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Many well-known theorems have lots of "different" proofs. Often new proofs of a theorem arise surprisingly from other branches of mathematics than the theorem itself.

When are two proofs really the same proof? What I mean is this. Suppose two different proofs of theorem are first presented formally and then expanded out so that the formal proofs are presented starting from first principles, that is, starting from the axioms. Then in some sense two proofs are the same if there are trivial operations on the sequence of steps of the first formal proof to transform that proof into the second formal proof. (I'm not sure what I mean by "trivial")

You've hit on an area of research that's picking up some momentum at the moment. It involves connections between proof theory, homotopy theory and higher categories. The idea is that a proof or deduction is something like a path (from the premiss to the conclusion), and when you "deform" one proof into another by a sequence of trivial steps, that's something like a homotopy between paths. Or, in the language of higher-dimensional categories, a deduction is a 1-morphism, and a deformation of deductions is a 2-morphism. You can keep going to higher deductions.

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