Terence Tao on Nostr: I remember when this story of two high school students discovering one or more new ...
I remember when this story of two high school students discovering one or more new proofs of the Pythagorean theorem came out some years ago, but (frustratingly) without any substantive details on what these proofs were and why they were new. Now there is a published paper: https://www.tandfonline.com/doi/full/10.1080/00029890.2024.2370240
The question of defining precisely when two proofs are "the same" (or whether a proof is "new" compared to existing proofs) is actually a very subtle and interesting one (see some discussion at https://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/ or https://mathoverflow.net/questions/3776/when-are-two-proofs-of-the-same-theorem-really-different-proofs). Here the authors take a largely syntactic approach: they are considering a proof to be "trigonometric" if it avoids the use of circles (or coordinates), but uses angles in an essential way. With these restrictions, they do find at least five proofs that do not obviously resemble any of the standard known proofs, for instance one that involves summing a geometric series.
In ptinciple there may be "semantic" ways to distinguish these proofs from other proofs, in that there may be exotic variants of Euclidean geometry in which one of the proofs in this paper is valid but other proofs are not, or vice versa. But even without having such a semantic way to make this distinction, this was a fun read and a reminder that even the most ancient and well-established foundational results in mathematics can sometimes be revisited from a fresh perspective.
The question of defining precisely when two proofs are "the same" (or whether a proof is "new" compared to existing proofs) is actually a very subtle and interesting one (see some discussion at https://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/ or https://mathoverflow.net/questions/3776/when-are-two-proofs-of-the-same-theorem-really-different-proofs). Here the authors take a largely syntactic approach: they are considering a proof to be "trigonometric" if it avoids the use of circles (or coordinates), but uses angles in an essential way. With these restrictions, they do find at least five proofs that do not obviously resemble any of the standard known proofs, for instance one that involves summing a geometric series.
In ptinciple there may be "semantic" ways to distinguish these proofs from other proofs, in that there may be exotic variants of Euclidean geometry in which one of the proofs in this paper is valid but other proofs are not, or vice versa. But even without having such a semantic way to make this distinction, this was a fun read and a reminder that even the most ancient and well-established foundational results in mathematics can sometimes be revisited from a fresh perspective.