Our discussion is couched in terms of the class (or better, the category) of mathematical images which a given natural system may have, and the relations between these images. Essentially, we define a simple system as: (a) a system whose mathematical images are dynamical systems (i.e. dual structures consisting of a set of states, and superimposed dynamical laws or equations of motion) and (b) this class contains a unique maximal image, which behaves like a free object (i.e. all other images are homomorphic images of the maximal one). A complex system is one which possesses mathematical images which are not dynamical systems. In particular, there is no maximal description of a complex system.

We suggest a number of specific examples of complex systems and their class of mathematical images. The behaviours of complex systems are contrasted with those of simple counterparts. We point out in particular that contemporary physics is essentially the science of simple systems, and is neither directly applicable nor adequate for natural systems, like organisms, which are not simple. In effect, we are arguing that the problems associated with complex systems, particulary in biology, pose problems for contemporary physics at least as serious as those posed by e.g. atomic spectra or chemical bonding in the last century.