Emergent Phenomena and Complexity∗

I seek to define rigorously the concept of an emergent phenomenon in a complex system, together with its implications for explanation, understanding and prediction in such systems. I argue that in a certain fundamental sense, emergent systems are those in which even perfect knowledge and understanding may give us no predictive information. In them the optimal means of prediction is simulation. I investigate the consequences of this for certain decidability and complexity issues, and then explain why these limitations do not preclude all means of doing interesting science in such systems. I touch upon some recent incorporation of this work into the investigation of self-organised criticalities. 1 Motivation and Objectives The calculations were so elaborate it was very difficult. Now, usually I was the expert at this; I could always tell you what the answer was going to look like, or when I got it I could explain why. But this thing was so complicated I couldn’t explain why