The Nature of Consciousness

Piero Scaruffi

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These are excerpts and elaborations from my book "The Nature of Consciousness"

Catastrophe Theory

Catastrophe theory, originally formulated in 1967 by the French mathematician René Thom and popularized ten years later by the work of the British mathematician Erich Zeeman, became a widely used tool for classifying the solutions of nonlinear systems in the neighborhood of stability breakdown.

In the beginning, Thom was interested in structural stability in topology (stability of topological form) and was convinced of the possibility of finding general laws of form evolution regardless of the underlying substance of form, as already stated at the beginning of the century by the British biologist D'Arcy Thompson.

Thom's goal was to explain the "succession of form". Our universe presents us with forms (that we can perceive and name). A form is defined, first and foremost, by its stability: a form lasts in space and time. Forms change. The history of the universe, insofar as we are concerned, is a ceaseless creation, destruction and transformation of form. Life itself is, ultimately, creation, growth and decaying of form. 

Every physical form is represented by a mathematical quantity called an "attractor" in a space of internal variables. If the attractor satisfies the mathematical property of being "structurally stable", then the physical form is the stable form of an object. Changes in form, or morphogenesis, are due to the capture of the attractors of the old form by the attractors of the new form. All morphogenesis is due to the conflict between attractors.

The universe of objects can be divided into domains of different attractors. Such domains are separated by shock waves.  Shock wave surfaces are singularities called "catastrophes". A catastrophe is a state beyond which the system is destroyed in an irreversible manner. Technically speaking, the "ensembles de catastrophes" are hypersurfaces that divide the parameter space in regions of completely different dynamics.

The bottom line is that dynamics and form become dual properties of nonlinear systems.

Thom proves that in a 4-dimensional space there exist seven types of elementary catastrophes.  Elementary catastrophes include: "fold", destruction of an attractor, which is captured by a lesser potential; "cusp", bifurcation of an attractor into two attractors; etc.  From these singularities, more and more complex catastrophes unfold, until the final catastrophe. Elementary catastrophes are "local accidents". The form of an object is due to the accumulation of many of these "accidents".

What catastrophe theory does is to "geometrize" the concept of "conflict". This theory is a purely geometric theory of morphogenesis, Its laws are independent of the substance, structure and internal forces of the system.

 


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