Relational biology and Church's thesis.

Church's Thesis is a metamathematical hypothesis that says the concepts of effective calculability and computability are coextensive. It is reasonable to consider everything that happens in the material world to be 'effective'. If Church's Thesis were true in the natural world, then it would mean that all material processes could be expressed in purely syntactic terms. A corollary in relational biology is that a living system must have noncomputable models. Thus the existence of living systems implies that Church's Thesis is false as a physical proposition. The paper begins with a review of the tenets of relational biology, which is the standpoint from which this exposition on the Foundations of Mathematics and Theoretical Biology is composed.

A Rosen etymology.

This essay contains a few of my interpretations of Robert Rosen's conception of Nature. I shall study the four notions that form the core of his whole-lifetime's scientific work: simple system, mechanism, complex system, and organism. Their set-theoretic interconnections culminate in Rosen's new taxonomy of natural systems.

A phenomenological calculus of Wiener description space.

The phenomenological calculus is a categorical example of Robert Rosen's modeling relation. This paper is an alligation of the phenomenological calculus and generalized harmonic analysis, another categorical example. Our epistemological exploration continues into the realm of Wiener description space, in which constitutive parameters are extended from vectors to vector-valued functions of a real variable. Inherent in the phenomenology are fundamental representations of time and nearness to equilibrium.

Differential geometry of proteins. Helical approximations.

We regard a protein molecule as a geometric object, and in a first approximation represent it as a regular parametrized space curve passing through its alpha-carbon atoms (the backbone). In an earlier paper we argued that the regular patterns of secondary structures of proteins (morphons) correspond to geodesics on minimal surfaces. In this paper we discuss methods of recognizing these morphons on space curves that represent the protein backbone conformation. The mathematical tool we employ is the differential geometry of curves and surfaces. We introduce a natural approximation of backbone space curves in terms of helical approximating elements and present a computer algorithm to implement the approximation. Simple recognition criteria are given for the various morphons of proteins. These are incorporated into our helical approximation algorithm, together with more non-local criteria for the recognition of beta-sheet topologies. The method and the algorithm are illustrated with several examples of representative proteins. Generalizations of the helical approximation method are considered and their possible implications for protein energetics are sketched.

Any material realization of the (M,R)-systems must have noncomputable models.

Robert Rosen's (M,R)-systems are a class of relational models with a structure that defines a necessary distinguishing feature of organisms. That feature is an impredicative hierarchy of constraint on the properties of the model that correspond to the closure of an organism's entailment relations with respect to efficient cause. As a consequence, a computable model cannot be an (M,R)-system. This has been mathematically proven, and hence indisputable. Nevertheless, "computable" implementations of the mappings in an (M,R)-system have been reported. This paper explains the logical impossibility of the existence of these "counterexamples." In particular, it examines the errors in the construction of one of the most interesting among them. The relevance of this result to neuroscientists is that the same structure of closure to efficient cause is observed in brain dynamics.

A living system must have noncomputable models.

Chu and Ho's recent article in Artificial Life is riddled with errors. In particular, they use a wrong definition of Robert Rosen's mechanism. This renders their "critical assessment" of Rosen's central proof null and void.