I think there is something to the nature of your addition example that is counter to your point. Addition being an "abstraction" doesn't make it destroy information it's just an algebraic property of addition. If you instead took multiplication of prime numbers as your "abstraction", no less abstract than addition, then every product would have a unique factorization in the primes. Whether either of these operations makes sense for your problem and thus whether their limitations apply doesn't have anything to do with what "abstraction" you choose. They are either isomorphic to some aspect of your problem or they aren't.