Richard Dawkins has extensively discussed arithmetic. The theme of The God Delusion is that there is an arithmetical solution to the improbability of evolution in a one-off event, namely gradualism, whereas there is no arithmetical solution to the improbability of God. Obviously, the ‘improbability’ of God cannot be solved by gradualism.

It is encouraging that Richard Dawkins is interested in mathematics. If he were to learn to correct his mistakes in math, he might do very well in re-educating those whom he has deceived in mathematics, science and philosophy due to his errors in arithmetic.

The following is a math quiz based on problems in arithmetic addressed by Richard Dawkins and his answers, whether implicit or explicit, in his public work. I present this as a helpful perspective in the delineation of Dawkins’ public errors in arithmetic.

1) What is the opposite of +1?
Correct Answer: -1
Student Dawkins: Zero. Let us then take the idea of a spectrum of probabilities seriously between extremes of opposite certainty. The spectrum is continuous, but can be represented by the following seven milestones along the way:
Strong positive, 100%; Short of 100%; Higher than 50%; 50%; Lower than 50%; Short of zero; Strong negative, 0% (p 50-51. Ref 1).
Those, who aver that we cannot say anything about the truth of the statement, should refuse to place themselves anywhere in the spectrum of probabilities, i.e. of certitude. (p 51, Ref 1)
Critique of Dawkins’ answer:
On page 51 of The God Delusion, Dawkins devotes a paragraph to discussing the fact that his spectrum of certitude, from positive to its negative opposite, does not accommodate ‘no opinion’. Yet, he fails to recognize what went so wrong that there is no place in his spectrum for ‘no opinion’. The reason, that there is no place, is that he has identified a negative opinion as zero, rather than as -1. If he had identified a negative opinion as -1, which is the opposite of his +1 for a positive opinion, then ‘no opinion’ would have had a place in his spectrum of certitude at its midpoint of zero. Instead Dawkins discusses the distinction between temporarily true zeros in practice and permanently true zeros in principle, neither of which are accommodated by his spectrum ‘between two extremes of opposite certainty’ in which the opposite extreme of positive is not negative, but a false zero.

2) Is probability in the sense of rating one’s personal certitude of the truth of a statement a synonym for mathematical probability, which is the fractional concentration of an element in a logical set? For example, is probability used univocally in these two concepts: (a) The probability of a hurricane’s assembling a Boeing 747 while sweeping through a scrapyard. (b) The probability of a multiple of three in the set of integers, one through six?
Correct Answer: No. The probability of a hurricane’s assembling a Boeing 747 does not identify, even implicitly, a set of elements, one of which is the assembling of a Boeing 747. Probability as a numerical rating of one’s personal certitude of the truth of such a proposition has nothing to do with mathematical probability. In contrast to the certitude of one’s opinion about the capacity of hurricanes to assemble 747’s, the probability of a multiple of 3 in the set of integers, one through six, namely 1/3, is entirely objective.
Student Dawkins: Probability as the rating of one’s personal certitude of the truth of a proposition, has the same spectrum of definition as mathematical probability, namely 0 to +1 (p 50, Ref. 1). The odds against assembling a fully functioning horse, beetle or ostrich by randomly shuffling its parts are up there in the 747 territory of the chance that a hurricane, sweeping through a scrapyard would have the luck to assemble a Boeing 747. (p 113, Ref. 1) There is only one meaning of probability, whether it is the probability of the existence of God, the probability of a hurricane’s assembling a Boeing 747, the probability of success of Darwinian evolution based on the random generation of mutations or the probability of seven among the mutations of the sum of the paired output of two generators of random numbers to the base, six.

3) In arithmetic, is there a distinction between the factors of a product and the parts of a sum? Is the probability of a series of probabilities, the product or the sum of the probabilities of the series?
Correct Answers: Yes; The product. The individual probabilities of the series are its factors.
Student Dawkins: Yes; The relationship of a series of probabilities is more easily assessed from the perspective of improbability. An improbability can be broken up into smaller pieces of improbability. Those, who insist that the probability of a series is the product of the probabilities of the series, don’t understand the power of accumulation. Only an obscurantist would point out that if a large piece of improbability can be broken up into smaller pieces of improbability as the parts of a sum, i.e. as parts of an accumulation, then it must be true that its complement, the corresponding small piece of probability, is concomitantly broken up into larger pieces of probability, where the larger pieces of probability are the parts, whose sum (accumulation) equals the small piece of probability. (p 121, Ref. 1).

