Daniel Dennett Presents: An Argument Against Naturalism

I was recently stunned to find, in an interview with Daniel Dennett published in Third Way (July/August 2013, p. 14), the following exchange:

Interviewer:  What might constitute an insuperable problem for naturalism, then?

Dennett:  Well, I thought of one thing in a little reverie just yesterday.  If you take all the integers – one, two, three, four – starting with zero and you arrange them in a sort of a square spiral and you put a red circle, say, around each prime number, you discover that there are some interesting patterns – it’s a very tantalizing fact about prime numbers.  Well, suppose someone said, ‘I’m going to arrange the number in a slightly different way and just see what happens,’ and they did it and, after a while, when they’d [arranged] enough millions of numbers, we saw that [the pattern that had emerged] was a crucifix… If something like that was embedded in the number system, my timbers would be shivered, no doubt about it – because you can’t fake the number system.

I must confess to being utterly baffled by this argument.  I have my doubts as to whether such an occurrence (or something analogous) actually would convince Dennett, should it turn out to be true, but one thing of which I am fairly certain is that it would not convince me nor the vast majority of professional theologians and philosophers I know and/or read.  It is a mere magic trick.  And if it proved the existence of anything, it would not be the God of Jesus Christ, but of simply one more thing among other things, albeit a thing of great power and, it would seem, whimsy.

(As an aside, the idea of a crucifix being the symbol for God in the number system is an interesting one.  It highlights something many have noted, namely that atheism does not emerge independently of religion.  It is utterly dependent upon religion.  Dennett isn’t interested in disproving the God of philosophy, but the God of the Bible.  In other words, for all the blustery appeal to ‘evidence’ and ‘common-sense,’ the new atheist project remains, in its essence, reactionary.  Furthermore, it is worth noting that it is precisely the God of the Bible, that One who has done more than any atheist philosopher in history to unmask the idols of this world and demonstrate that what we thought were the gods, were, in fact, nothing of the kind – it is this God who makes atheism possible by a relentless demythologization of the cosmos.)

All of my professional training, and indeed even my temperament, inclines me to put the best possible read on the comments of those with whom I disagree.  And I know, theoretically, at least, that Dennett is a bright guy.  But how does a bright guy find no trouble for his worldview whatsoever in the indisputable fact that there is something rather than nothing, and yet confess that a curious plant in the number system would shiver his timbers?
Oh, and in the same interview, Dennett refers to Jesus as “probably a mythical character.”  (p. 16)  Seriously?  It’s like arguing with a young-earth creationist.

Brett Salkeld is the incoming Archdiocesan Theologian for the Archdiocese of Regina, Saskatchewan. He is a father of four (so far) and husband of one.

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  • Frank M.

    Dennet isn’t the only person to come up with this sort of thing. In one of Carl Sagan’s books (I think it was Contact), he wrote about a math experiment wherein a character found unaccounted correlations in the digits of pi (the transcendental number which represents among many other things the circumference of a circle divided by its diameter in flat Euclidean geometry). In Sagan’s story, the experimenter finds that in a base-11 representation, the digits deep in the representation of pi form a pattern which when arranged into a square produce an image of a circle.

    IMHO, Sagan’s version shows much more sophistication than Dennet’s, which is purely anthropocentric and Euro-centric. Dennet is apparently looking for a message pointed rather specifically at him as a person whose faith in Science and Mathematics seems to replace in him any need to take Christianity seriously. He seems oddly similar to creationists who cling to “evidence” that evolution by natural selection can’t possibly account for the sophistication of terrestrial life; in their case it is their reliance on fundamentalism which replaces any need to take Science seriously. Sagan’s story looks for a message pointed at someone who is searching much more broadly than just “is Christianity as it has evolved today the one and only right thing to do, or not?”

    The implicit hope underlying Dennet’s and Sagan’s ideas echos the excitement about 10 years ago when the Shroud of Turin was subjected to carbon dating and spectroscopic analysis: All of these express the idea that God might somehow have embedded objectively measurable divine fingerprints in our world, with the expectation that we will discover them just as our technology advances and really seduces us into self-reliance. Then, we won’t have to ourselves and our relationship with God seriously at all, because “God” will become just one more object in our universe.

  • Mark , VA

    The mathematical phenomenon described above is called Ulam spiral, or Ulam cloth. Perhaps our resident mathematician will weigh in on this some more:


    Whether this points to the theological significance described by Mr. Dennett, seems like a curious turn of thought.

