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The Pythagorean Proposition states that there are 367 proofs of Pythagoras' theorem. Is there a limit for the number of proofs of this, or any, theorem? Are there provable theorems with infinitely many proofs, or do all theorems have a limit on the number of proofs for that theorem?

Suppose Theorem A is provable. Is there a way to find out the number of proofs for Theorem A? Now, suppose there is proof that there is a maximum number of proofs for Theorem A. Can we go deeper? Can we find out the limit of the number of proofs that prove that there exist a limited number of proofs for Theorem A? Or is it just not possible at all?

Edit: I guess the question really is, 'What makes proofs different if they prove the same thing'?

Bernard
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    This is too vague. When are two proofs the same? I'd be surprised if there were $367$ different ideas in those proofs you mention, for example. – lulu Jul 22 '21 at 15:10
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    You could stretch out the length of a proof of the Pythagorean Theorem to an arbitrary length by wasting some pages proving the equation $2+2=2 \cdot 2$ as many times as you want, over and over again. – Lee Mosher Jul 22 '21 at 15:13
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    All theorems have an infinite number of proofs if you don't exclude trivialities. The question is how to define a triviality – Amr Jul 22 '21 at 15:16
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    @username For silly examples, suppose we just change the variable names. Would you call those different proofs? But there are infinitely many possible variable names. – lulu Jul 22 '21 at 15:17
  • This question only makes sense if you mean 'proof' in the strict logical sense of "concatenation of axioms according to the rules of logic that end with your statement". Then one needs to come up with a notion of "equivalence of proofs", which would involve replacing any variable names, deletion/addition of statements already proved, deletion/addition of true statements that don't affect the validity of the proof as a whole... Sounds like a pain, but probably the sort of thing that someone like Russell or Gödel has already done. – Tom Sharpe Jul 22 '21 at 15:22
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    @TomSharpe There is currently no truly satisfying notion of equivalence of proofs in general (although in certain restricted contexts the situation can be different). Relatedly, there is no (and arguably can be no) fully satisfying notion of when two algorithms are the same. – Noah Schweber Jul 22 '21 at 15:30
  • @NoahSchweber That should probably go as an answer to the question, at least in some kind of expanded form. – Tom Sharpe Jul 22 '21 at 15:33
  • @TomSharpe I agree, I just don't have time to expand it appropriately at the moment - in a few hours if nobody's answered this I'll do so. – Noah Schweber Jul 22 '21 at 15:52

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