Working Around the Impasse in the God Debate

creation, creator, creature, genesis, being

 

creation, creator, creature, genesis

Almost 70 years ago a famous but now somewhat forgotten debate took place on BBC radio between Bertrand Russell and Rev. C. J. Copleston, S.J. The topic of the debate was whether or not God existed.

Russell, a British philosopher, logician, mathematician, historian, writer, social critic, political activist, and self-described atheist, argued against God’s existence. Copleston, a Jesuit priest, philosopher, and a historian of philosophy, who was best known for his influential multi-volume A History of Philosophy, argued that God did exist.

The transcript of a debate on the existence of God between Bertrand Russell and Rev. C. J. Copleston, S.J. was recently posted. The first part of the debate was the argument from contingency, which ended in an impasse. Russell could not admit that the philosophical principle of ‘sufficient reason’ applied to God’s existence.

The Impasse

The main problem during the debate was that was no common ground between the two sides. The pro side argument was based on the universality of the principle of sufficient reason. The con side approach to the debate was from a position of logical positivism. (Logical positivism is the philosophy that human intellectual knowledge of reality consists of (1) logic, including mathematics, and (2) the inference of mathematical relationships among the measurements of material properties.)

In the 70 years since the BBC broadcast one might think that by now we would have found some common ground for debating God’s existence. But it seems that we have not. Consider the stance of a prominent modern proponent of logical positivism, cosmologist and atheist Sean Carroll. He insists that reality fundamentally consists in elementary particles in motion in space and time.

Instead of continuing to look for a common ground, the pro side of the debate has simply adopted the pejorative ‘scientism’ as a label for the con side and declared that science can neither prove nor disprove God’s existence.

The Emerging Possibility for Common Ground

As the Copleston-Russell debate showed, philosophy could not provide a common ground for a debate about God’s existence. However, since mathematics is common to both science and logic/reason, it should be possible use mathematics as the common ground for such a debate.

In recent years the con side has proposed two explicit mathematical arguments that elucidate its position. The pro side needs to evaluate these mathematical arguments for their logical and mathematical validity.

The first mathematical argument is what Richard Dawkins calls ‘the problem of improbability.’ Here he concludes that God almost certainly does not exist. The second argument is the widely held tenet that the truth of a proposition or hypothesis, such as the existence of God, can be determined by applying the mathematical rationale of the equation known as Bayes’ theorem (see herehere and here).

Dawkins’ Problem of Improbability

I admire Richard Dawkins, despite his beliefs, for two reasons. First, he always seems ready to elucidate a problem through identifying its mathematical framework. Second, he identifies Darwinian evolution as a mathematical algorithm, which I also acknowledge it to be.

Before considering Dawkins’ solution, however, we should first consider if there can be a ‘problem of improbability.’ On page 121 of his book The God Delusion, Dawkins’ says there is a problem, not with improbability in general, but with one kind of improbability, namely prohibitive improbability. Dawkins’ solution entails breaking up over time a large piece of prohibitive improbability into smaller pieces of improbability, each of which is non-prohibitive. He claims that the problem of improbability is solved by such gradualism in the case of evolution, but that there is no solution in the case of God.

Elsewhere, Dawkins presents an astute analysis, which argues that there can be no such thing as a problem of improbability. There can be no problem because that would require distinguishing two kinds of improbability, such as prohibitive and non-prohibitive. Dawkins’ analysis invokes the principle that every two values of a continuous variable, defined over the range from zero to one, differ from one another, not in kind or type, but in degree. The two values are both fractions of the same thing. Improbability is such a variable.

Dawkins’ pejoratively labels a person who insists on distinguishing kinds within such a continuous variable, where the only possible distinction is of degree, as a person with a discontinuous mind. According to Dawkins, people of such a mind tend to exercise tyranny through the false distinction of kinds.

Dawkins’ Error

Dawkins’ argument against the ‘discontinuous mind’ is logically and mathematically valid. Therefore, there can be no problem of improbability. Defining the problem requires distinguishing two kinds of improbability. Most pertinently there can be no problem of improbability in Darwinian evolution.