  • David Cruz-Uribe, SFO

    The Ulam spiral is a curious phenomenon that shows that prime numbers, while “random” in some casual use of the word, are not really randomly distributed in the precise (mathematical) sense of the word. This particular pattern shows that there are so-called quadratic prime generators: if you choose integers a, b and c carefully, then the polynomial an^2+bn+c will produce lots of prime numbers when you plug in successive values of n: many more than you would get in a properly random collection of numbers.

    I find Dennett’s comment unsophisticated from a mathematical perspective. I would need to check this more carefully, but I believe the following is true: devise a way of arranging numbers into large rectangles, and assign a gray-scale or black-white according to the digits. For example, take 10000 numbers and arrange them into a 100×100 grid, and color the squares containing the number black or white depending on whether the number is odd or even. If you do this with the digits of pi, then I claim that if you go far enough out, you will get an image of a cross. If you go even further out, you will get an image of crucifix. If you go even further out, you will get a crucifix in which the crown of thorns has 27 visible thorns and Jesus has a hang-nail on his big toe. I claim this because the digits of pi appear to be truly random (in the precise sense that all patterns are equally likely). Therefore, if you go far enough out, you should be able to find any pattern you want. This is just a very sophisticated version of the 100 monkeys and 100 typewriters banging out Shakespeare. See http://www.askamathematician.com/2009/11/since-pi-is-infinite-can-i-draw-any-random-number-sequence-and-be-certain-that-it-exists-somewhere-in-the-digits-of-pi/ for a discussion of this.

    Dennett might claim that this is not comparable because pi truly is random, but the primes have a deep structure, and it would be “miraculous” if this deep structure yielded an image of a crucifix. However, note what he says: “suppose we try it some other way.” Well, I imagine that if you tried enough different patterns, and included enough prime numbers you could get pictures of a crucifix to emerge (of course, you could also get pictures of Donald Duck to emerge). Here is a simple way to think of it: if you give me any sequence of numbers, of any length, and ask me to predict the next number in the sequence. I can always answer 17 and be right. Because if I look hard enough, I can always find a generating function such that when I plug in successive integers (1,2,3,….) it generates the terms of your sequence, and then generates 17. You might complain that my generating function is “unnatural” but this would be a matter of aesthetics and not properly a mathematical criterion.

    If I remember something that Stephen Jay Gould wrote, there was a time when such naturalistic proofs were accepted by scientists as proofs of Christianity. It was not uncommon to find rocks and fossils that appeared to be crosses and other (Christian) religious symbols. These were not understood as being the result of random processes but rather as the handiwork of a benificient God who was creating natural evidence of religious truths. I think that the number patterns Dennett is looking for are quite similar.

  • Noah

    > But how does a bright guy find no trouble for his worldview whatsoever in the indisputable fact that there is something rather than nothing, and yet confess that a curious plant in the number system would shiver his timbers

    Hm, I guess you either missed a critical aspect of what Dennet was imagining or you skipped over considering the evidential weight of the “curious plant”.

    Dennet is obviously imagining a simple variation on the Ulam spiral that produced religious imagery (like a detailed crucifix) rather than either randomness or non-religious order. David’s explanation of the pseudo-patterns which can be found in the randomness of transcendental numbers like pi was neglecting the “rather than…” aspect, which is why he failed to recognize its evidential weight.

    If such a mathematical structure were real, it would indeed be an overwhelmingly strong piece of evidence for Christianity. If that’s not intuitively obvious to you, just do a quick rough estimate of the posterior odds to see. If Christianity is true, then the (Bayesian) probability of such a mathematical structure being real would be some very small but not terribly small number. Let’s imagine a Christian who was confident enough to bet as much as billions to one against there being such a thing. But if naturalism is true, then the probability of such a mathematical structure is gobsmackingly small. Dennet imagined the detail of the picture having millions of bits of information (i.e. millions of numbers either circled or not-circled), which would have a probability of 2^(-millions). The weight of the evidence would be a factor of (1/billions)/(2^(-millions)) = (2^millions) in favor of Christianity versus naturalism — disproving naturalism with virtually perfect certainty even if Dennet would have given prior odds against Christianity of billions-to-one.

