Monthly Archives: January 2014

I must begin with the disclaimer that I do not agree with the proponents of the Intelligent Design argument that it is a scientific argument or that it is even valid. However, I also do not agree with the common critiques of the ID argument especially in the contention that such critiques are themselves scientific.

This essay is to argue against the claim of John Derbyshire in the American Spectator ( that Michael Behe has not tried to get his scientific views expressed in the ‘scientific arena’. Derbyshire states, “You should at least try, as ID-ers like Behe obviously haven’t.”

Also, the animosity toward Michael Behe is so intense that he cannot state that the time interval between unexpected genetic mutations cannot be determined a priori but must be experimentally determined, without his opponents labelling his view as ‘naïve’ in a scientific journal.

My previous post on this blog noted the importance of distinguishing between probability and density. This post is another example. I composed the following essay some time ago, but it is timely in view of Derbyshire’s current essay:

Evolution in the Darwinian lexicon is a mathematical process which initially characterizes a particular numerical fraction as a random selection ratio and finally as a non-random selection ratio. Darwinian evolution itself has nothing to do with rates with respect to time. However, in some material applications the question of temporal rates will arise. It is therefore important to distinguish on the one hand, the two selection rates, which are essential to the concept of Darwinian evolution and are numerically equal to each other and, on the other hand, any temporal rates, which refer to particular material contexts.

Darwinian evolution consists of one or more stages of two selection processes applied essentially to the same logical set of n unique elements. They are random mutation and natural selection.

In each stage, the first selection process is random selection, at a ratio of 1/n. The selection is repeated multiple times randomly forming a pool of elements in which the probability of the pool’s containing each of the n elements is close to 1, due to the multiplicity of random selection. Randomness means that there is no rationale governing selection, so that the probability of selection, i.e. the rate of selection, is the fractional concentration of any element in the initial set, namely, 1/n. Note that random selection, which does not discriminate among the n different elements of the initial set, is a rate of selection with respect to the number of elements in the set. It is not the rate of selection with respect to time. Time is irrelevant.

In each stage, the second selection process is non-random (or if you prefer, ‘natural’) at a ratio of 1/n. This selection process is applied to each of the elements in the randomly formed pool. Since it follows a rationale, which recognizes each element in the initial set, it is the discrimination ratio of 1/n in its processing, by selection, of each of the elements in the random pool. Note that the non-random selection, which discriminates among the n different elements of the initial set, is a rate of selection with respect to the number of elements in the set. It is not the rate of selection with respect to time. Time is irrelevant.

These two mathematical selection processes constitute the mathematical transformation known as Darwinian evolution. In each stage, the process involves the initial labeling of a given numerical selection ratio as random and the final labeling of this same numerical selection ratio as non-random. Time and rates of selection with respect to time are intrinsically irrelevant.

Selection rates with respect to time can only be introduced alongside the completely distinct mathematical selection rates of Darwinian evolution by means of the analogical application of Darwinian evolution to material phenomena. The temporal rates of material changes can even lead to mistaking a non-Darwinian temporal material process for the purely logical process of Darwinian random mutation.

Random mutation is the random selection of an element from a logical set. Population genetics is concerned with the inheritance of a genetic marker based on its material density in a breeding population.

Consider a non-Darwinian scheme affecting two nucleotide sites in a genomic map. There are four different nucleotide pairs and consequently four possible variants for any site in a genomic map. Let some fraction of a population possess at one site a different specific base pair from the specific base pair of the rest of the population. Let the material density in the population for this inheritable marker be m1. For a second site, let the material density in the population for a second marker be m2. These material densities, m1 and m2 may be referred to as the respective probabilities of possessing these inheritable markers by a member of the population selected at random.

Contrast these probabilities with the probability of selecting a number at random to the base four. The probability is 1/4, which is the probability of the random selection from the set of four variant genomes identified by random mutation at a single site of a genomic map. Random mutation, of course, is random selection from a logical set. If the four variant genomes were a set of material elements, a random mutation would exist before the random mutation occurred.

Consider a population in which there are two distinct subsets, each identified by an inheritable marker. Over generations a subset of individuals in the population will form in which both of these markers are present. Durrett and Schmidt (Genetics 181: 821-822, 2009 showed that the mean waiting time for two inheritable markers to be inherited by the same individual of a population, one marker with a material density of m1 in the population and the other with a material density of m2 (when the first marker is neutral with respect to reproductive efficiency) is a function of m1 and a function of the square root of m2.

Contrast the mean generational waiting time for occurrence within a population with Darwinian random mutation of numbers to the base 16. Sixteen is the number of different genomes defined by random mutation at two genomic sites. Given a mere 72 Darwinian generators, at a probability of 99%, at least one copy of any of the 16 genomes, defined by variance at two nucleotide sites, will be generated in the length of time to generate one individual genome.

