"Representation and Change": extended summary

Purpose of the study:

The primary purpose of this framework would be to integrate existing concepts, theories and disciplines and thus to eliminate paradoxes and confusions. Its secondary purpose would be to allow new applications in the analysis and steering of processes in which new representations (i.e. new knowledge) are generated.

This framework would be the first stage in the development of a general theory of representation and representation dynamics. Such theory is called an "adaptive metarepresentation".

Scope and method of the study :

The representation problem formulated in this context leads to two more concrete research questions. In physics the main problem can be phrased as follows : "How to integrate classical and non-classical representations of the same dynamical systems ?" This difficulty can be illustrated by the well-known paradoxes of quantum mechanics. In cognitive science the basic question is : "What are the fundamental structures of representations and how do they evolve ?"

These problems are approached through an analysis consisting ofthe following subsequent steps :

1. analysing, comparing and synthesizing different representation concepts as they are used in the disciplines being studied.
2. using the synthetic concept constructed in this way as a framework for analysing the representations used in classical physics.

3. using the results of this analysis to derive a fundamental criterion which would define the "classicality" of a representation.

4. analysing the main non-classical representations used in physics in order to find out how, where and why the classical presuppositions are violated.

5. using the insights attained in this way to elucidate some paradoxes arising from the apparent inconsistency of classical and non-classical representations.

6. generalizing and coordinating the concepts conceived during the analysis so as to lay the foundations for an adaptive metarepresentation, and indicating how this framework can be formalized and operationalized.

Conclusions with respect to the different steps of the analysis.

2) the analysis of classical scientific theories (in particular classical mechanics) with the help of these concepts leads to a hierarchically structured, coherent and self-sufficient representation structure : the classical frame. Its subsequent levels are : objects, predicates, Boolean algebra, state space, topology, time as a continuous parameter, operator group, dynamical constraints. It is shown how these different substructures are interdependent, and how their use determines the world view of the subject who uses this frame. In particular it is shown by different examples how the unconscious bias of the classical frame leads to the rejection or to the ignorance of those phenomena which do not fit into its formal structure.

3) yet the classical frame appears very natural and stable so that we cannot dismiss it as just one of a multitude of possible representation structures. Therefore we should find the feature which makes classical representations unique and which distinguishes them from all other non-classical representations. This criterion is found by going back to the most fundamental mechanism of cognition : distinction. It is shown that the classical frame is characterized by the absolute invariance of all distinctions which determine the representation structure : the distinction between a proposition and its negation, the distinction between space (simultaneity) and time (precedence). Hence distinction conservation appears to be a necessary property for a representation to be classical. We should now also show that it is sufficient, i.e. that non-classical representations do not possess this property.

4)a) Quantum mechanics : it is shown that during the quantum observation process, which is described by "the collapse of the wave function", there is no conservation of certain distinctions describing the system. This expressed in the quantum formalism by the superposition principle and the projection postulate. It is shown how this structural feature of the quantum representation can be reduced to the existence of a non-trivial orthogonality relation between quantum states. This leads to a non-Boolean logic and to a non-Bayesian probability expression.

The cause of this non-conservation is analysed and found to be due to the impossibility of perfect self-determination for a macroscopical observation apparatus. This leads to the impossibility of perfectly distinguishing microstates. The non-disjointness of the resulting macrostates explains the non-Bayesian probability and hence the fundamental structure of the Hilbert space formalism.

b) Relativity theory : the relativity principle and the existence of a finite maximum speed for signal propagation entail the relativity of simultaneity and synchronization. Hence the distinction between simultaneous and non-simultaneous events loses its invariance. Instead we find a causal structure determined by an incomplete precedence relation.

It is shown that this space-time structure can be reconstructed by classifying the paths formed by locally distinction conservation connections between events. The cyclic paths are shown to be unable to transfer information : either they lead to static correlations, or to so-called "causal paradoxes" which can be eliminated by deleting the self-inconsistent distinction. The remaining global connections determine three relations which are proven to be sufficient to define a "causal structure" (in the sense of Kronheimer and Penrose). We may conclude that the non-classical space-time structure of relativity theory is the direct consequence of the principle of the impossibility of circular information transfer.

c) Theories of irreversible processes : complex systems are characterized by so-called irreversible processes in which the total internal information of a system can only diminish (2nd law of thermodynamics). In open systems this can be compensated by an external input of information. It is shown that this non-conservation of distinctions is due to the interdependence of macroscopical and microscopical distinctions : for a macroscopical observer it appears as though macroscopical distinctions simply dissappear (diffusion, entropy increase) or appear out of nothing (self-organization, bifurcation).

This non-classical irreversible evolution is a necessary prerequisite for the appearance of autonomous, adaptive, and cognitive systems which are able to create and to maintain their own boundary (i.e. their self-environment distinction) by internally processing the incoming information. Here also we may suppose that this irreversible information process is due to the principle of incomplete self-determination.

5) since we have shown in 4)b) that the topology of space-time and hence locality is dependent upon the non-circular transfer of distinctions (hence information) there is no longer any paradox in the fact that there are non-local influences during quantum observation experiments, since no information is transferred. The non-local correlations can be explained by the (non-local) creation of a distinction by the observation apparatus. Hence no local "hidden variables" are to be introduced.

6) The static scheme of distinctions which was found to form the base of classical representations can be formalized by means of the distinction algebra of Spencer-Brown, which can be realized as a Boolean algebra B. Classical (i.e. distinction-conserving) processes can then be represented by a group of automorphisms of the algebra. Non-classical processes can be represented by general morphisms sending the algebra B upon different algebras. These processes can also be interpreted as transformations sending the classical representation B upon a new classical representation B'. Hence this framework can be seen as a basis for an adaptive metarepresentation in which all classical and non-classical representations and their transformations can be expressed.