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Synonym or Antonym

Algorithmic and systematic are synonyms.

We typically think of systematic and random as antonyms, namely as ordered and non-ordered. Consistently, we superficially think of random selection as non-systematic mutation.

Yet, in the mathematics of sets, defining the process of random mutation is algorithmic. The primary conclusion to be demonstrated in this essay is: In the mathematics of sets, random mutation is algorithmically defined, i.e. it is systematic / ordered selection.

A set may be defined by the complement of its probabilities. Probability is the fractional concentration of an element in a logical set. Two sets having the same complement of probabilities may differ from one another by an integral multiple of their elements.

The simplest set defined in terms of its complement of probabilities is a set of one unique element. The probability of that element is one. The next simplest set consists of two unique elements for which the complement of probabilities consists of one-half and one-half.

In defining new sets based on the probabilities of a source set, the probabilities of the source set are retained, but the sets differ in that the ‘elements’ of the derived set are actually subsets composed of the elements of the source set.

The derived set, in its complement of probabilities, is derived by an algorithm, which identifies by ordered mutation, the contents of its subsets, where its subsets are viewed as the elements of the derived set.

In illustration, this is all very simple

Consider a source set of two unique elements, Heads and Tails. One set derived from the source set is a set consisting of subsets of three elements, which are ‘randomly selected’ from the source set. By this algorithm of ‘random mutation’ eight subsets are defined, if sequence is retained in the subsets. These eight subsets, which comprise the derived set, are: H,H,H; H,H,T; H,T,H; H,T,T; T,H,H; T,H,T; T,T,H and T,T,T. The complement of probabilities of the derived set are eight probabilities of 1/8 each.

If the algorithm of derivation does not retain sequence, then the derived set would contain only four unique subsets of three elements each. These would be one subset of three Heads, two subsets of two Heads and one Tail, two subsets of one Head and two Tails and one subset of three Tails. That is a total of six subsets, which are deemed the elements of the derived set. In this derived set, the complement of probabilities would be 1/6, 1/3, 1/3, and 1/6, respectively.

One Objection and Its Clarification

Suppose it is objected that a probability can be calculated on the basis of random selection from a source set without advertence to the composition of a derived set. For example, the probability of selecting H,T,H from the source set is (1/2) × (1/2) ×(1/2) = 1/8. It would seem that the probability is independent of a derived set. However, by definition probability is the fractional concentration of an element in a logical set. Consequently, the probability of 1/8 has meaning only in the context of a set of eight elements derived from the source set. This objection is based on shrinking the focus of attention, on not seeing the big picture.

Note that in any derivation of a new set from a source set by random mutation, if subsets are not viewed as elements, but the elements of the derived set are identified exactly as they are in the source set, then the complement of probabilities of the derived set is identical to the complement of probabilities of the source set. From this perspective the derived set is simply an integral multiple of the source set. In the derived set illustrated, there are a total of 12 Heads and 12 Tails. The probabilities of Heads and of Tails in both the source set and the derived set are one-half and one-half.

Another Objection and its Clarification

But doesn’t viewing random selection or mutation as a fully systematic algorithm contradict our common conception of random as non-ordered? Not really. Rather, it requires us to be more cognizant of what we mean by random.

Consider the conventional sequence of the ten Arabic numerical symbols: 0,1,2,3,4,5,6,7,8,9. We would consider that sequence and Sequences I through III as non-random. Sequence I: 0,2,4,6,8,1,3,5,7,9. Sequence II: 0,9,8,7,6,5,4,3,2,1. Sequence III: 8,6,4,2,0,1,3,5,7,9. However, we consider these four sequences non-random only because it is easy for us to impute a pattern to them. There are 10! = 3,628,800 different sequences of ten symbols. Our intellectual capacity is such that we cannot be equally familiar with each sequence to see each of them on a par with the others. It is only by familiarity with a convention which renders the sequences of this paragraph ‘ordered’ and roughly three million others ‘random’. In other words, our labelling anything as ‘random’ does not designate that which is labelled as non-ordered. The label, ‘random’, designates a limitation in our knowledge. It designates our ignorance. The lack of order or randomness is in our knowledge of the subject, not it the subject itself.

Each of the 3,628,800 different sequences of ten arbitrary symbols is an ordered sequence. It is only be defining an arbitrary conventional sequence that one or more of the sequences can be considered ordered and the others as non-ordered. In imposing such a convention, the symbols are no longer arbitrary, but are defined in relation to one another.

Other Sources of Confusion

In the mathematics of probability and randomness the IDs of the elements are purely nominal, i.e. they are purely arbitrary. The probability relationships of a set of six elements consisting of three elephants, two saxophones and one electron are identical to those of a set of three watermelons, two paperclips and one marble.

The fact that probability and randomness are purely logical concepts is befogged by the jargon employed in material emulations. We identify the probability as 1/4 for the top card’s being a diamond after shuffling a deck of cards. Also, the outcome of a diamond in two successive trials is 1/16. It appears as if probability characterizes an event or an outcome. We think that this probability has nothing to do with deriving a new set from a source set. We are oblivious to the fact that two diamonds, or two D’s is one out of sixteen subsets of a set derived from a source set of four elements, such as Spades, Hearts, Diamonds, Clubs or S,H,D,C. The sixteen subsets or elements of the derived set are: SS, SH, SD, SC; HS, HH, HD, HC; DS, DH, DD, DC; CS, CH, CD, CC. The fractional concentration of DD in this set is 1/16. Absent this logical set of sixteen elements, to say that the probability of the outcome of two diamonds in succession is 1/16 would be meaningless. Probability is the fractional concentration of an element in a logical set. It is not an event or outcome.

In material emulations of mathematical probability, a material outcome is truly an event, all by itself, in its materiality. The confusion arises because we tend to characterize the material event by the numerical value of probability, when probability is meaningful only in reference to a logical set and not to anything material. The analog, which is the basis for the material emulation of the logical concept, equates randomness with the absence of human knowledge of the material causes of the material outcome. For example, we are ignorant of the forces which determine the outcome of a coin flip and label the outcome random.

The fact that probability is meaningful only in reference to a fully defined logical set prompts the error of thinking that in material analogies the material set must exist materially to validate the analogy. For the material analogy, where the sequence of a deck of cards after shuffling represents a probability of one in 52! = 8.06 × 10^ 67, that many decks of playing cards cannot possibly have material existence. This error, of transferring a logical requirement into a material requirement to validate a material analogy, leads to the proposed existence of a multiverse to explain, e.g. the probability of the physical constants of the universe.

The fog, which diminishes our ability to see the concept of mathematical probability clearly, is compounded by the fact that the word, probability, has a meaning entirely apart from mathematics. In this other meaning, probability is qualitative. This other meaning of probability is the certitude with which one judges a proposition to be true. Human certitude of the truth of a proposition has nothing to do with the fractional concentrations of elements in logical sets.

Conclusion

In mathematics, probability is the fractional concentration of an element in a logical set. Therefore, random mutation is systematic selection in spite of (1) jargon, which deflects or shrinks the focus of our attention and (2) the non-mathematical use of the word, probability, which refers to the quality of human certitude.

Logic requires a fully defined logical set in order to specify a numerical value of probability as a fraction of that set. In a material analogy, this does not necessitate the material existence of a set, but only its logical definition, in order to specify a numerical value of probability. Such analogies are emulations of the logical concepts.

In analogy, the purely logical concepts of random mutation and probability are not properties inherent in material entities such as watermelons and snowflakes. This is in contrast to measureable properties, which, as the subject of science, are inherent in material entities.

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