While I understand the instinct to physically fight with someone over an argument, it seems not so reasonable to solve a dispute based on ideas with a physical fight, because the fight does not prove an argument more or less valid.
Since pistol duels in the past were used also by educated men, I wonder why those were considered "legit", instead of, say, using a judge or a court.
The very short answer is that the duel is about honor - a somewhat amorphous term that boils down to a mixture of respect, 'face', and masculine identity. In placing yourself on the "Field of Honor" you show to society that you are willing to put your life on the line, that proving your honor is literally worth more than life itself. You couldn't simply put your honor into the hands of others (the courts) to judge, a man had to be the defender of his own honor. For the longer answer, well, I hope you'll excuse me for punting you elsewhere, but dueling is something I've written about a good deal before and I just don't have the time to do this question proper justice, so would recommend you check out this list, with specific attention to this answer which talks a good deal about honor and the impulse to duel.