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In the Dawkins-Pell debate of April, 2012, Cardinal Pell asked, “Could you explain what non-random means?” Richard Dawkins replied, “Yes, of course, I could. It’s my life’s work. There is random genetic variation and non-random survival and non-random reproduction. . . . That is quintessentially non-random. . . . Darwinian evolution is a non-random process. . . . It is the opposite of a random process.”

Obviously, Dawkins did not explain the meaning of non-random or random. He merely gave examples of algorithmic processes labeled as one or the other. In the Darwinian scheme, a selection ratio, when applied to generation, is labeled random. That same numerical selection ratio, when applied to survival, is labeled non-random. When a mathematical algorithm or selection ratio is labeled random or non-random, as in the Darwinian scheme of random genetic variation and non-random survival, what do random and non-random mean?

The Meaning of (Non-) Random

In the context of mathematics, the terms random and non-random refer to the members of a set, in those instances where the set is defined as composed of equivalent members. (Note: a member may be a subset or a single element). When a member is identified with respect to its generic equivalency to the other members of the set, it is designated as random. Each member, when identified in its specificity, is designated as non-random. The mathematical distinction of random and non-random is the distinction between the genus and the species of a member of a set.

Mutation or selection in this context uses a variable to represent a member of the set. Mutation refers to a succession of variations of the variable. If the algorithm of variation identifies the members generically, the algorithmic process of variation is designated as random mutation or random selection. If the algorithm of variation identifies the members by their specificity, the algorithmic process is designated as non-random mutation or selection. Both algorithms are systematic, i.e. ordered.

The Definitions

In a set of generically equivalent members, random identifies a member in its generic membership in the set. In a set of generically equivalent members, non-random identifies a member in its specific identity.

Intuitive Assessment of the Definitions

Consider in illustration, the set defined as the set of subsets composed of each possible sequence of any five symbols of a set of ten ordered symbols, e.g. the ten digits, 0 through 9. Three of the one hundred thousand subsets are: 12345, 21212 and 40719. Each of these sequences, when viewed solely as a subset, i.e. identified in its generic equivalency, is random. Each of these sequences, when identified as a specific sequence, is non-random.

These identifications of random and non-random don’t sit well with our intuition.

Our natural prejudice would prompt us to classify 12345 as non-random, because we recognize in it the order of convention. Similarly, we would want to classify 21212 as non-random, because in that sequence we readily recognize an order of symmetry. We would be adverse to consider these two sequences random in any circumstance. In the sequence 40719, we do not easily recognize any order, so that our prejudice prompts us readily to accept the sequence, 40179, as random according to the mathematical definition. Notice, however, that if 40719 were the sequence of symbols on your auto license plate, you would be inclined not to consider it random. It wouldn’t overwhelm your capacity for the familiarity of specific order, as do most of the one hundred thousand specific sequences.

We thus have a hint from whence our prejudice arises to deny the mathematical definitions.

We intuitively identify order or pattern as non-random. This is in accord with philosophy, which not only identifies order in the quantifiable through measurement as non-random, but identifies all of material reality as intelligible and, therefore, ordered and non-random. It would seem then that philosophy leaves no room for anything to be random. Indeed, it does leave no room at the level of reality. However, it does leave room in the logic of mathematics.

Our knowledge of reality is limited and our intelligence is limited in itself. We are sometimes incapable of perceiving order in that which is within our purview. Three grains of sand and fifty grains of sand may be considered within our purview. We readily appreciate the order of the three, but are overwhelmed by quantity to see the order of the fifty. Also, things may not be fully within our purview. We know that the outcome of rolling a pair of dice is due to the forces to which the dice are subjected. Yet, we accept the outcome as random by convention, because those forces, in detail, are not within our purview. In such cases we equate human ignorance with randomness and apply the mathematics of randomness to compensate for our ignorance.

