Why are philosophers interested in the topic of death?

Thank you for your question. I'm not sure there's one reason why philosophers care about this, and I'm not in a position to exhaust all the various reasons why the topic might be of concern to them. However, here are some of the reasons of which I'm aware:

1. A long tradition dating all the way back to Plato suggests that the self survives the destruction of the body. Socrates (Plato's teacher), for instance, didn't fear his death at all, and gave as a reason for this attitude the belief that he would exist after his body ceases to be alive. That thought has cast a very long shadow, and philosophers have for a long time tried to figure out what it would be to survive the destruction of one's body. These discussions can generally be separated from questions about the existence of God, and have in recent centuries coalesced into discussions of what is known as "personal identity". One core issue for this topic is what it means for one and the same person to survive over time, even when his/her body is radically changed or entirely destroyed. John Locke, for instance, held that I could survive the destruction of my body so long as there is some later "person-stage" that remembers experiences had by an earlier person-stage, such as the one sitting in a chair typing these words. Others have doubted Locke's view, and a good deal of this debate turns on the question what conditions would suffice for the death of a person.

2. One other way in which philosophers concern themselves with death is via the question whether a current person or generation has any obligation to a future one, even when that future one is only around when the current person or generation is long gone. Do we have an obligation, for instance, to care for the Earth so that it may be used by others after we die? Many people would say that we do have such an obligation, but articulating the source and range of that obligation (how many generations into the future should we look, for instance) is not at all easy.

3. Death is sometimes discussed as a way into the question what, if anything, gives life meaning or value. If one holds that life has meaning or value, then it might be fruitful to examine what it is about death that is held to be bad so that we may get clearer on life's meaning or value. The line of thought might go like this: After my death I won't be able to do or have x, y, or z, and that is why I fear, or at least dislike the prospect of my death. Accordingly, things such as x, y and z are part of what gives my life value or meaning. On the other side, some (such as Bernard Williams) have argued that death is a blessing, in light of the prospect of eternal life: that prospect might seem less appealing than popular conceptions thereof make it out to be on fuller examination. For some reasons why someone might give up eternal life, you might enjoy seeing Wim Wenders' movie, Wings of Desire.

I recommend taking a look at Fred Feldman's book Confrontations with the Reaper: A Philosophical Study of the Nature and Value of Death (Oxford: Oxford University Press, 1992). It covers many of the issues mentioned by Mitch Green. But make sure you read the Preface, where Feldman explains the circumstances that motivated him to take up the issue.

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