4) Jack and Jill go to a carnival. They view one gambling stand where for one dollar, the gambler can punch out one dot of a 100 dot card, where each dot is a hidden number from 00 to 99. The 100 numbers are randomly distributed among the dots of each card. If the gambler punches out the dot containing 00, he wins a kewpie doll. Later they view another stand where for one dollar, the gambler gets one red and one blue card, each with 10 dots. The hidden numbers 0 to 9 are randomly distributed among the dots of each card. If in one punch of each card, the gambler punches out red 0 and blue 0, he wins a kewpie doll. This second stand has an interesting twist, lacking in the first stand. A gambler, of course, may buy as many sets of one red card and one blue card as he pleases at one dollar per set. However, he need not pair up the cards to win a kewpie doll until after he punches all of the cards and examines the results.
(a) If a gambler buys one card from stand one and one pair of cards from stand two, what are his respective probabilities of winning a kewpie doll?
Correct Answer: The probability is 1/100 for both.
Student Dawkins: The probability of winning in one try at the first stand is 1/100. At the second stand the probability of winning is smeared out into probabilities of 1/10 for each of the two tries.
(b) How many dollars’ worth of cards must a gambler buy from each stand to reach a level of probability of roughly 50% that he will win at least one kewpie doll?
Correct Answers: $69 worth or 69 cards from stand one yields a probability of 50.0%. $12 worth or 24 cards (12 sets) from stand two yields a probability of 51.5%. (A probability closer to 50% for the second stand is not conveniently defined.)
Student Dawkins: A maximum of $50 and 50 cards from stand one yields a probability of 50%. A maximum of $49 and 49 cards from stand one yields a probability of 49%.A maximum of $14 and 28 cards yields a probability of 49%. (A probability closer to 50% for the second stand is not conveniently defined.)
(c) In the case described in (b), is the probability of winning greater at stand two?
Correct Answer: No
Student Dawkins: Yes
(d) In the case described in (b), is winning more efficient or less efficient in terms of dollars and in terms of total cards at the second carnival stand?
Correct Answer: More efficient. The second stand is based on two sub-stages of Darwinian evolution compared to the first stand, which is based on one overall stage of Darwinian evolution. The gradualism of sub-stages is more efficient in the number of random mutations while having no effect on the probability of evolutionary success. Efficiency is seen in the lower input of $12 or 24 random mutations compared to $69 or 69 random mutations to produce the same output, namely the probability of success of roughly 50%.
Student Dawkins: Efficiency is irrelevant. It’s all about probability. The gradualism of stand two breaks up the improbability of stand one into smaller pieces of improbability. (p 121, Ref. 1)
This problem is an illustration of two mutation sites of ten mutations each. I analyzed these relationships in Ref. 2, using an illustration of three mutation sites of six mutations each. In that illustration, I introduced two other modifications. One modification was that the winning number was unknown to the gambler. The other was that the gambler could choose the specific numbers on which to bet, so his tries or mutations were non-random. With the latter deviation from the Darwinian algorithm, the probability of winning a kewpie doll required a maximum of 216 non-random tries for the first stand and a maximum of 18 non-random tries for the second stand. The gradualism of the second stand smears out the luck required by the first stand. The increased probability of winning a kewpie doll at the second stand is due to the fact that one need not get his luck in one big dollop, as one does at the first stand. He can get it in dribs and drabs. It takes, respectively at the two stands, maxima of 216 and 18 tries for a probability of 100% of winning a kewpie doll. Consequently and respectively, it would take maxima of 125 and 15 tries to achieve a probability of 57.9% of winning a kewpie doll. Whether one compares 216 tries for the first stand to 18 tries for the second stand or 125 tries to 15 tries, the probability of winning a kewpie doll is greater at the second stand because it takes fewer tries. (See also the Wikipedia explanation, which is in agreement with Student Dawkins, Ref. 3)
Another example of extreme improbability is the combination lock of a bank vault. A bank robber could get lucky and hit upon the combination by chance. In practice the lock is designed with enough improbability to make this tantamount to impossible. But imagine a poorly designed lock. When each dial approaches its correct setting the vault door opens another chink. The burglar would home in on the jackpot in no time. ‘In no time’ indicates greater probability than that of his opening the well-designed lock. Any distinction between probability of success and efficiency in time is irrelevant. Also, any distinction between the probability of success and efficiency in tries, whether the tries are random mutations or non-random mutations is irrelevant. (p 122. Ref. 1)

5) If packages of 4 marbles each are based on a density of 1 blue marble per 2 marbles, how many blue marbles will a package selected at random contain?
Correct Answer: 2 blue marbles
Student Dawkins: 2 blue marbles.

