From the perspective of neuroscience, personal identity is what happens when the brain forms a model of the environment that includes a first-person perspective and narrative history.
Eric Kandel, lead editor of the ubiquitous textbook Principles of Neural Science and winner of a Nobel Prize for work on the neural basis of memory, calls memory the "neural basis of individuation." And it is. For without memory, we could not each carry around a unique sense of who we are, derived from a differentiated personal life history.
If everyone on the planet woke up one day with amnesia, human beings would be a herd of mostly undifferentiated people. Without the ability to distinguish one person from another, or remember unique histories or events, everyone becomes a vague blur of humanity.
In addition to our sense of unique personal history, the brain also maintains a model of other people. "Theory of mind" in cognitive science refers to the brain's ability to model and track the goals, beliefs, and behavior patterns of other human beings around us in a social context. With introspection, this model of others can extend to ourselves. As has been quipped: "How can I know what I think until I hear what I say?"
Because everyone in society carries around a model of themselves and the others they know, all the brains in human society collectively comprise a substrate for the distributed representation of human identity. Our identity is shaped not only by our own beliefs about ourselves, but by what others think of us as well. Social roles are collectively determined, and personality is shaped by how others treat us as well as are predisposition to a certain character and temperament.
And lastly, while personal identity feels unique, unified, and permanent, it is not. Identical twins are often confused by acquaintances. In institutions, people are identified by role (e.g. sales representative for the western region, shop clerk) while the actual person may change. And someone's personality can change with mood. In children, we witness the formation of personal identity, and in senior dementia, we watch it unravel.
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