Most of the comments in this discussion focused on topics that are emotionally significant for your "opponent." But here's something that happened to me twice.

I was trying to explain to two intelligent people (separately) that mathematical induction should start with the second step, not the first. In my particular case, a homework assignment had us do induction on the rows of a lower triangular matrix as it was being multiplied by various vectors; the first row only had multiplication, the second row both multiplication and addition. I figured it was safer to start with a more representative row.

When a classmate disagreed with me, I found this example on Wikipedia. His counter-arguement was that this wasn't the case of induction failing at n=2. He argued that the hypothesis was worded incorrectly, akin to the proof that a cat has nine tails. I voiced my agreement with him, that “one horse of one color” is only semantically similar to “two horses of one color,” but are in fact as different as “No cat (1)” and “no cat (2).” I tried to get him to come to this conclusion on his own. Midway through, he caught me and said that I was misinterpreting what he was saying.

The second person is not a mathematician, but he understands the principles of mathematical induction (as I'd made sure before telling him about horses). And this led to one of the most frustrating arguments I'd ever had in my life. Here's the our approximate abridged dialogue (sans the colorful language):

Me: One horse is of one color. Suppose every n horses are of one color. Add the n+1st horse, and take n out of those horses. They’re all of one color by assumption. Remove 1 horse and take the one that’s been left out. You again have n horses, so they must be of one color. Therefore, all horses are of one color.

Him: This proof can't be right because its result is wrong.

Me: But then, suppose we do the same proof, but starting with on n=2 horses. This proof would be correct.

Him: No, it won’t be, because the result is still wrong. Horses have different colors.

Me: Fine, then. Suppose this is happening in a different world. For all you know, all horses there can be of one color.

Him: There’re no horses in a different world. This is pointless. (by this time, he was starting to get angry).

Me: Okay! It’s on someone’s ranch! In this world! If you go look at this person’s horses, every two you can possibly pick are of the same color. Therefore, all of his horses are of the same color.

Him: I don’t know anyone whose horses are of the same color. So they’re not all of one color, and your proof is wrong.

Me: It’s a hypothetical person. Do you agree, for this hypothetical person—

Him: No, I don’t agree because this is a hypothetical person, etc, etc. What kind of stupid problems do you do in math, anyway?

Me: (having difficulties inserting words).

Him: Since the result is wrong, the proof is wrong. Period. Stop wasting my time with this pointless stuff. This is stupid and pointless, etc, etc. Whoever teaches you this stuff should be fired.

Me: (still having difficulties inserting words) … Wikipe—…

Him: And Wikipedia is wrong all the time, and it’s created by regular idiots who have too much time on their hands and don’t actually know jack, etc, etc. Besides, one horse can have more than one color. Therefore, all math is stupid. QED.

THE END.

To the best of my knowledge, neither of these two people were emotionally involved with mathematical induction. Both of them were positively disposed at the beginning of the argument. Both of them are intelligent and curious. What on Earth went wrong here?

^One of the reasons why I shouldn’t start arguments about theism, if I can’t even convince people of this mathematical technicality.