6) If packages of 4 marbles each are based on a probability of blue marbles of 1/2, how many blue marbles will a package selected at random contain?
Correct Answer: Any number from 0 to 4 blue marbles.
Student Dawkins: 2 blue marbles. This conclusion is so surprising, I’ll say it again: 2 blue marbles. My calculation would predict that with the odds of success at 1 to 1, each package of 4 marbles would contain 2 blue marbles. (p 138, Ref. 1)

References:
1. The God Delusion
2. http://www.youtube.com/watch?v=JW1rVGgFzWU minute 4:25
3. http://en.wikipedia.org/wiki/Growing_Up_in_the_Universe#Part_3:_Climbing_Mount_Improbable

Calculations:
Where the total number of generated mutations, x, is random, and n is the number of different mutations, the Probability of success, P, equals 1- ((n – 1)/n)^x.
For n = 100 and x = 69, P = 50.0%
For n = 10 and x = 12, P = 71.7%. For P^2 = 51.5%, the sum of x = 24
Where the total number of generated mutations, x, is non-random, and n is the number of different mutations, the Probability of success, P, equals x/n
For n = 100 and x = 50, P = 50%
For n = 100 and x = 49, P = 49%
For n = 10 and x = 7, P = 70.0%. For P^2 = 49%, the sum of x = 14
For n = 216 and x = 216, P = 100%
For n = 6 and x = 6, P = 100%. For P^3 = 100%, the sum of x = 18
For n = 216 and x = 125, P = 57.9%
For n = 6 and x = 5, P = 83.3%. For P ^3 = 57.9%, the sum of x = 15

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.

In its recent decision in Derwin v. State U., the State Supreme Court ordered the State University to award Charles Derwin the degree of doctor of philosophy. Derwin admitted that his case, which he lost in all the lower courts, depended upon one sarcastic statement made in writing by Prof. Stickler of the faculty panel, which heard his defense of his graduate thesis. The bylaws of the University in awarding the degree of doctor of philosophy require unanimous approval of the faculty panel by written yes or no voting. The members of the panel are free to offer verbal or written criticism during and after the defense, but must mark their ballots simply yes or no. However, in casting the lone negative vote, Prof. Stickler wrote in addendum to his ‘No’, “If Derwin and his major advisor were to submit an article for publication reporting the experimental results of Derwin’s thesis, I suggest they submit it to The Journal of Random Results or to The Journal of Darwinian Evolution.”

Derwin’s legal team argued that Stickler violated the university bylaws by adding the written addendum as well as academic decorum by its sarcasm. Further, and most importantly, they argued that Prof. Stickler exposed his own incompetence to judge the thesis by his attempt to belittle Darwinian Evolution. By this Stickler had disqualified himself as a judge of the thesis panel. The State Supreme Court agreed and ordered the State University to award the degree in accord with the university bylaws requiring unanimous approval by the thesis panel. The Court noted that the university bylaws allow for a panel of six to eight faculty members. The panel, which heard the defense of his thesis by Derwin, consisted of seven, including Prof. Stickler. The Court also ruled that the academic level arguments presented in the lower courts by both sides regarding ‘random results’ in general and ‘random mutation’ in the particular case of Darwinian evolution, were simply of academic interest and irrelevant to the legal case.

For their academic interest those arguments are presented here.

Prof. Stickler stated that random experimental results are of no scientific value and that Derwin conceded the results he reported in his thesis could be characterized as random. Stickler argued that even those, who contend that genetic mutation is random, claim that Darwinian evolution is non-random, and therefore scientific, even though their claim is erroneous. Stickler attributed the following quote to Emeritus Prof. Richard Dawkins of Oxford University as his response to the question, “Could you explain the meaning of non-random?” Dawkins replied, “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 (evolution) is the opposite of a random process.” (Ref.1).

Stickler stated that Dawkins’ argument that Darwinian evolution is non-random and therefore, scientific, does not hold water. He noted that the pool of genetic mutants subjected to natural selection in Darwinian evolution is formed by random mutation. The pool’s containing the mutant capable of surviving natural selection is a matter of probability. Consequently, the success of natural selection cannot be 100%. Evolutionary success is equal to the probability of the presence of at least one copy of the survivable mutant in the pool subjected to natural selection and is therefore random.

In his rebuttal, Derwin agreed with Stickler that Darwinian evolution was indeed characterized by probability and randomness. However, the universal scientific acceptance of Darwinian evolution indicates that random results are indeed scientific, which he noted was the pertinent issue in the case.

Stickler’s counter argument was to note that Darwinian evolution is based on data consisting of a series of cycles, each cycle consisting of the proliferation of genetically variant forms and their diminishment to a single form. Darwinian evolution explains such cycles by the hypothesis of the random generation of genetic variants and their reduction to singularity by natural differential survival. Stickler claimed the data could also be explained by what he called ‘The inverse Darwinian Theory of Evolution’. The inverse theory explains the same cyclic data of the standard theory, but as the non-random, natural generation of genetic variants by scientifically identifiable material processes and the reduction of this pool of genetic variants to singularity by random, differential survival.

