4) I think it is another kind of thing altogether. You can create an informational thing that refers to the mapping, but the mapping itself is something deeper. The mapping is the instantiation of the informational thing in physical form. It's a realization, not just another reference.
5) That's fairly antithetical. I'm actually using "relationship" and "mapping" in distinct senses above.
In a great many cases, it's easy to determine whether physical thing A is an instance of informational thing B. There's always room for a sliver of doubt, but it's generally not hard to get below the threshold where we regard that sliver of uncertainty as negligible.
It's in the cases where part of the mapping is starting to break down that subjectivity comes into it. If I have three books sitting on my coffee table, it's clear that that's an instance of the informational entity '3 books'. We have complete correspondence between the informational thing and the physical thing, with (practially speaking) no room for argument.
But if I have two whole copies of Hamlet and a loose stack of pages from the first three acts... now it's harder to say. It's still 3, but the 'books' part is a little broken. Is it still a book when half of it is gone? When the pages are unbound? Does it matter that Hamlet is a play, not a novel? The mapping is flawed, the correspondence is partial, and the question of whether the physical thing still counts as an instantiation of the informational thing becomes subjective.
My first point of contention: it seems to me that in this case--the Hamlet example--you have certain elements of the informational thing that apply to the physical thing. However, I don't think it's really a subjective choice.
It seems to me that the informational object of "book" differs between two observers. Both agree that works of literature are on the table, both agree that three distinct pieces are there, both agree that one is unbound, and both agree that the work in question is a play. Now, what are the critical components of the informational object "three books" that allow the mapping to take place?
If the two people agree on what is critical about "three books" to make it map to an object, they will agree on whether what they see is three books. If they can't agree on what constitutes a map to the physical object, are they really talking about the same informational object?
They are talking about the same thing. Generally speaking, everyone means the same thing when they talk about books. The thing is, "book" is an object with fuzzy boundaries.
Most things (physical and informational) have fuzzy boundaries. Some are fuzzier than others. (Things that aren't fuzzy are incredibly rare.) For informational objects, the more referential the object, the fuzzier it is.
Most of the time, fuzziness doesn't matter. This is just a case where the context exposes the fuzziness, and makes the correspondence between informational and physical objects difficult to evaluate.
It's not that the two observers have fundamentally different notions of the concept "book". If you gave them the option, they'd probably both say "there are two books, and one partial book". But if the question is constrained to "are there 3 books, yes or no?" they have to take the partial correspondence and resolve it to either "book" or "not book", and to do that, they each have to move outside the non-fuzzy core of "bookness" and start invoking the network of other informational objects that constitutes their mental model of the universe, and those networks will differ, even though they have a great many components in common. Small differences will compound, the fuzziness multiplies, and you can end up with different answers.
The very notion of two things being "the same" is itself fuzzy. How fuzzy depends on the fuzziness of its operands.
(side note: if this stops being interesting, stop at any time, no offense taken)
This is interesting in itself: the idea of people dealing with the fuzziness of things in their world by "invoking ... their mental model of the universe."
Can you describe how this is done? Does this allow people to resolve fuzziness into distinction, or does it allow them to trick themselves that they resolve fuzziness when really they have no idea, or does it allow people to think "fuzzily" without the need to resolve the borders? In your example, it seems that it allows them to decide border cases, which seems to point toward an apparent resolution of the fuzziness of the borders based on these larger models.
Feel free to use or drop the three books example, whatever is useful.
Well, let me start by saying this is nothing strange or special. It's what you do all the time whenever you need to resolve an ambiguity. You just take the cloud of features that makes up the definition of a conceptual category, compare it to the features you observe on a physical thing, and see how well they match.
So (to blatantly rip off my example from someone else's blog), if you see something skittering through your kitchen out of the corner of your eye, and you think "is that a rat?" you take a better look at it (after pausing the TiVo) and compare: it's a small rodent, but look, it's got the wrong kind of tail. It's a squirrel, not a rat. The vast majority of the time, the matching is easy and obvious, so fuzziness doesn't matter.
Now, consider a fuzzier case: there's a big juniper in your yard. Is it a bush, or is it a tree? It's hard to tell. We could probably make a determination, but we don't need to if all we want to do is trim the thing; whether it's a tree or a bush is irrelevant. So we can handle fuzziness without addressing it in some cases.
In other cases, we have to resolve the ambiguity in order to make a decision or some such. And the need provides important context for the resolution, because if we didn't have a need to resolve it, we could just leave it ambiguous and address it in its naturally fuzzy state, right?
The stack of Hamlet pages on the coffeetable can be matched against any number of patterns. The ambiguity in matching it against the "book" pattern only matters if the question "is it a book?" has somehow been asked, implicitly or explicitly, in a way that does not allow "sort of" as a valid answer.
[Aside: I'm starting to think that the correct answer to many (most?) philosophical conundrums is "that's a bad question, because it depends on context that hasn't been provided".]
So let's consider this situation: Alan says to Bob "hey, would you go and grab the books off my coffeetable?" Bob sees two regular books and a stack of loose pages. Did Alan mean to grab that, too? Is that "a book"? The answer depends on context. If Bob knows that Alan is editing a manuscript, he might decide yes. If Alan is shelving things on a bookshelf, he might decide no. It depends on the circumstances surrounding the question.
If the immediate circumstances don't help, then Bob will start pulling on broader context, drawing on his personal understanding of the universe. If he works at a bookstore, the aspect of books as "things that are bound" comes more easily to mind for him, and he decides that the unbound pages are not a book. If he's been sorting through letters and forms all day, the aspect that is "a lengthy collection of text" might be prominent, and he decides yes, it is.
Now, in neither case has the fuzziness gone away. All that's happened is that Bob was constrained to deal with a borderline case in an all-or-nothing way, so he evaluated it in that context to determine whether the mapping between physical object and informational object held. When exterior context was insufficient to resolve the ambiguity, he called on his mental model of the universe to get more context so that he could make a subjective decision. In a different context, he might come to a different decision. A different person in the same context might come to a different decision.
The fuzziness remains, it's just a way of treating it as if resolved in a particular context.