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Values of percentage differ from one another by degree, not kind. Dawkins’ rule is similar. It states that discrete elements of a set, varying by the magnitude of some criterion, differ from one another by degree, not kind, with respect to that criterion of magnitude (http://www.richarddawkins.net/news_articles/2013/1/28/the-tyranny-of-the-discontinuous-mind#). A corollary of Dawkins’ rule is that a set of elements differing in degree cannot be logically divided into discrete groups, where the groups are differentiated as kinds, because differences of degree are continuous, not discrete.
In one application of his rule, according to Dawkins, genetic variants in an evolutionary line must vary by degree as a continuum because offspring do not vary in kind from their progenitors.
Dawkins applies his rule within and across evolutionary lines. He gives two examples within evolutionary lines. One is the evolutionary line extending from the common ancestor of both the pig and man to man. The other line extends from the common ancestor to the pig. Dawkins indicates that because man and the pig differ from their common ancestor by degree, man and the pig must differ from each other by degree, not kind. I assented to that expression of the conclusion, until I realized that Dawkins’ rule is not complete unless ‘not kind’ is completed by ‘with respect to that criterion of magnitude’.
If B differs from A in magnitude by some criterion and C differs from A in magnitude by that same criterion, then B and C differ (or are equal) in magnitude. Although B and C differ (or are equal) in degree, not kind, with respect to that criterion, they may differ in kind by other criteria.
Dawkins implies that in every evolutionary line, each subsequent variant differs in magnitude from its preceding variant by one or more criteria. However, this does not rule out differences other than by those criteria of magnitude. According to Dawkins, man and the pig differ (or are equal to) one another in some criterion because they differ in magnitude of that criterion from their common evolutionary ancestor. Nevertheless, they must differ from one another by at least one other criterion, because they are in different evolutionary lines.
If man and the pig differed only by degree for each criterion of a set, the two evolutionary lines could not diverge from the common ancestor. They would have to be a single evolutionary line, which by the evolutionary scheme, they are not. Therefore man and the pig differ by an additional criterion relevant to evolution, which admittedly could also be one of magnitude, but must be different from the criteria of magnitude by which they differ in degree, not kind, within their respective evolutionary lines from their common ancestor. The implication is that biological offspring can only differ in degree from their progenitors. If evolutionary lines are viewed solely as hereditary lines, evolutionary change can only be that of degree in some criteria of magnitude. This implication cannot explain a divergence into two different lines.
Explaining divergence into two evolutionary lines is not as important as identifying the criteria of magnitude by which variants in a single line differ. At the current stage of scientific development, genetic variants are distinguished from one another by their genomic maps, i.e. by discrete kind. Yet, Richard Dawkins assures us that all genetic variants in their evolutionary relationships differ from one another by continuous degree with respect to criteria of magnitude, not by discrete kind. The criteria must be identified. It is insufficient to argue that such criteria must exist because a biological progenitor and its offspring cannot differ in kind. What must be explained is how the differences between genomic maps of variants in the same evolutionary line are differences of degree, differences solely of magnitude and not differences of kind.
I am anxious to read an essay by Richard Dawkins identifying the criteria, varying in magnitude, which are the key to genomic evolutionary genetic variation. Realistically I should only expect the reply of Heraclitus of Ephesus that genomic maps are the prevalent forms currently undergoing evolutionary change in the course of the continuum of biological generation, not discrete entities in themselves; that this essay of mine is merely based on a false perspective, the false perspective of a discontinuous mind, like that of Zeno of Elea.
It is one thing to say that small changes in form, i.e. in kind, add up to big changes in kind, whether those morphological changes are identified as visual as in the days of Darwin or are identified as genomic and molecular as in the days of Dawkins. It is quite something else to say that evolutionary modification through descent is not morphological but one of magnitude because by definition biological reproduction admits of change only in degree.
Dawkins can be consistent. According to Dawkins, material complexity gives the illusion of design, when in fact design by definition is a human concept, which achieves material expression only in human artifacts. Similarly, according to Dawkins, morphological evolution, whether visual or genomic, is the illusion of a difference of kind, when we know that evolutionary modification through descent can be only of degree, not kind, by definition of reproductive descent.

Several of Richard Dawkins’ arguments are a Godsend to those with whom he disagrees and a potential blessing to himself.  This is because their initial rationale is that of the logic of simple arithmetic and because he articulates his arguments sufficiently well that, with a little study, it is apparent when they follow the tracks of the logic of arithmetic and when they jump the rails.  The study of Dawkins’ arguments can serve as a mental exercise and as an occasion for elucidating the truth. In his own review of them, Dawkins has the possibility of seeing where he has jumped the rails of logic and thus, in his review, he has the potential of discovering the truth for himself in two areas.  Dawkins stands to benefit from such a self-review in his avocation as philosopher of improbability.  Such a self-review should also result in Dawkins’ professional understanding of Darwinian evolution, which is arithmetically cogent in contrast to Dawkinsian evolution.  This essay deals with one fundamental instance in which Dawkins begins with the logic of arithmetic.  It identifies his point of departure at which he jumps the rails of the logic of arithmetic and enters into fantasy, his philosophy of improbability.

