Alex Tabarrok over at Marginal Revolution wonders Should we care if the human races goes extinct? CS Lewis states it is part of the Tao to care about our posterity, but he is also pretty damning of in Out of the Silent Planet Weston's drive to spread the human race as far as possible as an unquestioned destiny.
I have written many times that I care less and less about posterity the farther it extends. My children yes, and grandchildren. Presumably, their children as well, though even when elderly people have visible great-grandchildren they don't seem as deeply concerned about them - and I find I don't much care.
My church, or at least some expression of the Christian Church I would hope to see thriving, and there are ideas I would like to see go forward, which implies caring about the people who subscribe to them. Something along the continuum of New England - America - Western Civ culture I care about, though I'm not so worried about the land or the appearance.
Extinction in Tabarrok's sense does not fill me with horror, at any rate. And perhaps any extinction is not the terrible defeat we always consider it to be. If this world is not my home, then another type of continuance entirely is more important.