Does the individual consciousness depend on the actual atoms or only on the configuration of the atoms?
Suppose we have mastered cryo-freezing and atom-manipulation technology. We can freeze and unfreeze people at will.
We freeze Sarah. We replace Sarah's atoms one by one. With all atoms replaced, we wake her up. Is it the "same" Sarah? (the same to herself, not just to us).
If one thinks of 'atoms' as ultimate constituents of matter, indistinguishable apart from their spatial and temporal position, then it is hard to see how changing them for identical duplicates at the same location could make a difference. (I'm interpreting the experiment as replacing every atom with an identical duplicate: same atomic weight, same charge, etc.) After the experiment, one would have replaced every atom with a partner with exactly same causal powers at the same location, and it is hard to see how this could effect an overall difference in her consciousness. The only difference between Sarah before and after the operation is an historical one: the atoms in Sarah after the operation would have a different history to those in Sarah before the operation. But it is difficult to see how this difference could make a difference to Sarah's consciousness. (Going through a freezing process might make a difference to her consciousness though!)