In this work we discuss the notion of observable - both quantum and classical
- from a new point of view. In classical mechanics, an observable is
represented as a function (measurable, continuous or smooth), whereas in (von
Neumann's approach to) quantum physics, an observable is represented as a
bonded selfadjoint operator on Hilbert space. We will show in part II of this
work that there is a common structure behind these two different concepts. If
$\mathcal{R}$ is a von Neumann algebra, a selfadjoint element $A \in
\mathcal{R}$ induces a continuous function $f_{A} : \mathcal{Q}(\mathcal{P(R)})
\to \mathbb{R}$ defined on the \emph{Stone spectrum}
$\mathcal{Q}(\mathcal{P(R)})$ of the lattice $\mathcal{P(R)}$ of projections in
$\mathcal{R}$. The Stone spectrum $\mathcal{Q}(\mathbb{L})$ of a general
lattice $\mathbb{L}$ is the set of maximal dual ideals in $\mathbb{L}$,
equipped with a canonical topology. $\mathcal{Q}(\mathbb{L})$ coincides with
Stone's construction if $\mathbb{L}$ is a Boolean algebra (thereby ``Stone'')
and is homeomorphic to the Gelfand spectrum of an abelian von Neumann algebra
$\mathcal{R}$ in case of $\mathbb{L} = \mathcal{P(R)}$ (thereby ``spectrum'').