    • The math really has very little to do with it. The question is, is this the kind of thing that the God professed by Christianity does? Is he a meddler and a tweaker who plants silly proofs in the number system that may or may not be found or is he the ground of being who reveals himself first in the contingency of the created order, and secondly through his radical actions in history? (In point of fact, a god who does plants in the number system would be the god of gnosticism, a heresy that the Church has fought from its inception.) You may well doubt the resurrection, and that is another argument, but the point is that this kind of event is what Christians understand their God to be about, not Bible codes and number fudging. I mean, the simple fact that it is a crucifix that Dennett posits is based in a historical event, an act of public revelation. That is the nature of the God Christians believe in.
      In any case, disproving naturalism (that was the question, after all) is not the same thing as proving Christianity (far less some bizarre and gnostic form of Christianity that seeks a trickster in the number system to call the Father of Jesus Christ). That Dennett does not distinguish these two questions is very telling.
      The whole point is that Dennett has simply no idea what the word “God” means when Christians use it. (Of course, the same can be said of many Christians, who would also point to such nonsense as “proof.” But that is really neither here nor there.) In fact, not only does he not seem to know what Christians mean by God. He seems to find the whole question wildly uninteresting. This is bound to lead to poor criticism.

      • Frank M.

        “The whole point is that Dennett has simply no idea what the word “God” means when Christians use it. (Of course, the same can be said of many Christians…”

        Considering the percentage of Americans who still don’t believe in evolution by natural selection, and presuming they are motivated by “religious” bias, I think you’d be closer to the truth with “the same can be said of the vast majority of Christians…” (Hey, they aren’t reading VN!)

        Is it reasonable to expect that Prof. Dennet would hold a dramatically different perspective on the God he doesn’t believe in than the great majority of believers?

        • If he writes whole books on the subject? Yes, I think it is reasonable. I mean these guys all claim to have read Aquinas, whom they then mock as a 2nd rate commentator on Aristotle and nothing more. If they actually have read Aquinas and still don’t know what Christians mean by “God” (or at least what we don’t mean), then they are perfectly legitimate targets of criticisms such as mine.

        • Frank M.


          I’m not questioning the legitimacy of your criticism. I find it completely on point. What I question is the legitimacy of your surprise.

          I don’t think it’s hard to read Aquinas with a literalist perspective, and come away with a picture that supports Dennet’s criticisms. The popularity of Dennet and the other new atheists is a reaction to the form Christianity takes most visibly, not to the meaning of “God” in Aquinas or Lonergan. They rightly react to that majority of Christians who hold to a patriotic “God,” favoring their shining American City on a Hill, their way of life and their egoistic place in creation.

          Conservative and “moderate” Christians are just as likely to mock the idea of seeking relationship with the Ground of Being as Dennet is to mock Aquinas, and for very similar reasons. If there’s anything to cause surprise and dismay, it ought to be how few Christians know what Christians mean by “God.”

          • I am in substantial agreement here Frank. One thing that concerns me is that when Christians reach a kind of intellectual maturity that should lead them to seek deeper answers to such questions they often feel guilty for having “doubts” and say very little to anyone until they build up the courage to leave the Church without ever having seriously investigated the issues at hand. I am very keen on a catechesis that lets our people know that questions will inevitably come and that they are good things!

      • Noah

        Hi Brett:

        OK, I’ll grant that “the math has very little to do with it” for you, but nevertheless the math is clearly the whole of it for Dennett, so it doesn’t seem the least bit sensible to object to Dennett’s argument without understanding why the math means what it means.

        You write, “The question is, is this the kind of thing that the God professed by Christianity does?”. Quite right! That’s why I explicitly included in the discussion of why the math means what it means that a Christian would view the likelihood of God doing such a thing as “very small but not terribly small”, and being of the mind to bet something like “billions to one” against God doing such a thing. For the meaning of the math to change significantly, Christianity would have to make a prediction not much different than naturalism. But, although you dismiss the Christians who would take such miraculous evidence seriously, you cannot seriously claim that Christianity excludes with (2^millions)-to-1 certainty the possibility that God might do such an unexpected thing. There is no such dogma.

        You follow up with two other minor points that I’ll address. First, “In any case, disproving naturalism (that was the question, after all) is not the same thing as proving Christianity…” You make a perfectly valid point, of course! But don’t mistake it for an objection against Dennet’s argument, which was, after all, about what would be for Dennett “an insuperable problem for naturalism”. Second, “The whole point is that Dennett has simply no idea what the word ‘God’ means when Christians use it.” That’s not the whole point; rather, it’s none of the point. Dennett’s argument is independent of the meaning of “God” in all ways except one, which is that his argument requires a God capable of Creating reality the way He pleases. Dennett probably does take the commonplace expressions of religion to be “the main thing” rather than theologians’ nuanced expressions, but that’s an objection against other arguments he might make, not this one.

    • If such a mathematical structure were real, it would indeed be an overwhelmingly strong piece of evidence for Christianity. If that’s not intuitively obvious to you, just do a quick rough estimate of the posterior odds to see.

