I am looking book recommendations that offer a mathematical framework for answering the following questions. Given a problem - i.e a (mathematical) theorem to be proved or a (mathematical) computation to perform - and a reasonable definition for what counts as a proof or computation (p/c), can we answer such questions as:
- Given a p/c, can we prove that a shorter p/c exists without necessarily constructing it? Can we prove that our p/c already is the shortest proof? If we can prove there are shorter proofs without constructing them, can we prove the (non-)existence of shorter proofs still?
- Can we prove there are (or are not) finitely many proofs/computations that solve a problem, without constructing them? If yes, can we rank all of them in terms of length?
- Is there a way of quantifying the notion of “difficulty” of a p/c? For example one p/c may be shorter than others, but may only be understood by relatively fewer mathematicians, so we may suspect it to be a more “difficult” or more “abstract” p/c. Is there a way to make this idea precise? Can we rank “difficulty” of proofs or find relationships between p/c length and p/c difficulty?
