I think mismatched-axioms misunderstandings happen all the time and it really is a useful mental model to ask people about their axioms. In fact, I think it's e

I think mismatched-axioms misunderstandings happen all the time and it really is a useful mental model to ask people about their axioms. In fact, I think it's even helpful when people are arguing in bad faith. If they have to state their axioms you pin them down and there are then at least some things they can't redefine on the fly. I think even really high profile debates suffer from mismatched-axioms misunderstandings, such as the abortion debate. For instance, I think a lot of people have very different definitions of what constitutes a person. You're probably not going to have a meaningful discussion about abortion if you disagree about that, you probably have to argue about the definition of personhood first. I think your characterization of maths is spot-on and most people don't understand what "mathematical truth" means. But my impression is that the 2+2=5 guy understands all of this (I think he was a maths PhD student too) and his point was completely valid, including its application outside of mathematics. The reactionaries who caricaturized his argument, however, don't understand any of this and believe he was making a different point entirely, namely that all truth is relative and that every person's point of view is valid. Of course, this misunderstanding is on them.