The problem with discussing politics (and "changing someone's -- rarely one's own -- mind) are at least partially rooted in two facts:
We usually start with the end-point of an opinion, not its source. If we are interested in why people think the way they do, it helps to know what they ultimately hope the achievement/goal of their opinion is.
Most of the time, we will find that they are significantly similar to our own. Already then, we realize that we are talking 'merely' about the "how", not, in essence, about the "what". Assuming good intentions, or simply asking about them, if we are not sure, helps such discourse immensely. Doesn't get rid of the ideological part that makes conversing about these topics difficult, but minimizes its influence.
Not that I'm not as guilty as most of us in being inconsistent in applying this, but I try to assume (whenever level-headed enough), that those who hold political ideas contrary to mine, that they still hold similar social ideas. They wish to live in a society worth living. (Although even that, of course, can be an element of self-rationalization... along the lines of the patronizing idea: "s/he wants the same thing, it's just that I know better how to get there.)
The other aspect is the fact that politics is inseparable of ideology and our idea of self. In that, it's much closer to religion than science or music. If we expect someone to change their opinion, we are essentially asking them to change their idea of self. But the idea of self is deeply ingrained in us, obviously, and expresses itself in the way we dress, the kind of people we surround ourselves with, which coffee shop we go to, what music we listen to or profess to like. It's an immodest goal, to say the least, to ask someone else to change any aspect of that... especially knowing how we are usually loath to do the same.
And yet, from civilized discourse about such essential matters rises our ability as societies (be it a community or a country or a forum like this... all heterogeneous in different ways) to grow. And, perhaps, even to avoid mistakes. And no matter how much we disagree on such matters, at least in this forum we can all agree, in the end, that Bach is the greatest composer and that anyone who doesn't is a degenerate dummy with tin cans for ears.*
*
P.S. And yes, I am aware that this is essentially only rephrasing what others, for example like Mirror Image in his post just above, have said elsewhere in this thread.
