The observation and the discussion of the physical
reality of phenomena, leads to bring out concepts
which have to be described in a non ambiguous mathematical
language. Concerning Dirac's calculus we shall introduce,
besides the usual definitions for the concepts of point,
number, function etc ... , additional concepts for the physical
point, the physical equalities, physical infinities and
infinitesimals ... etc ... In particular we introduce a new
equality $=^{D}$, called Dirac-equality, which differs as well
from the classical equality as from the weak equalities
introduced in various theories of generalized functions. All these
definitions are based on a definition in the language of Relative
Set Theory of the metaconcepts of
improperness used by P.A.M. Dirac in [Di], when he claimed "
\textit{Strictly of course, $\delta(x)$ is not a proper function of
$x$, ... , ... $\delta'(x), \delta''(x)$.... are even more
discontinuous and less proper than $\delta(x)$ itself}". We
defined this way a concept of observed derivative which extends
the usual one to a large class of \textit{discontinuous} possibly
non-standard functions. All the multiplications of improper
or very improper functions, including the delta-functions and
their observed derivatives, are obviously allowed. Now the
problem of the multiplication is replaced by another one: under
which conditions is the Dirac-equality of two functions preserved
by a multiplication term by term?