Jargon and the Existence of God

Richard Dawkins presents his central argument in The God Delusion as not only rescuing Darwinian evolution from absurdity (p. 122) but as demonstrating “why there almost certainly is no God” (p. 111 ff.). That argument stands on the validity of one line of jargon:

… [Natural] selection is a cumulative process, which breaks the problem of improbability up into small pieces.

We can agree with Dawkins that such jargon would not apply to God. However, is “the problem of improbability” of Darwinian evolution solved by natural selection’s breaking up a piece of improbability into small pieces?

Jargon is acceptable when it serves the function of an analogy, when it clarifies a relationship. However, even when it is a poor expression, as it is in this case, it can expose the error in a proponent’s thought.

Common Ground

Dawkins’ jargon clearly alludes to a basic principle of addition in arithmetic: a whole is equal to the sum of its parts. Thus, Dawkins and I have common ground for a discussion because we agree on the principles of arithmetic.

In a recent comment on Catholic Stand, An Ordinary Papist proposed an excellent analogy to debate lacking common ground. He noted it was like two trees falling in a forest, which produced vibrations, but no sound, because no one was listening. Too often, in a debate, the sides talk past one another due to the lack of common ground.

One Meaning of “Breaking Up”

In the context of pages 122-123 of The God Delusion, it is clear that breaking up a piece of improbability refers to replacing a stage of evolution with a series of substages. In 1991, in accord with this meaning, Dawkins gave a numerical example of this replacement. A stage of Darwinian evolution affecting three mutation sites of six mutations each was replaced by a series of three substages. Each site mutated, but only in its own substage.  The comparison was between the probability of the two equations:

Equation 1: P = 1 – [(5/6) × (5/6) × (5/6)]^N

In Equation 1, all three sites mutate together, as is expressed in the single bracketed [] term.

Equation 2: P = [1 – (5/6)^N] × [1 – (5/6)^N] × [1 – (5/6)^N]

In this Equation 2, each site mutates independently of the others, as is expressed by the three separately bracketed [] terms, which correspond to three serial substages of evolution, each terminated by natural selection. The probability of the series equals the product of the probability of each substage.

A Numerical Example

For N = 19, in Equation 2, P = 0.9. The total number of mutations to achieve this probability in the process represented by Equation 2 is 3N = 57. In contrast, the probability of 0.9 requires 496 mutations in Equation 1. In this example, the arithmetical distinction between the equations is that Equation 1 achieves a probability of evolutionary success of 0.9 in one stage of 496 mutations. In contrast, Equation 2 achieves a probability of evolutionary success of 0.9 as the product of the probabilities of three substages. Each substage involves 19 mutations. Each substage has a probability of success of (0.9)^(1/3). The series of three substages requires only 57 mutations to achieve a probability of 0.9, while the single stage requires 496 mutations.

Note that these two equations are not of improbability but of probability, P.

In Equation 1, there is only one stage of mutation, so the probability, or its improbability, could not be said to be broken up into pieces. In Equation 2, the overall probability is the product of the probabilities of the three substages. Also, Dawkins’ comparison requires it to equal the probability of success of the single stage.

Using Dawkins’ jargon, one would say the overall probability of 0.9 was broken up into three larger pieces of probability of (0.9)^(1/3) = 0.9654 each. However, this makes no sense mathematically, and it is inelegant. Moreover, such jargon renders no insight into the relationship. You can’t break a piece the size of 0.9 into three pieces, each the size of 0.9654.

This breakup of probability into large pieces is concomitant to Dawkins’ claim of breaking improbability into small pieces. Both expressions are not only poor jargon; they are also nonsense. They express a relationship of multiplication using a jargon of addition.

A Second Possible Meaning of “Breaking Up”

That natural selection is doing the breaking-up could imply that any series of substages of Darwinian evolution is a series of sequences only and not, according to Equation 2, a series of probabilities whose product is the overall probability. This would mean that natural selection produces a clean break from every probability in the series. According to Dawkins, those who deny this don’t “understand the power of accumulation” (p. 121).

By such a claim, Dawkins is imposing his choice of perspective on others. This would be comparable to claiming that the probability of rolling three sixes with three dice was a probability of 1/216, but only if the dice were rolled simultaneously, not each die rolled successively, because rolling each die individually breaks up the problem of improbability into small pieces. That is nonsense.

The rolling of dice is an analogy of probability. Probability is defined solely within the context of abstract sets. Any illustration, such as the roll of three dice or the three mutation sites of Dawkins, is for illustrative purposes only. The illustrator’s perspective is foremost, but it cannot be exclusive. Furthermore, it must be coherent.

Material application of the mathematics of probability is by analogy. It is not a characteristic of material reality itself. The outcome of every roll of dice is due to the forces to which the dice are subjected when rolled. Its identity as a probability is analogical and is an admission of our ignorance of the underlying material forces involved which definitively produce each result.

The real problem with Dawkins’ claim is that he applies the nomenclature of addition directly to multiplication. But, unfortunately, he has also snuck into his argument this second meaning of a “breakup,” thereby denying Equation 2 and succumbing to equivocation.

The Accepted Terminology of a Fraction vs. the Jargon of a Piece

Instead of using the accepted terminology of a “fraction,” Dawkins uses the jargon of a “piece.” Dawkins claims that natural selection breaks up a piece of improbability into small pieces. However, he says this not while referring to an equation of improbability but an equation of probability. The equation expresses probability as the product of a series of probabilities. Adopting the jargon, this means that a  “piece” of probability, when expressed as the product of probabilities, is the breakup of that “piece” of probability into larger pieces. This renders the jargon of a “piece,” instead of a “fraction,” nonsensical and useless.

When Dawkins claims that “natural selection … breaks the problem of improbability up into small pieces” (p 121, The God Delusion), he is implicitly claiming by that jargon, with respect to the corresponding probability, that a fraction of a fraction is greater than either fraction. This is a mathematical and material impossibility. Yet, Dawkins claims this nonsensical error rescues Darwinian evolution from absurdity and is “why there almost certainly is no God” (p. 111 ff., The God Delusion).

Serial Probabilities Involve Multiplication

For a probability of 1/2, the improbability is 1/2, e.g., heads or tails. When serialized in a sequence of 2 substages, the probability of heads or tails becomes 1/4, while the corresponding improbabilities become 3/4. These are not relationships of addition amenable to Dawkins’ jargon, of a piece and its sub-pieces. A piece and its sub-pieces are related by addition, not multiplication, as are serial substages of probability.