I have given a definition for `emergence', based upon a notion of predictive complexity. Emergence implies computational irreducibility - which can be seen to be the finite analogue of formal undecidability. Hence, the strange lack of understanding in emergent systems has its roots in complexity theory, from which it would seem that emergence itself is an undecidable proposition.
I conclude that not only do emergent systems exist, but also that they match very closely our working definition of the term. Simulation is an optimal means of determining the outcome of such systems, and is thus an important means of investigation. Simulations can be coupled with any of a number of different analysis methods on the emergent state. Together they are suitable for carrying out very interesting, real science in emergent or near-emergent systems, and should lead to explanations of some of the many thought-provoking emergent phenomena we observe. I have elaborated on just one approach, of current personal interest, to reveal the application of these ideas to critical phenomena.
To summarise, my results suggest the best approach to take in studying emergent complexity is a feedback process of simulation and analysis of the actual emergent phenomena. These results should go some way towards legitimising the concept of a simulation as a real scientific tool for the investigation of emergence.