Before I continue, I need to make one disclaimer. I don't consider myself a Buddhist, I don't believe in karma or reincarnation in any literal sense, but I do believe there are many valid teachings that can be absorbed from the religion. I welcome any and all corrections or discussions to be had about my interpretations of Buddhist concepts, but keep in mind that these are my interpretations of the concepts and how I incorporate them into my life.
Take first the Buddhist concept of anattā, or "no-self". The notion is often referred to as the "illusion of the self", and as a result can sound a bit strange to many of us, particularly those with firm western mindsets. To alleviate some of you who are going through an existential crisis at the mere suggestion of "no-self", be assured that neither Buddhists nor I deny that your conscious experience is real, nor do we deny that your experience of consciousness is continuous (more on what this means later). Instead, the claim that we do make is that the consciousness that perceived the events of yesterday is not the same consciousness that perceives the events of today. This idea draws heavily on the Buddhist concept of impermanence, but can also be derived from western philosophy as well as the physical sciences.
This can be more readily fleshed out by pondering on the western philosophical thought experiment known as The Ship of Theseus. The experiment is based on the myth of Theseus (the details of which are unimportant), and goes something like this: While Theseus and his companions are adventuring on the Aegean, his ship becomes damaged little by little over time, and the damaged pieces are replaced with newer parts. Over the course of his adventure, however, the ship accrues so much damage that every part is eventually replaced, and some more than once. The question then materializes: Is the ship that Theseus originally sailed out on the same ship that he sailed home with? If so, by what merit is it the same ship if every piece of the original is no longer there? If it's not the same ship, then at what point did it stop being the same ship? When the last part of the original was replaced? When half of the original was replaced? When a single part of the original was replaced? When a single molecule of the original was replaced? The very concept of permanence and identity seem to go out the window.
The unsettling part of this thought experiment is apparent when it's applied to the very structure running the experiment: our minds and the physical brains that support them. While the exact figures are up to debate, the scientific consensus seems to be that every cell in our bodies--including seemingly "fixed" ones such as the ones in your teeth, bones, and even nervous system--are replaced with new ones every few years at the most (time varies based on who you ask). For more on the science of this, see Aebersold's seminal 1953 paper on the topic, or any recent paper on the insights into neuroplasticity.
The implication of this is that our physical bodies, including the brain which is the seat of the self, is exactly like Theseus's ship: constantly in flux, and in time completely replaced. So then, in what sense could your brain today be the same brain that you had last Tuesday, or when you were five? You might be tempted to argue that the physical structure of the brain is irrelevant, and that what matters is the software running on it. Analogously, we don't care about the nuts and bolts of our computers, you might say, so long as the information on them stays the same. We can swap out the RAM, the motherboard, the CPU, and even the hard-drives so long as the contents are transferred over to the new one. What matters is the information. The vacation photos, the work papers, etc. Similarly, what matters for our identity is our personality and our memories. These two things form the continuity of our consciousness: "I feel like the same person from last Tuesday, because I have all the memories and the personality of the person from last Tuesday." Perhaps it's this continuity of mind that I can call "I".
There are two main flaws with this line of thinking. First, the software that runs on your brain--your mind--is not fixed or unchanging. Much like the cells in your brain, the different structures of your mind are continuously in flux. Your sensory perception is almost never the same at any two arbitrary instants, your stream of thoughts never revisit the same exact thought pattern, your personality constantly undergoes changes however small, your memories fade and reemerge unexpectedly, and who knows what's going on with your subconscious mind? It's safe to conclude that your mind is, in a real sense, different from one instant in time to the next.
But does this mean that it's impossible to define "self" in a logical and meaningful way? What if we define "self" not as the concatenation of the individual snippets of consciousness that vary from moment to moment, but rather as the higher awareness that can unify those snippets and point to different states and say "that was me last Tuesday, and this is me right now." This definition is a necessary concession since the "self" can't be any one of those individual states, because each of them varies from one moment to the next. From a Zen Buddhist perspective, this definition changes literally nothing. This notion of a higher awareness that can differentiate between your past, present, and future states, is simply a single instance of your mind arising in the present, and in no way binds your previous experiences into a single cohesive "I", nor does it persist in time. It is no more "you" then the you of last Tuesday, or the you of when you were five.
Moreover, this higher awareness is not immune from impermanence either. For example, your current higher awareness might be able to glue together your current state to perhaps your ten year old self, if you're lucky enough to have a memory that works well enough. So then you can conclude that your ten year old counterpart shares an identity with your present self. Further, when you were ten, you were likely capable of remembering your mind states from earlier ages. So say when you were ten, you could have pointed to your five year old self and made the same conclusion that the present you just made about your ten year old self, and so on. Here's where the tricky bit comes in. We've already concluded that present you = ten year old you, and that ten year old you = five year old you. But, chances are you currently have little to no access to the mental states of the person that you believe you were when you were five, so current you != five year old you. In other words, your current self can claim to be the same person as you were when you were ten but not the same person as you were when you were five. Allow me to abstract this for clarity. What this line of reasoning suggests is that A can be the same as B, B can be the same as C, but A and C need not be the same! To cling to a notion of self, one must sacrifice transitivity of identity (mathematicians all over the world will cringe).
Instead of thinking of the mental machinery which processes your thoughts, awareness, and perception as "you", think of them as tools and organs at your disposal--like your hands and liver--to be used for personal growth and development. And much like your hands and liver, this mental machinery does not define "you" any more than do your hands and your liver.
To be perfectly clear, I understand that some people might indeed have intimate knowledge of their five year old selves. But certainly there are states of mind which you had that you no longer have access to, but that you did have access to at one point in time (not including the moment you were actually experiencing them). Furthermore, for those anxious to bring up a soul-related objection, all of the arguments above are just as applicable to the "soul" as they are to the mind. Simply interchange the words, and the arguments remain just as sound.
So where is there left to run? Where must we look to find identity? Well you wont find me answering these questions. I firmly believe that attempting to reason a "self" into existence when there isn't one is detrimental to spiritual progress because it's simply another form of clinging and attachment--attachment to the self. The illusion of self is so deeply ingrained in us as a species, that language itself makes it nearly impossible to point out the reality of the illusion, hence my inconsistent use of the words "self" and "I" throughout this writing. So instead of rationalizing a self into being, I'll attempt to redirect the language around it so that having and digesting this discussion is a bit easier. When we say "self" we need not refer to a fixed entity that persists through time. We can establish a new convention wherein "self" refers simply to the substrate of the conscious mind, and the flow of sensory experience through it. Like a canvas and an image formed out of paint strokes, respectively. Neither of these things need to be permanent, and in fact neither are. This is not a way of circumventing everything I said above, there is still no "self", this is merely a workable English definition that is congruent to intuition but free from the impediments and illusions that the old notion suffered from. I can now say "you" or "I" in an English sentence without contradiction, and you will know precisely what I mean.
By letting go of the old self, and embracing the reality of anattā, you open yourself to compassion, and liberate yourself from all sorts of delusions and misery. The exact nature of this liberation I will leave open-ended for now, because almost every future topic will require it as a prerequisite, and its utility will become apparent with every new post. What's important for now is that we can conclude that the idea of a fixed unchanging person that persists over time is an illusion. Armed with this realization, we can now move forward to higher more complicated ideas and discussions.