Your war against common sense though,...
I have no war against common sense. That would be like having a war against taste or smell. My claim is simple. Common sense just isn't synonymous with mathematics, or more precisely, with reasoned thought and analysis. In fact, the two domains share nothing in common, no pun intended. Regardless of the fact that reasoned thought can explain common sense and fathom its examples, it acts entirely in spite of common sense. And "commonplace" is certainly not synonymous with "common sense". That has a different meaning altogether.
Common sense is the perception of the concrete world that we all share. Common sense is devoid of reasoned thought and analysis. That is my definition and in the context of this discussion, neither unfair nor unwarranted. Mathematics on the other hand is analysis. It is a reasoned and imagined theory about abstract and imagined entities, most significantly, the real numbers. When we attempt to teach this imagined theory and its imagined elements to the unaware student, we use concrete examples (common sense) as an aid in that task but not as a substitute for that task and not as a substitute for the goal of the task, reasoned analysis and thought.