Observing complexity, seeing simplicity

This century has seen the formulation of a number of novel mathematical and computational frameworks for the study, characterization and control of various classes of complex phenomena. Most of these involve some non-trivial dynamics. In order to be of genuine use in the real world, it is essential that such theoretical developments are related to observed data. This paper is concerned with the question of how this might be achieved. In particular, it investigates how much information about a complex unknown system one can hope to recover from observations. The vast majority of theoretical analysis assumes that we have an accurate model of a system and that we know the variables that uniquely determine its state. In principle, the application of such a theory to real problems requires the simultaneous measurement of all these variables. This is rarely feasible in practice, where often we will not even know what the important variables are. All that we may be able to achieve is to make a sequence of repeated measurements of one or more observables. The relationship between such observations and the state of the system is often uncertain. It is therefore unclear how much information about the behaviour of the system we can deduce from such measurements. It turns out that for a certain class of mathematically idealized systems it is, in principle, possible to reconstruct the whole system from a sequence of measurements of just a single observable. As a consequence, we may be able to build remarkably simple models of apparently complex looking behaviour. We shall outline the theoretical framework behind this remarkable result, and discuss its limitations and its generalizations to more realistic systems. Finally, we shall speculate that the complexity of theoretical models may sometimes outstrip our ability to detect them in real data.