Quantifying Emergence

Several authors argue that in any attempts to formalize or quantify the concepts, true emergent properties should be observer independent. This does not necessarily mean that emergent properties should be observation-independent. Observations undertaken by different methods result in differences in acquired knowledge. This means that emergent properties can be defined as the differences in knowledge gained by the observation of a system by two different methods. This is partly reflected by the computational emergence.

It is this observer dependency that leaves a way open for the quantification of emergent properties. Emergent properties could then be expressed in a semiquantitative way by the use of an 'index' derived of Kullback's measure of information (Figure 3). This involves moving the normal reference frame in information theory assuming the a priori knowledge of the system to be zero, which is not necessarily the case.

Rather in ecology, we do possess some knowledge about the system and what we usually refer to are the deviations in what we observe in the systems or models of systems compared with our expectations built on previous knowledge. The way of quantifying emergence has to be built on the use of computers and models. If our knowledge gained hitherto is synthesized and treated in a

Subject/observer

Information gain

Object/system

Subject/observer

Information gain

Object/system

Hierarchical levels

Figure 3 Quantification of emergence, based on Kullback's measure of information, might be carried out from quantifying the difference between actual observed, a posteriori, behavior or composition of a system and what may be predicted from a priori knowledge about subsystems. The analysis may be carried out at various levels of hierarchy, differing in emergence value.

Hierarchical levels

Figure 3 Quantification of emergence, based on Kullback's measure of information, might be carried out from quantifying the difference between actual observed, a posteriori, behavior or composition of a system and what may be predicted from a priori knowledge about subsystems. The analysis may be carried out at various levels of hierarchy, differing in emergence value.

computer model (from traditional ecological science) is p*, and the outcome of an experiment or observations of a system differs by p** the emergent properties can be calculated by the following:

which correlates emergence to the concept of exergy. Emergence now is a consequence of information gained between observations.

The question is if emergence in this manner will, at the end, dissolve itself and disappear as knowledge increases, which refers to the above debate of reductionism versus holism.

Many of the concepts used to characterize ecosystems are based on various numerical treatments of data observed in the ecosystem. Since the concepts are immediately deducible (calculable) from certain knowledge about the components of the ecosystem, for example, numbers, species, biomass etc. such concepts cannot be coined as emergent property but rather as a 'collective' property of the system. An interesting corresponding analog in this context are the macroscopic properties from thermodynamics such as entropy and parallels in formulation of formulas. Reductionism cannot win the debate since it will be impossible to achieve enough knowledge. If not for anything else, then for thermodynamic reasons, since the achievement of more and more detailed knowledge becomes more and more expensive in terms of not only energy but also dissipation.

Meanwhile, what strikes is that such a traditional, vertical organization of systems is not mandatory in order to produce emergent behavior. Vertical, here, refers to levels being either higher or lower in the hierarchy. Rather only parts are needed, of which none have actual regulatory functions and therefore should be evaluated or ranked higher than the other(s). Emergent properties can occur also in horizontally organized systems, emergence appearing alone as a consequence of interactions at the same level. The study of these intra-level relationships and their consequences to the higher levels in the hierarchy may be important to investigate in the future.

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