Deciding which hypothesis, if either, was valid would require some ingenuity beyond the stipulated data of the proliferation and diminishment of variant genetic mutations. He noted, however, that the inverse hypothesis would be rejected a priori by the claim that we know at least some of the scientific factors affecting differential survival, so it could not be hypothesized that differential survival was random. This, Stickler claimed, demonstrates that randomness and probability cannot be proffered as a scientific explanation. He said, “If randomness is rejected a priori as scientifically untenable as an explanation of variant survival because differential survival is due to scientific material processes, then randomness must be rejected a priori as scientifically untenable as an explanation for the generation of genetic variants for the same reason. If randomness is sauce for the goose of scientifically variant generation, randomness must potentially be sauce for the gander of scientifically variant survival. In fact it is sauce, i.e. the mathematics of randomness and probability is a tool of ignorance to cover a gap in scientific knowledge. It is in the context of the absence of the scientific knowledge of genetics in the mid-nineteenth century that made it seem plausible at that time to propose ignorance of the scientific knowledge of material processes, that is, to propose random changes, as part and parcel of a scientific theory.”

In response, Derwin noted that quantum mechanics, perhaps the most basic of the sciences, is recognized as founded on probability and therefore randomness. (Ref.2).

References:
1) http://www.youtube.com/watch?v=tD1QHO_AVZA (minute 38:56)
2) http://theyhavenowine.wordpress.com/2014/03/26/quantum-or-wave-dilemma-of-the-imagination/

100_0023

Amaryllis is a form of lily and as such its petals are in sets of three. It has two such sets, one fore and one aft. One set may be roughly identified by direction as north, southeast and southwest. The other set as south, northeast and northwest. These sets are natural measureable properties of the amaryllis plant and are useful in the science of taxonomy.

The mathematics of sets may be characteristically embodied in nature and when so, it forms part of the base of science. However, that is not to say that the entirety of the mathematics of sets is characteristically embodied in nature. One important exception is the mathematics of probability.

Probability is the fractional concentration of an element in a logical set. The mathematics of probability is centered in the formation of other logical sets from a source set, where all sets are identified by their probabilities, i.e. their fractional compositions.

In discussing the mathematics of probability where the source set is one of two subsets of three unique elements, it would not be inappropriate to refer to the amaryllis flower as a visual aid in discussing what is essentially logical and in no way characteristic of the nature of the amaryllis flower.

In randomly forming sets of two petals where the source set is the amaryllis flower, what are the probabilities of the population of sets defined in terms of fore and aft petals? The population defined is of four sets of two petals each. One set consists of two fore petals. One set consists of two aft petals. Each of the other two sets of the population consist of one fore petal and one aft petal. The probability of sets of two fore or two aft petals in the population of sets is 25%, while the probability of a set of one fore and one aft petal is 50%.

One could imagine placing one set of six petals from an amaryllis flower in each of four hundred pairs of hats and blindly selecting a petal from each of two paired hats. One would expect the distribution of paired sets of petals to be roughly one hundred of two fore petals, one hundred of two aft petals and two hundred of one fore and one aft petal. This would represent a material simulation of the purely logical concept of probability.

What is the probability of a set of eighteen randomly selected amaryllis petals containing at least one ‘north’ petal? The answer, P, equals (1- ((n – 1)/n)^x), where n = 6 and x = 18. P = 66.5%.

Notice that these questions in the logic of probability have nothing to do with amaryllis flowers or petals or their manipulation. The petals and their visual (or actual) manipulation are entirely visual aids in a discussion of pure logic. It is materially impossible to select any material thing at random from a set of material things. Selection is always explicable in terms of the material forces involved. Thus, it is always non-random. It is by convention that we equate human ignorance of the details of the material process of selection with mathematical randomness.

We say that the probability of one sperm fertilizing a mammalian egg is one in millions. What we mean is that the fractional concentration of any one sperm is one in millions and that we are ignorant of the detailed non-random physical, chemical and biological processes by which one sperm of the natural set fertilizes the egg.

It is, of course, permissible to use the mathematics of probability in many instances when for any number of reasons we are ignorant of the scientific explanation of material processes. However, we must be constantly aware that the mathematics of probability characterizes human ignorance and not material reality when it is used as a tool to compensate for a lack of knowledge.

The mathematics of probability is an exercise in logic, unrelated to the nature of material things and their measurable properties. In contrast, science is the determination of the mathematical relationships among the measureable properties of things, which properties are characteristic of the nature of material things.

In my previous post (Ref. 1), I noted that John Lennox concurs with Richard Dawkins that replacing a single cycle of Darwinian evolution with a series of sub-cycles ‘drastically increases the probabilities’ (Ref. 2).

However, Lennox is not in full agreement with Dawkins’ view. Lennox’ primary criticism of Dawkins’ interpretation of the biological version of Darwinian evolution is that it is a circular argument, “And strangest of all, the very information that the mechanisms are supposed to produce is apparently already contained somewhere within the organism, whose genesis he claims to be simulating by the process.” Lennox misinterpretation of Dawkins’ view is ‘within the organism’ (Ref. 3). For Dawkins and Darwin, natural selection is essentially inanimate and external to the evolving organism. It is natural selection which defines the target.

