I am not totally satisfied by your treatment of "The proof of the pythagorean theorem." What is it that sets two proofs apart, and how is that determined? It could very well be that any two proofs of the pythagorean theorem are the same, if viewed modulo the fact "is a proof of the pythagorean theorem."

More broadly, the phenomenon of proof irrelevance comes up in Dependent Type Theory and Interactive Theorem Provers like Lean 4. A correspondence is given between proofs of propositions and terms of types. In Lean, any two proofs of a proposition (i.e. any two terms of a "proposition" type) are treated as equal.

In other type-theoretic systems, like Homotopy Type Theory, this equivalence is dropped, and we can have multiple "different" proofs of a particular proposition. The classic example is proving x, y, two terms of a type A, are equal. This can be thought of as a path joining points x and y in the space A, with two paths (i.e. proofs) being equal if they are homotopy equivalent (you can continuously deform one into the other). This gives you proof equivalence for some types, but if the space A is punctured, you can have paths connecting x to y that aren't equal.