I like Keith’s idea of a God whose messages are the strings of digits that DON’T appear in pi. That way the messages would only be read by really smart beings and not by dorks with big computers. Jim Propp On Wednesday, December 5, 2018, Keith F. Lynch <kfl@keithlynch.net> wrote:

James Propp <jamespropp@gmail.com> wrote:

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I can imagine a God who intervenes in ways that locally violate the laws of physics, or a God who chooses the laws of physics, but not a God who chooses the laws of mathematics. Is there anything that this might mean?

I don't know. I can imagine a finite being who can appear to choose the laws of physics, since the laws of physics we know might just be special cases of more general laws, with the special cases selectable by advanced technologies. But what then is left for an omniscient omnipotent Being to do that no finite being can do? Nothing?

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Hiding a message in the digits of the base ten expansion of pi is no different from hiding a message in the digits of the base ten expansion of seventeen.

What about in base eleven, the base mentioned in the novel? :-)

There isn't much room for messages in seventeen. There's plenty of room in pi, or any other irrational number.

Note that there's no claim that pi was *changed*, or that it might be different in different universes. To me, the implication is that it always had the value with the messages in it, in all possible universes. It's a transcendental number, not just in the mathematical sense, but also in the spiritual sense of the word. The Author's desire for those messages to exist is the cause of those messages existing, but it's not cause and effect in *time*.

Similarly, there's a powerful sense in which timelike cause and effect work within the primes. Integers are or aren't primes *because* of which smaller numbers are primes, as if the positive integers had come into existence one at a time, starting with the smallest. But that's obviously not cause and effect in time, as there was never a time when not all positive integers were yet prime or composite (or unit).

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(?Mathematicians will tell you that it?s all zeros after the decimal point, but how far out how they really checked??)

On the off chance that you're not joking, they're all zeros (or all nines, depending on how you write it) after the decimal point in seventeen and all other integers, not by calculation, but by definition. You might as well try to find patterns in defined physical constants such as the speed of light, which is a specific integer number of meters per second by definition. Currently, the value of the physical constant mu0, the permeability of free space, is a specific defined transcendental number. Perhaps it would be more constructive to search it for messages. But you'd better hurry, since on May 20th of next year it will change from a defined physical constant to a measurable physical constant. At the same time, the reduced Planck constant, h-bar, will change from a measurable physical constant to a defined physical constant with a transcendental value, so we can start searching it for messages.

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I know that the mathematician and science-fiction novelist Greg Egan plays thought experiments with the mutability of math, but I always get the sense with him that at least a fraction of his tongue is in his cheek when he does this, ....

I'm not convinced of that. He likes to take ideas seriously and see where they lead. For instance in _Quarantine_, he takes seriously the idea that it's observation that collapses the quantum state. When our astronomical instruments get good enough, we start collapsing distant quantum states that aliens need to remain non-collapsed, so the aliens block our view of everything beyond our solar system in self-defense. Biologists then work on removing the state-collapsing part of the human brain, resulting in people who, while otherwise perfectly normal, see Schrodinger's cat as both alive and dead when they open the box.

In "Dark Integers," Egan has people and aliens communicate with each other by reading and writing messages in mathematical constants.

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Trying to convey something beyond human comprehension is a tricky business; it?s akin to designing good technobabble, but harder.

Indeed. Much harder.

"David Wilson" <davidwwilson@comcast.net> wrote:

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Presuming pi is normal in all bases, aren't all possible (finite) messages somewhere in there?

Yes. Do a web search for "Do not calculate pi in binary" and you will find hundreds of copies of one of my most popular jokes.

The claim in Sagan's novel is that interesting patterns were very close to the beginning. For instance a section that, in base eleven, had a square number of consecutive digits that were all 0 or 1, such that when arranged in a geometric square showed an image of a circle.

Dan Asimov <dasimov@earthlink.net> wrote:

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Now I'm curious how *probable* it is, or complementarily, how probable it is that at least one finite string is missing from an infinite string of digits chosen at random.

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When doing calculations like this I tend to get stuck when trying to account for substrings that overlap. At least for finite calculations.

I'd think it's obvious that the probability of a finite string not appearing approaches zero as the random string within which you're searching for it grows longer. So if the string you're searching is infinite, the probability must be zero. (Which doesn't mean it can't happen, merely that it's infinitely unlikely. Like flipping a fair coin infinitely many times and having it come up heads every time. Or like choosing a point at random inside a circle and finding that you've chosen the center.)

Perhaps the omniscient omnipotent Being, knowing, of course, exactly what Sagan would write, arranged for the message Sagan described to appear *nowhere* in pi, which is much less likely than it appearing somewhere near the beginning. :-)

Allan Wechsler <acwacw@gmail.com> wrote:

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The miracle is of an extreme enough order that Divine intervention is the only "plausible" explanation.

It could also be an astonishing coincidence, or a hacker messing with the computer you did the calculation on.

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But the whole episode is a casual throwaway.

To me, it's by far the most interesting thing in the novel.

James Propp <jamespropp@gmail.com> wrote:

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Though I suppose there could be universes harboring intelligent life in which the laws of physics are such that pi isn't seen as a very interesting number until fairly late in a culture's development, and facts about pi are seen as arcane and boring.

Pi appears in the value of the Riemann zeta function at every even positive integer argument, and the Riemann zeta function is interesting because it involves primes. Also, pi appears in the natural log of -1. So pi would certainly be considered an important mathematical constant even in a world without geometry (e.g. a world that consists only of texts and their interactions).

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By the way, have any of you read the R. A. Lafferty short story in which some people discover some small integers (less than ten) that had hitherto gone unnoticed?

I don't recall reading it, but I'm fascinated by crackpot theories, and enjoy attempting to debunk them. A very few of them turn out to be true (e.g. continental drift). I enjoy the lesser-known ones, such as the HAB theory and Fomenko's "new chronology." (Conspiracy theories, like zombie movies, tend to be boring because there are so many of them, they're so widely known, and they're so much alike.)

One reason I like math is that it's relatively easy to confirm claims for myself. For the past 40 years I've been a fairly radical skeptic, for reasons I won't get into here (details in one-on-one email by request), and I try to confirm as much as possible directly. For instance tasting oceans to make sure they really are salty (they are), or watching cockroaches to see whether they lay eggs or give live birth. (They give live birth, which is *not* what books say happens.)

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