Comment by openasocket

You can think of the formal proofs that mathematicians write as essentially pseudocode, as compared to Lean which is actual code that can be compiled. As an example, it would be common to say “2*x + 4 = 0, therefore x is -2”, in a mathematical proof. But in a computer-readable proof, you have to actually go through the algebraic manipulations, citing what you are doing and what theorem/lemma/rule you are applying. Modern systems like Lean make that much easier, but that’s the essential difference.

Advanced proofs essentially just consist of a series of assertions “X, therefore Y, therefore Z, …” and the justification for that isn’t always laid out explicitly. As a result, when you read a formal proof, it often takes some work to “convince yourself” the proof is valid. And if a proof has a mistake, it’s probably not is one of those assertions, but rather in how you get from assertion X to assertion Y. It can often be really subtle.

Disclaimer: I have a math undergraduate, and have played around with the theorem proving language Coq, but haven’t worked with Lean