A quick and dirty response to Macula's Thought Experiment:
Hey Kyle, long time no see. Almost all my favorite people are back and are regularly posting! It reminds me of the good old days. Maybe I should stick around this time. :)
I think your question (of whether or not replacing our bodies, bit by bit, would compromise the identity of who we are) is a subset of a more interesting question regarding general physiological changes, in any context. Dealing with the relatively simple case of replacing parts first, I agree with you that I cannot see why replacing bits and pieces of human anatomy would compromise "who we are."
However, all of this seems to assume some kind of unchanging, inviolate "I": that is, that if our consciousness is somehow changed, we would stop being "ourselves." I suggest that this is completely not the case. When this truly becomes interesting is when we start combining parts. If you were to take a neuron out of your best friend and stick it somewhere in your frontal lobe, would "you" still be "you?" It's not a large change, and after all, what's one neuron among friends? In addition, if you were to put his neuron in place of one of yours that has the same function, there would be no chemical or electrical difference in your brain as a result of the transfer. Thus, can we say it makes any difference at all? To suggest that it does would assume that there is some cosmological standard out there, seperating "your neuron" from "his neuron." I certainly see no reason to think that this is the case.
Now: If you were to cut out your left hemisphere and replace it with your best friend's left hemisphere, would "you" still be "you?" How is this question fundamentally different than the one regarding one neuron? It's not. It's simply a matter of degree.
What makes this question so confounding is that it falls easy prey to reductionistic thought. If a large change in your brain would change your identity, so must a small one. If replacing half a hemisphere will make you stop being "yourself," that is, if your sensation of being conscious will no longer be "you," then replacing a neuron must also have the same effect, since we seem to be talking about a binary system here, of "you" versus "your friend." But why stop at neurons? Plucking out an atom at some arbitrary point in your brain must also compromise your consciousness, reductionistically speaking, as must arbitrarily putting an atom inside. When your brain recieves free radicals from the outside world, reacting with your engine of consciousness and forever altering it, in what way are "you" forever altered?
I seem to recall reading a story about a baby girl who, due to complications that don't come readily to mind, needed to have half of her brain surgically removed. Even with half a brain, the girl developed in a relatively normal way; she was reasonably intelligent, and on the surface indistinguishable from any other girl. If the other half had been healthy, had been kept alive and transferred into a tank somewhere, could you say that the other half contained a bit of the girl's consciousness within it? Or would it be a new entity, complete and with its own consciousness, and according to the theists, its own soul? I would argue, at the very least, that it would develop its own identity. (with no physical connection between the two, no communication can exist, thus they must exist independently.) If a brain can be seperated into two independent people, what does this mean for the ontological question "Who am I?"
This question all boils down to what you mean by "you," and more importantly, why we assume that our sensation of consciousness is exclusive, somehow seperate from the rest of the universe by some impenetrable wall. I'm suggesting that the binary terms "you" and "I" are arbitrary and really unsuited for this kind of discussion, particularly when we try to define the terms in this context. There is no conceptual wall seperating your consciousness from the consciousness of your friend as your brains as combined, just as there is no conceptual wall seperating the identity of two galaxies colliding. It is quite simply a matter of human interpretation. There is no cosmic law defining who you are any more than there is a cosmic law defining what a galaxy is. A galaxy is an island of matter, and you are an island of consciousness. Changing the consistency of the island does not change the fact that it is, as always, an island. In the same way that all galaxies added together represent the universe, I would submit that all consciousness added together comprise a sort of "universal consciousness," of which the entity you call yourself is only a part.
Boy, isn't reductionism fun? I'll write a more coherent essay on this subject later, lack of sleep and food have a tendency to make me babble.