Monthly Archives: January 2015


Gradualism in Darwinian evolution is identified as the replacement of a conceptual overall cycle of random mutation and natural selection with an actual, gradual series of sub-cycles. It is assumed that the series of sub-cycles randomly generates all the mutations defined by the conceptual overall cycle, but in stages rather than in one gigantic evolutionary cycle. The gigantic set of mutations is itself a graded set, not of sub-cycles, but of mutations.

The mutations defined in each sub-cycle is a subset of the graded set of mutations defined by the overall cycle. Each sub-cycle consists of subjecting its subset of mutations to random generation and the resulting pool of random mutations to natural selection.

The gradualism of sub-cycles is often taken to be synonymous with the gradualism represented by the entire graduated sequence of mutations defined by the associated conceptual overall cycle of evolution.

Everyone agrees that replacing a single cycle of Darwinian evolution by a series of sub-cycles yields a series of sub-cycles, each of which has a probability of evolutionary success greater than the probability of success overall. This is simple arithmetic. The product of a series of factors, each a fraction of 1, is less than any of its factors.

However, proponents of Intelligent Design claim that there are some biological structures that cannot be assembled gradually in a series of subsets because survival in the face of natural selection requires the full functionality of the surviving mutant in each subset.

There are in fact two distinct Intelligent Design arguments against Neo-Darwinism. One argument is entitled, irreducible complexity. The other argument is the argument of gradualism presented by Stephen Meyer (Ref. 1). Both of the Intelligent Design arguments cite complex biological structures such as the ‘motor assembly’ of the bacterial flagellum. In opposition to Neo-Darwinism, the Intelligent Design arguments claim that it is the integrity of the assembled unit that confers functionality and thereby survivability when subjected to natural selection. Those mutants, which have partial assemblies, have no functionality and therefore no survivability based on functionality.

Intelligent Design’s Irreducible Complexity Argument

This argument acknowledges that the gradualism of sub-cycles increases the probability of evolutionary success in terms of the probability of the individual sub-cycle. However, it is argued that the integrity of a cited biological assembly requires a single cycle of such a size that it thereby has a level of probability too low to be acceptable. The assembly is not reducible, by a series of sub-cycles, to a lower level of complexity without sacrificing survivability. The lower the complexity, the greater the probability of evolutionary success. Thus, the level of complexity is irreducible by means of sub-cycles to a low level of complexity, which would raise the probability of each sub-cycle to an acceptably high level of probability. According to this argument, Darwinian evolution fails on the basis of probability.

Intelligent Design’s Gradualism Argument

Stephen Meyer’s Intelligent Design argument (Ref. 1) ignores the numeric values of probability and an alleged value of probability above which probability becomes large enough to serve as an explanation. Rather, the argument concentrates on the proposition that the gradualism of Darwinian evolution requires the actual generation of the entire, graduated spectrum of mutations. If there is a sub-sequence of graduated mutations, which have no functionality and therefore have no survivable utility, then the terminal of this sub-sequence could never be generated. The ‘motor assembly’ of the bacterial flagellum is cited as one example. Consequently, the evolution of such a terminal mutation is incompatible with gradualism, which is a key characteristic of Darwinian evolution.

According to this argument Darwinian evolution fails on two grounds with respect to gradualism. (1) The complete biological assembly, in the cited instances, cannot be assembled gradually because the necessary precursors, namely incomplete biological assemblies, would not pass the test of natural selection to which each precursor would be subjected in its sub-stage of gradualism. (2) The fossil record has gaps in the spectrum of mutations, which gaps are incompatible with gradualism.

Critique of Both Irreducible Complexity and Darwinian Evolution with Respect to Probability

Probability is defined over the range of 0 to 1. There is no logical basis for dividing this continuous range of definition into two segments: one segment from 0 up to an arbitrary point, where probability is too small to serve as an explanation; a second segment from that point through 1, where probability is numerically large enough to serve as an explanation.

The Irreducible Complexity Argument claims there are cycles of evolution for which the probability is in the segment near zero. Because these cycles cannot be replaced by sub-cycles, the gradualism required by evolution cannot be achieved. The Neo-Darwinian response is that there are no such cycles. Any cycle, which due to its size would have too low a probability, superficially may appear to be indivisible into sub-cycles, but is not so in fact.

Both Irreducible Complexity and Darwinian evolution claim that it is the replacement of a cycle by a series of sub-cycles which solves the ‘problem of improbability’ (Ref. 2).

Granted that each probability of a series of probabilities is larger than the overall or net probability, the net probability remains constant. The net probability is the product of the probabilities in the series. Consequently, it is nonsensical to say ‘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). The individual probabilities of the sub-cycles in the series do not change the net probability of evolution.

