Dawkins on Understanding Fractions

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.

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6 comments
  1. Howard said:

    By this logic, unless the first humans were English (assuming we count only human ancestors), then no one is English. “Englishness” may take on fractional values, but that value is zero for everyone. It might be worth reconsidering whether this is a useful definition of “Englishness.”

    “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.” That’s just nonsense. Regarding nationality, governments have to make that kind of decision all the time. For example, the question “Is Obama an American?” requires a yes-or-no answer when he runs for office. It even has to be done with ethnicity sometimes, particularly in cases involving membership in a Native American tribe.

    I might as well also point out that the set of rational numbers between 0 and 1 is not, in fact, continuous.

    There are other problems with the article, but these are enough.

    • I accept Dawkins’ expansion of the concept of continuity to include ordered discrete sequences, where a gap or a point in the sequence is identified as a ‘discontinuity’. In his essay, the main example of Dawkins’ concern is the ‘continuity’ of the ordered sequence of biological variants in an evolutionary line.
      Like you, Dawkins also acknowledges the practical necessity of identifying arbitrary discontinuities such as the either or of being an American. To illustrate such practical necessity, Dawkins cites voting age in his essay, but he specifically denies the validity of identifying any ‘discontinuity’ in the case of black ethnicity, which Dawkins correctly insists is a matter of degree, not kind.
      Mathematical relationships can be limited in their extent of application. Of course, the Englishness of ancestry cannot open ended.

  2. Julie said:

    I feel so ashamed but although I am interested, I don’t think I’m educated enough to understand what is being said here. Sigh. 😦 My brain hurts…

    • Thanks. I’ll keep trying to write more clearly.
      In his essay on the Tyranny of the Discontinuous Mind, Dawkins is right in saying a continuous variable is all of one kind. It varies by size, not by type. In filling a dump truck with sand, whether it is half full or three-fourths full, it’s all sand. The same is true of ethnicity, an example Dawkins uses. The same is true of probability, which, like ethnicity, can have any value between zero and one. It’s all probability. However, in “The God Delusion” Dawkins violates this truth by claiming that values of probability close to zero are a different kind from larger values. Values close to zero are “prohibitive”, in contrast to larger “non-prohibitive” values. Such a distinction cannot be made because all values of probability are equally valid over its range of definition of zero to one. In his essay on the Tyranny of the Discontinuous Mind, Dawkins criticizes others for that of which he is guilty. Improbability is one minus the probability. The thesis of “The God Delusion” falls to pieces under Dawkins’ criticism in his essay on the Tyranny of the Discontinuous Mind. Dawkins’ thesis in “The God Delusion” is that the “problem of improbability” of Darwinian evolution has a mathematical solution, but there is no solution to the “problem of the improbability” of God. Dawkins’ “problem of improbability” is based on distinguishing two kinds of improbability, a distinction which Dawkins rightly judges to be ‘unreasonable’ and leading in some cases to ‘a type of tyranny’.

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