TITLE: Is there any way to define differentiablity without any reference to the Euclidean space?
QUESTION [5 upvotes]: We define metric spaces based on the properties of the real numbers $\Bbb{R}$. In the same spirit we define smooth manifolds. But there is a more general and elegant way to formulate our intuition of nearness without any reference to the real numbers which is called topology. Is there any way to do the same for smoothness? One might think of modeling smoothness on topological vector spaces instead of  $\Bbb{R}^n$ , but that is basically the same. So I am looking for something general, natural and at the same time useful like topology. And of course I know there isn't such a thing out there but I want to know what would you do if you wanted to lay down the theory of "manifolds"? \
Can't we define differentiability on an arbitrary ordered field?

REPLY [5 votes]: Sure. Frechet differentiability. That's not to say that you necessarily get very far without the real numbers, or that the results over a non-Archimedean field will make sense to anyone. But Frechet is just linear maps and an inequality.