Nonlinear systems: between a law and a definition

Abstract

A discussion relevant to the logical foundation of nonlinear dynamic systems theory and statistical theory is presented. The initial point of view is associated with the question of whether or not an equation can represent a physical law. It is argued that the answer depends on which of the associated concepts/processes - that of direct physical measurement or that of counting - is more important in the theory of the relevant system. Without the inclusion of the concept of counting in the set of the most basic physical concepts, consideration of the nonlinear dynamic (or statistical) systems which may have an `iterative mathematical nature' is seen to be logically unsatisfactory. The ergodic hypothesis is considered, from the points of view of direct physical measurement and counting. It is shown that for the foundation of statistical theory we must consider `measurement in principle' even when we cannot, by reason of the complexity of the system, actually perform this measurement. The discussion inevitably involves many concepts, which reflects the essence of the very complicated situation under discussion. This gives the work, which is intended, first of all, for a reader with a physics background, a logical-critical inclination. The latter is justified also by the opinion here that the solution to the fundamental problems of `nonlinear physics' cannot come from the existing physics. It is argued that the topic of intuitionistic logic has to be considered for axiomization of the nonlinear dynamic theory. Though the work is not a review of the results of the existing dynamic or statistical theories, the discussion of the logical problems in the foundation of these theories is relatively much more complete than it usually is in such reviews. The classical results of analytical mechanics are finally considered, including some nontrivial applications.