The State Space Approach to Evolution

Mathematical models of dynamical systems typically employed in modem science are based on a very simple paradigm: the system has a state and an environment, and the time rate of change of the state is a function of the state and the environment. This function is a known mathematical function, and the system evolution if possible is studied under the assumption that the environment stays constant. This paradigm, derived originally from Newton's Second Law, is one of the greatest achievements of science. It has been used with overwhelming success to describe a vast range of phenomena in nature. However, its apparent simplicity belies its true nature. The paradigm serves to unify extremely diverse conceptual structures, subjecting them to mathematical treatment in a common language while scarcely limiting their reach. In this paper we illustrate this fact by offering examples of the paradigm of different types, showing how widely it has been used, and how little restriction it imposes on nature. We also propose some generalizations that have rarely been seen outside pure mathematics. Finally we note that the essential value in this paradigm is to be found in both its malleability and its relation to mathematics and quantity.