P&M VII - Let Us Agree to Disagree

I strongly feel that whoever invented this "Let us agree to disagree" must have been an optimist. I think of him as an ivory tower type with a belief that real people actually could see that there was some validity in the other person's point of view and, though they believed in the opposite, they needed to be open-minded about the issue. As we all know, real people are incapable of seeing anything except from their own point of view.
Let us get one thing out of the way. Just as two of you are into the pith of an argument and you feel completely sure that you are going to rub the other person's nose in the dirt, there is always a third party busybody who pops in with a "Let us agree to disagree." In this case, there is no real 'us' here since Mr. Nosy Parker was never a party to the argument till he poked his proboscis in, thereby creating agreement in both your minds about how obnoxious he is. Be that as it may, we will concentrate on only those times when there is actually no 'agree' in your minds.
Of course, people have varied thoughts in their mind when they say this phrase and one cannot generalize about exactly what they mean. For example, you could be an ardent fan of Shahrukh Khan while your friend holds this insane belief that Salman Khan is God's gift to women-kind. You know, of course, that she is totally irrational on this issue. You, however, do not feel that this lunacy is about to exhibit itself in her taking the kitchen knife and slashing you into ribbons in the night. As long as you avoid taking her advice on what movies to see, her madness is harmless and, thus, you feel that you can brush aside this argument with a "Let us agree to disagree".
There are times, however, when you feel less inclined to be forbearing. In any corporate meeting you will find someone holding a view diametrically opposite to your own. Much as you may feel like grabbing him by the collar and shaking some sense into him, you are prohibited by the stupid rules of social engagement that are irrationally averse to settling arguments conclusively by physically silencing the opponent. You, of course, know that you can settle his hash logically over a three hour argument. The problem, however, is that your client is unlikely to see your victory over him as a sufficient return on the fees he is paying you (Clients are unreasonable that way). So, if it is a minor point and with a mental note to settle scores with him at a more appropriate time, you grit out, "Let us agree to disagree."
There are more serious issues, however, that crop up between people. When you are of the staunch opinion that the chicken came first and the other person insists that the egg did, there is what is called an impasse. Google is no help in conclusively proving your point and you just cannot let the other person go around propagating a wrong philosophy. The argument could go something like this
"How can there be an egg unless there is a chicken to lay it?"
"How can there be a chicken unless there is an egg to hatch out of?"
"How can the egg hatch unless there is an chicken to warm it?"
"You never heard of incubators?"
"You never heard of Darwin? Incubators came from men and men came after the chicken"
And so on and on into the esoteric realms of Science, Philosophy and any number of '..isms' without a conclusive end till one of you yells, "Let us agree to disagree."
And it is at that exact point that both of you are in total agreement - in concluding that the other person is an opinionated jackass who is incapable of listening to reason.
I am sure you are in agreement with me here. Otherwise, let us agree to disagree!