Ancient Greece, via Plutarch
Replace the planks one at a time. When none of the original is left, is it the same ship? This is the sauce-pot problem.
The hero Theseus’s ship is kept in the harbour as a memorial. Wood rots, so rotten planks get swapped for new ones, one at a time. Decades later, not one original plank is left.
Is it still the ship of Theseus? Most people say yes.
Then comes the nasty follow-up. What if you kept every discarded plank and rebuilt a second ship out of them? Every piece original. Two ships side by side. Which is the real one?
There are roughly three reasons a thing stays "the same thing".
Go by material and the rebuilt one wins. Go by form and function and the one still floating and being used wins. Go by unbroken history and the one that was never once not called "that ship" wins.
None of the three is wrong. They split here because in daily life all three agree, so you never have to pick. Take them apart and you find out which one you had been using all along.
Your cells get replaced too. A few years on, most of the material making you is not the material that made you then. You still count as the same person because memory, relationships and a name ran on unbroken.
So for people we already use the history rule. The split happens because we try to judge the ship by its material instead.
The eternal sauce pot
The sauce pot in an eel or kushikatsu place, "topped up since we opened". Because it is topped up, almost none of the original molecules survive. Nobody objects to calling it the founding sauce.
Everybody uses the history rule here — and the same people switch to the material rule the moment it is a ship.
Ise Shrine, rebuilt every twenty years
Every twenty years the shrine is rebuilt from scratch on the neighbouring plot. As a building it is brand new, and it is treated as a thing 1,300 years old.
Here rebuilding is the method of continuing. By material it is a different thing each time; by history it never changed.
The band with none of the originals
A band tours under the same name with none of its founding members. People split cleanly into "that is a different band now" and "same name, same songs, same band".
Companies, sports teams, your mother’s cooking. The same argument, over and over, in new clothes.
If you take one thing
Think of one thing you would insist is still the same thing. If your reason is not the material, you already have your answer.
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