Monday, October 28, 2013

Pi and the "signature of God" from Carl Sagan's Contact

While watching Peter Boghossian (Link to his new book, A Manual For Creating Atheists) in conversation with Richard Dawkins on October 11 (Link to RDFRS and video of the conversation), my interest was piqued by their discussion about what might constitute evidence for God's existence. At one point in that discussion, they talked about a scene from Carl Sagan's novel Contact in which the protagonist, Ellie, finds in the expansion of pi what she calls a signature, perhaps of God. I emailed Peter about this discussion, and I'm sharing some of what I had to say here, as it is related to my own upcoming publication, Dot, Dot, Dot, about infinity and God, currently in the final stages of preparation.

In very brief, the fictional "signature" referred to in Contact is effectively a very long string of 1s and 0s far out (after some 10^20 seemingly random numbers) in the base-11 expansion of pi that when arranged in a square of a specific size yields a clear drawing of a circle with diameter. The question that Dawkins and Boghossian discussed briefly is whether or not such a thing could be identified as a "signature of God." That is, they discussed what, if anything, it might mean. From my perspective as a mathematician, I immediately thought that what it would mean is "not much."

My thinking that I sent to Peter went like this: Riding on a rather significant conditional, if the "digits" of pi (in any base), are truly random, it's guaranteed that the so-called "signature of God" from Contact will occur at some point in the string of numbers--along with anything and everything else that could be rendered that way. It is not known, though, and may not be the case that the digits of pi are truly random. It is my opinion that finding such a thing somewhere in the decimal expansion of pi would not be surprising on its own.

Peter wrote back and asked a great question that serves as the impetus of this blog post. He asked: "What if the 'signature' in pi repeated itself only once? Would that be evidence?" The remainder here is adapted from my response.

I wrote:
That's likely to be unknowable.

There are two ways to guarantee that such a "signature" is there. The first is obvious: observe it somewhere. As I said, given that we ever look far enough and get the parameters right, I wouldn't actually be surprised to find out that it is there somewhere, so this could definitely be done, at least conceptually. It wouldn't prove anything, though--not anything to do with God and not anything assuring it is there only once.

The second guarantee would follow from finding out that the decimal digits of pi are truly random, and if they are, then the guarantee would extend to seeing the "signature" infinitely many times if we looked far enough (via the Infinite Monkey Theorem). I'll repeat: on the condition that the decimal digits of pi are truly random, the "signature" must appear infinitely many times. That closes that door to it appearing only once. Incidentally, I do not know how we could prove--or if we could prove--that the decimal digits of pi are truly random. We cannot simply by examining what we know. Some kind of order could always lurk beyond what we have seen (though it cannot be the kind of simple order that defines rational numbers).
Now, suppose we observe the "signature," whether we think the digits are random or not, and we observe many, many decimal places beyond and do not see it again. What can we say? We've seen what seems an unlikely thing and seen it only once, but there are infinitely many--infinitely many--more decimal digits we have not yet observed where the "signature" may appear again. Years pass. Supercomputers supercompute. Many more digits are examined, billions or trillions more. No signature is found. What can we say? There are infinitely many--infinitely many--more decimal digits we have not yet observed where the "signature" may appear again. When does this state of affairs change? Never.
The thing is, such an occurrence is fantastically unlikely in random digits (base-11 or otherwise). Thus, we already should expect it to take a very long string of such digits in order to see it even once, making it feel uncanny if we do see it. We should expect, starting after the "signature," yet another fantastically long string before we'd see it again, and at times, if random, we'd go far longer than average stretches without seeing it. With only finite computational abilities, there's only so much that could be expected here. Note that even the extraterrestrials in Sagan's Contact cannot say for sure if it is a statistical anomaly, something Sagan did well with by having it appear in base-11 and so far out in the string.

Now, if we could prove mathematically that the "signature" only appears once in pi, which is probably not possible to prove, it would be a curiosity, but I don't think it would prove anything about the existence of God, or at least it would not be anything like clear evidence. Think about it for a moment devoid of the context of someone calling it "the signature of God." What on Earth would lead you to conclude that it is that? If it were to be taken as a "sign" of God, it's a very bad one. Pi isn't even approximated in the Bible to standards known to predate that era! That wouldn't stop faith-heads, of course--John Loftus's comment that "faith is a parasite on the mysterious" immediately comes to mind.

There could be far more uncanny situations that crop up, and far clearer ones as well. The "signature" could appear immediately instead of at some uncertain distance out into the number. It could encode "I am the Lord, thy God, and this is my Signature" just before or after using our modern binary encoding of letters (which an omniscient God would know we would develop). It would be far more uncanny, in fact, if the expansion of pi did the "signature" repeatedly with only enough space between them to ensure the number is irrational, though this is still quite unclear as an "evidential" sign. We should note that such numbers exist, and they're useless to the point of being utterly uninteresting in every other capacity. It would be striking, if nothing else, if pi itself were one of those numbers, but it isn't.