In The God Delusion, Dawkins identifies gradualism as solving the problem of improbability in Darwinian evolution. What then is the role of gradualism in the mathematical algorithm of Darwinian evolution? Here, again, Dawkins provides an excellent analysis.

However, he misidentifies the variable, which is increased by the role of gradualism in Darwinian evolution. Dawkins demonstrates that a series of small stages of Darwinian evolution requires fewer mutations to achieve the same probability of evolutionary success as a single large stage. However, Dawkins identifies this as an increase in the probability of success of natural selection, when in fact it is an increase in the efficiency of mutation. Gradualism requires fewer mutations to achieve the same level of the probability of Darwinian evolutionary success.

Turning back to Dawkins’ claim that there is no solution to the problem of improbability of God, whereas there is a solution to the improbability of evolution in a single large stage of Darwinian evolution, we can see that his argument itself crumbles prior to any reference to the existence of God. The reason for this is twofold. First, the argument depends on distinguishing kinds of improbability, which Dawkins has shown to be untenable. Second, gradualism in Darwinian evolution does not increase the probability of success. Rather, as Dawkins has demonstrated, gradualism increases the efficiency of mutation.

The Relevance of Bayes’ Theorem to Working Around the Impasse

Any debate has traditionally been the argumentative contention over a premise and its negation. As referenced above, Bayes’ theorem has been invoked as relevant to the God Debate. This was central to the debate, “Does God Exist?” between Trent Horn and Raphael Lataster in May, 2016 at the University of Adelaide.

In the debate Horn, an apologist sponsored by Catholic Answers, proposed several philosophical arguments. Lataster, a researcher in religious studies at the university, dismissed those arguments because they were not probabilistic in the form of Bayes’ theorem.

Lataster implicitly proposed a way to work around the impasse in the God Debate that is due to the rejection by the con side of philosophical arguments. Horn, however, declined to accept the challenge to debate the relevance of the mathematics of Bayes’ theorem even though mathematics would have provided common ground. Such a debate would be ‘Is Bayes’ theorem relevant to debating?’ i.e. whether the con side’s argument was logically and mathematically valid.

Does Bayes’ Theorem Furnish a Format for Debate?

Had Horn accepted the challenge the debate might have been a short one. The question “is Bayes’ theorem apropos to the evaluation of antithetical propositions?” can be answered in three words: It is not. It solely addresses subsets as compatible in their complementarity.

In any debate, the pro side defends the validity of a proposition while the con side defends the validity of its antithesis. For example, the antithesis of the proposition, ‘Some sheep are black’, is, ‘No sheep is black’. In contrast, ‘Some sheep are non-black’ is compatible, but you can’t have a debate over compatible propositions, e.g. ‘Some cats are black’ and ‘Some cats are tan.’ Compatible propositions can even represent complementary subsets as in the case of black and non-black sheep. As a sum, complementary subsets cover the entire population of a set.

Bayes’ theorem is applicable to a population which is partitioned into complementary subsets by each of two criteria. Consider the population consisting of farmer Brown’s sheep and the crayons of the local kindergarten. One criterion partitions this population of elements into black elements and non-black elements. The other criterion partitions this population into sheep and non-sheep.

If we are given (1) the fraction of the population which is black, (2) the fraction of the population which is sheep and (3) the fraction of the black population which is sheep, we can calculate the fraction of the sheep population which is black. The equation of calculation has been named Bayes’ theorem.

The problem with Bayes theorem

Because the fraction of the sheep population that is black is the complement of the fraction of sheep that is non-black, we can calculate the relative size of both subsets. These subsets are ‘Some sheep are black’ and ‘Some sheep are non-black.’ But the relative sizes of subsets do not tell us whether an individual sheep is black or non-black. That would have to be determined by examining the individual sheep.