Critique of Both the Intelligent Design and the Darwinian Evolution Claims of Gradualism

Meyer’s Intelligent Design argument agrees with Neo-Darwinism that the gradualism of sub-cycles ensures the generation of the entire spectrum of graded mutations defined by the overall cycle of evolution (Refs. 1 and 3). To what they agree is false. The fact is that role of sub-cycles is to increase the efficiency of mutation by eliminating the possibility of the generation of most of the graded mutations in the defined spectrum. Although he misinterpreted what he was demonstrating, Dawkins did an admirable job of demonstrating this efficiency (Ref. 4).

In Ref. 4, Dawkins used an example of three mutation sites of six mutations each to illustrate the efficiency of sub-cycles, where efficiency is achieved by eliminating the possibility of the generation of most intermediate evolutionary forms. The excellence of his illustration of increased mutational efficiency is not vitiated by the fact that Dawkins mistakenly thought he was illustrating an increase in the probability of evolutionary success by natural selection. The net probability of success is unaffected by the introduction of sub-cycles (Ref. 5).

Three mutation sites of six mutations each defines 216 different graded mutations, i.e. 6 x 6 x 6 = 216. These mutations are the two end points and 214 intermediates. Let the 216 mutations of the graded spectrum be designated 000 to 555.

In a single cycle of Darwinian evolution, all 216 different mutations are liable to be generated randomly. In Dawkins’ illustration of gradualism, a series of three sub-cycles replaces the single overall cycle. Each of the three sub-cycles in the series subjects only one site out of the three to random generation and natural selection, independently of the other two sites. This entails only six different mutations per sub-cycle. In the first sub-cycle, the six mutations are between 00’0′ and 00’5′ inclusively. The six mutations of the second sub-cycle are between 0’0’5 and 0’5’5 inclusively. The six mutations of the third sub-cycle are between ‘0’55 and ‘5’55 inclusively.

Although there are six possible different mutations per sub-cycle, in the second sub-cycle mutation 005 is a duplicate of 005 of the first sub-cycle. In the third sub-cycle mutation 055 is a duplicate of 055 of the second sub-cycle. That yields only 16 different mutations in total, which are liable to random generation, not 18.

In the single overall cycle there are no missing links or missing gaps in the spectrum of 216 mutations which are liable to random mutation. These are 000 to 555, for a total of 216.

In the first sub-cycle of the illustration, all six graded mutations are liable to be randomly generated, i.e. 00’0′ to 00’5’. In the second sub-cycle the six mutations liable to be randomly generated are separated by gaps in the graded spectrum. The gaps are of 5 mutations each. The six mutations which can be generated in the second sub-cycle are 0’0’5, 0’1’5, 0’2’5, 0’3’5, 0’4’5 and 0’5’5. The first gap comprises the five mutations between 0’0’5 and 0’1’5. These are 010, 011, 012, 013 and 014. There are 5 gaps of 5 mutations each, for a total of 25 mutations of the overall spectrum which cannot be generated in the second sub-cycle due to the gradualism of sub-cycles.

In the third sub-cycle of the illustration, the six mutations which are liable to be randomly generated are ‘0’55, ‘1’55, ‘2’55, ‘3’55, ‘4’55 and ‘5’55. Between each of these mutations there is a gap of 35 graded mutations which cannot be generated due to the gradualism of sub-cycles. For the first gap, the 35 are 100 to 154, inclusive. The total of different graded mutations, which cannot be generated in the third sub-cycle, is 35 x 5 = 175.

The totals for the three sub-cycles of different mutations are: Sub-cycle one: 6 mutations possibly generated, 0 mutations in non-generated gaps; Sub-cycle two: 5 mutations possibly generated, 25 mutations in non-generated gaps; Sub-cycle three: 5 mutations possibly generated, 175 mutations in non-generated gaps. Totals for the three sub-cycles: 16 mutations possibly generated, 200 mutations in non-generated gaps.

For a critique of gradualism from the perspective of probabilities see Refs. 5 and 6.


Both its proponents (Ref. 3) and its critics (Ref. 1) assume that a key characteristic of Darwinian evolution is the generation of a complete spectrum of graded mutations. This shared view assumes that the generation of all mutations in this spectrum is facilitated by the gradualism of a series of sub-cycles of random mutation and natural selection. This is false. The Darwinian algorithm of random mutation and natural selection, applied in series, ensures that most of the mutations, defined by the overall graded spectrum, cannot be generated. The role of sub-staging in Darwinian evolution is the increased efficiency of mutation due to the non-generation of most of the mutations comprising the defined graded spectrum. This results in huge gaps in the spectrum of mutations actually generated.

To the typical bystander (Ref. 7), the debate between Intelligent Design and Neo-Darwinism appears to be one of science vs. science or, as the Dover Court ruled, faith vs. science. In fact, the arguments of both sides are based on their mutual misunderstanding of the arithmetical algorithm, which is Darwinian evolution.


1. “Darwin’s Doubt” with Stephen Meyer,
2. “The God Delusion”, page 121
4. minute 4:25

Note: The single quote marks are used simply to highlight the mutation site in question.