The 2010/11 Collingwood Lecture

Professor Robert S. MacKay FRS FInstP FIMA (Warwick University)

 "The Mathematics of Emergence" 

Abstract: "Emergence" has been a topic of discourse by philosophers for over 150 years and is one of the central themes in the currently popular science of complex systems: dynamic systems with many interdependent components. In short, it is the difference between the behaviour of a system and the sum of its parts, which was already appreciated by Aristotle. But how can one make this more precise?

I suggest to use the concept of "space-time phases": probability distributions over realisations of the system compatible with its dynamics and initialisation in the distant past. If the interaction between components is non-trivial, then its space-time phases are not products over time-phases for independent units, and this difference can be quantified by a suitable metric. Even more significantly, some systems can exhibit more than one space-time phase, a phenomenon that corresponds to the philosophers' notion of "strong emergence". Several examples where this has been proved will be illustrated, and directions for future research sketched.