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 it was like that.So I told Fermi I was doing this problem, and I started to describe the results. He said, ``Wait, before you tell me the result, let me think. It's going to come out like this (he was right), and it's going to come out like this because of so and so. And there's a perfectly obvious explanation for this-''
He was doing what I was supposed to be good at, ten times better. That was quite a lesson to me. Richard Feynman
The defining characteristic of a complex system is that some of its global
behaviours, which are the result of interactions between a large number
of relatively simple parts, cannot be predicted simply from the rules of
those underlying interactions
.
The word ``simply'' in the preceding paragraph is somewhat troublesome. I set out to explain, justify, understand and rigorise, in the context of such systems, what ``simply'' means.
With that in mind, let us hypothesise the concept of an `emergent phenomenon' as a large scale, group behaviour of a system, which doesn't seem to have any clear explanation in terms of the system's constituent parts. Rather than viewing this as the first step in the hierarchical evolution of hyperstructures[1], I am interested in `first-order emergence' in its own right, as it is currently more amenable to a precise formalisation than the elusive increase in complexity one observes in natural evolution.
Why is it that we seem to be incapable of answering the interesting questions about how and why the final state was reached? In the context of the above quotation, can we always find, or meaningfully postulate, a person like Fermi who can solve our problems through a deeper understanding than our own - side-stepping all of our complicated calculations? More succinctly, do truly emergent phenomena exist?
In order to address these issues, we shall need to clarify what we mean by a `clear explanation' and what we mean by `understanding'. This is especially warranted due to the misunderstandings inherent in different people's working definitions of such terms in different fields - complexity and emergence have become very much vogue labels for problems in fields as diverse as economics, artificial life, artificial intelligence, neuroscience, and even in the study of cultural change and development. I will use the term `emergent system' with the understanding that not all phenomena observed in that system will be emergent.