Bayes’ theorem solely concerns complementary subsets, and it cannot elucidate the validity of antithetical propositions, such as ‘Some sheep are black’ and ‘No sheep is black’ or ‘God exists’ and ‘God does not exist’. Antithetical propositions are not complementary subsets, so antithetical propositions and Bayes’ theorem are irrelevant to one another. As such, Bayes’ theorem cannot be invoked to validate one or the other of two antithetical propositions, which validation is the object of debate.

Also, it would be a trivial case of Bayes’ theorem, if in the illustration, it were given that the fraction of black elements that are sheep was zero. It could not be claimed that Bayes’ theorem proved the antithesis of ‘Some sheep are black’, namely that ‘No sheep is black’ by calculating that there are zero elements in the subset called ‘black sheep.’ There is no need to prove, by way of calculation, that which is given.

Black and non-black are obviously complementary and thereby compatible subsets. In contrast, true and false versions of a proposition are antithetical. However, in the context of Bayes’ theorem, true and false (as non-true) can refer to subsets, which are complementary in spite of any antithetical connotation. As complementary, their sum equals the entire set.

A simple example

An example of this would be the true data and the non-true data in an annual report. The sum of the true and non-true data is the total data. It is their complementary compatibility which can render nominally antithetical subsets subject to Bayes’ theorem. In contrast, antithetical propositions are subject to debate precisely because of their incompatibility.

Bayes’ theorem calculates the relative size of complementary subsets, but a debate does not seek to determine the relative size of the subset to which a proposition belongs. A debate seeks to determine the validity of a proposition, or, if you will, the subset, true or false, to which the proposition belongs by examining the proposition, itself.

Conclusion

When the con side of the God Debate suggests a mathematical argument as a work-around to the decades-long philosophical impasse in the God Debate, the pro side should accept the challenge to debate using mathematics as the common ground.

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5 thoughts on “Working Around the Impasse in the God Debate”

  1. Pingback: On Earth as it is in Heaven

  2. These type of arguments are of use only for those who are willing to be persuaded by them. Such people do exist, but my experience leads me to believe they are very, very rare. Ultimately, these arguments are only good for informing someone of the truth; they are of no use for someone who stubbornly insists on denying what he does not really doubt.

  3. I do not want a god whose existence can be proven with – and only with – human reason (unaided by faith). Such a lesser god would only be a figment of our human reason and not the God of the Trinity, the God of revelation, the God of the Cross. A 2+2 = 4 god is not much of a god.

    You might find interesting C.S. Lewis’s loss (“obliteration”) in a version of a “Does God Exist” debate to Catholic convert and philosopher Gertrude Elizabeth Margaret Anscombe (“GEM” Anscombe) – Lewis, in a way, arguing for God’s existence. Anscombe, who had seven children, scandalised liberal colleagues (Oxford and Cambridge) with articles in support of the Church’s views on contraception and twice she was arrested outside abortion businesses after abortion had been legalised.

    Regarding the debate that Lewis lost to her: “The subject was Lewis’s argument that naturalism (the view that the natural world is all that exists) is self-refuting, since “no thought is valid if it can be fully explained as the result of irrational causes” (Lewis, Miracles, ch. 3).” Anscombe devastated Lewis’s position.

    Lewis later wrote of GEM Anscombe:

    “The lady is quite right to refute what she thinks bad theistic arguments, but does this not almost oblige her as a Christian to find good ones in their place: having obliterated me as an Apologist ought she not to succeed me? (Letters 3:35; emphasis mine.)”

    Guy McClung, San Antonio, Texas

    1. Thanks for the heads-up. After reading the 1960 revision of chapter 3 of Lewis’ “Miracles” and Arend Smilde’s essay, “A Philosophical Layman’s Attempt to Understand the Anscombe Affair”, a 2 + 2 = 4 god begins to sound quite attractive. However, the God of philosophy cannot be a 2 + 2 = 4 god, because it is not possible to draw a philosophical conclusion from a mathematical argument. The Catechism (CCC 32) cites St. Paul, who, lucidly identifies the God of philosophy as the creator of the things of our everyday experience and implies that this is the same God he preaches through the gift of revelation.

    2. My understanding is that Miracles, as it exists today, incorporates changes in response to Anscombe’s objections.

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