Dawkins has identified the generation of biological forms (at whatever level, such as that of the genome or the morphology of a bird’s wing) as random, while natural selection is non-random, thereby rendering, in Dawkins’ judgment, evolution overall as ‘quintessentially non-random’ (Ref. 4). Thus, consistently with the algorithm of Darwinian evolution, it is random mutation, which is ‘a blind, mindless, unguided process’, not natural selection, as Lennox alleges (Ref. 3).

Natural selection is entirely external to the organism and essentially physical, not biological. Natural selection may be described as biologically blind, mindless and unguided, while being physically sighted, intelligent and guided. Natural selection is an ecological niche, defined in purely inanimate, physical and chemical terms.

In the biological simulation of the Darwinian algorithm, life is plastic by means of random mutation. This plasticity is allowed survivable, discrete expressions only within the molds of environmental constraints. These biological expressions appear to us as biological norms when in fact they are environmental norms.

Typically we think in terms of biological norms, such as mice, amoebae, algae and tulips. According to Darwinian evolution and natural selection in particular, these are not biological norms. What is biological is simply potency working through random generation (mutation). What appears to us to be biological norms are really environmental, i.e. physical, norms, evidenced in biological terms. We are aware of the existence of the inanimate environmental niches by the existence of the biological forms, once randomly generated, which now fill them. The environmental norms are physically defined and physically formed. They are merely filled with living matter, which coincidentally expresses a biological form compatible with the determining environment.

In the biological simulation of the Darwinian algorithm, natural selection is completely external to the organism and may indeed be viewed as an existent target, where the culling of ‘failed’ mutations is one aspect of the overall physical processes of natural selection. The Darwinian algorithm, in itself and as presented by Dawkins, is not circular because the target is not within the organism as alleged by Lennox (Ref. 3).

It should be noted that Dawkins is in error in claiming that the overall algorithm of Darwinian evolution is ‘quintessentially non-random’ simply because the last part of the algorithm, namely natural selection is non-random. Though natural selection is non-random in itself, the probability of success of natural selection depends upon the probability of the presence of the survivable mutant in the pool of mutants randomly generated. Thus, the end result of the Darwinian algorithm is random. Darwinian evolution is quintessentially random. Darwinian random mutation is the generation of random numbers as Dawkins has illustrated in his excellent metaphor of the combination lock (Ref. 5).

References:

1. http://theyhavenowine.wordpress.com/2014/04/10/smearing-out-the-luck/
2. “God’s Undertaker”, page 165
3. “God’s Undertaker”, page 167
4. http://www.youtube.com/watch?v=tD1QHO_AVZA Minute 39:10
5. “The God Delusion”, page 122

My previous post (Ref. 1), may have given the false impression that no one agreed with Richard Dawkins’ explanation of smearing out the luck of Darwinian evolution. This post hopefully corrects that impression.

Dawkins has stated “. . . natural selection is a cumulative process, which breaks the problem of improbability up into small pieces. Each of the small pieces is slightly improbable, but not prohibitively so.” (Ref. 2) What does this mean? We can tell from his illustrations (Ref. 3 – 4).

It would seem that Dawkins is saying that the probability of the generation of a given number by a random numbers generator, is increased by the introduction of natural selection. This doesn’t fly. Natural selection doesn’t generate mutations. It culls mutations. It permits only copies of one particular mutation to survive. It doesn’t affect the generation of the survivable mutation from which arises its probability.

Consider a single mutation site defining six different mutations. The six faces of a die define six different mutations. In this example of a total of six defined mutations, the probability of the random generation of at least one copy of the number, 6, for a total of one randomly generated mutation is 1/6 = 16.7%, with or without natural selection. Similarly the probability of the random generation of at least one copy of 6 for a total of six randomly generated mutations is 80.6%, with or without natural selection. In Darwinian evolution natural selection has no effect on probability (Ref. 1). It merely eliminates superfluous mutations, whether the superfluous mutations have been generated randomly or non-randomly.

It is apparent that Dawkins is not assessing the role of natural selection, but analyzing the replacement of a single cycle of random mutation and natural selection with several sub-cycles. In Ref. 3, he compares a single cycle affecting three mutation sites of six mutations each to three sub-cycles, each affecting a single site. The replacement of a single cycle with a series of sub-cycles has no effect on probability. Rather, it increases the efficiency of random mutation. Yet, Dawkins does not identify this as efficiency in mutation due to sub-cycles. He calls it ‘smearing out the luck’, as if the probability of success changed from 1/216 to 1/18. Dawkins is comparing 216 non-random mutations to 18 non-random mutations at a probability of success of 100% (Ref. 4).

Some of Dawkins’ reviewers have agreed with him. The Wikipedia review says the comparison is between probabilities of 1/216 and 1/18 (Ref. 5). More remarkably, in referring to a set of twenty-eight mutation sites of twenty-seven mutations each, John Lennox cites a probability of 10^(-31) and one billion mutations for a single cycle compared to the probability and the number of mutations for a series of 28 sub-cycles (Ref. 6). A computer simulation of the sub-cycles reached a probability of 1 in a maximum of 43 mutations per sub-cycle. In accord with Dawkins, Lennox refers to this as drastically increasing the probabilities. Superficially, Lennox’ comparison appears to imply an increase in probability due to the introduction of sub-cycles.

