At one extreme one could say that to know math means to know the axioms and to know how to derive the theorems from those axioms.
I prefer (which may be the other extreme) to say that to know math is to remember enough of "mathematical facts" (theorems, formulas, algorithmic procedures, etc.) and to have enough of mathematical thinking skills to be able to derive/solve mathematical problems.
Notice that "to remember enough mathematical facts" and "to have sufficient thinking skills" are, to some extent, mutually replaceable by each other. At "vocational" end of the spectrum one relies mostly on memorization, at "scientific" end of the spectrum one relies mostly on thinking.
For example, one can memorize the volume formula (for the solid of revolution) or one can derive it if one needs it. The same with most other "facts".