[qubitly]

Reducing two mathematical proofs to being 'essentially the same' just because they reach the same conclusion overlooks something crucial: each proof isn’t merely a path to a result but a unique expression of understanding. A proof has its own logical and conceptual structure, and that structure isn’t interchangeable without losing some of its inherent value. Comparing proofs shouldn’t just focus on a shared outcome: the path taken, the relationships it establishes, and the concepts it explores are as fundamental as the conclusion itself. Perhaps it’s time to view mathematics not just as calculation, but as a real act of knowledge that in its diversity deepens our grasp of reality

[samatman]

But where is the line?

Proof A and Proof A' are identical, except that Proof A says "and therefore" where Proof A' says "and so we see that". Different proofs?

Proof A'' is a faithful translation of Proof A into French, is it now different? Or is it a trivial translation of the same proof into different language?

This is, in fact, the topic of the Fine Article. The layman (myself included) sees easily that proof is something more durable than the exact words chosen, or even the language the proof is written in. Mathematicians (and patzers such as yours truly) will tend to view trivial transformations of a step in a proof, or trivially equivalent tactics, as resulting in the same proof.

What makes such a transformation trivial? Good question.

[smfjaw]

computational reducibility/irreducibility is a big topic in computer science and is incredibly interesting. It allows us to prove that certain "computers" are the same through proofs and that they can carry out the same tasks regardless of the actions that take place within them. I would suggest looking into that as it really opens your eyes to just how similar computationally so many things are

[red_trumpet]

The result of a prove is a theorem. I don't see any claim in the article that any two proofs of the same theorem are essentially the same?