When thinking about one cares about, there are two general positions: to care about something extent, or to care about something not yet extent. Both, in some sense, are future oriented: the first orients toward the future experiences and condition of a thing (even if in the very near, immediate future), the latter to the existence or non-existence of a thing, its conditions, and its experiences.
Those of us with children, often funnel our care through our children: they are the first locus of what we care about (outside our own future experiences). Our children’s lives, like ours, culminate in death. We care past the horizon of the death of our children to the extent that we care about their care: perhaps (and most easily) their children. We can imagine a future-chain of caring about children, in which we take concern in generation n+1 as part of our care for the interests of generation n. There will still, as far we understand, still be an end case, a last generation – the end of the universe itself, at least according to current cosmological models, but probably well before that; there are also possibilities of a post-generational humanity.
Care is contingent on time. Conceptual historian Reinhart Koselleck distinguished two categories of time. I have only recently understood how materialist his conception of human historical time. He distinguishes between the space of experience (what we might call “past and present”, although by understanding it as experience, we recognizing that the past is not a “place/time” where things exist, but the possibility of producing a representation or other enactment of something experienced) and the horizon of expectation (the winnowing of possibilities to probabilities and then to expectations.)
Memory is an active process: this means that when we remember something, we are assembling a mental representation that we then identify as located in the space of the past. Even archival footage is a residue or after-effect, one which creates variably isomorphic sensory experiences (“if you had been behind the camera, you would have had the same visual experience your are having now” – very different from “this is what it was / this is what it was like.”) The technologies of history and memory – writing/reading, building statues, repeating stories, various archaeological methods – are material practices in the current moment which produce effects of past experience.
(It seems that memory is such an active, present process, that the act of recalling a memory may disrupt the mechanisms by which the memory was encoded, and then rewrite them – with revisions. The consequence of this is that the thing long-forgotten and then remember may actually be better remembered than the memory often-returned-to: the latter is a copy of a copy of a copy etc.)
The horizon of expectation includes the immediately-dawning conclusion of a process that one starts. To even begin thinking a phrase, one has the expectation of concluding it: the form of the end of a sentence one has started to think is not present to consciousness, but is held in the imminent future as the inevitable consequence of the process of thinking (or speaking etc.) Care about anything, too, is care about an expectation of its conditions and experiences, however far in the horizon of time that is. But the expectation is also a process of representation, at least when it extends beyond the completion of a current process. (The end of the sentence, the destination of a step: these are unmediated even if they extend outside the present moment – this can be said of many processes which are underway, including those with longer timescales.)
The question for the extension of care into the n+1 generation is its relationship to our ability to represent generation n when it extends past the threshold of the representable. We know a pre-hominid primate would care for its offspring, and for its offspring’s offspring. Would it care about us, even if we were it’s nth offspring? When our offspring’s offsprings aren’t human, or are alien from us in some other sense, will we care? But can we not care if n-1 does?