Thinking fodder

Since there are a few inquiring minds online at the moment, I shall grace you all with a trivial little post concerned with (very) basic philosophy of identity, with great potential for lots of quasi-intelligent comments.

Evidently, I cannot conceive of me being anyone else but me. Nonetheless, I would like to know where my me-ness begins and ends; for just as evidently, if you gave me a prosthetic arm, I would still be me, and if you gave me a liver transplant, I would still be me. Am I still me if I had a brain transplant? Surely, evidence argues that the brain is an organ, just like the liver, or a part of the body, just like the arm. Others, of course, would argue that the brain is the source of all identity, but I suppose that is precisely the issue at hand here.

And what if I changed my entire body and kept my brain intact?

Or changed my brain, and kept my entire body intact?

Where does the soul fit into all of this?

Or, more interestingly, is "me" not a constant, but in a state of constant flux? That is, if you believe that we are constantly changing from minute to minute and second to second, it follows naturally that at any point between life and death, we are never the same "me" for very long at all.

Can I ever know who "me" is?