However, the Darwinian algorithm with sub-cycles, as well as this example of it, does not increase probability. In fact, the comparison in Lennox’ example, implies efficiency in the number of mutations due to sub-cycling with no change to the probability. A more appropriate comparison would have been at a probability of 90% for both the single overall cycle and for the series of 28 sub-cycles. This would compare 2.3 x (27)^28, i.e. roughly 2.7 x 10^40 mutations for the single cycle to 4144 mutations for the series of 28 sub-cycles at the same probability of success, namely 90%.. The 4144 mutations are 148 mutations for each of 28 sub-cycles, where the probability of success for each sub-cycle is 99.6%. This yields a probability of 90% for the series of 28.

Contrasting non-random vs random mutation, within the algorithm of Darwinian evolution for a single cycle, also shows that natural selection has no effect upon probability. For non-random mutation, one mutation yields a probability of 1/n. This increases linearly to a probability of 1 as the number of non-random mutations reaches n. Natural selection merely culls the superfluous mutants. Similarly, random mutation starts out at a probability of 1/n with one mutation and asymptotically approaches 1 as the number of random mutations increases. When the number of non-random mutations is respectively, n, 2.3n, 4.6n and 11.5n, then the respective probabilities are 63%, 90%, 99% and 99.999%. Here too, natural selection merely culls the superfluous mutants.

Another common error in the evaluation of Darwinian evolution is to attribute temporal and material constraints to random mutation. Due to the fact that Darwinian evolution is strictly a logical algorithm of random mutation and natural selection, it is not subject to any temporal or material constraints. It is material simulations, not the logical algorithm, which can exceed such constraints. Also, in a material simulation, there is no increase in time or material due to a random mutation compared to a non-random mutation.

There are 52 factorial or 8.06 x 10^67 different sequences of 52 elements. The inverse of this is the probability of any sequence. In a material simulation, how many decks of cards and how long does it take to generate randomly any sequence, if shuffling for five seconds is granted to be a random selection? The answer is one deck and five seconds. Granted this, how many decks of cards and how long would it take to generate a pool of decks of cards containing at least one copy of a particular sequence at a probability of 90%? The answer is 2.3 x 8.06 x 10^67 decks and 5 x 2.3 x 8.06 x 10^67 seconds. If we apply the Darwinian algorithm of a single cycle of random mutation and natural selection, these paired numbers of decks and seconds are required for a probability of success of evolution of 90%. This exceeds by far any practical temporal and material limits. However, if we are content with any value of probability, then we would be content with one random mutation. Natural selection does not affect probability. It merely culls superfluous, randomly generated mutants. If we trust success to just one random mutation, then there is no need for natural selection, while the material and temporal requirements of the simulation are insignificant, namely one deck and five seconds.

Indeed, we must be content with any and every value of probability. I have argued that no value of probability represents a ‘problem of improbability’. To claim that ‘the probability of this outcome is so close to zero that it could not be due to chance’ is a self-contradiction. Of course, I am not claiming that probability is to be accepted as an explanation. Rather, if probability is accepted in any instance as an explanation, then in no instance can it be rejected as an explanation on the basis of its numerical value, irrespective of how close it is to zero. (Ref. 7). Similarly, the acceptance of probability as an explanation is not bolstered by a value of probability closer to 1.

References:
(1) http://theyhavenowine.wordpress.com/2014/04/04/dawkins-on-gradualism/
(2) “The God Delusion”, page 121
(3) “The God Delusion”, page 122
(4) http://www.youtube.com/watch?v=JW1rVGgFzWU minute 4:25
(5) Growing Up in the Universe, Part 3 http://en.wikipedia.org/wiki/Growing_Up_in_the_Universe
(6) “God’s Undertaker Has Science Buried God?” Page 165-167
(7) http://theyhavenowine.wordpress.com/2013/12/30/too-improbable-to-be-due-to-chance/

According to Richard Dawkins the problem which any theory of life must solve is how to escape from chance (Ref. 1), how to solve the problem of improbability (Ref. 2).

Darwinian evolution consists not simply in a single cycle of random mutation and natural selection, but the gradualism of a series of such cycles. According to Dawkins, the problem of improbability of a single cycle is solved by the gradualism due to cycles, each of which is terminated by natural selection. Although Dawkins has declared that the meaning of non-random is his life’s work (Ref 3), it is really gradualism which is his key to explaining the randomness and improbability of Darwinian evolution.

What follows are analyses of four explanations Dawkins has offered to elucidate the role of gradualism in Darwinian evolution. The explanations focus on: (1) a numerical illustration involving three mutation sites of six mutations each, (2) the breakup of a large piece of improbability into smaller pieces, (3) the analysis of the vector of evolution into its component vectors of improbability and gradualism and (4) the gradualism of mutant forms in an ordered sequence approaching mathematical continuity.

