Perhaps you have heard the observation that, since our bodies are continuously changing as we live and breathe, with sloughing off of skin cells, the retirement of blood cells and their replacement, hair growth, and the like, the person we were 10 years ago is gone in all essential matters today. The food we have eaten over these past years has provided the molecular basis of the substitution of older materials in our bodies with new materials. We heal by regrowing damaged tissues, our joints and organs get replaced, and in so many other ways. We might call this physical evolution process the Body Theory of Personal Identity.
Another way to look at this change process is by comparing a person's interests and recognition of themselves. Our attitudes change, our sense of self shifts moment to moment so that we become a new person continuously. This is called the Illusion Theory which says that to think that we stay the same is the illusion. In fact, we continuously change.
A third perspective states that we are the same person as long as we have a continuous memory of our past, and possess a set of overlapping memories. When memories are gone, we are no longer the same person we were. Those of us dealing with our elders with dementia feel this loss of continuity deeply and may consider the loss of memory to be a way of saying that our loved one is not who they once were, that their identity has been lost. I've read stories dealing with dementia where the author describes the person with memory loss as dying twice, once mentally and the second time physically when the body can no longer function. It sounds like Rousseau and the mind/body problem he so famously summarized with "I think, therefore I am." When we can no longer think, we are not.
The last perspective is one a bit more sacred, as it addresses the living self through identity with a soul. This theory posits that we are our soul in a very literal sense. Same soul = same person. It would seem that the changes discussed above are considered to be ephemeral to the central identity of our soul, which forms our being and is unchanging. The problem is that the soul is non-physical and eternal, and exists associated with but separate from the bodies we inhabit. Various religions suggest that the soul is put into our bodies by some force that we cannot see or measure, and that upon death the soul rejoins the larger pool of souls from which it came (heaven is one model, the over-soul in Eastern religions is another). Philosophers are not fond of mystical identities like this, since there is no way of measuring or identifying a "soul" in any meaningful way.
This idea that everything changes constantly is an ancient theme. Heraclitus famously observed that it is not possible to step into the same river twice, as the water is in constant motion. The philosopher William James talks about our constantly changing consciousness. David Hume, the Scottish philosopher whose picture is at the top of this page that I took in Edinburgh, wrote that it was not possible to observe a permanent self.
If all this great philosophy concludes that we all are in a constant state of flux, and the world around us is continuously changing, how can we hold onto an idea of our personal identity?