Futurists often argue that artificial intelligences will be effectively immortal, able to file copy themselves into replacement machines as the older ones wear out. Similarly, if the brain replacement argument (Clark Glymour, Hans Moravec, and others) allows us to upload human intelligences into machines they suggest we too could become immortal (Mind Children, Harvard U. Press, 1988).
But in order to reduce memory size and speed up search an AI needs to delete older, less useful memories in
order to make room for newer more useful ones. Over a long period of time you simply wouldn't be the same
"person" anymore, even if you inhabited the "same" body.
It would be good for humans to live longer. We spend too much of our lives preparing for our careers
(typically 22-28 years) and too long in retirement (15-20 years or so). But the only sort of immortality that
seems possible might be recurrence (Recurrences and Discrete Dynamical Systems, Gumowski and Mira,
Springer, 1980). (One would want recurrence of that portion of the world that is "you" but NOT recurrence of the rest of reality. You don't want to live the same life over again.)