In the famous Only Fools and Horses episode "Heroes and Villains", Road sweeper Trigger wins an award from his employers for having owned the same broom for 20 years. He reveals that it has had 17 new heads and 14 new handles, but insists it is still the same broom. The writer John Sullivan was using one of the oldest problems in philosophy: The Ship of Theseus. The Roman philosopher Plutarch wrote: "The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians [for many years] for they took away the old planks as they decayed, putting in new and stronger timber in their places, insomuch that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same."
Many examples are seen in fiction, especially science fiction. In Star Trek, when Captain Kirk is beamed up via the transporter, the description seems to show that all his atoms are somehow separated and recreated elsewhere. Is this therefore a different Captain Kirk? The writers explored this in "The Enemy Within" where two Kirks are created in a transporter malfunction. In the Wizard of Oz, the Tin man is a lumberjack who has lost all his body parts bit by bit and has had them recreated in tin, and in the Hitchhiker's Guide, Marvin the Paranoid Android has had every part of him replaced at least fifty times (except for the diodes down his left side, which hurt). We should also remember that our cells are constantly being replaced and over a few years all are completely replaced with new ones, so in what sense are we the same person over time?
We know that Ancient Greek philosophers Plato and Heraclitus discussed the same problems and it is one of 'identity'. Do the parts that make up a thing constitute that thing and when they are changed or replaced, does that then make it a different thing? Or is the thing more than just the sum of its parts? is there something else, some essence or spirit added to the parts to make it the 'thing'? Variations on the theme exist, if the original brush and handle could be put back together would there now be two Trigger's brooms or would they both have new identities? Various solutions have been proposed by philosophers over the centuries but all have issues around them so there is no one definitive answer. The problem is that we create internal models of what something 'is' and that relies on human memory, consciousness, language and culture which are themselves transient and inconsistent. You can watch the wonderful, late, Roger Lloyd-Pack playing Trigger here: