Smooth Infinitesimal Analysis

High Quality Content by WIKIPEDIA articles!Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities. As a theory, it is a subset of synthetic differential geometry. The nilsquare or nilpotent infinitesimals are numbers where ² = 0 is true, but = 0 need not be true at the same time. Despite this fact, one could attempt to define a discontinuous function f(x) by specifying that f(x) = 1 for x = 0, and f(x) = 0 for x 0. If the law of the excluded middle held, then this would be a fully defined, discontinuous function. However, there are plenty of x, namely the infinitesimals, such that neither x = 0 nor x 0 hold, so the function is not defined on the extended real numbers.... Mehr