Your wrongness has to do with your view that one perspective exhausts the subject. You do not have a hairy body because you are mammal, though some people easily fall into the trap of thinking that way. Addition and multiplication on N thru C are not what they are because those things are classified with many disparate algebraic sructures as ringoids, either. Ringoid theory will tell you many things about those structures, but not everything about some particular structure. There is more to be learned about the particular relationships of those operations in the unique individual structures they are studied in. For example, ringoid theory says nothing about logical derivability of one operation from the other in some particular structure, or about computational perspectives.