What makes me the same person today as I was any time in the past? I have new memories and experiences, so why aren't I someone else?

You have changed. But for anything to change, it has to still be there after the change--otherwise how can it differ from how it was? In our jargon, you differ qualitatively---you have different features from before---but you are the same numerically---you are the very individual who existed in the past.

Puzzle: is the gain or loss of the feature of existing (when something comes or ceases to be) a change? Can something gain or lose that feature?

Most philosophers would I think say that what makes you the same person over time is not that there is any one thing present at each time in your life, but rather that there is the right pattern of change and the relations between different life stages. Just which patterns and relations make for a single person and which do not is controversial, but some of the relevant factors seem to be that, in a single person, features of later stages are caused by features of earlier stages in suitably normal ways. If you think about non-human cases, it seems pretty clear that identity over time does not require any 'golden kernel' present at each time. Thus we are happy to talk about a single tree that has been alive for many years, even if there were not a single atom in the tree today that was in that tree ten years ago.

There is a little book by John Perry called A Dialogue on Personal Identity and Immortality that contains a good introduction to many of the issues about personal identity over time that are raised by your question.

This is, indeed a matter of great controversy, and one that has generated a vast literature. There are those who argue that it simply wrong to say that you are REALLY the same person you were before, because there are so many differences; others who argue that because it is correct to say that you are that there must be SOMETHING about you, or the continuum that constitutes you that undergirds that fact. Others suggest that you identity is a kind of very useful fiction or construct, and so that while it is true that you are the same person now as you were when you were 2, that is true in the same sense that it is true that Ahab was captain of the Pequod, that is because we say so, and this is the kind of thing that can be made true by say-so, as opposed to by discovery. You might enjoy reading, from the Buddhist literature, the QUESTIONS OF KING MELINDA, relating a lovely dialogue between a king and a Buddhist philospher regarding this point, or David Hume's discussion in A TREATISE OF HUMAN NATURE, or for more recent treatments, Derek Parfit's REASONS AND PERSONS or Mark Siderits' PERSONAL IDENTITY.

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