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 14 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”