What Do Mathematicians Do?

Solution 1:

This is more of a philosophical answer and I don't know if this is what you were looking for, but in general what a mathematician does is finding an abstract representation of reality (or some reality). This is why mathematics finds so many applications in physics, for example.

Through symbolic manipulation, mathematics creates representations that, respecting some fixed rules (axioms), define the world and make predictions about its laws.

Please, note that "reality" and "the world" do not necessarily imply our reality and our world. I can conjecture a reality in infinite dimensions for example. However, even if that reality does not exist, I can still create its representation, and it will hold true should that world actually exist. Minkowsky created a metric used by Einstein way before it was needed, for example.

So, mathematics abstracts concepts creating symbols that can be manipulated, and by so doing we can derive rules and make predictions.

Everything else is a redefinition of these concepts, or an application.

As another mathematician, whose name I forget, said regarding mathematical physics: "To summarize, the end of mathematical physics is not merely to facilitate the numerical calculation of certain constants or the integration of certain differential equations. It is more, it is above all to disclose to the physicist the concealed harmonies of things by furnishing him with a new point of view".

This holds for other sciences besides physics, of course.