Academic journal article Emergence: Complexity and Organization

Emergence and Computability

Academic journal article Emergence: Complexity and Organization

Emergence and Computability

Article excerpt

This paper presents a discussion of the possible influence of incomputability and the incompleteness of mathematics as a source of apparent emergence in complex systems. The suggestion is made that the analysis of complex systems as a specific instance of a complex process may be subject to inaccessible 'emergence'. "We discuss models of computation associated with transcending the limits of traditional Turing systems, and suggest that inquiry into complex systems in the light of the potential limitations of incomputability and incompleteness may be worthwhile.

Introduction

We suggest that what we intuitively define as (strongly) emergent systems may include processes which are not computable in a classical sense. We ask how incomputable processes would appear to an observer and, via a thought experiment, show that they would display features normally defined as 'emergent'.

If this conjecture is correct, then two important corollaries follow: first, some emergent phenomena can neither be studied nor modelled via classical computer simulations and second, there may be classes of emergent phenomena which cannot be detected via standard physical measurements unless the process of measurement exhibits super-Turing properties in its own right. Borrowing from recent literature in computer science we then show that tools which enable us to break the classical computational barrier are already available and suggest some directions for a novel approach to the problem.

Emergence

Implicit in most approaches to the study of emergence are 3 concepts:

1. Multiple levels of representation: there are classes of natural phenomena which, when observed at different levels or resolution, display behaviors which appear fundamentally different (Shazili, 2001; Crutchfield, 1994a, 1994b; Rabinowitz, 2005; Laughlin, 2005; Laughlin & Pines, 2000; Goldstein, 2002);

2. Novelty: for most complex systems, while we expect the properties of higher levels to causally arise from lower levels of representation, how this happens appears somehow inexplicable (Bickhard, 2000; Bedau, 1997; Darley, 1994; Rosen, 1985; Heylighen, 1991; Anderson, 1972);

3. Inherent causality: while we expect causality to arise solely from lower levels, for most complex systems the higher levels also appear to possess inherent and independent causal power (Bickhard, 2000; Campbell, 1974; see also Pattee, 1997; Goldstein, 2002; Rabinowitz, 2005; and Laughlin, 2005 for a discussion of the role of causation in complex systems).

The dilemma which has kept scientists and philosophers busy for decades is whether this novelty and inherent causality are real physical phenomena or merely lie in the eyes of the observer; said differently, whether reductionism is the only tool we need to understand Nature.

The limits of mathematics

The most efficient language we possess to study Nature is Mathematics. This is used not only to describe processes but also, by using mathematical transformations rules, to deduce, extrapolate and manipulate novel processes. It is thus crucial to be sure that the mathematical machinery we use is consistent and correct. It is also important that it is as exhaustive as possible, since the more mathematical rules (theorems) we discover, the more options are available to us to interpret and manipulate Nature's workings. These needs motivated mathematicians at the end of the 1 9th century who dreamt of devising a set of axi- oms and transformation rules from which all other mathematical truths could be deduced as theorems. In Hubert's dream, this would be achieved simply by mechanical manipulation of symbols devoid of external meaning (Chaitin, 1993, 1997: 1-5). Basically, Hilbert was seeking a consistent and complete formal system which would guarantee that all theorems of Mathematics could be proved. The dream was famously shattered by the work of Godei (1931) who proved that no formal system in which we are able to do integer arithmetic can be both complete and consistent. …

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