## Saturday, December 3, 2011

### Infinitesimal analysis

In the absence of known discrete foundations for some mathematical system we can produce some other foundational abstraction. In particular, if elements occur at extremely small scales, the extent of which is also unknown, then infinitesimals are the particular class of abstraction that should be considered.

Once we have some notion of infinitesimals in our grasp, the next step is to describe the effects of infinitesimal changes in input has to a function. In particular, a function is called smooth if infinitesimal changes in input only result in infinitesimal changes in output.

Derivatives are just the slope of an infinitesimal secant line of a function and integration is the process of calculating areas of infinitesimal points.