To label a change as random, where randomness implies the random selection of an element from a set of n unique elements, is to identify the change as the product of a random numbers generator to the base n. To label such a change as a random mutation, without adverting to it as the product of a random numbers generator, can lead to misunderstanding. Abandoning the concept of Darwinian random numbers generation at nucleotide sites in genomes, yet persisting with the terminology of random mutation at these sites, can be misleading.

Durrett and Schmidt, whose field of interest is population genetics, characterized the mathematics of Darwinian random mutation, i.e. the mathematics of random numbers generation, as naïve. Yet, they implied that Darwinian random mutation was their topic by the phrase, ‘the mean waiting time for two mutations to occur in the same individual’. Their actual topic was one of population genetics, which would have been evident, if they had used the phrase, ‘the mean generational time for two genetic markers to be inherited by the same individual, given their initial densities in the population’. Similarly, they alluded to the size of a subpopulation using the misleading phrase, ‘the probability of a mutation’, rather than the more clearly descriptive, ‘the material density or fractional concentration of an inheritable genetic marker within the breeding population’. The density of an inheritable marker in a population is not the probability of a mutation.


It is important to distinguish between pure mathematics and its applications. This distinction is pertinent to the distinction between probability and density.

Probability is the fractional concentration of an element in a purely logical set. Density is the fractional concentration of a physical element in a physical set. Let the probability of blue marbles in a set of logical marbles be 0.5. For a comparable physical set of marbles the density of blue marbles is 1 blue marble per 2 marbles. In contrast to the probability, the density of blue marbles is not 0.5, but 0.5 blue marbles per marble.

The necessity of such a distinction is seen in the following example. If we form (even logical) packages of 2 marbles each on the basis of a density of 0.5 blue marbles per marble, all packages will have exactly one and only one blue marble per package. If we form logical packages of two marbles each on the basis of a probability of blue marbles of 0.5, we will have defined a logical population of packages in which the proportions of packages with 0, 1 and 2 blue marbles are 1:2:1.

Note: the formation of material sets on the basis of the probabilities of a source set can only be analogous to or a simulation of the purely logical concepts of mathematical probability.

Probability is purely a logical concept. The formation of new sets based on the probabilities of a source set is called random selection from the source set. This is strictly logical. No physical thing can be selected with a total disregard to its physical properties, yet this is what is required by the concept of random selection. Thus the formation of new physical sets based on the probabilities of a physical source set, can only be an analogy of the mathematics. In such analogies, human ignorance of the IDs during selection is equated with randomness.

In contrast to probability, which is solely logical, density can be characteristic of material reality. An example is the density of carbon atoms in glucose, which is one carbon atom per four atoms.

An example of confusing density and probability is the calculation of the number of earthlike planets in the universe by Richard Dawkins in The God Delusion, page 138. He conservatively estimates the number of planets in the universe as one billion, billion. Assuming a density of 1 earthlike planet per billion planets, he correctly calculates the number of earthlike planets in the universe as 1 billion.

That, however, is not how Dawkins describes his calculation. Instead of assuming a density of 1 earthlike planet per billion planets, Dawkins claims he is assuming a staggeringly, absurdly, stupefyingly low probability of earthlike planets of 1 per billion. If Dawkins were assuming a probability rather than a density, he would not have been calculating the number of earthlike planets in the universe. He would have been defining a population of universes, which would cover the spectrum from 0 earthlike planets per universe to 1 billion, billion earthlike planets per universe. In the defined population, modal universes would contain exactly I billion earthlike planets. However, this mode would be a very small percentage of the total population of universes. This very small percentage would be the probability that a randomly selected universe would contain exactly 1 billion planets.

In my next post I will present a more subtle and intriguing instance of conflating probability and density.

Richard Dawkins has identified the framework of Darwinian evolution as a series of stages or cycles of random numbers generation and discriminate integer filtering. He has used a multiple dial lock to demonstrate Darwinian evolution, both in The God Delusion and in a video lecture.

In the video lecture, Richard Dawkins demonstrates the efficiency of mutation due to sub-staging in Darwinian evolution ( I noticed a discrepancy in the model of the 3-dial lock, which is used for illustration. The lock is verbally described as having six different mutation sites for each dial. However, pausing at 5:15 of the video reveals that the digits adjacent to the 6 on the dials are 5 and 0, rather than 5 and 1. There are 7 symbols in the range 0 to 6, whereas there are 6 symbols in the range 1 to 6. The number of mutations defined by 0 0 0 to 6 6 6 is 7^3 = 343. The number of mutations defined by 1 1 1 to 6 6 6 is 216, which is the number discussed in the audio of the lecture.

