Representational formalisms : what they are and why we haven ’ t had any

Currently, the only discipline that has dealt with scientific representations— albeit non-structural ones—is mathematics (as distinct from logic). I suggest that it is this discipline, only vastly expanded based on a new, structural, foundation, that will also deal with structural representations. Logic (including computability theory) is not concerned with the issues of various representations useful in natural sciences. Artificial intelligence was supposed to address these issues but has, in fact, hardly advanced them at all. How do we, then, approach the development of representational formalisms? It appears that the only reasonable starting point is the primordial point at which all of mathematics began, i.e. we should start with the generalization of the process of construction of natural numbers, replacing the identical structureless units, out of which numbers are built, by structural ones, each signifying an atomic “transforming” event. This paper is conceived as a companion to [1], and is a revised version of [2]. Mathematics is the science of the infinite, its goal is the symbolic comprehension of the infinite with human, that is finite, means.