Source: RJ Lipton blog, Oct 2019
Proofs in mathematics and not just formal arguments that show that a theorem is correct. They are much more. They must show why and how something is true.
They must explain and extend our understanding of why something is true. They must do more than just demonstrate that something is correct.
They must also make it clear what they claim to prove.
A difficulty we felt, then, was that care must be given to what one is claiming to prove. In mathematics often what is being proved is simple to state. In practice that is less clear. A long complex statement may not correctly capture what one is trying to prove.
a great proof is a proof that helps create new proofs of something else. …
What mean is a great proof is one that enables new insights, that enables further progress, that advances the field. Not just a result that “checks” for correctness.