Similarly, the causes of genetic variation are micro or smaller, while some of those of death are readily observable. We are inclined to treat genetic variation as random and survival as non-random as did Charles Darwin in the 19th century. Seven years after the publication of the Origin of Species, Gregor Mendel presented his work in which he employed the mathematics of randomness. He did not infer that genetic inheritance was random. Rather, his results established what would be expected at the micro level, namely that genetic inheritance, based as it is on two sexes, involves binary division and recombination.

The Mathematics

The mathematics of randomness is algorithmic and thereby fully ordered. Material application of the mathematics is by analogy. It is by way of illustration only. The mathematics is not inferred from measurement as are the mathematical formulae of the physical sciences.

Thus, within the context of the logic of mathematics, we have fully proper definitions of random and non-random. Random designates the generic identity of a member of a set. Non-random designates the specific identity of a member of a set.

Probability

Probability is the fractional concentration of an element in a logical set. Probability is the ratio of a subset to a set. Probability is the ratio of the specific to the generic within a set. Probability is the ratio of the non-random to the random within a set.

In contrast to the four different expressions of the singular definition above, the common definition of probability is the likelihood or chance of an event. However, such is not a definition. It is a list of synonyms. One could just as well define chance or likelihood as the probability of an event.

Illustration of Random and Non-random Mutation/Selection

Consider the set used by Richard Dawkins in illustration. It consists of the sequences of three digits, each of which varies over the range, 1 through 6. The set consists of the 216 specific sequences from 111 to 666.

A three-digit variable would generically represent the specific members of this set. If successive mutations or selections may be any of the 216, the variation is generic and identified as random mutation or random selection. If the variation is constrained by an algorithm of specific order, it is identified as non-random mutation or non-random selection. The result of variation, whether random or non-random, may be viewed as a pool of mutants or variants.

The pool of mutants formed by random mutation would also be random, unless it was constrained by some ordering algorithm, such as being limited to one copy each of the 216 mutations.

If a pool is subjected to an algorithm, which culls mutants based on specific identity, then the culling is non-random mutation/selection.

In his illustration (minute 7:50), Dawkins indicates that the maximum number of variants in a pool, needed to insure that one copy of the specific variant 651 is present, is 216. This means that, although the sequence of generation of its elements may have been random, the resultant pool is non-random. It is constrained to be of the exact same composition as the set from which it was formed by mutation. When this generated pool is subjected to non-random selection of variant 651, the probability of success is 100%.

In contrast, if the pool size was 216 and if the pool was random, the probability of its containing at least one copy of the specific element 651, would be 63.3%. We can readily calculate the probability of not selecting the specific element 651 in 216 random mutations. It is (215/216)^216. The probability of selecting at least one copy of 651 would be 1 minus this value. P = 1 – (215/216)^216 = 0.633

When this generically generated pool of mutants is subjected to the specific selection filter for 651, it would have a success rate of 63.3%. The random generation of pools of size 216 for each pool, defines a population of such pools, where the size of the population is 216^216 = 1.75 x 10^504. Of this population of defined pools, 63.3% contain at least one copy of the specific element 651.

Richard Dawkins is correct that Darwinian evolution consists of two algorithms. The first is labeled random. It is random numbers generation, a generic process. The second is labeled non-random. It is the culling of the pool of generated numbers through a specific number filter. The filtering process has a probability of less than one. Consequently, Darwinian evolution must be characterized as random as Dawkins has done on page 121 of The God Delusion. He identifies the result of each sub-stage of Darwinian evolution as “slightly improbable, but not prohibitively so”. Dawkins was right in 2006 in his characterization of Darwinian evolution as random in The God Delusion and wrong in his statement in the Dawkins-Pell debate of 2012, “Darwinian evolution is a non-random process.”

Summary

The algorithmic processing of members of a set, based on the member’s specific identity, is non-random. The algorithmic processing of members of a set, based on the member’s generic identity as a member of the set, is random. Colloquially, generic means ‘one’s as good as another’. In both instances, the algorithm, as an algorithm, is necessarily systematic, i.e. orderly.

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