It seems that no one can live forever.
Is there anything that can rightfully be called "immortality", which is both desirable and a genuine response to the problem of mortality?
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The challenge is defining "immortality."
One must be careful in defining it as "something which does not change." The only way to manage that is to (a) defeat thermodynamics and (b) remove onesself from everything governed by thermodynamics. This basically means you have to discover something we believe to be impossible, and remove yourself from everything that isn't you. That's a really high price to pay. You literally could not have any influence in the world, or you would be subject to thermodynamics yourself. Worse, if you judged wrong, and actually were still under the effects of thermodynamics, you would have removed yourself from anything else in the world that could potentially teach you how to truly overcome it!
If we shift viewpoints a bit, we could look at eigenvectors of the time function. Eigenvectors are a neat thing that show up in linear algebra. They are a value, call it X, which when thrown through a particular transformation, result in a value which is kX, where k is some real number. For values which are not eigenvectors, the result is never so easy. This provides an interesting potential definition of immortality, because the "flavor" of X remains, as long as k is nonzero.
If you could find the "time function" which governs the rules of the universe as time progresses, you could identify the eigenvectors of that transformation (assuming it is linear. QM assumes it is linear, and there are equivalents for eigenvectors in non-linear systems, they're simply harder to talk about). As long as you always ended up with a non-zero k constant, the "flavor" of your eigenvector could remain forever.
I wont go into the math, but if you make the huge assumption that the world is a linear system (which particle physics seems to suggest, but it's a big leap to go from particle physics to life philosophies), you find that these eigenvectors must actually reveal themselves as rotations. The only non-rotation term allowed by this linear system is a vector of magnitude 1. However, this acts more space-like, and less time like; less like "immortality" and more like "omnipresence."
Thus, if you see anything which holds fast, and tries not to rotate in any way, either it is not immortal, or the world is nonlinear.
The human body is a very complex machine. In principle, when it is damaged that damage can be undone provided that we know about the damage and it hasn't gone too far. At present, the technology required to do this doesn't exist. If it did, then your body could be kept alive indefinitely in a youthful state. That means there would no upper bound on how long you could live barring accidents and murder. I don't know whether you would want to call that immortality.
A researcher called Aubrey de Grey has come up with an idea about how to repair age related damage before it goes too far that he calls Strategies for Engineered Negligible Senescence (SENS):