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'Equals' has more than one meaning in math

There are some pretty nebulous concepts in mathematics that can be hard to wrap your head around, but the meaning of 'equals' was one we thought we had covered.

It turns out that mathematicians actually can't agree on the definition of what makes two things equal, and that could cause some headaches for computer programs that are increasingly being used to check mathematical proofs. However, taking canonical isomorphism to mean equality is now causing "some real trouble", Buzzard writes, for mathematicians trying to formalize proofs – including decades-old foundational concepts – using computers. He thinks the incongruence between mathematicians and machines should prompt math minds to rethink what exactly they mean by mathematical concepts as foundational as equality so computers can understand them.

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