Good point! I'll try to think about a better way to phrase this (happy to hear suggestions)

"Although mathematicians have been able to prove this, the proof is too complex to verify without computer assistance."

Or some such.

Nah, actually I agree with you. What counts as believe and what as fact is rather abitrary. Is 2+2=4 a fact? Is global warming a fact? What about man-made global warming? Ask 100 people whether something is a fact or a believe.

To top that up, it's fact that there have been "proves" that were wrong (or maybe that's just my believe? :^]) even for a long time.

Hence, I think we can say that there are 4 options for a theorem:

1) Some mathematician believes the theorem is correct (but can't prove it)

2) Some mathematician believes the theorem is incorrect (but can't prove it)

3) Some mathematician believes the proof of a theorem is correct

4) Some mathematician believes the proof of a theorem is incorrect

Proving that a proof is correct is kind of meaningless. At that point it's all believe anyways.

^ Exhibit A why using "believe" is a bad choice of words.

Mathematical poofs are either correct or false. There is no middle ground.

Well.. there is. Middle ground being a very complex, but somehow convincing argument that no one can reasonably check. There was one of these cases in number theory some years ago, can't remember the details. Proofs can be only true or false, but accepting proofs is in the end a social process.

A convincing argument that cannot be checked is not a proof. If you want to extend the definition of proofs you're welcome to do that, but for academic mathematics the meaning of proof doesn't contain a middle ground.

Why would it not be a proof?

What is your criteria of "can be checked then"? If a proof for "sqrt(2) is not a rational number" can't be checked by a 5yo, it's still a proof no?

A couple come to mind

* The proof of the classification of simple groups[0]

* The work on topological four manifolds by M. Freedman [1]

[0]: https://en.m.wikipedia.org/wiki/Classification_of_finite_sim...

[1]: https://news.ycombinator.com/item?id=28471159

> Proofs can be only true or false

Yes.

The fact that we don't know the truth doesn't mean there isn't one.

Brilliant episode of the BBC’s In Our Time that goes down this rabbit hole. https://www.bbc.co.uk/programmes/b04v59gz

s/believe/know

That Coq proof is not "without computer assistance". No Coq proof is, as Coq is literally an "assistant" that runs on a computer.

And those many jobXtoY.v and taskXtoY.v files sure look like they also do the same as the Appel and Haken proof, namely enumerate lots and lots of cases that are then machine-checked. So I don't think the computerized Coq proof is really qualitatively different from other computerized proofs that enumerate so many cases that a manual check would be impractical.

I would consider a proof to be a "repeatable argument". One you could show to someone and expect them to be convinced by it. I think it is a defensible viewpoint that a proof in coq is not 100% convincing. If you think otherwise, then how can you reconcile the existence of falso (a coq verified proof of "false")?

https://github.com/clarus/falso

Proof and belief I think are pretty strongly intertwined, but I'm not going to pretend to have a particularly rigorous philosophy on the matter. Similarly, when the proof of Fermat's last theorem was published, I don't know if I should consider that to be a proof because it is well beyond my comprehension. I have no reason to question it, but should I consider it a proof? I know that people smarter than me (e.g. Wiles) thought the original version of it was a proof, but it had a subtle error in it which required a fix. While I haven't looked at the proof and revision, I would be surprised if I could look at the two versions as labelled and tell which one is the correct version.