And Slaters Go Plop

Imagine a super-duper clever machine that can work its way through your brain and, one cell at a time, remove a cell and create a new cell in its place that is physically identical in every way. I know that the brain has a lot of cells but let’s say one cell is replaced every hour and time isn’t an issue (this is, after all, a thought experiment).

When the machine was finished, what would you experience? That you are still there or that you have gradually died?

Or would there be a crucial part of the brain in which you cannot swap even a single cell without destroying what is “you”? If so, where might this be? And would this be remedied by replacing atom-by-atom?