And it can lead to constructs like abstract algebras in which we define new sets of "numbers" and alternative definitions of "+" and "*" to create things like multi-valued logics, fuzzy logics, rings, fields, etc. for nonmonotonic reasoning, cryptography, state machine design, computer circuit design, etc.
A number need not represent a quantity. It could be an index on to a more abstract set. (But, of course, that must be clearly stated so that everyone understands how the number-symbols and operator-symbols have been overloaded with different functionality for the purposes of a given research effort or field of math).