  In this manuscript we discuss the notion of (statistical) independence
embedded in its historical context. We focus in particular on its appearance
and role in number theory, concomitantly exploring the intimate connection of
independence and the famous Gaussian law of errors. As we shall see, this at
times requires us to go adrift from the celebrated Kolmogorov axioms, which
give the appearance of being ultimate ever since they have been introduced in
the $1930$s. While these insights are known to many a mathematician, we feel it
is time for both a reminder and renewed awareness. We present the independence
of the coefficients in a binary expansion, the independence of divisibility by
primes, and the resulting, famous central limit theorem of Paul Erd\H{o}s and
Mark Kac on the number of different prime factors of a number $n\in\mathbb N$.
We shall also present some of the (modern) developments in the framework of
lacunary series that have its origin in a work of Rapha\"el Salem and Antoni
Zygmund.
