Monday, January 07, 2008
Universal Basis for Science and Religion: Part 4, Mathematical Interlude
Universal Basis for Science and Religion: Part 4, 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. It is given on [http://homepage.mac.com/karlek/.Public/WE/WEA1P01.html].
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 in [http://we.karleklund.net] 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 in terms of events we can observe if we divide the decision point into two steps: a stimulus to the entity results in a response; and then that response acts as a stmulus to the entity's environment, whose response acts as the next stumulus to the entity. The life-sequence is thus 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 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, the elements of the P-matrix will be "fuzzy" and it makes sense to represent the fuzziness by quantizing the P-space and define 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 the 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 reference point was much stronger than the tendency toward individuality.
We can expect, therefore, that with some adjustment for the effects of the environment, we would expect human behavior to 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.
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. It is given on [http://homepage.mac.com/karlek/.Public/WE/WEA1P01.html].
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 in [http://we.karleklund.net] 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 in terms of events we can observe if we divide the decision point into two steps: a stimulus to the entity results in a response; and then that response acts as a stmulus to the entity's environment, whose response acts as the next stumulus to the entity. The life-sequence is thus 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 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, the elements of the P-matrix will be "fuzzy" and it makes sense to represent the fuzziness by quantizing the P-space and define 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 the 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 reference point was much stronger than the tendency toward individuality.
We can expect, therefore, that with some adjustment for the effects of the environment, we would expect human behavior to 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.
# posted by Karl Eklund @ 10:10 AM 
