Dear Noson,
I'm happy to see you posting an entry to this contest. I find much---on a first, cursory reading---that I agree with, but also some points where I differ. Let me offer some thoughts your essay elicited within me.
Identity is not a singular notion. There are many notions of identity, appropriate to different contexts, and neither is inherently more 'appropriate' than others. So Theseus' ship may retain its identity as an object of veneration by the Athenians, but loose it as a particular conglomeration of wood, nails, and canvas.
Questions of this kind---also those of mereology---are, in my view, intimately connected to the notion of modeling, which is what's been sort of the lynchpin of my own ideas. We don't engage with the world as such, the 'things in themselves', to bend this whole thing back to Kant. Rather, we meet the world via models of our own construction, and Kant was right that, in a sense, the structure of our mind---or, as I would put it, the constraints implicit in the possibility of constructing models---has a role to play in how this happens.
So whether something is a part, or a whole, whether it looses or keeps its identity is in this sense dependent on who asks---and how.
However, I think that considering all of this as just being subjective goes a step too far. There are evidently constraints on how we may consider things; we can't consider Theseus ship as the moon, or a piece of cheese.
Furthermore, there is intersubjective agreement between different observers about what's out there; the theories we formulate must get something about the world right, otherwise, their success would seem miraculous (the so-called 'no-miracles argument' for scientific realism).
There is something 'out there' that constrains what sort of mental models are applicable to a given object.
I think the best candidate for that something is the notion of structure. One object supports many different structures, and fixing a structure does not suffice to completely specify the object bearing it; nevertheless, the choice of structure is not wholly arbitrary. Then, an object, and every possible model of it, hence the answer to whether it keeps its identity or looses it, whether it should be considered as a part or a whole, share at least one possible structure; the mind chooses what structure to focus on, while the object supplies constraints on the appropriate structures.
So ultimately, I think the mind has to meet the world half-way: it's neither quite true that 'there are no objects', nor that some external given simply dictates the way the world appears to us. It's a creative process, and the world as we experience it is its output.
The limitations of modeling are, I believe, inherent, and of the same type as those encountered by Gödel, Tarski, Turing et al. As you might remember, my work in that area owes a great debt to yours, and in fact, I have argued that the value indefiniteness of quantum mechanics is exactly a consequence of this sort of limitation (I give a brief outline of this in my essay in this contest). This has interesting (well, to me, at any rate) consequences for the measurement problem and the notion of entanglement, which is my main topic in this essay. I'd love to hear your opinion!
Regarding entanglement, at first sight, one might consider it a counterexample to the idea that the world decomposes neatly into local particular matters of fact, in the Humean sense---each subsystem considered on its own will fail to yield the complete description of the system (two maximally mixed density matrices don't suffice to reconstruct the Bell state the total system is in). Thus, the whole system seems to have a stronger notion of 'objectness' than its parts do, hence arguing against the notion that what we consider, for instance, part and whole is ultimately up to us.
But this argument is a bit circular. We put in a particular decomposition of the system, by imposing a certain tensor product structure onto the Hilbert space; but this is not provided by the mathematics of the theory, but rather, by the physics---say, the decomposition into particle 'over here' and 'over there'. In a different basis, with a different tensor product decomposition, the entanglement structure may be very different.
So this, too, can be viewed as being a result of how the mind meets the world, rather than constituting an argument for some objective holism out there.
I wish you the best of luck in this contest!
