Incomplete information is both new and old. New because scientists always claimed, in constructing physics theories, that their theories contain all necessary information for specifying the systems under consideration. This is the case of all the conventional physical theories : from Newtonian to quantum physics, in passing by Einstein, Boltzmann and Shannon (certainly, a theory containing only partial information about the system of interest is a little bit discouraging). Old because since the discovery of irrational numbers by Pythagoras, mathematicians, including Pythagoras himself always claiming that all things are numbers, know that within arithmetical system, they loss some information about the world and that one cannot know everything with infinite precision. In 1931, G�del already shows that mathematics systems (e.g. arithmetic) are incomplete in the sense that within any such axiomatic system there is never sufficient information to prove all possible statements of the theory. From the epistemological point of view, mathematics is a theory of the simple world felt and imagined by human beings, a fragmented world containing only isolated and independent parts. So one should ask how far he can go with the "incomplete mathematics" in the real messy world or in a complex system with interacting, entangled and overlapped parts when the interactions can no more be neglected. In my opinion, it would be modest to say that mathematics is an approximate theory containing finite amount of information for the description of a world needing more and even infinite information to be described. This is just a lesson we can get from the incompleteness theorem of G�del.
Complete information is possible whenever all possible states are well known so that we can count them to carry out the calculation of probability and information. In physics, this requires that we can find the exact hamiltonian and also the exact solutions of the equation of motion to know all the possible states and to obtain the exact values of physical quantities dependent on the hamiltonian. Unfortunately, these two ``exact conditions of complete information are impossible to satisfy.
Let us forget the incompleteness of the mathematical structure of physical theory. Suppose that mathematics can give sufficiently precise results if the physical model can handle all the necessary information. We will see that, due to the omnipresent complexity in the world, we cannot have access to all the necessary information in order to completely describe a system. Here ``complexity" means that the systems show nonlinear behaviors which are extremely sensible to initial conditions and unpredictable. This is the famous chaos observed almost everywhere in the world.
A complex system is not necessarily a complicated system with a large number of freedoms. A one dimensional oscillator with well known nonlinear interaction (with potential proportional to x4, for example) or a three body system with gravitation (1/r) can behave chaotically. These two cases are just very good examples of the impossibility of the two ``exact" conditions of complete information mentioned above. In the case of the three body problem, we know (at least we believe that we know) the exact interaction of the system (Newtonian gravitation). But Poincar� showed that the exact solution of the equation of motion was not possible. The movement is chaotic. This means that we never know all possible states of the system. Complete information treatment becomes impossible. We even have to redefine probability distribution in order to calculate it in chaotic or fractal phase space.
When the hamiltonian cannot be exactly written, the situation is more complicated. Complete information description of such systems is only a science fiction. Although we cannot say that all these systems have chaotic or fractal nature, but a common feature of them is that a part (or some points) of their phase space is unknown so that exhaustive exploit of the phase space is impossible. The calculable information in this case will be incomplete.
To my mind, the incompleteness of information for complex systems is evident. The treatments of these systems based on complete information and probability distribution are not well founded. A question arises : how to introduce information incompleteness into physics?