  1. "The second guarantee would follow from finding out that the decimal digits of pi are truly random, and if they are, then the guarantee would extend to seeing the "signature" infinitely many times if we looked far enough (via the Infinite Monkey Theorem)."

    This should then also guarantee that there's a string that translates to "Zeus is god", "There is no god", or any other possible string you wish to search for.

    1. Indeed, along with "Oy! Apologists! Get a new hobby!" ;)

  2. Trying to decide whether the digits of pi are random is a little strange. Suppose I put 200 pennies down in sequence on a table. Can you decide whether I flipped each coin, or whether I deliberately placed them, while ensuring the appearance of randomness? If it's the second one, and I do a reasonably good job, then you can't. If the coins are discovered mortared into a temple wall (immutable like pi) then the question may become more important, but the answer is just as ill-defined.

    1. Statistically speaking, since most people are indeed very bad at it, usually we can tell with high confidence whether you flipped the pennies or not (or, as it turns out, made up the social security number you put on a document). Indeed, this is a little fun experiment we used to do in a class that I taught--some of the class was assigned to flip a coin 100 times and report the results and the rest were to make it up. I wasn't to know how the assignment went, and then by rather simple tests that took me very little time, I could tell with what worked out to better than 95% accuracy who was in which group. Again, this was with a *very* simple test that took literally only minutes per page to make a determination.

      Indeed, if you did a *very* good job of it, I wouldn't be able to tell, at least not with such a small sample and without much more sophisticated tools. Analyzing the digits of pi suffers neither problem--indeed, the lack of shortness of a sample is one of the key difficulties involved since, after whatever point, we don't know what comes next at least without some very crafty proof.

      Ultimately it's beside the point, though. It's merely a conditional: *if* the decimal digits of pi are random, then yadda, yadda. I think many people would find it a fascinating curiosity if it were determined that they are, indeed, truly random (and to the first few billion decimal places, they appear to be quite random).

      Perhaps it is strange to wonder about, but is it not more strange and fun that here we have a simple question with a simple true or false return that we have no idea how to answer and may not be able to answer?

    2. It is strange and fun.
      I'd quibble with one thing: " and without much more sophisticated tools. " Surely at some point, as the quality of the random-emulation improves, you reach a point where it's just too vague to call, no matter what tools you have? Even something obvious like 12 coins the same in a row - 11 copies of the previous - is a 1 in 2048 chance at a given point but I've got 189 places to do that so it should happen in almost 10% of random trials, too large to dismiss.
      I guess what you'd be looking for is predictors that 'hit' throughout the whole sequence, leading to a cumulatively tiny likelihood of randomness. So it would be harder still to detect where a group of 20, say, are placed deliberately via an undisclosed rule, and the rest are truly random - which may be more relevant to the pi question, And the real issue - as I see it - is that yes, you find special patterns and are uncertain about them- how much of the overall uncertainty is due to one's openness to detect patterns to begin with? You can interpret an interesting detected pattern on the premise it's significant - are you then tempted to assign a likelihood - perhaps a very high one - that the pattern you found in fact really did have that particular significance, thus affecting via Bayes analysis the likelihood that it appeared other than by chance? This is a big part of the numerologist's trap, of course. But if you don't do that, then no pattern you find is any different from any other pattern so the whole question is a non-starter.

      One way this would be convincing: if there was a predictor found which reliably (or even better-than-chance-reliably) predicts all digits of pi - thus showing it to be in fact non-random -- up to the 10^20 digit, and then it fails ... and you find that pattern there ... and then the predictor works after that.
      (in fact, the test of the pattern being there only once: the predictor "the god signature does not occur" technically qualifies here, but seems much too weak and confirmation-biasy, and impractical to fully test)

    3. I'm not sure if you realize a few things, since you're picking this nit. First of all, it's fairly immaterial to the topic at hand. Second, this is a well-developed field of mathematics that I'm certainly not expert in. A quick google search indicated for me that there's a copious amount of what appears to be good literature available about detecting randomness and nonrandomness.

      In the event that your example took place, the questions remain, though: is it a signature or a coincidence? what is it a signature of? God? Why? And what would lead any rational person to conclude that? And why would a benevolent deity use such an obscure sign (seems more capricious than anything)?

    4. "And why would a benevolent deity use such an obscure sign?"

      ...well because, it's all part of his nature for us to "know and discover him", right? If he just came out and appeared on international news media outlets for the whole world to see, it would ruin the "chase" to discover him, eh?

      Love your blog by the way, thanks


    5. The point to think is a god´s signature are not just in the sequence itself but in how soon it appeared! Statisticaly it´s known that somewhere you must find that sequence, what is odd is that´s appear completely improbably soon.

    6. It's not known that any particular sequence must show up in pi. That's not proven yet, though it's probably the case.

      "Soon" is a funny word here. It's so funny, in fact, that it doesn't really mean anything. It wouldn't constitute a "signature of God."


      This image does not really explain the simple message found in pi but it is there in the first 13 decimal places. I do not know if this will reapeat but it happened in the first bit. "If God be E=MCC ie 8." Or If God be E=MCC "that is" infinite.