The fact that we are even having this conversation proves that math is not grounded in common sense. Everyone has common sense. That is why they call it "common" sense. Obviously, this is not a sufficient requirement for mathematics. Q.E.D.
Lou actually answered this awhile back...
I did what I ask my students to do: I studied, and I thought about what I was studying. And "we" can't duplicate that; students must.
Many of them are willing to study. But few are willing to think about what they are studying.
It is very easy to convince students that they are thinking deeply, and most of them will love you for doing so. But it very difficult to get them to actually think deeply. And almost all of them will hate you for doing so.
- --Lou Talman
This is my focus lately. How do we get the students to be pedantic (for lack of a better term). I am using "pedantic" instead of "think" because I want to clear my head of foolish misconceptions (mine included) of the relationship between common sense and formal reasoning (thinking). I don't think the case is always that the student is unwilling. I think the case is often that they simply don't know how. We take it for granted, but if you have ever taught a young student (the same student) for many years, you realize it is nothing short of fantastic that any of us make that leap to the abstract.
You (Joe) and Clyde want them to go to the light (common sense). I want them to go away from it.