In order to write a valid proof today, it needs to follow a very specific structure and it needs to pass peer review. There are very high standards to pass through. However, there are proofs and theorems that were written thousands of years ago that are still considered valid today. While I understand that math is unchanging and that which is mathematically valid will always be mathematically valid, how do these ancient proofs stay structurally and logically valid? Is there a stable proofmaking tradition stretching back thousands of years? Is the modern proof structure based on the ancient structure? Were many ancient proofs retroactively struck down with the establishment of an organized system of theoretical mathematics or are theorems such as the pythagorean theorem simply "grandfathered in"?
What an interesting question!
Turns out, a lot of history’s constraints apply on the history of mathematics too. We may know that some theorems were being used in some form in other cultures, but we have to go by the first attested proof.
To take your example, we do know that Mesopotamia and India among other places were using some application of Pythagoras theorem but we do not know of a general case proof. The proof is attested to pythagoras through the writings of a later mathematician/philosopher. The technique he allegedly used is very much accepted - it’s called proof by rearrangement - a technique frequently used in teaching geometry. Later on a few more elegant, rigorous proofs were found and those are what we studied ( using similarity and constructing two right angled triangles within the bigger right angled triangle ) but that does not take credit away from earlier proof.
Then there is Euclidean geometry - which is pretty open and shut as it were. But turns out you can create completely new geometries by postulating differently. Thus, the work remained incomplete. Plus, he erred in assuming some things like existence of intersection points - which were corrected later on by Hibbert and finally Tarski.
Attempts at calculating Pi ( and thereby defining area of circle ), finding a formula for prime numbers etc are also example of ever evolving process ( and in these cases we have simply turned to the computers now ).
Hope this gives some insight into what you were looking for. :)
Hi, great question! I have hesitated between posting an answer as not a historian and potentially breaking the rules of the sub; and leaving the question unanswered. So instead I would encourage you to repost to /r/math if you want an answer from a mathematician's point of view.
>In order to write a valid proof today, it needs to follow a very specific structure and it needs to pass peer review.
well, speaking as a mathematician, the proof is valid even before it's been peer reviewed. Just a matter of whether the community accepts the proof or not. And 'specific structure' is in practice not very specific, in practice (it can depend on the audience the proof is for: a top expert speaking to a top expert can merely outline the key idea and convey the entire proof). I cannot comment on the historical aspects of your question.