Richard Dawkins’ argument for ‘Why there almost certainly is no God’ (Chap 4, The God Delusion), is mathematical. He proposes that there is no mathematical solution to ‘the problem of improbability’ of God, whereas there is a mathematical solution to ‘the problem of improbability’ of a large stage of Darwinian evolution.
The problem occurs when an improbability is prohibitively improbable. According to Dawkins, the solution to a problem of improbability is to replace its complementary probability with a series of the factors of its complementary probability. Each improbability, complementary to the probability of a factor, is ‘slightly improbable, but not prohibitively so’ (p. 121, The God Delusion).
The problem of improbability of the success of natural selection for a large stage of Darwinian evolution is solved by replacing the large stage with a series of smaller sub-stages. The series of sub-stages represents the staged evolution of the ultimate mutation. In this gradualism, for each sub-stage there is a mutation, which survives the test of natural selection for that sub-stage. In contrast, for the overall, large stage, all mutations are subjected to but one, the ultimate, test of natural selection. This test of natural selection represents a much larger and prohibitive improbability than that of the test of natural selection within each sub-stage in the series.
In order for there to be a solution to the improbability of God, God would have to come into being by a series of sub-stages, where each sub-stage improbability is not prohibitively large as is the single stage improbability of the existence of God. Obviously, God would not be God, if he came into existence gradually. Therefore, there is no solution to the improbability of God.
Dawkins’ Elucidation of the Role of Gradualism Is Self-Criticism
In delineating the role of gradualism in Darwinian evolution, Dawkins’ demonstrated that it has no effect on the probability of the evolutionary success of natural selection. He showed that the role of gradualism is to increase the efficiency of mutation. Thus, he disproved that gradualism is the solution to the mathematical problem of improbability. In criticizing his own solution to the problem of improbability, Dawkins disproved his rationale for why there almost certainly is no God.
To illustrate the role of gradualism in Darwinian evolution, Dawkins chose an example of three mutation sites of three mutations each. He accordingly noted that this defines 6x6x6 = 216 different mutations. If the one mutation of these 216, which is capable of surviving natural selection, is unknown, then a minimum of one copy of each would have to be generated non-randomly to ensure 100% evolutionary success of natural selection.
He compared this large stage of non-random mutation and natural selection with a series of three sub-stages, each affecting one of the three mutation sites. Each sub-stage would require the generation of six non-random mutations to ensure 100% evolutionary success. That would be a total of 18 non-random mutations for 100% overall success of natural selection for the series of sub-stages.
The difference between the single, overall stage and the series of sub-stages is not in the success of natural selection. For both, the success of natural selection is 100%. The difference is in the total number of non-random mutations to ensure 100% success. The difference is that of 216 and 18 total mutations. The series is mutationally more efficient by a factor of 216/18 = 12, at 100% probability of success of natural selection.
If the pools of mutations subjected to natural selection in the illustration are generated by random mutation, rather than non-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 be required to 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 90.9% 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.
Thus, as his own critic, Dawkins has disproved his claim that the problem of improbability of success of a large stage of Darwinian evolution is solved by replacing it with a series of sub-stages.
Dawkins’ Criticism of ‘The Problem of Improbability’
Dawkins has also demonstrated that there is no such problem as ‘the problem of improbability’. He has labeled those who propose such a problem as being persons of a ‘discontinuous mind‘.
Dawkins noted that some variables, which are defined over a range of 0 to 1, are essentially fractions of a whole of some thing or property. The range is 0% to 100%. Implicitly, any two values of such a variable differ from one another by degree, not by kind. Dawkins claims that only a person with a ‘discontinuous mind’ would propose that there is some value in the range which distinguishes two kinds of the variable. One kind would be 0% to the arbitrary point marking the discontinuity from the second kind, defined from the point of discontinuity to 100%.
Of course, wearing his mathematician’s hat, Dawkins is correct. In being correct he has demonstrated that there is no valid definition of a ‘problem of improbability’. Defining the problem requires an arbitrary point demarcating a discontinuity in the range of improbability of 0% to 100%, thereby forming two kinds of improbability, non-prohibitive and prohibitive. It is these two kinds of improbability, which form the basis of his discussion of ‘the problem of improbability’ of Darwinian evolution and of why there almost certainly is no God.
Dawkins has proved
1. Gradualism does not solve ‘the problem of improbability’ of the success of natural selection in a large stage of Darwinian evolution. It has no effect on probability. It merely increases the efficiency of mutation.
2. ‘The problem of improbability’ is a self-contradiction. It proposes a distinction of kind between two subsets, within the defined range of a continuous variable whose values vary by degree, not kind.