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.

Friday, October 11, 2013

Dot, Dot, Dot is almost here!

Just passing along heads-up about a post on Jonathan MS Pearce's blog on his corner of Skeptic Ink where he details a bit about my upcoming Dot, Dot, Dot: Infinity Plus God Equals Folly (Link). For those uncomfortable that a book about mathematics might be difficult, let me quote him to ease your mind and pique your interest.
One book I am editing is in the final stages. The book is by James A. Lindsay and is called Dot, Dot, Dot: Infinity Plus God Equals Folly. I have managed to get Vic Stenger to write the foreword for it, too, which is great. The book is an easy-to-digest mathematical treatise on infinity and its application to the concept of God, which given the nature of infinity, becomes more and more abstract to the point of non-existence (looking also at ontological arguments). I am really enjoying the book, which has now largely been re-written to make a coherent narrative as opposed to its original form of unconnected essays. James has done a great job on making it a greatly readable and fascinating book.
Philosopher Peter Boghossian--author of the upcoming and potentially game-changing book A Manual for Creating Atheists (Amazon link)--had kind words about it as well, saying, 
A short and engaging read on the meeting of two huge ideas, infinity and God, that leaves us seeing both as abstract notions that may have nothing to do with reality. Honest and accessible, Dot, Dot, Dot is a great little book to stretch your thinking.
It's coming soon, probably by the end of the month at the latest, and it's very exciting. Keep your eyes and ears open for it and for upcoming news about it!