Can Arithmetic Set an Atheist Free?

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We may be wary of the truth. Rather than setting us free, we may fear it will impose an obligation. We may be reluctant to reach a conclusion as true, simply because we view its implications as onerous. Take the formation of a new individual of the species by means of biological sex as an example.

No Spur to Intellectual Prejudice, in One Case, If

The process of sexual replication in the flowering plants is of the same form as that of animals, including humans. Suppose, however, sexual replication was unique to flowering plants, having no implications to human biology.

Research has revealed that male and female germ cells are produced in flowering plants by meiosis, the binary division of genetic material resulting in pollen and ovules. The recombination of such halved genetic material, through the fertilization of an ovule by a pollen cell would be commonly recognized as the formation of a new individual of the plant species.

The fertilized ovule would be identified as a new individual, in light of its newly formed genetic complement, characteristic of the species. No one would question this because this truth is no skin off of anyone’s teeth.

In this supposition, the truth of sexual reproduction would have no consequences in human moral behavior. Consequently, we would all feel free readily to admit the truth. There would be no spur to twist the truth to serve our desires.

This If Is Not True, Prejudice Enters

But the formation of a new individual of the species is questioned because the process of sexual reproduction in the flowering plants and in animals is of the same form.

If I were to admit that fertilization of the female germ cell in plants by the male germ cell resulted in a new individual of the species I would have to admit the same is true in humans. That could impose a burden on my behavior, so I am inclined to deny the truth.

Philosophy Is a Touchy Subject, Arithmetic Is Not

The particulars of most human virtues derive from the nature of man and can be elucidated without direct recourse to the existence of God. The existence of God completes the big picture of human morality. Atheism per se does not eliminate morality, but atheism tends to provide as much wiggle room in ethics as one may desire.

The philosophical arguments which lead to the existence of God and the details of ethics are sufficiently complicated to allow their denial without extreme intellectual embarrassment. Thus, apologetics in these areas is no easy task. However, in his justification of what is labelled the new atheism, its most notable proponent, Richard Dawkins, has based his philosophical conclusions on arithmetic.

The Arithmetic of Dawkins’ Atheism

Although arithmetic has no immediate philosophical consequences, we may, for the sake of argument, concede that it does. Dawkins’ typical argument is:
Premise 1: If arithmetical calculation A, is valid, then philosophical fact B is true.
Premise 2: Arithmetical calculation A is valid. Conclusion: Therefore, the philosophical fact B is true.

What Dawkins calls ‘the anthropic argument: planetary version’ (The God Delusion, page 134) is that the number of earth-like planets in our universe is sufficiently large to explain the origin of life. In Dawkins’ judgment, this obviates the existence of God, which Dawkins views as the alternative postulation for the origin of life.

The form of the argument is:

P1: If the number of earthlike planets in our universe is large, then God almost certainly does not exist because that number is sufficient to explain the origin of life.
P2: Based on the validity of the arithmetical calculation that the number of earthlike planets equals the probability of earthlike planets times the number of planets in a universe, the number of earthlike planets in our universe is large.
C: Therefore, God almost certainly does not exist.

From a contrarian perspective, it is convenient that Dawkins draws his philosophical conclusions from arithmetical arguments. If the arithmetic is false, then the philosophical conclusion does not follow. Most importantly, everyone is typically disinterested in arithmetic. No one is emotionally invested in arithmetic.

In this particular instance, Dawkins is so pleased with the outcome of his arithmetic, that he states it three times in three consecutive paragraphs.

“And yet . . . even with such absurdly long odds, life will still have arisen on a billion planets. . .

“This conclusion is so surprising, I’ll say it again. If the odds of life originating spontaneously on a planet were a billion to one against, nevertheless that stupefyingly improbable event would still happen on a billion planets.”

“But my earlier calculation demonstrated that even a chemical model with odds of success as low as one in a billion would still predict that life would arise on a billion planets in the universe.”

In his arithmetical calculation, Dawkins takes the probability of earth-like planets as one per billion and the number of planets in the universe as one billion, billion. His arithmetical formula is: The number, E, of earth-like planets in the universe equals, the probability of an earth-like planet, P, times the number, N, of planets in the universe: E = P × N

In this calculation, P = one per billion and N = one billion, billion planets. Therefore, E = one billion planets, not one billion earth-like planets. In other words, material units are part and parcel of an arithmetical equation.

