So, even though normally all pure mathematical issues should be able to be settled objectively with careful, strict mathematical treatment. This is what proofs are about.
But it's not always the case that they get settled here. But the argument, the back-and-forth, the debate does serve some educational value for those observing.
How can we have such a debate in this (or any) SE?
The problem I have with just moving it to chat is that the chat does not support $\LaTeX$ or MathJax or whatever. And the argument is over mathematics, so we should be able to express the equations and mathematical claims that we are arguing about.
What's wrong with publicly displaying our dirty laundry in the comments (where MathJax does work) and everyone can see the content of the dispute and judge for themselves?