  Mathematical theories are classified in two distinct classes : {\it rigid},
and on the other hand, {\it non-rigid} ones. Rigid theories, like group theory,
topology, category theory, etc., have a basic concept - given for instance by a
set of axioms - from which all the other concepts are defined in a unique way.
Non-rigid theories, like ring theory, certain general enough pseudo-topologies,
etc., have a number of their concepts defined in a more free or relatively
independent manner of one another, namely, with {\it compatibility} conditions
between them only. As an example, it is shown that the usual ring structure on
the integers $\mathbb{Z}$ is not rigid, however, it is nearly rigid.
