If I recall correctly, there's also some neat stuff on limits, so that in some cases what 'looks like' 0/0 (for example) - in the limit - is well defined. In (at least some of) these cases 'division by 0' may 'make sense'.

More generally, 'math' is a formal system, so why certain things 'make sense' and others don't is entirely determined by the axioms of the piece of math you are working with.