Richard Dawkins claims that the solution to the ‘problem of improbability’ is gradualism. A large stage of probability is replaced by a series of substages of probability. Each substage is slightly improbable, but not prohibitively so (p 121, The God Delusion). In a lecture, “Climbing Mount Improbable” (beginning at min 4:25), Dawkins uses a set of three mutation sites of six mutations each to illustrate this. Mathematically, this is analogous to the mutations of three dice.
In Chapter 4 of The God Delusion, Dawkins uses this mathematical analysis as his central argument of ‘why there almost certainly is no God’. Obviously, God could not develop gradually. Thus, the mathematical solution to ‘the problem of improbability’ does not solve the improbability of God.
The Problem and Dawkins’ Numerical Solution
In a lecture, “Climbing Mount Improbable”, Dawkins notes that three mutation sites (e.g. three dice) of six mutations each (six faces), defines 6 x 6 x 6 = 216 possible mutations. If, instead of subjecting the three dice to mutation simultaneously, each is subjected individually, the result is a series of three substages of six mutations each. Dawkins characterizes the single stage as a probability of 1/216 and the three substages as a probability of 1/6 each.
In Dawkins’ jargon, the gradualism of the series breaks up the big improbability, 215/216, of the single stage into three smaller improbabilities of 5/6 per substage. But, he is not that explicit. His illustrated comparison is between the 216 defined mutations of the single stage to the sum of 6 mutations each, for a total of 18 defined mutations for the series. His comparison is essentially of these two listings of total mutations, 216 vs. 18.
Dawkins’ Error
What Dawkins is comparing are not probabilities or improbabilities but total defined mutations. The total listing of defined mutations for the single, overall stage is 216. The total listing of defined mutations for the series of three substage is 18. BOTH lists contain the same three digit sequence of interest. The single list is 111 to 666. The series of three substages yields three sub-lists of 1 to 6, 1 to 6, and 1 to 6. The digits of the three sub-lists, correspond to the three digits of the single stage listing. Dawkins’ comparison is not of probability or improbability, but the total number of mutations in lists, i.e. the one list of 216 compared to the sum of the three sub-lists, which in total is 18. Consequently, both the single stage and the series of substages represent a probability of success of 100% of containing the special 3-digit sequence. The two processes do differ. They differ in mutational efficiency, the series of three substages requiring fewer mutations by the mutational efficiency factor of 216/18 = 12.
The Same Analysis Using the Analogy of Three Coins
Let us make the problem simpler and more easily understood by illustrating it with the three mutation sites as three coins: P, a penny; N, a nickel: and D, a dime. Each site is of two mutations: heads and tails, or 1 and 2.
Instead of a list of 216 defined mutations, we have a list of 8 defined mutations for the three mutation sites subjected to mutation together. These 8 are: P1,N1,D1; P1,N1,D2; P1,N2,D1; P1,N2,D2; P2,N1,D1; P2,N1,D2 ; P2,N2,D1; P2,N2,D2.
Instead of a list of 18 defined mutations, we have a list of 6 defined mutations for the three mutation sites subjected to mutation individually in series. These are: P1,P2; N1,N2; D1,D2. Both lists contain every possible combination and both, therefore, have a probability of 100%. The series of three substages is more efficient in mutations at achieving the 100% probability than the single stage by a mutational efficiency factor of 8/6 = 1.33.
The gradualism of Dawkins’ solution does not increase the probability of success, thereby solving his ‘problem of improbability’. Dawkins’ gradualism increases mutational efficiency, while having no effect on the probability of success.
Dawkins’ argument is irrelevant to the existence of God. However, it renders Darwinian evolution ‘absurd’, which is the label Dawkins applied to Darwinian evolution in a single, overall stage (p 122, The God Delusion). Dawkins did not solve his ‘problem of improbability’, which, in his judgment, renders Darwinian evolution in a single, large stage ‘absurd’. The series of substages and the single stage have the same probability of success as one another, namely 100%. They differ in mutational efficiency. It is noteworthy that Dawkins uses nonrandom mutation in his comparison of a single stage vs its series of substages.
The Mutational Efficiency of Gradualism in the Case of Random Mutation
Three Mutation Sites of Six Mutations Each (Dice)
With random mutation, The single overall stage would require 497 random mutations to achieve a probability of success of 90%. However, it would take only 19 random mutations per substage to achieve an overall probability of 90%. That is a total of 57 random mutations for the series of substages. Thus the series is 497/57 = 8.7 times mutationally more efficient than the single stage without any effect on the probability of success.
Three Mutation Sites of Two Mutations Each (Coins)
Five random mutation in each substage of the series would yield a probability of success per substage of 96.875% and an overall probability for the series of 90.9%. That would be a total of 15 random mutations. A probability of success of 90.9% for the single stage would require 18 random mutations. Thus the series is 18/15 = 1.2 times mutationally more efficient than the single stage without any effect on the probability of success.
Summary
In The God Delusion and in a Lecture, “Climbing Mount Improbable”, Richard Dawkins claims that low values of probability pose a mathematical problem which is solved by replacing a single stage of probability with a series of substages. The solution, which is that of gradualism cannot apply to God, who cannot be subject to gradualism.
Dawkins uses a set of three mutation sites of six mutation each to illustrate his mathematical solution. What he has demonstrated is not, as he claims, an increase in the probability of success, but an increase in mutational efficiency.
This essay demonstrates the mutational efficiency of the series of substages. It also demonstrates that the gradualism of a series of substages has no effect on the probability of success. Dawkins’ claim is false. This is illustrated using 3 mutation sites of 6 mutations each (dice) and 3 mutation sites of 2 mutations each (coins).
Dawkins has not presented a God Delusion, but a Mathematical Self-Delusion, his personal error in mathematics, which is mistaking efficiency for probability.
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