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There are around 370 different ways to prove the Pythagorean Theorem, but what does that exactly mean? For instance, if your proof states that $x^2+y^2=z^2$, I could construct a different one by claiming that $a^2+b^2=c^2$. On the other hand, some proofs just take a different turn or a shortcut at one step (often arguably the same), others could almost be considered corollaries of more generalised ones but, still, are considered to be disjoint.

I hope we can at least agree that changing the names of the variables of a proof wouldn't make it different, but then here goes my question: where do you draw the line? Is there some sort of equivalence relation that classifies the types of proofs or is it just something vaguely subjective, all in all?

nuuusxd
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    Usually, when people say that a proof is different to another one, this is a subjective statement. My understanding is that there is some active interest amongst logicians about coming up with a precise formulation for what it means for two proofs to be distinct. My suspicion is that none of these precise formulations will entirely capture the intuitive notion, however. (Which is not to say that this a fruitless activity.) – Joe Nov 28 '23 at 11:21
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    Once I started reading a book on homotopy type theory. Somewhere in the first few chapters there was a precise formulation/explanation of exactly this question. I only have some very rough memory about this - statements (types) are considered as some sort of spaces where equality proofs work like "paths" between points in them, so that two proofs are equivalent if their paths are homotopically equivalent. Anyway, the Q seems like a dupe of https://math.stackexchange.com/questions/1242043/when-are-two-proofs-the-same ? And also https://math.stackexchange.com/q/67954 ? – Al.G. Nov 28 '23 at 11:54
  • @Al.G. oh my, that's rougher that what I thought... Anyway, thanks for the references :) – nuuusxd Nov 28 '23 at 12:10
  • And the same question on MO: https://mathoverflow.net/questions/3776/when-are-two-proofs-of-the-same-theorem-really-different-proofs – Al.G. Nov 28 '23 at 12:46

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