Today’s theory of reference:
Only statements (beliefs), not smaller phrases like nouns, are “connected to reality”. You sense something, that leads to belief in a statement (“the house is on fire”); you believe a statement (“I should run”), that leads to action. Noun phrases are meaningful only as parts of statements. The idea of attaching an “identifier” or other noun phrase to reality by some kind of mystical “reference” or “denotation” relationship is a fiction, a theory, a manner of speaking. Some version of this story may be testable, as a complex property of a speaker of a language, but like other theoretical constructs (“energy”, “phoneme”, etc.) it is never provable – at best it can only be useful in predicting what someone might say or do.
When you say ‘X refers to Y’ you are making a statement that links a theory involving ‘X’ to a theory involving ‘Y’. That enables inference – e.g. of new statements in which X occurs (maybe you now “know what X is”) – and inference permits new beliefs. So the word “reference” is not useless, any more than the word “and” is useless. But it’s the beliefs that are connected to reality through the apparatus of operational interpretation, not the noun phrases. Reference is just a subroutine of inference – part of a theory of how an agent links sensation to action through belief and inference.
This is especially true in model theory. The framework of logical semantics is that statements in a formal theory lead to new statements in the theory through formal proof. Formal proofs are painful to construct and verify (for people at least), and thinking about the existence or nonexistence of formal proofs directly is hard. So we invent the idea of interpretation. Interpretation trades formal proofs for the informal, natural language proofs we do in mathematics, which are much easier. In some cases (e.g. satisfiability, as opposed to entailment) we may want to think about some particular interpretation into a more specific theory. But this “particular interpretation” is of arbitrary nature – it could be into a theory of platonic things, or real things, or some other kind – that’s the beauty of logic and mathematics, they don’t care what they’re talking about. In any case the interpretation is just the link between the formal theory and the informal one – all we get is another, bigger theory that includes both formal and informal statements. It’s all just talk! until we get around to taking some belief, formal or informal, and connecting it to sensation or action.
(Apologies again to Pat Hayes and Willard Quine for mangling the story.)