Probability as “Secondary Shadows of Being and Causes”

In a recent essay in the National Catholic Register, author Joseph Pearce identified the atheism championed by Richard Dawkins in The God Delusion as easy to debunk because it is irrational, relying on rhetoric, polemic, and sophistry. He cited Answering the New Atheism: Dismantling Dawkins’ Case Against God, by Scott Hahn and Benjamin Wiker, as debunking Dawkins’ case with relative ease.

I disagree. Such may be true of Hahn and Wiker’s analysis of Dawkins’ other arguments, but not of Dawkins’ central argument, which is based on probability. That argument is a mathematical assertion that the probability based on one well-defined algorithm is larger than that of a second well-defined algorithm. The evaluation of Dawkins’ argument requires an understanding of arithmetic at the level of fractions and factors, i.e., division and multiplication.

Hahn and Wiker define chance, i.e., probability, as a secondary shadow of other beings and causes (p. 21). That is meaningless. Because they failed to explain mathematical probability lucidly, Hahn and Wiker failed to address Dawkins’ central argument in The God Delusion.

An Outline of the Central Argument from Dawkins’ Perspective

Probability is a fraction between 0 and 1. Near-zero values of probability render probability an insufficient explanation of an outcome. Accepting low values of probability is accepting chance as an explanation (p. 119). This “problem of improbability,” which is chance, may be solved by replacing a large stage of mutations with an equivalent series of substages. The probability of each substage is greater than the probability of the single step because each factor of a fraction of 1 must be greater than the fraction itself, e.g., 1/30 = (1/5) × (1/6). In Dawkins’ argument, the fraction is the probability of the unfactored stage, and each factor is the probability of each substage. Thus, each substage in the series can have a probability large enough to serve as a sufficient explanation of that substage, even though the probability of the singular overall stage is too low to serve as an explanation.

This solution to values of low probability is evident in the gradualism of the substages of Darwinian evolution. But, implicitly, the solution is inapplicable to the improbability of God because gradual development is incompatible with the concept of God.

Hahn and Wiker’s Counter-Argument

Hahn and Wiker’s counter-argument is that probability is not a being or a cause. Therefore, it cannot serve as an explanation.

Then what is probability? Hahn and Wiker give an example: The probability of a 5 in the set of integers 1 through 6 is the ratio 1/6. This implies that probability is the ratio of a subset to a set. However, according to Hahn and Wiker, this is not the definition of probability, although it is implicitly the definition upon which Dawkins based his argument. Instead, they define probability as “a secondary shadow of other beings and causes” (p. 21).

To substantiate their definition, Hahn and Wiker claim that unless the faces of a cube, such as a die, are labeled with six unique marks, the probability of one face out of six is undefined. In other words, the six faces of a cube are not unique unless labeled. Thus, implicitly, according to Hahn and Wiker, one errs mathematically if he judges it the other way round; that is, if he thinks that one can label the six faces of a cube with six unique labels because the six faces are geometrically unique.

To bolster their ironic position,  Hahn and Wiker claim that the probability 1/6 of one face of a die can be eliminated by filing off the labels that identify each of its six faces. Unfortunately, Hahn and Wiker are wrong. Such erasing of labels does not affect the geometry of a cube, by which each face in a set of six is unique.

From the context, it is clear that Hahn and Wiker are considering the labeling of the six faces as a matter of geometrical principle and not as a practical matter of distinguishing one face from another (p. 21). Despite the claim of Hahn and Wiker to the contrary, the six faces are unique in principle geometrically, whether or not as a practical matter they are uniquely marked. The probability of one face of a cube is 1/6, whether or not the six faces are labeled.

This error in the geometry of a cube prompts Hahn and Wiker to define probability as a “secondary shadow of other beings and causes” (p. 21). According to Hahn and Wiker, erasing the labels erases the secondary shadow.

A Valid Critique of Dawkins’ Central Argument

Dawkins failed to demonstrate that the algorithm defining a series of substages of probability increases the probability of success of Darwinian evolution compared to the single stage. Instead, he has demonstrated something else. He has demonstrated that the Darwinian algorithm defining the series of substages increases mutation efficiency without affecting the probability of success.

In contrast to Hahn and Wiker, Dawkins has correctly identified probability as the ratio of a subset to a set. His error is mistaking an increase in mutation efficiency for an increase in the probability of success. Dawkins’ error is rather subtle and, in that sense, excusable when compared to the blunder in definition by Hahn and Wiker.

This critical insight into efficiency in mutation as the role of gradualism in Darwinian evolution depends upon recognizing the definition of probability as a ratio, that of a subset to a set. As noted above, Hahn and Wiker implicitly accept this definition on page 21 of Answering the New Atheism when they note that the probability of 5 in the set of integers, 1 through 6, is 1/6. However, they quickly fall out of the boat of rationality by defining probability as “a secondary shadow of other beings and causes.” It took Hahn and Wiker three sentences to go from the implicit and valid definition of probability as a ratio to the gibberish of a “secondary shadow.”

The Source of Confusion

The source of Hahn and Wiker’s confusion is the error of thinking that mathematical probability is a material property rather than solely a concept in the logic of sets. The application of probability to material sets is analogical. This can be seen in the equal validity of a probability of 1/52 of the ace of spades in a deck of playing cards and the probability of 1/52! (i.e., 1/52 factorial) of the sequence of a deck of playing cards.

In the analogy of a probability of 1/52, the subset, the ace, is material, as is the set, a deck of playing cards. Shuffling the deck is the material analog of random mutation/selection.

In the analogy of a probability of 1/52!, the subset is also material, a deck of playing cards as a sequence. Shuffling the deck is the analog of random mutation/selection. However, the set of 52! different sequenced decks, from which random selection is made, has no material analog. The set cannot exist. That number of decks of cards is just too big to exist materially. It is 8 × 10⁶⁷ decks.

If the set, the denominator of a valid probability, cannot exist materially, how can probability be a material property? It can’t. However, the probability of 1/52! is valid because probability is not a material property. Probability is purely logical. The set of 52!, or 8 × 10⁶⁷ decks of playing cards, from which random selection is made, cannot exist, except logically. Indeed, it is logical, and the probability, 1/52!, is valid, but the logical set has no material analog.

(Suggestion: Calculate the mass of 52! decks of cards, each deck having the mass of a hydrogen atom. I got a mass equivalent to 22 quadrillion times that of the earth.)

Summary

In his recent essay, Joseph Pearce, a respected Catholic author, recently commended Hahn and Wiker’s attempted rebuttal of Richard Dawkins’ The God Delusion. However, because they failed to define mathematical probability lucidly in Answering the New Atheism, Hahn and Wiker failed miserably in their attempt to address Dawkins’ central thesis. Dawkins’ theory identifies the solution to “the problem of improbability” as replacing a single large stage of low probability with a series of substages. Implicitly, this solution, said to apply to Darwinian evolution, is inapplicable to the improbability of God because God cannot develop gradually.

Suppose Hahn and Wiker had persisted in the correct definition of probability as a ratio. In that case, they might have demonstrated that the gradualism of Darwinian evolution in a series of substages increases mutational efficiency without affecting probability. They would have thereby refuted Dawkins’ central argument by exposing its erroneous arithmetic.

The stumbling block is the concept of probability, which is not a material property. It is a logical concept within the mathematics of sets. Probability is not a material property. It applies fully to logical sets and nominally material sets that cannot exist materially due to the constraints of material size. Probability is certainly not a shadow. The general public deserves better arguments from Catholic apologists than those based on “shadows.”