What is a structural representation ? Fourth variation ∗

We outline a formalism for structural, or symbolic, representation, the necessity of which has been acutely felt in all sciences, particularly biology, for quite some time now. At the same time, biology has been gradually edging to the forefront of sciences, although the reasons obviously have nothing to do with its state of formalization or maturity—which is quite primitive as compared, for example, to that of physics. Rather, the reasons have to do with the growing realization that the objects of biology are not only more important and interesting, but that they also more explicitly exhibit the evolving nature of all objects in the Universe. It is this view of objects as evolving structural processes that we aim to address here, in contrast to the ubiquitous mathematical view of objects as points in some abstract space. One can gain an initial intuitive understanding of the proposed representation by generalizing the (Peano) process of construction of natural numbers: replace the single structureless unit out of which a number is built by multiple structural ones. An immediate but critical consequence of the distinguishability/multiplicity of units in the construction process is that we can now see which unit was attached and when. Hence, the resulting representation for the first time embodies temporal structural information in the form of a formative, or generative, history. ∗ This paper (UNB Faculty of Computer Science Technical Report TR05-174) is a substantially modified variation of [1], [2] (and also [3]). In informal segments of this paper, we often use single quotes to identify standard terms in ETS and other technical areas.