This is an interesting topic, but your exposition is sorely lacking layperson examples. The closest it comes to one is

they could agree about the standard numbers, how the successor and addition operator works, and yet disagree on multiplication

but it is still unclear what exactly they disagree on. Does one of them state that 2 times 2 is 5? Probably not.

When I explain complicated physics to a non-physicist, I strive to give examples which are as simple as possible. I wish mathematicians did that with math.