James Propp <jamespropp(a)gmail.com> wrote:
> 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?
> 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).
> (?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.
> 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.
> 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(a)comcast.net> wrote:
> 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(a)earthlink.net> wrote:
> 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.
> 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(a)gmail.com> wrote:
> 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.
> But the whole episode is a casual throwaway.
To me, it's by far the most interesting thing in the novel.
James Propp <jamespropp(a)gmail.com> wrote:
> 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).
> 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.)