Dawkins’ formula is false. The number, E, of earth-like planets in a universe of N planets equals the density of earth-like planets, D, times N: E = D × N

In this calculation, D = one earth-like planet per billion planets and N = one billion, billion planets. Therefore, E = one billion earth-like planets. Material units matter.

In contrast to a basis of density, the number, E, of earth-like planets in a universe of N planets, based on a probability, P, of earth-like planets varies from zero to N earth-like planets

Based on a density, every universe of a given size will have the same number of earth-like planets. This is not true for populations of universes defined on the basis of probabilities. Small numbers will serve in illustration.

For numerical values of density, D, equaling 1/2, 1/3, 1/4 and 1/10 and the corresponding numerical values of N equaling D^2, each universe of numerical size D^2, respectively will have 2, 3, 4 and 10 earth-like planets.

In contrast, based on probabilities of P = 1/N of 1/2, 1/3, 1/4 and 1/10, then for the population of universes of universe size N^2, not 100%, but 37.5%, 27.3%, 22.5% and 13.2% of universes comprising a population of universes, will have 2, 3, 4 and 10 earth-like planets, respectively. Some universes in a population would have as few as zero, while some would have as many as N^2 earth-like planets. The defined populations of universes will consist of N^(N^2), or 16; 19,683; 4,294,967,296 and ten thousand, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion universes, respectively.

One very simple illustration of defining a population of sets based on probability is that of sets of two elements each and a probability of 1/2 for a specific element, say heads. This defines three distinctive sets containing 0, 1 and 2 heads. The quantity of each of these sets in the defined population of sets is respectively, 1, 2 and 1, for a total of 4 sets. Two, which is 50% of the sets, contain exactly 1 head. One, which is 25%, contains 0 and the other one,  25%, contains 2.In contrast, Dawkins’ calculation fails to distinguish probability from density, claiming that any one, i.e. each set of the population, contains exactly 1 head.

For P = 1/N, where N = one billion = 10^9, and a set size of N^2, i.e. one billion, billion = 10^18, the population of sets comprises N^(N^2) sets. That is 10^162, or one million, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion sets.

Yet, Dawkins assures us that according to his calculation, our universe, which is just one out of these many trillions of trillions of universes, contains not any number from zero to one billion, billion earth-like planets, but exactly one billion.

Because Dawkins’ arithmetic is false, his philosophical conclusion that God almost certainly does not exist is entirely gratuitous. It doesn’t follow from his stated premises.

Conclusion

The bad news is that conflicting philosophical arguments and their conclusions are difficult to resolve. Even Dawkins’ human epistemology, as the inference of mathematical probability, yields a moderately self-coherent system of philosophy.

The good news is that many of Dawkins’ philosophical arguments are based directly on arithmetic. His anthropic argument: planetary version, which concludes that God almost certainly does not exist, is one of these. It is based on populations of sets defined by a value of probability and a value of set size.

Arithmetic is common ground. No one is arithmetically prejudiced. Thus the admission of arithmetical errors need not be intellectually embarrassing.

It would take very little study of arithmetic for Richard Dawkins to recognize his arithmetical errors. Such a recognition would be the admission that his philosophical conclusions were gratuitous in being based on erroneous arithmetic.

The better news is that we can pray for Richard Dawkins that, coming to know arithmetical truth, he might receive the grace to know the Eternal Truth.

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24 thoughts on “Can Arithmetic Set an Atheist Free?”

  1. “Dawkins’ typical argument is:
    Premise 1: If arithmetical calculation A, is valid, then philosophical fact B is true.
    Premise 2: Arithmetical calculation A is valid. Conclusion: Therefore, the philosophical fact B is true.”

    I very much doubt that is a “typical” Dawkins argument. The man is not an idiot, whatever some people would like to think. The “premises” above are both utterly patently, obviously, stupid – so much so there’s no point in even bothering to explain why.

    ” Therefore, the philosophical fact B is true.
    What, logical, grammatical, gibberish is that, for example?

