对不起 literally means “sorry”. It`s a formal and straightforward way to admit you made a mistake and apologize for it. It`s often used when you’ve made a mistake, caused inconvenience, or hurt someone’s feelings. The tone is sincere, can be used in both serious and casual contexts...
…Two weeks ago I quoted Jasmin Blanchette, a Dutch computer scientist who complained about “proofs [that] look more like LSD trips than coherent mathematical arguments.” I thank Alex Best for informing me that the expression goes back to the very interestingTeaching Statementthat Scott Aaronson...