Different Kinds of Infinity

On Aug 2, 10:26 pm, "R. Baldwin" <res0k...@xxxxxxxxxxxxxxxxxxxx>
wrote:

What I said, and pay close
attention this time, is that if the algorithm does happen to work in
that it does actually make correct predictions over time, the
hypothesis that it is in fact the correct algorithm and that it has
started at the correct position [in a normal number sequence]
gains predictive value with each
successful prediction.

And, this is wrong. Remember, the context here is normal numbers. As I
showed in the proof that you snipped below, even if your algorithm appears
to find a match to a segment of the normal number, the conditional
probability that it really is a match is effectively zero. No matter how
long a finite test you allow for your algorithm, there are infinitely many
other substrings within the normal number that match that finite sequence.

That's true, but you forget that there are different kinds of
infinity. Some infinities are actually "bigger" than other
infinities. As it turns out, the infinity of "correct" sequence
predictions decreases exponentially relative to the increase in the
infinity of non-correct possibilities with each increase in the size
of the finite portion of the normal sequence. Therefore, the odds of
actually landing on one of the correct infinities drop exponentially
with each successful prediction.

It is like continually hitting an exponentially shrinking target that
approaches zero in the limit with each successful prediction or throw
of the dart. Very quickly the prediction algorithm gains so much
predictive power that it is essentially equivalent to the predictive
power of having picked both the perfect algorithm *and* the perfect
starting point. There is no essential statistical difference in the
limit.

< snip >

If my formula for pi happened to successfully predict a few million
digits in an incoming sequence, without error, would you actually call
this result "trivial" as a "prediction algorithm"? You might think
this result impossible for all practical purposes, but if such a
result were actually realized, it would be far from "trivial". I dare
say you would be all by yourself with this conclusion of yours if SETI
scientists actually did find such a match in a radiosignal proved to
be coming from outer space.

Yes, I would call it trivial.

Then you hold a very different position compared to various statements
of SETI scientists. I mean really, what would you say to the
following comment by Seth Shostak, a senior astronomer at the SETI
Institute:
"Perhaps the extraterrestrials will preface their message with a
string of prime numbers, or maybe the first fifty terms of the ever-
popular Fibonacci series. Well, there's no doubt that such tags would
convey intelligence."
- - Seth Shostak, How to Sort Signs of Artificial Life from the Real
Thing, SETI Thursday, Space.com, January 30, 2003
In short, it seems very much like you would be forced to call any such
signal, to include a million repeats of the first 100 digits of pi,
"trivial" and "worthless". Even if such a signal came from a
terrestrial source you wouldn't be able to detect it as a likely
biased signal because of your argument that it could have been the
product of any random origin in pi or any portion of any other normal
number - to include all random normal numbers.

Assuming the conjecture about irrational
numbers holds, those several million digits may be matched within pi at an
infinite number of places. They may be matched within e at an infinite
number of places. They may be matched within any other irrational number at
an infinite number of places. They may be matched within any real number at
an infinite number of places. Having no idea of the alien broadcast's
starting point, and no idea of their numeral system, and no idea how they
map signal modulation to their numeral system, there isn't much point in
getting excited about several million digits that I can match to just about
anything (and which I can't see arriving on a short snapshot of a narrow
band extrasolar signal, anyway).

You can't match several million digits of a randomly produced binary
sequence to just about anything. That's the whole point. Even though
there is a theoretical infinity to the possible number of different
matches to an infinite number of normal numbers, the infinity of a
non- match is much much more infinite. Therein lies the strength of
scientific induction.

On the other hands, if I found a short number of decimal digits matching the
starting sequence of pi lying around in a human computer network, I would
conclude it is an artifact simply because we have so many copies of short
starting sequences of pi in our data processing environment, and very few
copies of intermediate substrings of normal numbers, and most of them are
either in our conventional decimal or binary numeral systems, which we know
humans use, and all that makes for good odds.

So, basically, you can detect humans because you know something about
humans, but it would be impossible for you to detect an alien signal
because you don't know anything else about aliens? Try explaining that
to SETI scientists. I'm afraid you are quite alone on this one . . .

Beyond this, you only think you know something about humans because
you have experienced a certain pattern in humans that has held true
for you in the past. This doesn't mean that it will always hold true
in the future. In fact, your history with humans can itself be
represented by a string. You've associated a certain pattern in your
finite experience with a different pattern in your finite experience
is all you've done. You've associated the "human pattern" with a
tendency to produce a certain favored portion of the "pi pattern".
This is, in essence, no different from having a longer finite string
with more repetitions of the first finite digits of pi or in
repetitive association with another "tag sequence". Regardless
though, all such patterns are repeated an infinite number of times in
the infinity of pi or any other infinite "normal" number.

The same thing is true about a roulette wheel. You assume you know
how a roulette wheel works, but this is just an assumption for any
particular roulette wheel in action. You just assume that the
particular wheel matches your previous experience or "string" when it
comes to similar patterns called "roulette wheels". The fact of the
matter is that any particular roulette wheel may have a completely
different mechanical system controlling it of which you are unaware.
It could be controlled by deliberate design for each play for all you
know. Yet, based just on the sequence of numbers generated, many have
made millions of dollars betting against the House's assumption of a
normal distribution and in favor of a particular biased non-normal
distribution.

It is just too bad that, by your own logic, you can't conclude human
generated bias or alien bias given a radiosignal with the first 100
digits of pi or the Fibonacci series repeated a few million times. I
guess you'd just be sittin on your thumbs if such a signal were ever
found? - especially if it were proved conclusively to be coming from
outer space (like from Mars, or a galaxy far far away)?

Sean Pitman
www.DetectingDesign.com

.

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