  Emergence is a pregnant property in various fields. It is the fact for a
phenomenon to appear surprisingly and to be such that it seems at first sight
that it is not possible to predict its apparition. That is the reason why it
has often been said that emergence is a subjective property relative to the
observer. Some mathematical systems having very simple and deterministic rules
nevertheless show emergent behavior. Studying these systems shed a new light on
the subject and allows to define a new concept, computational irreducibility,
which deals with behaviors that even though they are totally deterministic
cannot be predicted without simulating them. Computational irreducibility is
then a key for understanding emergent phenomena from an objective point of view
that does not need the mention of any observer.