    1. I have cited five different errors in arithmetic by Dawkins, some of which he makes repetitively. https://theyhavenowine.wordpress.com/2014/09/15/the-anwers-of-student-dawkins-to-a-high-school-math-quiz/
      These errors in arithmetic are foundational to his philosophical conclusions. His central argument in “The God Delusion” is that there is an arithmetical solution to the improbability of a large scale evolutionary event, but there is no solution to the improbability of God. The validity of his philosophical conclusion about the existence of God depends, within the very context of his argument, upon the validity of his arithmetic. That arithmetic is erroneous. It is not a solution to the improbability of a large scale evolutionary event. In his arithmetic, Dawkins mistakes an increase in the efficiency of mutation for an increase in the probability of evolutionary success. https://theyhavenowine.wordpress.com/2015/09/01/evolution-probability-vs-efficiency-revisited/

    2. “True Facts” are tautological nonsense.
      That’s it, Bob. Finish.
      I also suspect Dawkins is aware of that.

      ” His (Dawkins) central argument in “The God Delusion” is that there is an arithmetical solution to the improbability of a large scale evolutionary event,”
      I’ve not read the book and will not (can’t be bothered, I already know the arguments for, and against, God). But, if he really thinks there are any arithmetical solutions to any philosophical problems – (an eventuality I seriously doubt, by the way – as I said before, he’s not daft,) you are wasting your time second-guessing him at such prolix length… and would be far better employed taking your dog for a walk.

    3. I was shocked to think that I let the expression, true facts, slip past me in my earlier reply to you. I always re-read what I have typed before hitting the post button. I will still be extremely embarrassed, if you can cite any instance in which I have used the expression, true facts, without quotation marks to highlight its redundancy.
      You are in good company, such as that of Edward Feser, in claiming that Richard Dawkins et al have nothing new to offer philosophically. In contrast, I think Dawkins and others, such as Sean Carrol, have proposed a fundamentally new philosophy, https://theyhavenowine.wordpress.com/2015/03/01/dawkins-philosophy-of-probability-a-p. One principle of their new philosophy is that human knowledge is the inference of mathematical probability. Another is that existential power is inherent in mathematical probability.
      Though superficially disagreeing with Dawkins, Stephen Meyer, e.g., concurs in the existential power of mathematical probability. Like Dawkins, Meyer restricts that power to a portion of the range of definition of probability. Ironically, Dawkins got the arithmetic right when he argued that you can’t divide such a range of definition into qualitatively distinct subranges, https://richarddawkins.net/2013/01/the-tyranny-of-the-discontinuous-mind-christmas-2011/. Dawkins violates this principle elsewhere, e.g. in dividing improbability into two qualitatively different subranges, namely a prohibitive subrange and a non-prohibitive subrange (See p 121 of “The God Delusion”).
      I admire Dawkins for his appeals to mathematics, through which he expresses his views leaving nothing to doubt.

    4. A really good reply, Bob. I will give it as serious thought as I am capable of mustering, read the links, and write more later. Personally I have so far looked to Dawkins for information regarding biology. not mathematics or philosophy. Maybe I should look again.

    1. That is nothing to get excited about. It is neither a real conversion story nor a solid intellectual achievement, just the intersection of fad (of the “are we living in a computer simulation?” variety) and publicity (which is good to attract funding).

  2. Dawkins speculates wildly; it is childish scientism dressed in pseudo-logical language unworthy of attention. Do wish you would get off your Dawkins thing Bob Drury.

    1. Arius was “unworthy of attention.” But he needed to be, or Arianism needed to be, cut down whenever and wherever it popped up. Similarly today if someone were to declare Jesus got it wrong when He said some goats would suffer eternal damnation for a long long everlasting time. And so with Dawkins. Guy McClung San Antonio TX

    2. Dawkins is not a Christian heretic. He is a hard hearted know-it-all who dismisses the faithful as insane. Like Jesus advised – shake the dust from your feet.

    3. Science is the determination of mathematical relationships among measurements of material properties. I think Dawkins’ intuition of science is concurrent with that. His arguments are basically mathematical and lucid. For this I admire Dawkins. Another reason I so highly admire him is that I disagree with the criticisms of Dawkins which I have read. He makes errors in logic. Yet, many of his critics concur in those errors. I believe it is his persona, which gives a childish aura, not his mathematical arguments.

