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The distinction between random and non-random is a distinction in mathematical logic. Is it also a distinction in nature, a distinction between natural principles?

Consider one mutation site of fifty-two different mutations. An analogy would be a playing card.
(1) Let each of the fifty-two different mutations be generated deliberately and one mutation be selected randomly, discarding the rest.
(2) Let fifty-two mutations be generated randomly and the selection of a specified mutation, if generated, be deliberate, discarding the rest.

These two algorithms are grossly the same. They present a proliferation of mutations followed by its reduction to a single mutation. They differ in whether the proliferation is identified as random or non-random and whether the reduction to a single mutation is identified as random or non-random.

Apply these two mathematical algorithms analogically to playing cards.

For the evolution of the Ace of Spades, the first algorithm would begin with a deck of fifty-two cards followed by selecting one card at random from the deck. If it is the Ace of Spades, it is kept. If not, it is discarded. The probability of evolutionary success would be 1/52 = 1.9%.

For the evolution of the Ace of Spades by the second algorithm, fifty-two decks of cards would be used to select randomly one card from each deck. The resulting pool of fifty-two cards would be sorted, discarding all cards except for copies of the Ace of Spades, if any. The probability of evolutionary success would be 1 – (51/52)^52 = 63.6%.

The probability of success of the second algorithm can be increased by increasing the number of random mutations generated. If 118 mutations are generated randomly, the probability of this pool’s containing at least one copy of the Ace of Spades is 90%.

Notice of the two processes, the generation of mutations and their differential survival, that either process is arbitrarily represented as mathematically random and the other is arbitrarily represented as mathematically non-random.

Also, notice that in the material analogy of the mathematics, the analog of randomness is human ignorance and lack of knowledgeable control. In its materiality, ‘random selection’ of a playing card is a natural, scientifically delineable, non-random, material process.

In the mathematics of probability, random selection is solely a logical relationship of logical elements of logical sets. It is only analogically applied to material elements and sets of material elements. The IDs of the elements are purely nominal. Measurable properties, which are the subject of science, and which are implicitly associated with the IDs, are completely irrelevant to the mathematical relationships. A set of seven elements consisting of four sheep and three roofing nails has the exact same mathematical relationships of randomness and probability as a set consisting of four elephants and three sodium ions.

In the logic of the mathematics of probability, the elementary composition of sets is arbitrary. The logic does not apply to material things as such because the IDs of elements and the IDs of sets can only be nominal due to the logical relationships defined by the mathematics. This is in contrast to the logic of the syllogism in which the elementary composition of sets is not arbitrary. The logic of the syllogism does apply to material things, but only if the material things are not arbitrarily assigned as elements to sets, but are assigned as elements to sets according to their natural properties, which properties are irrelevant to the mathematics of probability. The logic of the syllogism applies to material things, if the IDs are natural rather than nominal.

Charles Darwin published The Origin of Species in 1859. Meiosis, which is essential to the detailed modern scientific knowledge of genetic inheritance, was discovered in 1876 by Oscar Hertwig. In the interim, Gergor Mendel applied mathematical probability as a tool of ignorance of the details of genetics to the inheritance of flower color in peas. The conclusion was not that the material processes of genetics are random. The conclusion was that the material processes involved binary division of genetic material in parents and its recombination in their offspring. The binary division and recombination are now known in scientific detail as meiosis and fertilization.

The mathematics of randomness and probability, which can be applied only analogically to material, serves as a math of the gaps in the scientific knowledge of the details of material processes.

Consider the following two propositions. Can both be accepted as compatible, as applying the mathematics of randomness and probability optionally to one process or the other? Can either be rejected as scientifically untenable in principle, without rejecting the other by that same principle?
(I) The generation of biological mutations is random, while their differential survival is due to natural, non-random, scientifically delineable, material processes.
(II) The generation of biological mutations is due to natural, non-random, scientifically delineable, material processes, while their differential survival is random.

My detailed answers are contained in the essay, “The Imposition of Belief by Government”, Delta Epsilon Sigma Journal, Vol. LIII, p 44-53 (2008). My answers are also readily inferred from the context in which I have presented the questions in this essay.