  We calculate the multiplicity function of matter condensations by directly
considering the actual, deeply non-linear density field, which we compare to
the popular Press-Schechter approximation. We show the mass function is a
function of a unique parameter that contains all the dependence on the mass and
radius of the object. It enables us to derive the mass function of general
astrophysical objects (in addition to just-virialized halos) which may be
defined by any density threshold, that may even vary with the mass of the
object. This is beyond the reach of usual formulations based on the initial
gaussian field and gives a clear illustration of the advantages of our
approach. We explain why numerical tests seem to favor both Press-Schechter and
non-linear prescriptions even though the two approximations differ by their
scaling as a function of mass as well as a function of redshift. The difference
between the two is due to the fact that the Press-Schechter prescription
assumes that present-day mass fluctuations can be recognized in the early
linear universe, and that their number is conserved, while the non-linear
approach takes into account their evolution which leads to an increase of the
number of highly non-linear objects (very large or small masses). This
difference is seen to be of the same magnitude as the difference obtained by
varying the initial spectrum of density fluctuations. Earlier conclusions drawn
about the relevance of the latter using analytical approximations to the mass
function must thus be reexamined.