Dawkins presents these explanations as in the mainstream of the scientific acceptance of Darwinian evolution. He is not attempting to re-interpret Darwinian evolution or to present any departure from it. His explanations are sufficiently explicit that they are a clear testimony to the coherent mathematics underlying Darwinian evolution in spite of Dawkins’ errors in explaining the mathematics.

The Role of Gradualism, a Numerical Illustration (Ref 4)

A set of three mutation sites of six mutations each defines 216 different mutations, which is 6 x 6 x 6. The 216 mutations may be viewed as an ordered sequence consisting of the initial mutation, the final mutation and 214 ordered intermediate mutations. Let a pool of one copy of each mutation be generated and subjected to natural selection. Natural selection would cull all but one mutation, the final mutation. The pool of 216 would have been non-randomly generated. The probability of success of natural selection would be 100%.

Let this single cycle of mutation and natural selection be replaced by the gradualism of three cycles, where each cycle affects one of the mutation sites independently of the other two. In each cycle a pool of six non-randomly generated mutations would be subjected to natural selection. A total of 18 mutations would be non-randomly generated. The probability of success of natural selection for each cycle would be 100% and the overall probability of success of natural selection would be 100%, i.e. 100% x 100% x 100%.

The gradualism of sub-cycles would generate a total of 18 mutations consisting of the two endpoints, but only 16 of the ordered 214 intermediates defined by evolution overall. Gradualism introduces large gaps, not just missing links, in the actually generated spectrum of ordered mutations in comparison to the ordered spectrum defined by the mutation sites. Gradualism introduces an efficiency factor of 216/18 = 12 in non-random mutations without any change in the probability of success of natural selection, which is 100% for both the overall cycle and the series of sub-cycles of gradualism.

Although such is Dawkins’ illustration of gradualism, it is not his interpretation. Dawkins refers to the non-random mutations as ‘tries’ implying that they are random mutations. He refers to a one in 216 chance compared to three chances of one in 6. He calls gradualism smearing out the luck into cumulative small dribs and drabs of luck. Dawkins erroneously believes he has illustrated random mutation rather than non-random mutation. Dawkins thinks he has illustrated, not an increase in efficiency in the number of mutations due to gradualism. Dawkins erroneously claims to have illustrated an increase in the probability of success of natural selection, the probability of success of Darwinian evolution due to gradualism.

If the pools of mutations subjected to natural selection in the illustration are generated by random mutation, a similar efficiency in the number of mutations is achieved, without any change in the probability of success of natural selection. A pool of 19 randomly generated mutations in each of three sub-cycles would yield a probability of success of natural selection of 96.9% for each cycle and an overall probability of success of natural selection of 90.9%. For a single cycle of random mutation involving all three mutation sites, a pool of 516 random mutations would yield a probability of success of natural selection of 90.9%. The efficiency factor in random mutations would be 516/57 = 9 due to gradualism with no change in the probability of success of natural selection.

The probability, P, of at least one copy of the mutation surviving natural selection in a pool of x randomly generated mutations with a base of n different mutations is: P = 1 – ((n -1)/n)^x.

The Breakup of a Piece of Improbability (Ref. 5)

Dawkins claims that natural selection, which terminates each cycle of random mutation and natural selection, increases the probability of success of natural selection by forming a series of cycles. The overall improbability of evolutionary success in a single cycle is broken up into smaller pieces of improbability.

To break up a large piece of something into smaller pieces implies that the smaller pieces add up to the larger piece.

Probability and improbability are complements of one. They add up to one. The overall probability of a series of probabilities is the product of the probabilities forming the series, not their sum. Each value of probability in a series, if less than one, is greater than the overall probability.

The probability of any face of a red die is 1/6 = 16.7%. The improbability is 5/6 = 83.3%. The same is true of a green die. The probability of any combination of the faces of the two dice, one red one green, is 1/36 = 2.8%. Its improbability is 35/36= 97.2%. Suppose it were mathematically valid to say that rolling the dice individually in a series, rather than both together, ‘breaks the improbability of 97.2% up into two smaller pieces of improbability of 83.3% each’. It would then be mathematically valid to say that rolling the dice in series rather than both together, ‘breaks up the small piece of probability of 2.8% into two bigger pieces of probability of 16.7% each’. Both statements are nonsense. They confuse multiplication and division with addition and subtraction. Yet, Richard Dawkins claims that those, who insist that the overall probability of a series is the product of the probabilities, don’t understand ‘the power of accumulation’, i.e. the power of addition. He claims that ‘natural selection is a cumulative process, which breaks the problem of improbability up into small pieces. Each piece is slightly improbable, but not prohibitively so.’ (Ref. 5) In other words, gradualism increases the probability of evolutionary success.