Nevertheless, the demonstration, as narrated, is an excellent illustration of efficiency in the number of mutations due to replacing a single cycle of non-random mutation and non-random selection with sub-cycles.

The video illustrates the evolution of the sequence 6 5 1, but based on non-random rather than random mutation. A better illustration of Darwinian mutational efficiency would have been the use of three dice, each of a different color. Rolling dice is conventionally recognized as random numbers generation. As an illustration of Darwinian random mutational generation, the rolling of three dice 479 times would generate a pool of random numbers to the base 216. Subjecting such a pool of random numbers to Darwinian non-random selection would yield a probability of 89.17% success of natural selection of the sequence 6 5 1. This can also be accomplished by subjecting each of the three mutation sites individually to Darwinian evolution, which is the theme of the video illustration. Generating three pools of 18 random numbers to the base 6 would yield a probability of 96.25% evolutionary success for each site and a probability of overall evolutionary success of 89.17%.

Although replacing a single stage of Darwinian random mutation and natural selection with a series of sub-stages does not affect the probability of Darwinian evolutionary success, it does increase the efficiency of Darwinian random mutation. For an 89.17% probability of the evolution of the sequence 6 5 1, the total of random numbers generated drops from 479 for a single stage to 54 for the series of three sub-stages. That is an efficiency factor of 479/54 = 8.87 for random mutations due to sub-staging. There is no change in the probability of success of 87.19% = 96.25^3.

Three dice would serve better to illustrate what Dawkins has dubbed ‘the smearing out of the luck’. In Darwinian evolution ‘the luck is smeared out’ due to sub-staging, but there is no change in ‘the luck’ overall. There is no change to the overall probability of Darwinian evolutionary success. However, there is an increase in the efficiency of random mutation.

Three dice, each of a different color would be an excellent model of three mutations sites of 6 mutations each, in contrast to the 3-dial lock of 6 mutations per dial. The reason is that the dice lend themselves very well to random mutation rather than serve simply to illustrate sequential, non-random mutation, to which Dawkins was constrained in the video by the model of the 3-dial lock. He could have spun the dials rather than turning them methodically, but rolling dice would teach the lesson more clearly. One drawback to spinning the dials is that in the video the dials were analog rather than digital. This difficulty could have been overcome by constructing the dials as prize wheels.

Of course, an illustration employing tetrahedral dice would be preferable to one employing cubic dice because each mutation site in a genome is of four, not six, mutations. The chimpanzee genome can be visualized as a map of 3 billion tetrahedral dice where each tetrahedron represents a site of 4 possible mutations.

One drawback to using tetrahedral dice is that the outcome of a roll is the face-down triangle. Another drawback is that they are hard to roll. These two difficulties may be avoided by using four sided rolling pins.

Whatever the model used for illustration, it should serve to elucidate and not to obscure the distinction between the efficiency of random mutation and the probability of success of natural selection in Darwinian evolution. In the video, Dawkins compares the 216 non-random mutations of a single stage with the total of 18 non-random mutations, i.e. 6 for each of three sub-stages. That is an efficiency factor of 216/18 = 12 for non-random mutations. The probability of the evolution of the sequence 6 5 1 is 100% in both cases. Yet, it is easy to get the impression that the lecture illustrates an increase in the probability of success of natural selection and not an increase in the efficiency of mutation. A synopsis of the video in Wikipedia gives such a false impression, “In Dawkins’ case we have 3 dials, with 6 positions each, so the probability that you open the lock by sheer luck is one to two hundred and sixteen. Dawkins then shows the mechanism of the lock with a big model: Each dial has to be in the correct position in order to open up the lock. The model is then adapted to demonstrate a staged or gradualist solution to finding the right combination to open the lock. The probability of unlocking the combination in three separate phases falls (sic) to one in eighteen.” ‘Falls’ is inadvertent. An increase to 1/18 from 1/216 would be a rise, not a fall. However, what matters is that the writer is under the false impression that the 3-dial lock serves to illustrate an increase in the probability of evolutionary success.

Similarly, in The God Delusion, pages 121-122, Dawkins gives the impression that replacing a single cycle of Darwinian evolution with a series of sub-cycles increases the probability of success of natural selection. In fact such a replacement increases the efficiency of random mutation, while having no effect on the probability of evolutionary success. Dawkins states, “What is it that makes natural selection succeed as a solution to the problem of improbability, where chance and design both fail at the starting gate? The answer is that natural selection is a cumulative process, which breaks up the problem of improbability up into small pieces. Each of the small pieces is slightly improbable, but not prohibitively so.” The overall probability of a series is the product of multiplication. Dawkins’ explanation could only make sense, if the overall probability of a series was the sum of addition.

See also, “Too improbable to be due to chance” (