      While Brett is right that the mathematics is essentially irrelevant to the matter, I have to say that this is a thoroughly bizarre use of Bayesian epistemology. The existence of some crucifix structure somewhere in mathematics producible by some mathematical operation is already completely certain, since, no matter how detailed, the crucifix structure admits of a mathematical description. The whole supposed argument boils down, then, to whether there is some operation on prime numbers specifically that would result in such a structure, and that this specific prime number scenario is somehow especially significant in comparison with others; but nobody has any particular reason to think there isn’t some operation that would somewhere in the infinite series of primes yield something as crucifix-like as you please. And even if that were not so, simply confining it to primes is already cheating, for precisely the reason David mentioned: the ‘rather than’ weakens, rather than strengthens, the force of the argument, since, unless there is a definite reason for it, it amounts to rigging the description to get the probabilities one wants. It’s very similar, in fact to the sorts of arguments given by anti-evolutionists that it would be a miracle for evolution to produce this-very-precise-and-therefore-very-improbable-thing. It’s the precision, not anything else, that is creating the weird probability. Indeed, this is precisely what the overall reasoning is: it’s just a standard creationist / intelligent design line, but with Dennett trying to push the threshold to even more extremely high levels in the hope of getting them so high that he has nothing to fear from them.

      Subjective Bayesianism is a dynamic, not a static, approach: it gets its real bite from long-term convergence over a lot of new evidence, not from any specific stage. At a single stage, one can plug numbers in to get virtually any result one wants, and the formal structure will fit some possible hypothetical reasoners; we can think of reasoners who would at a given stage have reason to think that the probability of such a structure is nearly certain regardless, nearly zero regardless, or anything in between, and this will affect the conclusions drawn for any single-stage reasoning. And all this is assuming a Bayesian framework rather than a competing framework like frequentist error statistics or objective Bayesianism, which would yield a completely different results.

      • it’s just a standard creationist / intelligent design line, but with Dennett trying to push the threshold to even more extremely high levels in the hope of getting them so high that he has nothing to fear from them.


      • Noah

        Hi Brandon:

        Your first paragraph repeats the erroneous oversight or confusion that I responded to, which is that simplicity is critical to what Dennett was imagining. Your paragraph is mostly correct factually, but the facts are not being used against the argument that Dennett was actually making, nor against my explanation of it.

        Your second paragraph is too vague (or maybe too heavy on jargon I don’t know) for me to understand what your objection is. Subjective Bayesianism is just using the probability axioms with regard to betting ratios. How to best calibrate your sense of “subjective probability” or how to select the theories to compare are not part of it.

  • Mark , VA


    I find these kinds of “proofs” peculiar, something akin to numerology, “Bible codes”, and such. Have you ever had a chance to read the biography of John Napier?

    By the way, on the subject of randomness, is the following true or false?

    With the professor absent form the building where the below experiment takes place:

    A group of students, who know very little about probability, divides itself, as they please, into two groups (i.e. physically separate into different classrooms). One group performs a hundred tosses of a fair coin, and records the results.

    At the same time, the other group never does the coin tosses, and only writes down what they imagine the outcomes of such tosses would be.

    Afterwards, both groups write down their outcomes on a blackboard (without identifying which group produced which set, just write the two sets of heads and tails down), and call in the professor. Or they can just e-mail the two sets to the professor.

    The professor will have no problem identifying the set produced by the actual coin toss, from the other set. No tricks or any other professorial shenanigans are implied.

    True or False?

    (I’ve first heard of such an experiment on a radio program)

    • David Cruz-Uribe, SFO

      Mark, this experiment is commonly done in introductory probability or statistics courses. The point of the exercise is that such students do not have a good understanding of what “random” means, and therefore create sequences that are in fact highly non-random. I have a colleague who claims that in a class of 30-40 students, she can have (without her knowledge of who) half the students make up “random” strings of 20 coin flips and the other half actually create random coin flips, and she can with over 99% success, distinguish the two. I have never tried this myself.

      • Mark , VA

        Mr. David Cruz-Uribe, SFO:

        To be able to tell the difference between a real and an imagined random sequence from just twenty coin tosses is, in my opinion, very good. A tip of the hat to your professor colleague.

        If there were a philosophical angle or two to this, I would go out on a limb and say this:

        1) The uninitiated don’t have a good grasp of what random means. Since all of us have to pass thru this stage (well, perhaps geniuses like Euler are excluded by a singular grace), it is interesting to note that we all tend to err in a similar way. For example, I’ve never heard of anyone erring by producing too robust streaks or sequences– we tend to under-predict and wimpify them;

        2) This characteristic of randomness speaks for, rather than against, some aspects of the theory of evolution. Perhaps this is God’s method of creating the sequences needed to refine the living matter to a desired stage. Maybe Mr. Salkeld would care to comment on this;

        3) We are witnessing a rare occurrence – except perhaps for Noah, we are all in agreement regarding Mr. Dennett’s statement.

