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Random anecdote that this reminded me of:

In Polish an ID card is called literally "proof of identity". The conversations happened in high school.

A: Do you have the proof?
B: What proof? (Thinking: proof of what theorem?)
A: Well, proof of identity.
B: Which identity?
A: Yours!
A: What is my identity? (Thinking: what was the mathematical identity that someone would associate with me?)

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