On closer inspection though, more subtle notes emerge. Despite the dreams of formalists, the overwhelming majority of proofs in use and circulation are not minute low-level logical derivations, but rather intuitive high-level explanations. Some of which turn out to be false even after peer-review. Similarly, the level of details varies enormously across accepted proofs.
Those high level explanations are not what I understand as “proof” in the mathematical sense. I get that “please, back up your claims” is sometimes framed as “Where is the proof?”.
Proof is something that is properly axiomatised and that is purely mechanical where there is no ambiguity which formal moves are allowed and which ones are not.
Now the most common use case for a proof is “this proofs axioms seem reasonable so its theorems are true (of the actual world)”. This is a detailed backing of claims but it is not a proof in itself because there is no mechanical way to determine which axioms are reasonable.
Proofs preserve the certainty of their axioms but any axiom adopted means assuming more things and you can not get an axiomatic system going without assuming anything. Thus any axiomatic system must refer outside of itself on how solid its theorems are to apply elsewhere.
The form of reasoning where each step “seems reasonable” can be important but it is distinct from proofs.
To a very, very good approximation no proofs in actual published mathematical papers and books are “proofs” in the sense you describe here.
The intention is that they could be turned into such proofs without substantial creative effort, but it’s almost unknown for anyone to bother doing it and those who do typically find that it’s a lot of work.
Those high level explanations are not what I understand as “proof” in the mathematical sense. I get that “please, back up your claims” is sometimes framed as “Where is the proof?”.
Proof is something that is properly axiomatised and that is purely mechanical where there is no ambiguity which formal moves are allowed and which ones are not.
Now the most common use case for a proof is “this proofs axioms seem reasonable so its theorems are true (of the actual world)”. This is a detailed backing of claims but it is not a proof in itself because there is no mechanical way to determine which axioms are reasonable.
Proofs preserve the certainty of their axioms but any axiom adopted means assuming more things and you can not get an axiomatic system going without assuming anything. Thus any axiomatic system must refer outside of itself on how solid its theorems are to apply elsewhere.
The form of reasoning where each step “seems reasonable” can be important but it is distinct from proofs.
To a very, very good approximation no proofs in actual published mathematical papers and books are “proofs” in the sense you describe here.
The intention is that they could be turned into such proofs without substantial creative effort, but it’s almost unknown for anyone to bother doing it and those who do typically find that it’s a lot of work.
https://proofwiki.org/wiki/ProofWiki:Jokes/Physicist_Mathematician_and_Engineer_Jokes/Burning_Hotel
The diffrence between thinking you have a solution and having a solution. And indeed the former admits great degrees of uncertainty.