[abdullahkhalids]

Mathematical proofs are checked by noisy finite computational machines (humans). Even computer proofs' inputs-outputs are interpreted by humans. Your uncertainty in a theorem is lower bounded by the inherent error rate of human brains.

[griffzhowl]

I agree we can't be absolutely certain of anything (maybe none of our memories are real and we just popped into existence etc.)

But we can be more sure of the deductive validity of a proof than we can be of any of the claims you make in these sentences, so I don't think they can serve to establish any doubt. If we're wrong about deductive logic, then we can only be more wrong about any empirical claims, which rely on deductive logic plus empirical observations

[goatlover]

Plenty of mathematical proofs have been proven true with 100% certainty. Complicated proofs that involve a lot of steps and checking can have errors. They can also be proven true if exhaustively checked.

[naasking]

> Plenty of mathematical proofs have been proven true with 100% certainty

Solipsists would like to have a word with you...

[griffzhowl]

Tell them to mind their own business ;

[drdeca]

This may be, but not, I think, in a way that is particularly worth modeling?

When we try to model something probabilistically, it is usually not a great idea to model the probability that we made an error in our probability calculations as part of our calculations of the probability.

Ultimately, we must act. It does no good to suppose that “perhaps all of our beliefs are incoherent and we are utterly incapable of reason”.

[hedgedoops2]

You're saying maybe people have mistakenly accepted incorrect proofs now and again, so some theorems that people think are proven are unproven. I agree that this seems very likely.

In practice when proofs of research mathematics are checked, they go out to like 4 grad students. This isn't a very glamorous job for those grad students. If they agree then it's considered correct...

But note this is just the bleeding edge stuff. The basic stuff is checked and reproven by every math undergrad that learns math. Literally millions of people have checked all the proofs. As long as something is taught in university somewhere, all the people who are learning it (well, all the ones who do it well) are proving / checking the theory.

Anyway, when the scientific community accepts a bad proof what effectively happens is that we've just added an extra axiom.

Like when you deliberately add new axioms, there are 3 cases

- Axiom is redundant: it can be proven from the other axioms. (this is ... relatively fine? we tricked ourselves into believing something that is true is true, the reason is just bad.)

This can get discovered when people try to adapt the bad proof to prove other things and fail.

Also people find and publish and "more interesting", "different" proofs for old theorems all the time. Now you have redundancy.

- Axiom contradicts other axioms: We can now prove p and not p.

I wonder if this has ever happened? I.e. people proving contradictions, leading them to discover that a generally accepted theorem's proof is incorrect. It must have happened a few times in history, no?

o/c maybe the reason this hasn't happened is that the whole logical foundation of mathematics is new, dating back to the hilbert program (1920s).

There are well known instances of "proofs" being overturned before that, but they're not strictly logically proofs in the hilbert-program sense, just arguments. (Of course they contain most of the work and ideas that would go into a correct proof, and if you understand them you can do a modern proof)

e.g. https://mathoverflow.net/a/35558

Cauchys proof that, if a sequence of continuous functions converges [pointwise] to a function, the limit function is also continuous (cauchys proof only holds for uniform convergence, not pointwise convergence - but people didnt really know the difference at the time)

- Axiom is independent of other axioms: You can't prove or disprove the theorem.

English doesn't have a "I'm just hypothesizing all of this" voice, if it did exist this post should be in it. I didn't do enough research to answer your question. Some of the above may be wrong, e.g. the part about the 4 grad students. One should probably look for historical examples.