Not to be a smartass, but some might find this point of view interesting:
Mathematics never pays attention to what objects ARE, but rather what they DO.
Mathematics never pays attention to what objects ARE, but rather what they DO.
Similarly, mathematicians study objects by looking how they act on other objects. So if it turns out that two differently defined objects have the same actions on other objects, we call them isomorphic and don't really distinguish between them.
So for instance, we say that the set of rigid motions that preserve the triangle and it's orientation is the same as the set of permutations of the roots of say: x^3-3x+1 even if the two sets are absolutely not defined in the same way.
Hope that makes some sense.
Matrix is not defined just by operations like a ring is, but also by structure - you have N independent axes in a specified space.
> If you can do numeric arithmetic on it, it's a number.
>you have N independent axes in a specified space.
Only if you have an operation that maps the matrices to that space.
For example, if we are considering sets and functions between them, we generally don't care about the exact names of the elements of the set, only the fact that they are distinct elements. The important aspects of a function in this case are properties such as injectivity, surjectivity, etc, not that it sends one particular element of one set to another particular element of another set.
Another example is in linear algebra: we really care about linear transformations on an abstract vector space more than we care about what that linear transformation looks like relative to a specific set of coordinates.
This point of view is espoused in category theory, where the important information is carried in the morphisms between objects, not really the objects themselves.