Values of fractional variables defined over the range 0 to 1, as percentages, differ from one another by degree, not kind.  Such variables include ethnicity, probability and improbability.

Are you English?  Richard Dawkins gave the correct answer: It’s a matter of degree.  You’re as English as the fractional concentration of Englishmen among your ancestors.  If only one of your parents is English, you are half English and half non-English.  In this instance two of your four grandparents, i.e. half of your grandparents, are English. Similarly half of your eight great-grandparents are English.  Your English ethnicity plus your non-English ethnicity adds up to one.  Going back to your great, great grandparents, if only one of these ancestors was English, you’re 1/16 English and 15/16 non-English.  If none of your ancestors was English, you are zero English and the fractional concentration of non-Englishmen among your ancestors is one.  No matter what fractional concentration of your ancestors is English, that fraction plus the fractional concentration of non-Englishmen among your ancestors equals one.

Richard Dawkins has used this arithmetic as a stick to rap the knuckles of those who disagree with him (http://www.richarddawkins.net/news_articles/2013/1/28/the-tyranny-of-the-discontinuous-mind#).  Funny thing, it turned out to be a sledge hammer with which he hit himself smack between the eyes.  By the valid analysis in his essay, The Tyranny of the Discontinuous Mind, Dawkins established the inanity of the thesis of his book, The God Delusion.  Its thesis is that whereas there is a solution to the problem of improbability of evolution in a one-off event, there is no solution to the problem of the improbability of God.  However, his essay demonstrates that the problem of improbability, as a concept, is illogical.

The fractional concentration of English ethnicity covers the continuous numerical range from 0 to 1.  It can have any value within this range.  The fractional concentration of English ethnicity between any two people differs in degree, not in kind.  Dawkins correctly identifies the measure of ethnicity as one of degree, not kind, in his essay, The Tyranny of the Discontinuous Mind.  He says some people can’t see the continuity of the degree of ethnicity as well as the continuity of degree in other variables. Dawkins labels such people as having a Discontinuous Mind.  Persons of a discontinuous mind insist that such variation is black and white, differing in kind, not degree.  One such person of a discontinuous mind might say that anyone whose fractional concentration of English ethnicity is less than 25% is not English.  Another person of a discontinuous mind might say that anyone whose fractional concentration of non-English ethnicity is greater than 75% is not English.  Notice they are in perfect agreement.  More importantly, as Dawkins points out, they have chosen an arbitrary point of discontinuity in the continuous spectrum from 0 to 1 to divide the spectrum into a dichotomy, demarcating two different kinds when in fact there is only one kind differing in degree.

Who could have such a Discontinuous Mind and make such a blunder of seeing a dichotomy of kind, when there is only a continuous spectrum over the range of 0 to 1, where values differ only in degree?  How about Professor Richard Dawkins?  An appropriate alternative title page for his book, The God Delusion, would be:

My Discontinuous Mind

by

Richard Dawkins

Its Solution to Its

Problem of Improbability

One must have a discontinuous mind in order to identify the problem of improbability.  If one understands fractions as correctly explained by Dawkins in the case of ethnicity, then there can be no problem of improbability as identified by Dawkins in the case of improbability.

Just like ethnicity, probability is a fractional concentration over the range 0 to 1, where values of probability differ in degree, not kind.  However, Dawkins sees values of probability as differing in kind.  According to Dawkins there are two kinds: values of probability that are scientific explanations and values of probability that are insane chance.

Probability is the fractional concentration of an element in a logical set.  The probability of heads in the flip of a coin is one-half, because there are two faces in the set defined by the flip of a coin, one of which is heads.  The probability of two heads in the flip of two coins is 1/4 (25%) because only one of the elements out of the set of four different two-coin flips is two heads.  A probability can be any value over the range of 0 to 1.  A probability plus its improbability equals one.  The improbability of two heads in two coin flips is 3/4 (75%).

What about the Law of Averages?  Isn’t the probability of two heads the average number of times one would experimentally achieve two heads in two coin flips?  The answer is no.  If I performed 1000 two-coin flips and of these 400 were two heads, I would not have experimentally proven that the probability of two heads is 2/5 (40%), rather than 1/4 (25%).  If one billion people performed the same experiment and got that same result, it wouldn’t matter because probability is the fractional concentration of an element in a logical set.  Skewed material results would simply indicate that the coin flipping was not a good emulation of the purely logical, arithmetical definition of probability.  Such results would not be indicative of the poor quality of an experiment.  They would indicate the poor material emulation of a purely logical concept.  Good experimental results lead to a posteriori knowledge of material reality.  Probability is strictly a priori knowledge within the scope of logical definition.  It is immune from experimentation just as is every other purely mathematical definition.

The spectrum of definition of probability is 0 to 1.  The arithmetic complement of probability, namely improbability, has the same range of definition, 0 to 1.  Any two values of probability differ from one another by degree, not kind.  Any two values of improbability differ from one another by degree, not kind.

