If we dismantle a ship one plank at a time and replace the rotting wood with new planks of wood slowly over a few decades and then wake up to find that the ship no longer has any of the original parts left is it the same ship?  This is the puzzle of the Ship of Theseus. During this episode questions about identity over time will float past ships and be applied to persons as well.

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Art Attribution: Rémi Kaupp, CC BY-SA 3.0 , via Wikimedia Commons