Good question. My own take on this is inspired by works of Richard Wollheim, “The Thread of Life”, Derek Parfit , “Personal Identity” and Bernard Williams “The Self and the Future” (though Williams argues for an opposed view). I distinguish between on the one hand, my body and brain, and on the other, my mind. I choose to think of myself as my mind. I think my mind is made of, or, as we say, ‘realized by’, my brain, in something roughly like the way a statue is made out of, or realized by, a piece of clay. The piece of clay is not identical with the statue. It existed before the statue came to be. And if the clay in the statue were very gradually replaced with new clay, over time, the statue would survive without the help of the original clay from which it was made. I also think my mind is, or is like, a set of computer programs. In manmade computers the programs are usually realized by patterns of silicon chips. The chips could exist without making the patterns, as the clay could exist without being...
Why is the continuation of the human atomic structure an insufficient explanation for continued personal identity of an individual?
If subject "a" remains subject "a" on an atomic level surely that constitutes the continuation of that subject. Arguably the atoms change over time, but not all at once. If say one atom changes on Monday, and then next on Tuesday, the very fact that an atom from Monday remains on Tuesday (even if it was the new atom on Monday) allows for the continuation of that subject.
This simplistic example shows how on a basic level something of the person remains prior to the present moment.
Our atomic structures don't remain the same over time. Some changes in structure sustain personal identity. Others don't, such as death. And some would argue that certain types of changes in neural structure would result in person a being replaced by a different person, b, as owner of the same body (say if b's neural structure were copied in a's brain).