I’m going to use words, rather than equations, so this is going to be a bit sloppy.
For our purposes, aging is the gradual decline of biological function that leads to increasing mortality with time. Threescore and ten, and all that.
There are two senses in which aging is inevitable, but there is another sense in which it is not inevitable at all.
First, humans age. You can’t find anybody who doesn’t. Individuals from every population age, although not in exactly the same way -there are some racial differences.
Theoretical biology makes it quite clear that individuals ought to age. Every organism faces tradeoffs between reproduction and repair. In a world with hazards, such that every individual has a decreasing chance of survival over time, the force of natural selection decreases with increasing age. This means that perfect repair has a finite value, and organisms that skimp on repair and instead apply those resources to increased reproduction will have a greater reproductive rate – and so will win out.
Creatures in which there is no distinction between soma and germ line, such as prokaryotes, cannot make such tradeoffs between repair and reproduction – and apparently do not age. Which should be a hint.
Again, thinking about hazards and the declining power of selection with age: there’s not much point in paying extra for a car that never wears out if you’re about to enter it in a demo derby.
In practice, this means that animals that face low exogenous hazards tend to age more slowly. Turtles live a long time. Porcupines live a good deal longer than other rodents. Mainland opossums live only two years, but those on Sapelo Island, off the cost of Georgia, which has been predator-free for thousands of years, live three years. Organisms whose reproductive output increases strongly with time, like sturgeons or trees, tend to live longer.
The third way of looking at things is thermodynamics. Is aging inevitable? Certainly not. As long as you have an external source of free energy, you can reduce entropy with enthalpy. In other words, despite what your kids may claim, they really can clean up their rooms, as long as you feed them. Disorder decreases locally. It increases in the universe as a whole, mainly in the form of high-entropy radiation going into outer space, but who really cares about that? In principle there is no reason why people couldn’t live to be a billion years old, although that might entail some major modifications (and an extremely cautious lifestyle).
The third way of looking at things trumps the other two. People age, and evolutionary theory indicates that natural selection won’t produce ageless organisms, at least if their germ cells and body are distinct – but we could make it happen.
This might take a lot of work. If so, don’t count on seeing effective immortality any time soon, because society doesn’t put much effort into it. In part, this is because the powers that be don’t know understand the points I just made. Sometimes I wonder what they do understand.