This has come up in another thread, but I find it is something that needs more attention:
Why do non-theists engage in dicsussions with theists?
What is hoped for?
What is to be gained?
Do non-theists think that if they outargue the theists, the theists will leave them alone?
Do non-theists think that if they outargue the theists, the theists will admit the error of their ways and change?
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M*W: IMO, the two groups have little in common, but I don't think there is a problem with who engages whom in the discussion of belief and/or non-belief. I think it is more likely that it is theists who engage (i.e. provoke) atheists into discussions. Atheists already know the dialog. From the dialog on this forum, I think the theists are on a "scorch-the-earth" mission to convert atheists, and the atheists responses mostly point out theists' fallacious beliefs. As for atheists trying to convert theists to non-belief, that is not possible to do. Reaching the state of non-belief cannot be preached, pressured nor provoked. Becoming an atheist is a personal journey which usually requires many years of solid reading, research and revelation. However, to answer your question, "Why do non-theists engage in dicsussions with theists?", I suspect that it might be a desire to question or to learn more about the opposing position. In the case of atheists wanting to learn more about theists' viewpoint, it amounts to re-educating them. When theists engage atheists in dialog, it's because their aim is to convert us to their viewpoint. In short, atheists aim to re-educate and theists aim to convert.
"What is hoped for?", certainly not a mutual ground! I don't think there is one. Although it is possible to reach a mutual agreement on each one's right to his/her own belief, it is never right to force one's own belief on another. Atheists cannot be blamed for doing such a thing. There is nothing atheists can do to "convert" the religious. All we can do is re-educate those who have been lied to. We can't bring them to atheism nor is that our goal.
"What is to be gained?", I think mutual understanding of the opposition's viewpoint. However, most atheists I would think came out of christianity or some other religion, so our knowledge of the opposition's viewpoint is pretty well established. The truth is that atheists know more about christianity that its own adherents do! I am specifically using christianity as an example, because that's what I evolved from. Other than a mutual understanding, and hopefully a re-education, I don't think there is much more to be gained.
As we all can see on the Religion forum, both sides go 'round and 'round. One main difference is the approach to each side's belief. Becoming an atheist can take years and years, almost a life time of learning, but to become a christian (and most are born into it and, therfore, have no choice), all one needs to do is be slam dunked in a pool of water and voila they've gained entrance to a place that was never there! It's that instantaneous! So, which side carries with it the most logical and reasonable philosophy? Freedom from oppression, mis-education, and force fed lies is what is to be gained when the two sides converse.
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M*W's Friendly Atheist Quote of the Day:
"All thinking men are atheists." ~ Ernest Hemingway