  The real numbers, it is taught at universities, correspond to our idea of a
continuum, although the hyperreal numbers are located ``in between'' the real
numbers. The number $x + dx$, where $dx$ should be an infinitesimal number and
$x$ real, is infinitesimally close to $x$ but ``infinitely'' far away from all
other real numbers. Analogously: If $f'(x_0)$ and $f(x_0)$ are given for a
differentiable function $f:\mathbb{R}\rightarrow\mathbb{R}$ at
$x_0\in\mathbb{R}$, we can not determine $f(x)$ at {\em any} point $x\in
\mathbb{R}$ different from $x_0$. These points seem to be ``infinitely'' far
away. That is one conceptual problem of solving differential equations in
numerical mathematics. In this article, we will present a numerical algorithm
to solve very simple initial value problems. However, the change of paradigm
is, that we will not ``leave'' the point $x_0$. Solving ordinary differential
equations is like searching for ``recipes'' $f$. Instead of trying to find
these recipes for values $x\in\mathbb{R}$, we will learn them from special
relations in the ``monad'' of $x_0$.
