(1) A=A . . . Why a definition may not help.
(2). What is it to be the same thing over time? Forget you and me: what is it for any sort of thing, a table, a boat, a river, a car, to be the same over time?
(3) Your car is linked by a series of spatio-temporal events over time from its manufacture in Detroit to its sitting in your driveway So, continuity over time appears to be key.
(4). But take another example: the New York Yankees or the Seattle Mariners. Say they are playing each other and then there's a baseball strike and to keep playing each team calls up players from the Minor Leagues and resume their series where they left off. Are the same two times still at play? Perhaps the resumption of the series should begin after the baseball strike?
(5) Locke's Oak Tree. An oak tree sits in your front yard when you are young and it is still there when you inherit the house and come back to live in Yonkers when you retire from the dry cleaning business. Same tree? The cells, molecules sap and leaves which have come and gone have all changed. The organic material of my body (except my brain) is completely replaced roughly every seven years and now that neuroscience has shown that the brain can, after all, regenerate, the same may hold true of the brain as well. Locke says, the oak tree, this oak tree, has all the while been "partaking of the same life" Its identity might more simply be called "an organizational unity." So what do you think? Is Oak Junior the same tree as Oak Senior?
(6) The ship of Theseus: How much change can the identity of a thing tolerate? Theseus goes on a long sea voyage and various bits of his ship need replacing, some planks, the rigging, then the sails, until the whole ship has been replaced with new materials after three years at sea: it sails back into port. The same ship? Say an early entrepreneur has been trailing Theseus' ship and picking up the pieces and he reconstructs the ship and docks it down the coast and sells tickets to the real (ancient) ship of Theseus. Who has Theseus' ship? Theseus or the entrepreneur? See THESEUS' SHIP