In the context of a technological assay, such as a screening test for a disease, a false positive is a failure, but, if its frequency is low, it does not invalidate the utility of the assay. Such a case is a failure, but in one sense only partially so. In philosophy, a false positive, in the sense of being right for the wrong reason, is a complete failure. It is intriguing that today the same false argument is employed by both sides, pro and con, with respect to the existence of God. The stated conclusion of the one argument is true, the other false. But, the core of both arguments is invalid.
In his essay, “How Contemporary Physics Points to God”, Fr. Robert Spitzer states, “The odds against all five of the anthropic coincidences happening randomly is exceedingly and almost unimaginably improbable. Most reasonable and responsible individuals would not attribute this to random occurrence (because the odds are so overwhelmingly against it), and so, they look for another explanation which is more reasonable and responsible.”
Similarly, Richard Dawkins claims the problem that any theory of life must solve is how to escape from chance. Dawkins’ solution, within the context of evolution in a one-off event, is to increase the probability (pages 120-122, The God Delusion). He characterizes the odds of a billion to one, not as overwhelming, but as staggering, absurd and stupefying (page 138, The God Delusion).
Dawkins claims that the probability can be increased in the case of evolution by breaking up the improbability into smaller pieces (page 121, The God Delusion). Dawkins claims that this is not possible in the case of the improbability of God (page 113, The God Delusion). Appropriately, Spitzer does not consider this possibility in the case of the anthropic coincidences. However, both men envision the same problem, namely, that a value of probability is too close to zero for probability to be an explanation. Implicitly, if the value of probability were close to one, neither would object to probability’s serving as an explanation.
This is the same view as proponents of Intelligent Design, who consider biological complexities. Some are at sufficiently low levels of complexity for which the probability is close enough to one to serve as an explanation. Of course, the proponents of Intelligent Design are not interested in these, but in the occurrence of biological complexities that are irreducible, in the context of natural selection, to low levels of complexity. For these irreducible complexities, it is argued that the probability is too close to zero to serve as an explanation.
It is not their solutions or lack thereof, but the basic problem as they envision it, which is the focus of this essay. As envisioned by Spitzer in the anthropic coincidences by Dawkins, both in evolution in a one-off event and in the improbability of God, and by its proponents in the argument of irreducible complexity, the problem of improbability at its core is, “The probability of this outcome is so close to zero that it could not be due to chance.”
One might just as well say, “My chance of winning the lottery is so close to zero, that, if I did win the lottery, it could not be due to chance.”
One might just as well say, “The fractional concentration of this element in this set is so close to zero, that it cannot be the fractional concentration of this element in this set.”
Probability is the fractional concentration of an element in a logical set. It is a definition within the mathematics of sets.
The fractional concentration of this element in this set is the fractional concentration of this element in this set, no matter how close this probability is to zero. Most people would accept the stipulation that the top card of a shuffled deck simulates a probability of 1/52. Notice that the top card is in no way special compared to the other locations in the deck. This implies accepting the stipulation of the simulation of a probability of 1/52 for each card in the deck. Of course, this implies stipulating that the shuffled deck simulates a random sequence of 52 cards at a probability of 1.2 x 10^(-68). That is a probability of less than one in a trillion, trillion, trillion, trillion, trillion. The assent to the simulation of a probability of 1/52 by shuffling the deck is essentially assent to the simulation of a probability of 1.2 x 10^(-68). This is most appropriate because all values of probability are of equal validity over its range of definition, no matter how close to zero.
It should be apparent that there is no such problem as a ‘problem of improbability’. To reject or to accept probability as an explanation based on its numerical value is illogical. If probability is accepted at some value as an explanation, then at some other value it cannot be rejected as an explanation on the basis of its numerical value, irrespective of how close it is to zero. Similarly, the acceptance of probability as an explanation is not bolstered by a value of probability closer to one.
The concept of a problem of improbability, i.e. a problem due to the numerical value of probability within its range of definition, is meaningless. Dawkins unwittingly clarifies the issue by identifying some values of probability within its range of definition as ‘prohibitive’. Of course, the prohibition cannot be inherent in the mathematical definition itself. The prohibition is simply by fiat of the one who envisions the problem, where the problem is in fact his own fiat of prohibition. There can be no solution because there is no problem. The discussion of sets and subsets at the Mad Tea Party makes more sense than a discussion of sets, which proscribes numerical values of mathematical probability within its range of definition.
Probability is solely a matter of mathematical logic. Material simulations of the mathematics of probability, such as dice and playing cards, are visual analogies. The logic, which the material simulations are employed to visualize, is not inherent in the nature of the material employed. This is in contrast to the measurable properties of material things, which properties are inherent in their natures and which are the subject of science.
Alice, the Hatter, the March Hare and the Dormouse held a reasonable and responsible discussion of sets and subsets. The Hatter, e.g. noted, “You might just as well say, ‘I see what I eat’ is the same as “I eat what I see’!” Fr. Spitzer, Richard Dawkins, the proponents of the argument of irreducible complexity, et. al. are not reasonable and responsible in citing outcomes which cannot be attributed to random occurrence ‘because the odds are so overwhelmingly against it’.
It would be reasonable and responsible for Spitzer, Dawkins et. al. to reject low values of probability as an explanation, not because the odds are overwhelming, but because mathematical probability is never the explanation of anything, irrespective of its numerical value. They, and we, merely have to be consistent by also rejecting intermediate and high values of probability as an explanation.
Though one may define an unimaginably low value of probability, its significance is solely logical and therefore cannot buttress an argument about the existence of anything. An argument pro or con the existence of God, based on probability, is an argument in which the premises address purely logical concepts. Existence, or lack thereof, cannot be the conclusion of a purely logical argument. Mathematical probability, which numerically characterizes the composition of logical sets, has nothing to do with the existence of anything.
Those who base their arguments, pro and con, about the existence of God on the ‘problem of improbability’, do not have the recourse of going back to the drawing board of mathematics. The discussion of the relationships of logical concepts does not lead to a conclusion about existence. They do have the recourse of reviewing the principles of Aristotelian-Thomism: The characteristics of everything within our experience is explained by the nature of that thing. Nothing within our experience explains its own existence or that of another.
Outside of the context of mathematical sets, probability expresses a person’s degree of certitude: ‘That statement is probably true.’ Statistical probability is a hybrid of mathematical probability and personal certitude. It details a commonly accepted convention for expressing the degree of personal certitude in the mean of measurements based on assumptions of the distribution of measurements about their mean.
Also, it is philosophy which underpins both the proof of the existence of God and the validity of science, including modern physics. If modern physics pointed to God, it would be by a circular argument, namely: Philosophy justifies modern science which justifies philosophy. This is not to deny that the beauty of science is a specific case within the general argument of beauty for the existence of God.