Einstein's God 0516


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Page : EG0516 Title: Evolution 16

The Theory of Evolution that we have been describing depended on the notion that survivability could be measured as a probability function of parameters that depend on the characteristics of the organism. We used that idea in a qualitative way that made general sense, but it is a good idea to look at that with a bit more precision. The difference between the implications of 'survival of the fittest' and 'survival of the just-barely-fit' are so great that we should try to remove any doubt about the method we used.

We implicitly used a geometric description of the process. We can try an alternative.

The Theory of Evolution of Species describes the changes in the characteristics of a population of living entities (who are "the same" in the sense that they can interbreed) under the influence of environmental effects such as climate, predation, etc. A basic assumption in the theory is that there is some factor in the individual entity that causes the characteristics of the offspring to be related to the characteristics of the parent. Without that there would be nothing to cause the characteristics of a species to have any stability.

But one of those characteristics is the ability to have offspring. Like the other aspects of transmitting characteristics between generations, the ability to have offspring involves an element of chance.

The basic mathematical element in a theory of evolution is the probability that a single individual will be able to survive long enough to have at least one viable offspring that will carry its genetic constituents forward in time. The trick to using mathematics as a basis for a theory of evolution is to represent the entities and the process well enough to make a mathematical relationship between predecessor and successor populations reasonable.

We will assume that there is some validity to the notion of genetics, i.e., that, at least to some degree of approximation, the characteristics of the organism are relatable to the sequence of amino acids in the DNA present in the reproductive cells. Since these can be specified in terms of an ordered sequence of a finite number of labels the information contained in the genetic material can be represented by a vector, which is an ordered sequence of numbers.

We will represent these vectors by letters in square brackets like [v]. When it is necessary to identify a parameter, such as the generation number, it will be placed in parenthesis after the vector designation as [v(i)].

A vector can be represented geometrically as well as algebraically. The ordered numbers comprising a vector of order 2 can be represented as the vertical and horizontal coordinates of a point in two-dimensional space, and thus as a point on a grid inscribed on a flat surface. Similarly, the ordered numbers comprising a vector of order [n] can be thought of as the coordinates in an n-dimensional vector space.

If we consider an individual characterized by vector [a(0)] we can characterize its offspring as generation "1" with vector [a(1)]. Then the "family history" of that individual will be characterized by the sequence:

[a(0)], [a(1)], [a(2)],..etc.

The notion of genetics is such that the character of the individual is transmitted from one generation to another, i.e. that the characteristics of each generation are related to the characteristics of the immediately prior generation. This relationship is represented by a matrix operator {E} which transforms one vector to another. Thus if there is a consistent pattern of transformation from generation to generation we will have the equations:

[a(1)] = E [a(0)],

[a(2)] = E [a(1)] = E [E [a(0)]], etc.

or, in general:

[a(n+k)] = E(exp k) [a(n)], where E(exp k) represents the repeated application of the operator E "k" times. If the vector is represented by an ordered sequence of [m] numbers, the operator is represented by a square matrix containing [m.m] algebraic functions and the operation is matrix multiplication. Corresponding transformation operators are definable in the geometric representation.

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