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The Wikipedia article is ambivalent about this, stating "...Since then the proof has gained wider acceptance, although doubts remain". The MathWorld entry isn't much more reassuring.

This is notwithstanding the rigors of the mechanical proof by Gonthier.

Question: Has the Four-Color Theorem been accepted or are there still reasonable doubts?

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Yes, it is completely proven, however part of the proof entails a computer program that goes through a few thousand special cases. Without that program the proof would be incomplete.(*)

Now, some mathematicians at that time said (or even now say), proofs should be manually verifiable, therefore they did not accept this proof. Their reason for that is that it is known that computers DO miscalculate some very-very little amount, so they cannot (or at the time they could not) give full 100.00000...% certainty.

In my view though this should be put in contrast with human logic errors on verifying proofs: we can't be sure that we currently don't have false theorems. (In fact, some human-proven theorems already turned out to be false in the past others will come. ;) ) Also, there are such complex theorems now which can barely can be verified by other than the author, see the famous Poincaré conjecture, where Perelman refused to accept his price as nobody has rigorously checked this entire proof(!).

(*)(Actually, there are more proofs but every single proof relies on computers from a certain aspect: either special cases are checked OR the logical proof is a program, so you must trust the proof checker program's code. =) )

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    $\begingroup$ To be honest, I would rather trust a calculator than a mathematician with verifying few thousand cases... $\endgroup$
    – N. S.
    Dec 11, 2012 at 5:03
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    $\begingroup$ This might be a better reference than the yahoo answers link. The reasons for Perelmans refusal to accept the prize seemed to be motivated by different reasons. Mostly because some people checked the whole proof, made it more explicit and then claimed they proved the conjecture. There are complete expositions that mahematicians working in the area can check fully. $\endgroup$
    – Michael Greinecker
    Dec 11, 2012 at 7:00
  • $\begingroup$ So the currently accepted proof for FCT is probabilistic, and thus most likely true? $\endgroup$
    – TheEnvironmentalist
    Jul 17, 2017 at 23:17
  • $\begingroup$ no, it is proven by computers by exhaustive search. But if you don't accept a computer program as proof, then for you it is unproven. :) As much as computers do make mistakes as much it is unproven. My point is that humans also make mistakes so you cannot ever know for sure if something is proven or not. $\endgroup$
    – Barney Szabolcs
    Jul 25, 2017 at 8:30

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