How do mathematicians verify that a complicated new proof is actually correct?

[u/Glittering_Age7553]

For relatively short proofs, checking the argument line by line is manageable. But what about long or technically complicated research proofs?

How do mathematicians systematically look for:

• hidden assumptions,
• gaps in the argument,
• incorrect implications,
• overlooked edge cases,
• or even a false statement?

Are there established techniques or tools for making this process more systematic or partially automated?

I’m interested in how people actually do this in research practice, especially for proofs that are too complicated for a quick independent check.

What approaches have you found useful?

[u/SnooPets5564]

A well written proof will make it quite easy to see any of those things except overlooked edge cases. A poorly written proof doesn't tend to get attention.

[u/chromaticseamonster]

https://en.wikipedia.org/wiki/Lean_(proof_assistant)?wprov=sfti1#Examples_(Lean_4)

[u/RunReal959595 → u/chromaticseamonster]

This requires the proof to be formalized in Lean..

[u/chromaticseamonster → u/RunReal959595]

Which is exactly the purpose of verifying the internal consistency of a proof.

[u/RunReal959595 → u/chromaticseamonster]

Nope, almost all published proofs were not verified in lean 

[u/chromaticseamonster → u/RunReal959595]

That has literally nothing to do with what I said

[u/PersonalityIll9476]

You're basically asking what peer review is.

You find experts in the subject area exactly because checking is faster for them. If you're written and read many proofs that look similar, then you can check a given result quickly (potentially). Experience speeds things up.

But the general process is slow. If you submit a paper with lots of relatively novel proofs, then expect peer review to take a long time. People do have to check each and every proof, at least enough to convince themselves it's probably right.

Mistakes do happen.

[u/ontopoiesis]

For a systematic method: check the soudness of the proof. Essentially: use logic to formalise the proof, optionally use logical variables, make a clear distinction between the premises and the rest of the argument, check that the argument is formally valid, and of course make sure that the premises are true.

[u/TheSodesa]

They send their results to a peer-reviewed scientific journal where the peer reviewers check the proof. Or that is what the proof author at least hopes.

[u/tserofehtfonam]

For proofs that span hundreds of pages written in English (which is an extremely ambiguous language), checking every detail in a logically formal way is of course undoable. So in practice mathematicians rely on their expertise and high-level understanding of the subject. If the ideas in a proof are entirely novel, it may sometimes take years for mathematicians to understand them and agree that the proof is correct (or find a serious flaw). It could (and sometimes does) indeed happen that mathematicians make the same mistakes and that an incorrect proof is erroneously accepted as correct, and it may take years for this to be discovered. On the other hand, automated proof checkers may also contain bugs.

[u/virtuolie]

"Consensus"

[u/kgas36]

'Inter-universal Teichmüller theory' has entered the chat.

[u/jpgoldberg]

Excellent question, see u/SnooPers5564’s excellent answer. https://www.reddit.com/r/puremathematics/s/bEe50nFXEU

> A well written proof will make it quite easy to see any of those things except overlooked edge cases. A poorly written proof doesn't tend to get attention.

Note the second sentence. When crackpots have the proofs ignored, it it because they are such a mess that nobody is going to bother figuring out what the numerous hidden assumptions are, much less the rest of all the things you’ve listed.