Enter the Discontinuous Mind of the philosopher, Richard Dawkins: There is a problem of improbability with Darwinian evolution in a one-off event such as the evolution of the mammalian eye in a single cycle of random mutation and natural selection.  That improbability is so close to 1 that it is far beyond the reach of chance.  Chance is not a solution to the problem of improbability and no sane biologist ever said that it was.  The problem is how to escape from chance.  Chance IS the problem which must be solved.  The problem is solved by breaking the improbability up into small pieces, each of which is slightly improbable, but not prohibitively so (The God Delusion, pages 119-122; aka My Discontinuous Mind, pages 119-122).

In other words, there is an arbitrary value of improbability close to 1, which demarcates a discontinuity in the range of definition of improbability.  This point of discontinuity divides the spectrum into black and white, into a segment of the range of improbability that is a scientific explanation and a segment close to 1, where improbability is not a scientific explanation, but is insane chance.  Values below the point, which was arbitrarily established by the Discontinuous Mind, are of the non-prohibited kind.  They are scientific explanations.  Values greater than the point are of the prohibited kind.  They are scientifically inexplicable chance.  Not only does it take a Discontinuous Mind to see the two different kinds of improbability, it takes a Discontinuous Mind, like that of Richard Dawkins, to see a problem with one of the kinds of improbability.  It takes a Discontinuous Mind to issue the philosophical fiat of prohibition.  Dawkins’ philosophically whimsical identification of a discontinuity trumps the logic of the arithmetic of degree not kind for the variable, improbability.

Such nonsense is only rivaled by the arithmetically inane claim that the improbability of a series of probabilities can be ‘broken up’ into smaller ‘pieces’ of improbability!  Due to the complementarity of probability and improbability, this alleged arithmetical breakup of the improbability of a series into smaller pieces of improbability would require the concomitant arithmetical breakup of a small piece of probability into bigger pieces.

You are stupidly illogical if you divide the continuous range of definition of ethnicity from 0 to 1 by some arbitrary point of discontinuity into English and non-English.  Richard Dawkins is brilliantly logical if he divides the continuous range of definition of improbability from 0 to 1 by some arbitrary point of discontinuity into scientific explicability and insane chance.

The fact is that Dawkins is right about ethnicity and the principle of continuity, which is the theme of his essay, The Tyranny of the Discontinuous Mind.  However, Dawkins is wrong in The God Delusion because its thesis violates this valid principle.  Due to the principle of continuity, The God Delusion’s problem of improbability is logically indefinable.  There is no problem of improbability in Darwinian evolution solved by gradualism.  There is no problem of the improbability of God, whether solvable or unsolvable.  Consequently, The God Delusion (2006) is a not a book arguing in favor of the non-existence of God.  It is a book arguing in favor of the existence of a non-existent problem in the logic of arithmetic, which Dawkins now admits is illogical in his essay, The Tyranny of the Discontinuous Mind, posted in January 2013, first published in 2011.

Either his book is illogical or his essay is illogical.  Either improbability is of two kinds, which are separated from one another at some point within its range of definition according to the book or, according to the essay, any two values of improbability over its range of definition of 0% to 100% differ from one another by degree not kind.

It is evident that philosopher and scientist, Richard Dawkins has no clear, and thereby, no consistent understanding of the arithmetic of fractions.  Consequently he has no clear understanding of the three ratios, i.e. fractions, which define Darwinian evolution (See https://theyhavenowine.wordpress.com/2013/04/04/the-discriminatin-ratio-of-natural-selection/).

Philosopher Richard Dawkins is suffering angst.  In his gut, he believes in rational explanations, but sometimes he doesn’t want to believe.  Those times, he wants to be believe in the arithmetical fraction, probability, as a scientific explanation, while not believing in its arithmetic synonym, chance, which, due to their mathematical identity, has the identical spectrum of definition of 0 to 1 (Except in the discontinuous mind of Richard Dawkins).  Oh the agony of the problem of improbability, the problem of how to escape from chance, the curse of the Discontinuous Mind.

For a review of the philosophy of angst of Richard Dawkins over the problem of improbability defined by his discontinuous mind, see “What is modern in the new atheism? –The inference of probability”, Part 1 Volume LVII Fall 2012 Number 2; Part 2 Volume LVIII Spring 2013 Number 1 (the Delta Epsilon Sigma Journal).

In another aspect of continuity/discontinuity, the gradualism of Darwinian evolution produces gaping discontinuities in the spectrum of generated variants.  However, scientist Richard Dawkins is mentally content with the verbal impression of continuity suggested by the word, gradualism.  Why should he bother to put forth the mental effort to understand the arithmetic of gradualism in Darwinian evolution with its gaping discontinuities (see “The Discontinuous Mind, Now and Then”, https://theyhavenowine.wordpress.com/2013/03/21/3/)?  It takes less effort to pretend that the spectrum of variants generated by the gradualism of Darwinian evolution has no gaps, which Dawkins has falsely claimed to be the case in The Tyranny of the Discontinuous Mind.