The author presented a reflection on the concept of essential equality of proofs, particularly focusing on the question of when two proofs are considered the same. They provided examples of different-seeming proofs that are actually based on the same idea, such as the factorization of numbers and the irrationality of √2. They also explored the idea of conversion techniques to transform one proof into another. The author posed questions about whether essential equivalence is always present between proofs and whether it is possible to demonstrate that two proofs are genuinely different. They invited readers to share examples and insights on this topic.
https://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/