The depth of an argument is limited by the shallowest person in the argument. Which means arguments are futile if there is no decisive conclusion reachable at a shallower level. If the proof takes calculus and your opponent only knows algebra your only real option is to arrange an empirical demonstration that will seem like magic to them.

Blogger at ribbonfarm.com, running summerofprotocols.com, host of /gloom ☀️

Making up magic tricks just to persuade someone you know something they’re incapable of learning is almost never worth it.

Identifying depths is hard. I can remember a couple of debates where we were talking past each other. 

Perhaps this is different but orthogonal planes of depth? Any mental model to suggest who is “deeper”? Or is it on the deeper person to get shallower and if they can’t …  they’re not deep?

🤦‍♂️So a prereq of understanding is similar depth (but this is rare)?

Your basic assumption seems to be that people must understand each other. They mostly don’t.