Part of me finds this sort of thing a mite insulting. What's more likely: that in two and a half thousand years mathematicians have not worked out the kinks in mathematics this basic, or that you don't understand what you're talking about?
But I share that fascination too, and I've spent quite a while trying to puzzle out why laypeople hold the odd mathematical beliefs they do. I think the general and usually profound ignorance that most people have of mathematics, born of having never been taught how to do mathematical reasoning (or even really that such a thing as "mathematical reasoning" exists), manifests in two ways here.
One is that people fill in the gaps by resorting to meaningless word games or rigidly adhering to fixed rules they don't understand, akin to a mathematical cargo cult (like Matt McIrvin (npub12gn…d572)'s teacher insisting that "an absolute value can never be a negative" instead of actually thinking about the statement she wrote down).
The second is that people seem to think, even unconsciously, that mathematics is fundamentally arbitrary; that the standards by which mathematicians evaluate their claims are just a matter of subjective opinion and thus ignorable when those standards and their intuition come into conflict, as they frequently do.
I suppose it's not surprising that neglecting to teach people that there is a real subject called mathematics results in people thinking that there is not a real such subject as mathematics, but again, you'd hope that people would have the humility to recognise that we do actually know better than them about our own subject.
