I was gripped by your question some years ago until I realized something that I think solves a whole host of philosophical problems in one fell swoop. This question is indeed a good one and can lead to interesting things. I like the way it is formulated here:
From Mind and Materialism, by Geoffrey Madell; Edinburgh University Press, 1988. 151 pgs., page 103
V. Indexicality
It has been clearly recognised by some that the fact of indexical
thought presents a special problem for physicalism. This problem is
most clearly seen in relation to the first person. Thomas Nagel put his
finger on it in his paper 'Physicalism'. 1 Let us envisage the most
complete objective description of the world and everyone in it which
it is possible to have, couched in the objective terminology of the
physical sciences. However complete we make this description,
'there remains one thing I cannot say in this fashion -- namely, which
of the various persons in the world I am'. No amount of information
non-indexically expressed can be equivalent to the first person asser-
tion, 'I am G.M.'. How can one accommodate the existence of the
first-person perspective in a wholly material world? A complete objec-
tive description of a particular person is one thing; the assertion,
'The person thus described is me' is something in addition, and
conveys more information. But this extra information isn't of a
character which physical science could recognise. If reality com-
prises assemblies of physical entities only, it appears utterly mysteri-
ous that some arbitrary element of that objective order should be me.
But this doesn't really address the question of why you find yourself being the particular person you are rather than someone else.
I used to ask what the odds are that I would find myself being a human. I had an intuition that I could have been anything. And there is so much more that is lifeless! How did I get so lucky?
Many people would say that you are your brain, that you literally are identical with your brain. This doesn't seem so problematic until you start to think about it. It isn't the identity with the brain itself that is problematic, in my view, but rather the extent of what "you" are. It is the
problem of indexical extent. When people say you are your brain, what they are really saying needs to be clarified. They are saying that you literally are identical with this particular finite collection of particles and no more and no less. That's what "your brain" is. And if you are identical with your brain, that's what they are saying.
If what these people say is true, then I literally am a three pound hunk of matter. And I am only
this three pound hunk of matter. This is really strange! Why so little? Why so much? Why this particular collection of particles? Why am I not just one quark? Clearly, my identity spans multiple things. How? My identity seems restricted but extensive. I don't find myself being a whole population of people. What I am seems bounded somehow, as if there is a line drawn around this brain that designates it, and no more and no less, as belonging to
me, whatever that is.
Think now about the fact that these particles that compose your brain were once scattered all over, maybe a few particles in a carrot somewhere, some others in a rock, some others in a cloud, and so on. Were you these same particles then? Absurd, isn't it? What are the odds that your particles would just happen to come together in a brain like that?
Consider that objectively, this boundary around what you seem to be does not exist. There are no magical membranes. Out in the world itself, there is no clear separation of a person from their environment. There is nothing special about the matter composing a brain.
Let's return for a moment to the idea that you are lucky to find yourself being a human. While thinking about this, try to keep in mind that it is important whether you consider it from a first-person perspective or a third-person perspective. Objectively, it is silly to suggest that a certain banana is lucky to find itself being that banana, right?! Of course! But subjectively, matters seem different. The first-person perspective is what presents the puzzle. If you abandon it and try to solve the problem from the objective, third-person by declaring, "Of course this banana is this banana!", you miss the point!
Think about the odds of winning a lottery. Suppose that we are to randomly select one person out of seven billion to win a trillion dollars. It is assured that one person will win. When the winner has been determined, objectively, it is not at all surprising that someone won. That was always assured. And if you are not among these people and are just seeing it all strictly objectively, no matter who wins, there is nothing surprising. But if instead, you are one of these people and you find that you are the winner, you will naturally be surprised! You certainly should not have expected to find yourself the winner!
The situation with our identity as humans seems somehow similar to the lottery. Objectively, it isn't surprising that these creatures should be identical with themselves and should declare that they are themselves. Nothing puzzling at all. But if you find yourself occupying such a perspective, it seems different. There is the sense that you could be seeing the world from the perspective of anything. And if you were to draw one three-pound collection of particles out of a hat, the odds are overwhelming that you would end up with some lifeless material. Isn't it a bit surprising that you should find yourself in such a privileged position? Even if you can only find yourself being something alive, humans are vastly outnumbered by other possibilities. Why are you a human and not a mouse?
Let's get to what I think is the solution to all of this. That you are yourself is not the problem. The problem is the belief that your identity is restricted, period. Drop the boundary. That's it. You are everything. You occupy all perspectives. There is only one. The world is itself. That's it. There are no demarcation lines separating this from that. There are no true individuals, no separate objects. The first-person perspective finds itself everywhere simultaneously, and likely at all times as well. The world is everywhere present to itself.
There is only one thing to explain, and that is why our identity seems limited, why we aren't aware of being everything all at once. The answer to that lies in how information gets integrated. That which finds itself being me is the very same one as that which finds itself being you. But from over here, I don't know anything about being you because your memories are not in this brain. It's that simple. It is a question of access to information.
Consider an amnesiac named Bob who uses a chalkboard in a room as a substitute for his lacking memory. If we show him something, he records his observations on the chalkboard. If we ask him a question about what he has observed, he consults the chalkboard. Suppose we move him to another room with another chalkboard. He doesn't know he has been moved! If we ask him about what we showed him in the other room, he consults the chalkboard in this room and finds nothing. He has no way of integrating information between the two rooms. He might integrate information between them if he has a mechanism for this, such as a notebook, a way of carrying information back and forth.
This is analogous to what happens in experiments with split-brain patients, where it seems that by cutting the corpus callosum, we have turned one person into two, where it can be demonstrated that what is observed from only one hemisphere cannot be reported by the other.
If we show that in room B, Bob cannot report observations made in room A, we have not thereby demonstrated that Bob in room A and Bob in room B are two different subjects. The situation with you and me is similar. From my brain (think room with chalkboard), I cannot report your memories. And there is the illusion that who I am is restricted to the information I have access to.
Notice an interesting asymmetry with respect to time. You can remember the past, but not the future. When you look back, you feel identical with that past self because you remember those experiences. You have access. Not so with the future. Your future self is hypothetical and isn't really included in your sense of self. But once that future has arrived, you will feel that you are both that person and this now-past person.
In reality, your relation to your future self is not much different than your relation to me. It is a question of access.
The analogy of Bob in the rooms fails in a very important sense. Bob is someone who is separable from the rooms and moves between them. We are not similarly separable. It isn't that there is one little homunculus that runs around and occupies all the perspectives. No. There is nothing separable. There is just the whole world being identical with itself. There is one 'I', and it
is everything. There is nothing from which it can be separated.
Your body is experienced simultaneously by this one from the perspectives of all that interacts with it. You as 'other' and you as 'my body' are just what that particular body is like from two different angles. Both angles are experienced by the one subject simultaneously. But the information from the two perspectives is not integrated in such a way that there is a structure of experience that involves knowing that you are both at the same time.
So why do you find yourself being you, that particular human? It because you find yourself being everything. If one person is sure to hold the winning lottery ticket, and you are all of the people, you should expect to find yourself the winner, as well as all those who didn't win.
I think that people should rethink all anthropic principle stuff in light of this way of looking at things.
Also, if you think I am crazy, just some guy on the Internet, consider that many important thinkers have held a very similar view, Erwin Schrodinger and Arthur Schopenhauer among them. Also, Daniel Kolak, a living philosopher, has written an interesting book called
I am You:
link