In dealing with emergent phenomena, a common task is to identify useful descriptions of them in terms of the underlying atomic processes, and to extract enough computational content from these descriptions to enable predictions to be made. Generally, the underlying atomic processes are quite well understood, and (with important exceptions) captured by mathematics from which it is relatively easy to extract algorithmic content. A widespread view is that the difficulty in describing transitions from algorithmic activity to the emergence associated with chaotic situations is a simple case of complexity outstripping computational resources and human ingenuity. Or, on the other hand, that phenomena transcending the standard Turing model of computation, if they exist, must necessarily lie outside the domain of classical computability theory. In this talk we suggest that much of the current confusion arises from conceptual gaps and the lack of a suitably fundamental model within which to situate emergence. We examine the potential for placing emergent relations in a familiar context based on Turing’s 1939 model for interactive computation over structures described in terms of reals. The explanatory power of this model is explored, formalising informal descriptions in terms of mathematical definability and invariance, and relating a range of basic scientific puzzles to results and intractable problems in computability theory. In this talk

Abstract. Computability concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. We look at ways in which computabilitytheoretic structure emerges in natural contexts. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. The adequacy of the classical Turing model of computation — as first presented in [18] — is in question in many contexts. There is widespread doubt concerning the reducibility to this model of a broad spectrum of real-world processes and natural phenomena, from basic quantum mechanics to aspects of evolutionary development, or human mental activity. In 1939 Turing [19] described an extended model providing mathematical form to the algorithmic content of structures which are presented in terms of real numbers. Most scientific laws with a computational content can be framed