If you wish to publish a result that contradicts an already published one, that automatically means that one of them is wrong. By putting forward your own work for publication, you are saying that you believe it to be correct, so you believe the existing published work to be wrong. That is an extraordinary claim (in the literal sense of "out of the ordinary"), which requires extraordinary evidence. It would be very hard to publish such a result without finding the error in the existing work or finding a counterexample to it.

@DavidRicherby isn't it a statement of the completeness theorem that sooner or later a set of axioms will lead to a self-contradiction? Having two correct results that contradict each other might be a manifestation of that.

The completeness theorem states that an axiomatic system can be complete (you're able to prove every theorem constructed in it) or consistent (a theorem can only be true or false but not both). But we cannot build an axiomatic system that is complete and consistent.

@LLlAMnYP No. Completeness theorems say that, for every X, we can prove at least one of X and not-X. Most theories we're interested in are powerful enough to be incomplete, meaning that there are statements X such that we can prove neither X nor not-X (assuming we can't prove both, which would be a huge problem because, then, we can prove literally everything). But, here, we have a purported proof of both X and not-X.

Note that the question is about mathematics, not about science.

My point is that there's no guarantee that an accepted axiomatic system is consistent. (I understand that you are saying that if not complete, we usually use consistent systems?)

@LLlAMnYP Sure, but if you want to claim that mathematics is inconsistent, that's an OMG-huge-catastrophe-end-of-the-world kind of claim, so you need really, really, really, really, really, really, really, really good evidence. I mean, really good.

I need more caffeine to think about mathematics. I'm not sure if ZFC is consistent. I thought it was. But yes; if the axiomatic system is NOT consistent then it is possible to get these contradicting results with BOTH being correct.

David, that the axiomatic system isn't entirely consistent does not mean that all of mathematics, etc goes out the window. Also, it's a little unclear what result is being contradicted here: if it is a mathematical result or scientific one. Regardless, math proofs can be wrong. There's no need NOT to move forward. But again, I gave advice on how to move forward.

This answer does not fit the context of mathematics. Note that the point is not that we all believe the original paper is correct, we believe that the question who is correct needs to be settled before another paper is published.

It really doesn't, especially if it cannot be. This suggests that if the conflict cannot be reconciled, we should just fail to bring to light this new piece of information. That's absurd.

I updated my answer to explain why one should not necessarily wait until AFTER it has been shown who is right.

If the conflict cannot be reconciled at all, we are talking about an earth-shattering, completely different paper. If it is just the case that the author cannot manage, then they certainly should communicate that problem. However, this communication shouldnt be done via a published paper (not at least because one wouldnt want to wait the couple of years this takes).

Maybe it can be, but neither party knows how. Regardless, that's another reason to publish. Once the academic community is aware of the conflict, THEY can work to figure out why. Otherwise it's just two people working on it.

This is a sidetrack, and it is silly to wait until you figure out which is wrong, s long as you've done your due diligence to make sure that your proof is not wrong.

@DanielGoldman "and it is silly to wait until you figure out which is wrong, s long as you've done your due diligence to make sure that your proof is not wrong." I disagree. One should do "due diligence" to make sure that every proof that one publishes is not wrong. If one wishes to claim that existing work is wrong, the bar is set higher. That doesn't mean you can't publish until the contradiction is resolved. But it does mean you need to do a lot more work than you would for a "regular" paper.

David, I have added a rebuttal. As your comment stands, you are suggesting that we turn academia and turn it into a cult.

In mathematics, the base assumption is that a published result is correct. I understand this is not the case in most other academic fields, where either correctness is not so clear, or where establishing it requires far more effort than in mathematics.

Yes; in ALL fields the base assumption is that what is published is correct. However, that does not mean that it is reasonable to make something else which contradicts it HARDER to publish. That turns whatever is published into doctrine. To say that one should refrain from publishing until he is absolutely certain who is wrong is actually quite... disgusting. It takes academia and turns it into a cult.

The advice given in the answer by @Gro-Tsen allows for Simpson's paper as an exception: his proof is a limited counterexample, with a proof much simpler than that in the original paper. (That Simpson also gives a conjecture as to how much the original paper's result is true is a bonus.)

@DanielGoldman It has nothing to do with doctrine. Work that is published in mathematics is published with proof. That is, when mathematics is published, it comes with a convincing argument that it is correct. It's not "I will not believe you because you contradict what I already believe" but "I will not believe you unless you can convince me that you're right. Part of convincing me that you're right is to convince me that the guy who convinced me of the opposite is wrong."

@DavidRicherby, my advice included the suggestion that due diligence be used in making sure there are no errors in the new result. In other words, both results should, in theory, have the same level of justification. You are saying that because the new result, call it B, came after the existing result, call it A, B should be held to a higher level of scrutiny.

In other words, you are suggesting that older results be given higher prior probabilities, which is exactly what doctrine, dogma, etc is. You are putting more weight on that which is already "accepted as true" simply because it is "accepted as true."