notation) in cases where C is such an independent statement.'
This idea can perhaps be motivated by the good Wittgenstinean intuition that your understanding of a word consists in something like your ability to use it correctly. But, it doesn't strictly follow from that idea (at most, what follows is that meaning supervenes on use, not that there is some use which verifies or falsifies every statement one understands). And, indeed, this verificationist argument can't be right.
1. 'Verifying' a claim means, becoming completely certain of that claim - as it were, assigning probability 1 to it, and hence being unwilling to ever later question it, whatever your future experience is. In this case, what could possibly verify ordinary scientific claims like 'There is at least one black raven?' Whatever experience you have with a black raven, there's always some further experience which could give you reason to doubt this claim.
2. `Verifying' a claim includes being inclined to treat something as non-decisive evidence for it, having some experience that makes you guess that C (or not C), even though you'd be willing to revise it. But in this case, the independence of a statement, hardly establishes that there's nothing that would make us more likely to guess that P or that ~P.
There are lots of things we take as giving us strong reason to believe a mathematical claim, without quite amounting to a proof of it (think about how many people believe P !=NP, but don't have a proof!).
In fact, if you allow facts about how we respond to seeming to see a proof or hearing that there's a proof of various related propositions, it's pretty much trivial that for any mathematical proposition there is something which we would take as evidence that it's true (e.g. seeming to see a proof of it from ZFC, or a proof of some other claim that generalizes it, or of some special case, where we expected counterexamples to be lurking).
So, if our neo-verificationist takes verification to require certainty, even the simplest empirical statements will be unverifiable. But, on the other hand, if he is only saying that in order to understand a claim there must be some possible experience that we would take as (strong) evidence for it, then a)this seems to be true and b)he has given us no reason to doubt it.
If, as I recently read 'verificationism has never been decisively refuted', I think this is only because verificationism has never been stated reasonably clearly. Once you actually try to say what you require for a statement to count as having been verified, the project completely crashes.
Do any fans of verification out there care to answer this challenge, or any Dummett fans care to propose a better interpretation?