So, on the off chance this is a relatively original idea, I decided that I should get it written up.
For those unfamiliar, the Ship of Theseus poses this problem: If you take a ship, and replace every component of the ship, until nothing original remains, is it the same ship? At what point does it cease to be the same ship?
Common sense tells us, of course, that it is the same ship. And everyone from Aristotle forward has tried to explain why that is, with the occasional jerk that takes the stance that it isn't the same ship.
The Aristotelian approach was to separate the formal cause and the material cause of the thing. It's a fairly Platonic approach, which isn't surprising: there's the "idea" of the ship, and the "material" of the ship- even as the material changes, the idea stays the same. Pirsig's concept of patterns is a less Platonic approach to the same idea: there's the pattern of a ship which is made up of subpatterns in the form of the different components; changes to the subpatterns don't change the over-pattern.
The other big approach is "four dimensionalism"- which, at first glance, was my preferred approach to the subject. If you view the ship not as a three dimensional object, but a four dimensional one, where its history and future are one long line, and the 3D ship is just one temporal slice out of that line, you can easily resolve this conundrum.
Except, of course, that to be complete, one would have to have a 4D timeline for all of the components, and suddenly the ship is less a ship, than a generalized area where the timelines for other objects occasionally merge.
I'm not a fan of any of these approaches, in part because they all share an unspoken assumption: that the world is composed of objects, and that these objects are distinct from one another.
Our sense organs receive inputs from the world, a whole hierarchy of processing centers pick up those signals and organize them into structures. When you look at the pathway of just the visual processing centers, you can clearly see the evolutionary steps that built the rather complex visual system we have. At low levels, we have simple abilities like recognizing lines, and increasingly complex abstractions get built up from what is, at the lowest level, the stimulation of light sensitive cells.
Photons excite your rods and cones, which in turn send signals across your nervous system, which get processed by various neurological systems and result in a mental model of the world, where those photons are interpreted as "a ship".
A large purpose of our brains is to convert the inputs to our senses into models that can be manipulated and acted upon. If my brain is able to identify that ship as Theseus's ship, and not my ship, I know not to trespass, less I earn myself an ass-kicking. And by being able to distinguish those sensory impressions as "a ship" and not, "ocean", I can do nifty things like not drown because I thought I was standing on a boat's deck while really I'm on the ocean floor.
It sounds absurd, but we know that, when failures occur in this processing system, really bizarre results can occur. Like the rather famous story of a man who thought his wife was a hat.
The point I'm trying to dig down to is this: the world is not a collection of objects that we can interact with. The world is continuous. The identity of objects as distinct objects is not a fact of the world, but a fact about our understanding of the world. The pile of matter that constitutes Theseus's ship is just a pile of matter.
The problem of Theseus's ship is not one of the identity of objects: objects don't have an identity. The identity of objects is a perceptual thing, born from our own minds' organizing principle. Our brains are object oriented. The world can be understood in these terms, but as the Ship of Theseus problem shows, there are edge cases that can give us bizarre results when we start trying to analyze what about an object provides it with its identity.
God, I hope that made sense.