Saturday, August 29, 2009

Evolving Utopia Part 8

Part 8: Mathematical Interlude

We have used geometric arguments to show that the Darwinian Model (Survival of the Just-Barely-Fit-and-fitter) makes the Intelligent Design model unnecessary because it provides a mechanism for evolving complexity; and it is better at explaining observed evolutionary phenomena than the Spencerian Model (Survival of the Fittest). It is also possible to use algebraic arguments but that is too complicated for this format. The argument is given elsewhere (see note).

We will, in the next part, apply the Darwinian model to homo sapiens sapiens (i.e., us), but we will be implicitly using a mathematical description of behavior. The full argument is given elsewhere and it, too, is too complicated for this format; so I will just give a description of the argument without showing the mathematical and physical details. "Physical", because some parts are deliberate analogies to quantum physics.

We start with the representation of behavior.

As we noted earlier it is possible to represent the life experienced by an entity as a sequence of events observed by a Deus ex Machina or "DeM". Each of the decision points has a number of possible inputs and a number of possible outputs. We can represent these as vectors whose elements constitute the repertoire of events observeable by the DeM.

We can make this sequence in terms of the kind of events we can observe if we divide each decision point into two steps: first, a stimulus to the entity will result in a response; and the response acts as a stimulus to the entity's environment. This stimulus provokes a response from the environment which acts as the next stumulus to the entity. This is the sequence of events we call the "life-sequence" of the entity. The Life-sequence is a sequence of vectors that are related to one another by two transformation matrices: the P-matrix describes the behavior of the entity and the N-matrix describes the behavior of its environment.

We can also use this to define a space in which the P-matrix is a point, so that a change in the pattern of behavior represented by the P-matrix is represented by motion in that space. We note that since observations are finite, our knowledge of the probability of an action (represented by the elements of the P-matrix) will be uncertain or "fuzzy". Because of that it makes sense to represent the fuzziness by quantizing the P-space and define the constraints on the dynamics of the entities by occupation rules for the quantum "boxes". This will provide two situations analogous to the rules for particles:

the Fermi-Dirac rule which says that one, and only one, particle can occupy a box; and
the Bose-Einstein rule which says that any number of particles can occupy a box.

Aside from this occupation rule, the other factor that will cause (or prevent) a change in behavior pattern is the attractve force between two occupied points in P-space; where the force is observed as a pattern of conformity. The Fermi-Dirac rule would apply where there was a tendency for mutual conformity and also a tendency for individuality. The Bose-Einstein rule would hold where the force toward conformity with a fixed reference point (e.g., the dogma of an ideology) is much stronger than the tendency toward individuality.

We can expect, therefore, that with some adjustment for the effects of the environment, human behavior will be characterized by Fermi-Dirac or Bose-Einstein rules; depending on what environmental factors strongly affect survival.

The first situation is where the humans have a relatively primitive economy, i.e., are primarily dependent on gathering and scavanging.

(Note: The more complete argument, including the algebraic version, is made in an html file titled "Wholly Holistic Evolution, Mr. Darwin". That can be found in the folder labeled "WE".)

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