Probability and Gullibility in Dawkin’s Lock

So what if Richard Dawkins can’t count to seven? He is still a mathematician in accordance with the adage, “There are three kinds of mathematicians: Those that can add and those that can’t.”

In a recent Catholic Stand article that spurred many comments, Richard Dawkins, a prominent atheist, was cited for his recently expressed intellectual disdain for the Real Presence of Christ in the Eucharist. But why should we oppose his atheism at the intellectually sophisticated level of the real presence? He has fortified his atheism with a mathematical error at a level not much more sophisticated than counting to seven.

The central argument for Dawkins’ atheism is mathematical: There is a mathematical solution to the “problem of improbability,” but the answer does not apply to the improbability of God (The God Delusion, Chapter 4, “Why there almost certainly is no God”).

Wheels of Chance and Dials of Choice

In his 1991 lecture “Climbing Mount Improbable” and again in The God Delusion (pp. 121-122), Richard Dawkins presented his solution to “the problem of improbability.”

In the lecture, he considers three mutation sites of six mutations each. In the illustration, he compares a lock controlled by three combined sites of variation to a set of three locks, each controlled by just one site of variation. His jargon implies the control sites are wheels of chance of six stops each. However, his analysis treats them as dials of choice.

Dawkins states that the first set of wheels defines 216 different mutational sequences, namely 6 × 6 × 6. However, a still frame of the wheelset at minute 7:36 of the video indicates that there are seven stops on each wheel, not six, thereby defining 7 × 7 × 7 = 343 different sequences, namely 0,0,0 to 6,6,6, not just 216, namely 1,1,1 to 6,6,6. [Editor’s note: Due to size limitations, the still frame could not be included.]

This error (or as I have labeled it, Dawkins’ inability to count to 7) is trivial and in no way affects Dawkins’ mathematical argument.

Dawkins’ Mathematical Argument

Let us take the number of stops per wheel (or dial) as six despite the prop used by Dawkins in his video. In Dawkins’ illustration, the three-digit sequence of interest is 6,5,1. The deliberate generation of a set of sequences in the case of the combination lock, which would include the particular three-digit sequence of interest, whatever it may be, requires generating a minimum of 216 different 3-digit sequences, 1,1,1 … 6,5,1 … 6,6,6.

However, the deliberate generation of a set of sequences, including the particular three-digit sequence of interest, by the three independent locks requires only 18 choices or 6 for each lock. These are three sets of 1,2,3,4,5,6. By including the “6,” the first lock is unlocked; including the “5”’ unlocks the second; including the “1” unlocks the third.

Both methods of deliberate generation are equally successful in their inclusion of the unlocking sequence 6,5,1. They differ in efficiency. The process for the combination requires the generation of 216 deliberate (i.e., methodical) choices or “mutations.” The independent method involves the generation of 18 deliberate choices or “mutations.” The independent process is 12 times more mutationally efficient than the process for the combination. Both unlocking methods are successful at 100% probability. The efficiency factor is the ratio of the two efficiencies, where each efficiency is output/input:

[(100% success/18 mutations) / (100% success/216 mutations)] = 12

Spinning Wheels

If we introduce random selection using the analogy of spinning wheels of chance rather than deliberately turned dials of choice, the difference between the two processes is still one of mutational efficiency. In such a comparison, the efficiency would be one of random mutational efficiency rather than Dawkins’s nonrandom mutational efficiency.

An example would be at the 90% level of the probability of success of including the sequence 6,5,1. For the combination lock and the spinning of wheels of chance, the number of mutations (spins) required to achieve a 90% probability of success is 497. In contrast, for the three independent locks and the spinning of wheels of chance, the number of mutations (spins) required to achieve a 90% probability of success is 57, 19 for each independent lock. The independent method is 497/57 = 8.7 times more efficient at the 90% level of probability of success of including the sequence 6,5,1.

Further Evaluation

Dawkins has demonstrated that the independent set of three dials of choice is more mutationally efficient than the combined set of three dials of choice. He has implicitly shown that an independent set of wheels of chance is more mutationally efficient than a combined set of wheels of chance. But this does not affect the probability of success.

