The chemicals are only different in the sense that it is a newer source producing them, but the chemical type is still the same, because it still follows the same blue print instruction set. So the neurons, which remain unchanged during the mitosis process, still process the data the same.

If there are malfunctions during cell replication then I can see how identity can be affected because the chemical type itself may be different and can impact the nervous system in ways that induces change.

Maybe I am not following, I guess after you elaborate I can truly assess your position.

So going to this:

” We know that parts of who we are changes from year to year, but we still think we’re the same people.”

Our neurons do not get replicated, which in my opinion, the brain cells are the defining aspect of our personal identity. So because the neurons remain constant for the most part, our identity is maintained. Once the neurons malfunction, then we lose portions of our identity. Now if they can every replicate neurons, I guess that is when the ship will really come into play.

Yes, semantics has a part to do with identity, but my intended point was identity is based in belief. Even though all the parts of Theseus’s ship were replaced, Theseus believes it is his ship, and I’m sure no one on his ship would dissent. The museum believes only the original components of the ship constitute the real ship, therefore if pieces were replaced one-by-one, in the museum’s belief (and probably in the patron’s as well), the ship essentially becomes a forgery, even though the design and the essence of the ship has not changed.

Consider these additional examples:

Over the course of a year the average human consumes 1.5 tons of water, food, and oxygen. Assuming he has not gained 3000 pounds, this stuff (most of it anyway) has to be assimilated into the body, pushing out “old” stuff. It is estimated 98% of the body’s molecules have been replaced over a year. Over a year-and-a-half, over 99% of the body’s molecules have been replaced. So, do you believe yourself to be a completely different person than you were a year-and-a-half ago? Probably not – you believe you are the same because there there is continuity in your “self”, your ideas, your identity. It is your belief.

Contrarily, if I could collect all those molecules from your body that were expunged over that year-and-a-half, and reassemble them as they were in your body, would that new creation be you? I’m sure you wouldn’t think so. But maybe I do; maybe I believe this because I believe I have captured the “essence” of what is you. At that point our beliefs are at odds. Once our beliefs are at odds, the only thing left to distinguish between our beliefs is semantics. This is essentially what I was facetiously trying to convey in example number 2, above.

Then consider the “Washington’s axe” example. A museum owns Washington’s axe, but over the years, the handle has rotted so it was replaced. The head then rusted, and then was replaced. Is it still Washington’s axe? It is still an axe, the concept, design, and intent were preserved, but would you believe that it was still Washington’s axe? My belief tells me no, Washington never touched any of it’s component parts. It is a replica. The “essence” was lost.

There are three examples. The first argues that identity has little to do with component parts. The last argues that component parts are identity. The middle example, the best example, argues that identity is based upon both/neither. All show identity is based on perception, on belief. And when beliefs are at odds (like in the second example), semantics are all that’s left to sort things out.

One final example to ponder: The Gold Temple Pavilion in Kyoto, Japan was built in the 14th century. Over the course of 7 centuries, it has been destroyed and rebuilt several times. But to the Japanese, there is no paradox – they believe the temple that stands today is the temple.
“The idea of the building, the intention of it, its design, are all immutable and are the essence of the building. The intention of the original builders is what survives. The wood of which the design is constructed decays and is replaced when necessary. To be overly concerned with the original materials, which are merely sentimental souvenirs of the past, is to fail to see the living building itself.”
—Douglas Adams

Though Mr. Adams didn’t at first believe that the rebuilt building was the building, once he talked to the Japanese, he got it. It is all about belief.

(Note: this example was appropriated from Wikipedia, where there is an excellent discourse on the Ship of Theseus.)

The American pragmatist in me wants to accept the semantic orientation to this problem since that’s what really matters in most of our discussions, but the (conceptual) metaphysical query still stands. ;p

1) Theseus’ ship is completely replaced, timber by timber, over the course of its use – Even though the original components are replaced, it still functions as Theseus’ ship, therefore is the true ship. Even if some of the component parts differ somewhat from their originals (which must be the case, since “fresh” timbers replace “rotting” ones), the intent is to preserve the original ship as conceived by the builders, designers, and owner, namely the ship of Theseus. In this case, to be overly concerned with the materials is to fail to see the ship itself.

2) A new ship is build from the “old” components of the original ship – Again, context is key. The intent is not to build Theseus’s ship (which already exists and is owned by Theseus), but to build a new ship (not belonging to Theseus) entirely out of the components removed from Theseus’ ship. It is not intended to be Theseus’ ship, it is therefore a new ship. The line becomes grey if Theseus owns this “new” ship – at that point the dilemma becomes entirely Theseus’.

3) Theseus’ ship is now on display at a museum, and over time, thieves steal all the component parts of the ship (one at a time) and replace them with replicas – In this case, the component parts define the ship, as the intent of the museum is to display the ship that was given to them (in a sense it has stopped being Theseus’s ship, and is now the museum’s historic relic of Theseus’ ship). The theft/replacement of the component parts was not authorized by the owner of the ship (the museum); the thieves own a stolen ship, and the museum owns a forgery. The line here gets fuzzy if the museum replaces the component parts since a) the intent is to preserve the ship as it was given to them, and b) it stopped functioning as Theseus’ ship once Theseus stopped owning it. (I would argue that if the museum were to replace components of the original ship in this case, their action renders the ship un-original since the intent is to display/preserve the ship as it existed during the time-slice at which the museum acquired it. They would no longer be able to claim the “original” ship, but a “restored” ship. At which point replacing parts changes the “restored” ship into a “replica” ship, I’ll leave that for you to ponder.)

4) Theseus is dead, and a new ship is built entirely out of components replaced in the original ship – Similar to number 2, except that this ship is built after Theseus expired. For historical value, you could probably claim that this is a version of Theseus’ ship, but good luck finding all the parts, and proving they’re originals.