Sometimes an argument is impossible to resolve one way or the other, at least without new evidence. Often the best outcome is to Agree To Disagree, where each side acknowledges that the other has a different point of view and that, with most or all of the relevant information already on the table, continuing to argue serves no useful purpose.
The Hegelian Dialectic seems to be the main ingredient of growth on this Wiki. All exploration (and expansion of thought) occurs in the imperfect overlap among the views of the participants. It's "beneficent difficulty" again. Agreeing to disagree can be an expression of the need for rest, because constructive disagreement is Hard Work. So, Agree To Disagree, but not necessarily permanently.
Related is Aumanns Agreement Theorem (1976)
Aumann's agreement theorem, informally stated, says that two people acting rationally (in a certain precise sense) and with common knowledge of each other's beliefs cannot agree to disagree. -- en.wikipedia.org 
Some people (even on this Wiki) utilize "Agree To Disagree" to end arguments without having considered the available evidence (possibly ignoring it when it has been clearly presented or screaming "Book Stop" like a deranged troll). This is an abuse of this concept - it becomes a way of hedging, dodging, weaseling away from answering difficult questions. In this situation, you should NOT Agree To Disagree. When you Agree To Disagree, it says that you believe that the opinion of your opponent has an equal validity to your own (at least within the context of presentable evidence). Agree To Disagree only when you DO, in fact, agree that the argument is unresolvable (at least without new evidence) or where the argument is subjective (subject to a different point-of-view); otherwise you're being dishonest with yourself and all people in the audience.
Re: "When you Agree To Disagree, it says that you believe that the opinion of your opponent has an equal validity to your own". - Not necessarily. It could be that there is no agreed-upon way to test either option any further. Short of a mathematical proof in which there is no dispute about the givens, Agree To Disagree may be the only way to end an argument stuck in iteration.
When you Agree To Disagree, it tells everyone in the audience that the difference of opinion is unresolvable because it is either subjective or there is no evidence to clearly support one opinion over the other. It's a political thing. Agree To Disagree tells everyone else that you believe that the opinion of your opponent has an equal validity to your own... and Agree To Disagree says this REGARDLESS of what you intended Agree To Disagree to say. People will only see the words "Agree To Disagree", not your intent.
And Mathematical proofs are just as subject to the weaseling use of "Agree To Disagree" as any other: even if there are no disputes about the givens, some people simply can't be bothered to learn or understand the necessary mathematics to understand a formal proof, and they may attempt use "Agree To Disagree" to shore up their unsubstantiated belief as being equal (politically) to said proofs. That is exactly the sort of situation that calls for rejecting the "Agree To Disagree". Outside math, it also happens in science and history and a number of other fields, where a person will cherry-pick their evidence and blind themselves to any you present... thus creating breeding-grounds for pseudo-science, conspiracy theories, and new, cult religions. Reason and rationality will not win over a fundamentalist of any sort, but you should NOT Agree To Disagree when the primary reason you can't come to an agreement is that the opponent is using fallacy or sophistry to defend his position.
"Agree To Disagree" communicates a position of mutual respect. It should only be used where mutual respect exists.
What is the alternative, name-calling? That acheives no useful purpose.
Seek answers regarding alternatives below. And name calling, like violence, can achieve useful purpose (especially in the politics of convincing an audience). It's just not very civil or mature.
"Can", perhaps. Likely? No. The ratio of those who use name-calling skillfully to those who use it out of emotion and spite is very small.
Re: "some people simply can't be bothered to learn or understand the necessary mathematics to understand a formal proof"
Where has that happened on this wiki? For example, where has somebody used a formal proof to prove that language/tool/paradigm X is "better than" all others in a macro sense?
How should one turn down a proposition to "Agree To Disagree" when one cannot offer that much respect to the opponent's position? Obviously it can be done bluntly. Any suggestions on how to do so while remaining polite and civil? We need an alternative to "Agree To Disagree"... something that communicates (by just its title) that you can't even accept the premises or logic of the opponent, but also that the argument needs to end because it's going nowhere. Any ideas? Anyone?
Declared Truce? Argument Going Nowhere? Endless Quibble? Avoid Endless Quibble? Disagree But Tired Of Arguing? Cease Fire?
--- Might I be so bold as to suggest that Agree To Disagree** might to the trick? Thus we are done, but the reader is cautioned that the dispute is still an issue. It would then be up to the reader to decide if both sides are equal-- in the end they are the only ones who can make that judgment anyway.
I have I think a better idea. For years I have been using the radio 10 code '10-7' ( out of service ) to end pointless online dialogs without implying agreement. It is never clear to me if others understand the meaning or even the reference ( by way of full disclosure I am not a ham-- never have been-- but as a geek some of this information was absorbed ) which also served me well as the other party could not take offense. Of course if they did I had already made the decision that I did not care. So it stronger than goodbye, weaker than __ you. 10-7 is not a Wiki Word so I propose Ten Seven.
What makes your idea better than the other suggestions above? I like Cease Fire because I think it captures the essence of the situation better than anything else, and it doesn't assume any knowledge of ham lingo.
I guess I would relax the claim to personal preference. But there are some subtle differences:
Ten Seven is completely unilateral, which may be required in certain situations. Cease Fire will work for all cases were civility is still in play, but possibly at the cost of implying more respect than you care to do. There are some cases where it is in fact not reasonable to avoid insult at all costs. It is a last resort, and should not be used until just before, or perhaps even just after civility is no longer a useful option.
For example, after being insulted I might say -- ' I care no more what you think of my intellect, than if a cockroach should see me naked. 10-7.' I do not think Cease Fire will do it in this case. Now I do see that ceasefire is kinder, and more likely to preserve the relationship, and these are all good things. But I claim that they are not always useful.
My concern is that a offer of Cease Fire may be interpreted as surrender. In many cases you may not care, but for some kinds of issues you can not allow even that fig leaf. I certainly do not claim that Cease Fire should not be used, but to be used the other side needs to at least be respectable in polite company ( E.G not a Racist, not a Holocaust Denier etc) least you give away to much. Neither do I claim that Ten Seven must be used, and it is likely that on this wiki it may not be needed-- so much the better.
I'm with you on this one, Marc. There needs to be a way to indicate a disdain for continuing a pointless discussion (I'm not mentioning any names, but Top comes immediately to mind) without ceding any ground. Ten Seven indicates that you have made your point (Nuff Said) and you aren't willing to invest any more time discussing a matter that the other party is not discussing in good faith. Sad, but this is sometimes the case. -- Marty Schrader
I protest. See: Top Is Not Evil. --top
Yes, top. And on that page I agree that you are not evil. However, evilness was not the focus of this page in the first place; the need to dispense with troll feeding is. It is perhaps not smart to even mention it here for fear of this becoming Yet Another Useless Chat With Top that everyone is so tired of.
Evidence based on stated experience or anecdotes is hard to verify, sometimes leading to frustration because it cannot be directly studied and analyzed by the reader. It is "closed box" evidence, in a way. Agree To Disagree may be the only way out (i.e. when it is not practical to seek more experimental and verifiable evidence).
Contrast: Argumentum Ad Infinitum
See original on c2.com