Dawkins claims that the difference in the two processes is not in mutational efficiency, which he has demonstrated, but in the probability of success, in which there actually is no difference. This mathematical error of mistaking mutational efficiency for the probability of success vitiates Dawkins’ central argument for atheism. On his own terms, Dawkins has not only failed to present a lucid argument for atheism, but he has characterized his professional area of expertise, namely, Darwinian evolution, as “absurd.”

(On page 122 of The God Delusion, “absurd” is the term Dawkins applies to Darwinian evolution in a single large stage or “one-off event” in his parable “Climbing Mount Improbable,” pages 121 ff.)

There is little point in arguing the theologically and philosophically sophisticated topic of the real presence with Richard Dawkins. He is self-convinced of atheism due to his lack of understanding of the mathematical distinctions among the three ratios of density, probability, and efficiency. His argument for atheism is based on a mathematical “solution” to the “problem of improbability.” However, Dawkins’ mathematical errors regarding these three ratios nullify his solution. These errors are not excusable. In contrast, Dawkins’ inability to count to 7 within his ‘solution’ is excusable. His inability to count to 7 does not impinge upon his argument’s mathematical (in)validity.

Theists should argue atheism with Dawkins on Dawkins’ turf,  which is overtly mathematical but easily demonstrated to be erroneous. The mathematical character of Dawkins’ errors in his argument supporting atheism can be discussed objectively pro and con because both sides accept the same principles of mathematics. In contrast, both sides of philosophical discussions of atheism (and of the real presence of Christ in the Eucharist) do not share common philosophical ground.

If Dawkins were to admit that his solution to “the problem of improbability” was mathematically invalid, he would not necessarily have to give up atheism, but he would have to admit that, according to his own rationale and characterization, Darwinian evolution is “absurd.” His characterization of Darwinian evolution as “absurd” follows from his failure to solve “the problem of improbability” of Darwinian evolution.

Probability and Gullibility

Several years ago, on Catholic Stand, I argued that because probability is purely logical, it cannot be the basis of deciding upon the existence or non-existence of anything, including God. Questions of existence are fundamentally existential, i.e., philosophical. They are not mathematical and therein purely logical. Probability is a static concept in logic, not a dynamic material property. Dawkins’ “problem of improbability” is meaningless even within the context of mathematics because all probability values are equally valid within its continuous range of definition from zero to one. To accept any value of probability as a material explanation is gullibility.

For example, in games, we apply the ranking of poker hands based on probability and the roll of dice as a probability by mutual convention for the sake of a game. We do not deny that the outcome of dealt cards or the roll of dice is due to the material forces to which the cards and dice are subject. We do not take probability as a material explanation. We take it analogically as a simulation of purely logical relationships by way of common consent. Dawkins’ gullibility is in implicitly accepting probability as an explanation at high values of probability.

Dawkins displays the same gullibility in attempting to present a “solution” to “the problem of improbability” of Darwinian evolution. However, in the naivete of accepting probability, in principle, as an explanation, Dawkins has plenty of company. He has the company of some theists, who look for an answer other than probability in cases of low probability. In that gullibility, he also has the company of those biologists who argue that chance cannot explain the development of biological morphology in all cases, specifically in cases of irreducible complexity.

Conclusion

Probability is either an explanation of reality throughout its range of definition, or it is not an explanation at all. It is the latter. Probability is a purely conceptual definition within the logic of mathematical sets. It is not an explanation of anything material or existent.

Dawkins has correctly argued that it is mathematically self-contradictory to distinguish two discrete kinds of a continuous variable over its continuous and valid range of definition. However, he has violated this principle of mathematical logic when it comes to probability. Dawkins claims that very low values of probability, within its continuous range of definition, present a “problem of improbability,” while larger values do not. He has thereby divided the continuous range of the definition of probability into two discrete kinds of probability, “prohibitive” and “non-prohibitive” (The God Delusion, page 121).

Even without a mathematical background, one can see the distinction between an analogy based on a wheel of chance, typical of a carnival, and an analogy based on a dial of choice, typical of a combination lock. Dawkins confuses the two, using the jargon of a wheel of chance while explaining an analogy based on a dial of choice. Consequently, he misidentifies an increase in efficiency as an increase in probability.