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

A mathmatician's view of part of the conversation between Richard Dawkins and Peter Boghossian

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.
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