One of my favorite examples of this is the "q-calculus", which is like a multiplicative version of the classical subject of calculus of finite differences. One can, using suitably defined "q" versions of the derivative, integral, and so on, recover analogues of most of the usual theorems in calculus. But what's more interesting is that this all ties in with noncommutative geometry and the field with one element.

One of my favorite examples of this is the "q-calculus", which is like a multiplicative version of the classical subject of calculus of finite differences. One can, using suitably defined "q" versions of the derivative, integral, and so on, recover analogues of most of the usual theorems in calculus. But what's more interesting is that this all ties in with noncommutative geometry and the field with one element (see John Baez's This Weeks Finds in Mathematical Physics).