        Now I feel a ditty coming on:

        Pi is a number like no other,
        On account of having a round father,
        And a mother who cut papa thru the middle
        With a string from an old fiddle.

  • Thales

    I agree with the post and the comments, so I’m not commenting on any of that.

    But my first reaction to Dennett’s comment was different: Simply that I don’t believe him. With documented miracles, Eucharistic miracles, the Shroud, etc.,… there are plenty of things already that would cause the timbers to be shivered of a person who was honest and serious about investigating the truth of these unexplained-by-science matters. Supposing the numbers revealed the crucifix, I’d bet Dennett would ignore it or dismiss it or not think about it, just like he must do already with the other signs of God in our world.

    It’s like the Pharisees asking Jesus for another sign, and then another sign, and then another sign, with none of them being adequate, and Jesus finally saying that even if someone died and rose from the dead, they wouldn’t be satisfied. Likewise, I’d bet that Dennett wouldn’t be satisfied if we found a crucifix in the numbers.

    • One wonders what Dennett would say to Mr. Adamus from 53:30 to 54:30 in the following presentation:

      Actually, I think it’s pretty easy to guess what Dennett would say. I think his timbers would be toasty and dry.

  • David Cruz-Uribe, SFO

    I think Brett has successfully refocused the discussion on theology and Dennett’s understanding of God, but I want to respond to Noah’s argument about probabilities. My short response is this: probabilistic arguments that do not have a well defined probability space underlying them (i.e., a space of all possible outcomes and measure of their likelihood) are inherently weak. They may possibly be suggestive, but prove nothing.

    With regards to patterns in, as Noah puts it, “a simple variation” of the Ulam spiral, my original objection still stands: what constitutes a simple variation? To rephrase my argument: if you shoot an arrow and then paint the target, you will always hit the bullseye.

    But let’s suppose in fact that some variation of the Ulam spiral (or the Ulam spiral itself), if carried sufficiently far, yields an image of a crucifix. My response would be to consider one of several possibilities:

    1) This is a remarkable one off: pseudo-random data often produces striking things that appear non-random or intentional. Again, think of rocks that form patterns that look like crosses, etc. These are the result of pseudo-random processes.

    2) This new variation of the Ulam spiral reveals as yet unknown random behavior in the sequence of primes that is hidden behind the structure of the prime number theorem.

    Neither of these suggests anything about the veracity or falsity of Christianity, which should rise and fall, as Brett notes, on the resurrection of Jesus.

    • Noah

      Oops, I expected a notification of responses to my post that never came. I’ll start here as David was the only person I mentioned by name above.

      David: I take the meat of your above comment to be the objection against ambiguity in “simplicity”. There need not be any ambiguity, though. If the bettors are satisfied with an informal notion of “simplicity”, they need merely reach some agreement. If they insisted on holding out for a rigorous definition of “simplicity”, they could use the Solomonoff universal prior with an arbitrary universal turing machine.

      The subsequent text in your comment above repeats the initial oversight or confusion that I first responded to, the critical error of neglecting simplicity.

      Your first paragraph is more substantive. It hints at but does not explicity include a objection against the common formalization of probability in terms of betting ratios. I’m not going to engage in that philosophical argument as I have no horse in it. Rather, I’ll simply use the math wherever it fits. Betting ratios are one of the things that fit.

  • This notion is also found in Carl Sagan’s novel Contact, in which a message is discovered very deep within the transcendental number pi. The message, properly understood, creates a picture of a circle.

    I do not understand your objection to Dennett’s argument. Such a discovery would not prove merely the existence of “one more thing among other things,” but would seem to indicate something greater than mathematics itself. To put a message inside a number, one would have to be the creator of logic itself. Actually, I’m not sure that even God can do that. I feel quite sure that he cannot, for instance, make a different set of prime numbers than the ones that really exist, and hiding a message within a number would seem to amount to the same thing.

    • Adam,
      I think your last sentence answers your question about my objection . . . at least form one angle. God does not exactly create numbers. God creates a reality which numbers are very useful for describing. Asking whether God could make different prime numbers than there are is probably as nonsensical as asking if God could make a four-sided triangle. The thing itself is only describable because of ambiguities in human language. It has no real meaning. Consequently it is of no moment that God cannot do it. As Frank Sheed playfully puts it, it is nothing, literally NO-thing, and that is the one thing God can’t do.