The fact that the audience is so far is part of why I think that the attitudes of skepticism are more important than any particular set of facts. But it is als

The fact that the audience is so far is part of why I think that the attitudes of skepticism are more important than any particular set of facts. But it is also important that you make it easy to follow your explanation without having to challenge emotionally held beliefs. For example, consider those religious beliefs. A big part of why we hold them is that trying to give up those beliefs involves accepting a lot of pain. Both from external causes, like the condemnation of your religious community, and internal ones, like the gut level belief that even thinking about rejecting Jesus puts you at risk of going to Hell. It is very hard to engage in quiet contemplation while feeling that you're risking eternal damnation. As for where you tell people what to believe: And indeed it is a counter-example to the idea that mathematical truths are grounded in actual objective reality, but that is not news -- we already established that with the example of negative numbers and imaginary numbers. You claim to have shown that the ideas were not grounded in actual objective reality. But that isn't true. You only showed that they aren't grounded in the first way that you tried to ground it. Both of those examples are things that can easily be grounded in objective reality in other ways. But the Banach-Tarski construction is different. It depends on the axiom of choice. Which in turn is a statement about a process that cannot be actually done in objective reality. And without being able to do that process, we're unable to do Banach-Tarski in objective reality. I think that's an important difference. And it is one that is missed when you tell them that this is like that other thing which you just saw.