> it seems completely reasonable to consider a mind "alive" even in the absence of all else.
So AI is alive too if you program it to understand it's AI? Such arbitrary definitions of "life" is not interesting. What if I define life as "abstract existence in anyone's mind" - that way a person, for example, would be considered "alive" as long as he is "alive" in our memory.
Also counterexample for life with only consciousness - let's imagine that we advanced technology to be able to replace neurons with analogous mechanisms that perform same function. Now imagine your cells are slowly, gradually being replaced with these mechanisms, all while your consciousness is full intact. That means that this imaginary technology replaces your live cells with machine cells without you losing consciousness, and at the end you are entirely machine. Are you alive? I say no. If you say yes, we disagree.
Thing is, consciousness and mind is nothing but a continuity of electrical signals. No magic here. While life is entirely different concept. They are tangential. Neither of them require one another.
> a person who has lost the ability to reproduce is still alive.
I am talking about multi generational life. If I remove your brain, thus mind and consciousness, you are alive too for some time.
> I certainly don't see any obvious reason why a biological definition of life needs to include an end.
It doesn't. But an organism that doesn't die cannot adapt to environment through DNA shuffling. But there are organisms that are considered immortal, not nearly as complicated as humans though.