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If X and Y are one and the same object, they are said to be numerically identical even though people may not know that they are. Numerical identity is often contrasted with Qualitative identity which describes that X and Y share share all the same properties excepts for their relational properties. (Properties are characteristic, attribute to qualities of objects, including abstract properties such like fame, memory, ability...)

An example of two things that are numerically identical: Superman and Clark Kent. Even though they typically wear different clothes, behave very differently, and people don't know they are the same person, they are in fact the same person.

The ship of Theseus might be used to argue that an object would still be that same object over time despite changes (though whether it actually shows this is debated). According to Plutarch's description of this philosophical challenge,

"The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much 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."

— Plutarch, Theseus[1]

One might argue that the two ships at different times (before and after replacement of all the parts) are numerically the same, even though the parts of the ship have all been replaced with new parts. The way people argue for this varies, but one example is that the ship occupies a continuous spatio-temporal trajectory through time.

People could make a similar argument that a person as a child and a person as an adult are the same person. Because human body cells are constantly being replaced, overt time most of the body cells which made up the person as the child would be gone. But people may still argue that even though all the cells have been changed (like the planks of the ship), they are still the one and the same object. That is, the child and the adult are numerically identical, perhaps also because there is a continuous spatio-temporal trajectory through time.

[1] Plutarch. "Theseus". The Internet Classics Archive. Retrieved 2008-07-15.