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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.

Here's another thought experiment that philosophers sometimes appeal to in this context. Suppose someone invents a teleportation machine (like in Star Trek). The machine scans your body, vaporizes it, and then recreates a molecule for molecule duplicate somewhere on Mars. Would you survive this process? That is, would the person on Mars be the same person who stepped into the machine on earth? Or would you cease to exist, only to be replaced on Mars by someone who is exactly like you? If personal identity is just a matter of physical continuity, then you probably don't want mess around with teleportation.