It seems we are at a point, where we have to define what is a 'number'. More precisely: Can you tell me the difference between numbers and "more complex mathematical objects"? Is a complex number a number? Is a quaternion a number? Is a residue class a number? We can calculate with integers modulo some other integer like with integers - is that considered computation with numbers? Shall we distinguish between matrices of numbers and matrices of more complex mathematical objects? In signal theory matrices of polynomials are very common.
My question would be why is it so important to determine what is or isn't a number? Whether something is a number or not does not determine what operations and properties it has. Rather, we should try to determine what is a field, a ring, a group, etc. If we know that matrices of polynomials form a group, then we can perform the operations of the group on those objects. That being said, I'll have to play the other side of the coin: it would probably be a little bit of a pain to have to define instances of each data declaration (Integer, Int, Float, Matrix, Complex, etc.) on each of these seperate classes--especially when being in a certain class usually implies being in another (ie, the definition of a set being a field requires that that set is a group, right?) And another problem I can see is that, for example, the Integers are a group over addition, and also a group over multiplication; and in my small bit of thinking about this, it seems that having to keep track of all of this might get a bit unruly. Bryan Burgers