In my view complex systems science is branching off in two separate directions. Along one branch, ideas and tools from complexity research will be refined and applied in an increasingly wide variety of specific areas. In this book we’ve seen ways in which similar ideas and tools are being used in fields as disparate as physics, biology, epidemiology, sociology, political science, and computer science, among others. Some areas I didn’t cover in which these ideas are gaining increasing prominence include neuroscience, economics, ecology, climatology, and medicine—the seeds of complexity and interdisciplinary science are being widely sowed.

The second branch, more controversial, is to view all these fields from a higher level, so as to pursue explanatory and predictive mathematical theories that make commonalities among complex systems more rigorous, and that can describe and predict emergent phenomena.

At one complexity meeting I attended, a heated discussion took place about what direction the field should take. One of the participants, in a moment of frustration, said, “ ‘Complexity’ was once an exciting thing and is now a cliché. We should start over.”

What should we call it? It is probably clear by now that this is the crux of the problem—we don’t have the right vocabulary to precisely describe what we’re studying. We use words such as complexity, self-organization, and emergence to represent phenomena common to the systems in which we’re interested but we can’t yet characterize the commonalities in a more rigorous way. We need a new vocabulary that not only captures the conceptual building blocks of self-organization and emergence but that can also describe how these come to encompass what we call functionality, purpose, or meaning (cf. chapter 12). These ill-defined terms need to be replaced by new, better-defined terms that reflect increased understanding of the phenomena in question. As I have illustrated in this book, much work in complex systems involves the integration of concepts from dynamics, information, computation, and evolution. A new conceptual vocabulary and a new kind of mathematics will have to be forged from this integration. The mathematician Steven Strogatz puts it this way: “I think we may be missing the conceptual equivalent of calculus, a way of seeing the consequences of myriad interactions that define a complex system. It could be that this ultracalculus, if it were handed to us, would be forever beyond human comprehension. We just don’t know.”

Having the right conceptual vocabulary and the right mathematics is essential for being able to understand, predict, and in some cases, direct or control self-organizing systems with emergent properties. Developing such concepts and mathematical tools has been, and remains, the greatest challenge facing the sciences of complex systems.