    4. For someone who has actually done some real science, Dawkins makes some shockingly naive assertions. Take

      P1: If the number of earthlike planets in our universe is large, then God almost certainly does not exist because that number is sufficient to explain the origin of life.

      This is presumably meant to be self-evident, but it really cannot be on the basis of the form of the assertion:

      If one potential cause of an effect exists, other potential causes almost certainly do not exist.

      It is too easy to find instances of the generalization that are obviously wrong, for example,

      If mechanical problems are sufficient to explain the crash of EgyptAir MS804, terrorist organizations almost certainly do not exist,

      or even

      If paint brushes existed, Michelangelo almost certainly did not exist, because paint brushes are sufficient to explain paint being on the ceiling of the Sistine Chapel.

    5. good post. Drury idolizes Dawkins for some reason – perhaps it is reflective posturing.

    6. I don’t see it that way. That is, similar charges could probably be brought against me on the basis of several things I have written against the microculture of “ghost hunting”, which combines really, really bad theology with really, really bad physics. For example, trying to argue from the conservation of energy to the survival of the soul, or arguing that water — perhaps due to its electrical properties — has something to do with hauntings. Unlike Sam Kinison, they seem to overlook the fact that for the most part, we don’t live in deserts, and a living person is necessary to report a ghost. I’ve written even more against the crowd that believes that there is an indigenous North American ape. With very few exceptions, the Bigfoot crowd combines a cartoonish idea of how science works with a strong gnostic desire to be among the few perfecti who are able to know and embrace a reality denied by the hoi polloi. Both ghost hunters and Bigfoot hunters have some unhealthy thought patterns with potential spiritual problems, and I dearly wish I could get them to be more reasonable.

      Atheists, at least those of the noisy variety, are much more like ghost hunters and Bigfoot hunters than they would ever admit. Of course there are countless individuals who live lives of practical atheism, sometimes because they deeply hope there is no God to give commandments or to judge them after death, and in other cases because they are thoroughly indifferent to religion; both these kind tend to be comparatively quiet about their atheism, though. The ones who loudly proclaim their atheism are the ones who are feeling the gnostic desire to be among the perfecti.

      Your advice is to follow Proverbs 26:4; Drury prefers to follow Proverbs 26:5. There are pluses and minuses to both approaches, which is why both verses are in Scripture.

    7. My guess is your comment was intended for some other thread, since there is no quotation in the canonical gospels in which Jesus talked about Bob Drury idolizing Dawkins.

    8. Science is the approximate determination of physical realities which may be characterized by mathematical equations assuming the relationships are constant in time and space ….or not. I think Stephen Myers does a pretty good job on Dawkins faulty logic.

  3. Kurt Goedel’s studies of arithmetic and the logic of math lead to his famous Incompleteness Theorem in 1931, which is basically the mathematical proof of the limits of scientific knowledge. Folks like Dawkins confronted with this say that they prefer such limited science to the dogmas of belief. In other words, they have their Religion Of Science and don’t confuse them with the limits of any system based on their dogma of “fact.” Guy McClung, San Antonio TX

    1. The Incompleteness Theorem is really about systems with more rigor than is possible in the empirical sciences.

    2. Yes. For more re the limitations of science, its mistakes, and its deficiencies, and yet its use/abuse by those in power, see my Credo Scientific Dogma Part I and Part II on this site,

    3. Anything can be abused or have mistakes made about it, not least of all the Bible and Catholic Tradition. That’s not at all what I’m talking about. I’m talking about the fact that empirical sciences always involve some limited precision in every measurement, so that (for example) there is no distinction between a rational real number and in irrational real number as far as physics is concerned, though they are very different in math. Also, the empirical sciences are based on inductive reasoning, which is inherently less rigorous than deductive reasoning.

  4. I would think that if no one prayed for Dawkins his chances of reversing this error are just as good since
    God has His own plans for him – and if they are contrary to our prayers, Dawkins hasn’t a chance. It took a bolt of light to unseat Paul from his folly no matter how many terrified Christians wished him saved and error free..

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