One of philosophy's most fascinating puzzles is the question of personal identity. What is it about me that makes me me? Is it my body? Is it my brain? If the body is constantly changing then in what sense does my self perdure through time? If my identity is just the contents of my brain how do I remain the same self over time as those contents change? What about me remains the same over time that keeps me the same person?
Suppose we say that it's our brains and their contents that make us who we are. Imagine that your body is dying but your mind is working well. Doctors have, through amazing leaps in technology, developed the ability to transplant brains into different bodies. Suppose your brain is transplanted, at your request, into the body of a person named John who suffered a catastrophic brain injury. When you awaken from the surgery, who would you be, you or John?
Brain scientists know that if they cut the corpus callosum, the band of fibers that connects the two hemispheres of the brain, two different centers of consciousness can be created. If either hemisphere is destroyed it's possible that a person could live on as a conscious being. Suppose your brain is transplanted in such a way that one hemisphere is placed in the body of John and the other hemisphere is placed in the body of Mary.
Have you survived the operation? If so, are you now two people? If you're only one, which one are you? Is it possible to be more than one person simultaneously? If so, if you committed a crime before the operation, should both John and Mary go to jail for it?
If we adopt a skeptical view and say that there is no personal identity but rather that the self evolves over time and we're not the same person today that we were ten years ago, then how can anyone be held responsible for promises they made or crimes they committed ten years ago? If we are not the same person who committed the crime then to punish us would be to punish an innocent person, would it not?
A theist might partially resolve this problem by claiming that our identity resides in our soul, and that our soul is independent of whatever body or bodies it possesses. But how would a materialist or naturalist who has no belief in any non-material constituents to the self, who has no belief in souls, resolve it?