The probability of the outcome of the roll of three dice, one red, one green and one blue is 1/216. If each die is rolled separately the probability of each roll is 1/6 and the overall probability is 1/216. If the probabilities of the three individual rolls were cumulative as Dawkins indicates (Ref. 5), the overall probability of the series would be 1/6 + 1/6 +1/6 = 1/2. Yet, Dawkins implies the accumulation would be a probability of 1/18 (Ref. 4). Similarly, Wikipedia states, ‘The probability of unlocking the combination, in three separate phases, falls to one in eighteen.’ (Ref. 6). The non-random mutations of three individual sites of six mutations each, e.g. those of three dice, add up to 18. The fraction, 1/18, is not the probability of the series of three individual probabilities. It is simply the mathematical inverse of the 18 total mutations.

The Analysis of the Vector of Evolution (Ref. 7)

Dawkins proposes a parable of Darwinian evolution, ‘Climbing Mt. Improbable’. In the parable, evolution is a vector slope, the sum of a vertical vector, improbability, and a horizontal vector, gradualism. If gradualism is zero, then the vector of evolution is solely a vector of improbability. In the parable, the role of gradualism in Darwinian evolution is to change the slope of the vector of evolution from infinity when gradualism equals zero to some finite value, when gradualism is greater than zero. Dawkins implies that gradualism in the parable decreases the improbability. It does not. Both in Dawkins’ parable and in Darwinian evolution, the overall improbability is unaffected by gradualism, whether or not the vectors of evolution and gradualism are incremental. Also, improbability is not a vector. Consequently, Dawkins’ metaphor of Darwinian evolution as a vector slope, equaling the sum of its horizontal and vertical vectors, is irrelevant both to improbability and to Darwinian evolution.

The Spectrum of Ordered Mutations as Approaching Mathematical Continuity (Ref. 8)

In Darwinian evolution, each cycle of random mutation and natural selection defines a finite set of different mutations, n, which is the base of random numbers generation. The pool of x generated random numbers is subjected to natural selection, i.e. the pool is subjected to a filter at a discrimination ratio of 1/n, which culls all mutations except copies of the one mutation which survives natural selection. The set of n different mutations is an ordered set, where the initial mutation is the lowest and the mutation, surviving natural selection, is the highest mutation in the ordered sequence.

Dawkins notes that, if all of the intermediates between the lowest and the highest mutation in a biological evolutionary sequence survived, the ordered sequence would be recognized as a sequence approaching mathematical continuity (Ref. 8). Furthermore, Dawkins claims that all of the intermediates have been biologically and randomly generated, but are typically extinct. Man and the pig have a common evolutionary ancestor. ‘But for the extinction of the intermediates which connect humans to the ancestor we share with pigs (it pursued its shrew-like existence 85 million years ago in the shadow of the dinosaurs), and but for the extinction of the intermediates that connect the same ancestor to modern pigs, there would be no clear separation between Homo sapiens and Sus scrofa.’ (Ref. 8).

Dawkins would be right within the context of Darwinian evolution, except for one feature of Darwinian evolution which he demonstrated in Reference 4. Gradualism in Darwinian evolution is highly efficient in the generation of random mutations, by not generating most of the intermediates. Due to the efficiency of gradualism, as Dawkins has shown, there are gaps in the actually generated set of intermediates in comparison to the mathematically defined spectrum of ordered mutations. Most of the mathematically defined intermediates, which connect humans to the ancestor we share with pigs, are not absent due to extinction, they are absent because they were never biologically generated, thanks to the efficiency of gradualism of Darwinian evolution. Darwinian evolution is not characterized by missing links in the fossil record. In its mathematical definition, it is characterized by gapping discontinuities in the sequence of ordered mutations actually generated in comparison to the sequence of ordered mutations defined, but not generated.

In 1991 (Ref. 4), Dawkins demonstrated that there are discontinuities in the spectrum of generated mutants due to gradualism. In 2011 (Ref. 8), he claimed that gradualism insures the generation of the complete spectrum, that any discontinuities are due to extinction, not to a lack of generation.

Conclusion

Dawkins does not understand the mathematics of gradualism or its role in Darwinian evolution. Yet, understanding the meaning of random/non-random, which he has identified as his life’s work, is integral to understanding the mathematics of and the role of gradualism in Darwinian evolution. The role of gradualism in Darwinian evolution is to increase the efficiency of random mutation. It has no effect on probability.

References:
1. Page 120, “The God Delusion”
2. Page 121, “The God Delusion”
3. http://www.youtube.com/watch?v=tD1QHO_AVZA Minute 38:55
4. http://www.youtube.com/watch?v=JW1rVGgFzWU minute 4:25
5. Page 121, “The God Delusion”
6. Growing Up in the Universe, Part 3 http://en.wikipedia.org/wiki/Growing_Up_in_the_Universe
7. Page 121-2, “The God Delusion”
8. http://www.richarddawkins.net/news_articles/2013/1/28/the-tyranny-of-the-discontinuous-mind#

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