Patterns in pi, copyright law, and philosophy
- From: shepherdmoon@xxxxxxxxx
- Date: Sun, 25 Jan 2009 16:14:21 -0800 (PST)
Hello,
The subject of the post is absurdly grandiose, of course. But I found
this interesting link while playing around with searching for number
patterns in pi:
http://www.dublish.com/articles/31.html
I saw three general topics discussed a lot. And, although I did not
read all of them in their entirety, I also think a fourth topic is of
interest.
The best thing about the link is the comments section, which has some
fascinating back-and-forth discussion about the following topics:
1. Whether the offset of a string found in pi can be used as a form on
compression.
2. Whether the fact that a given string is found in pi negates
copyright law.
3. Whether pi itself can appear "in" pi.
4. (my own musings, not necessarily in the comments): What are the
implications for ideas of free will and creativity if it can be
demonstrated that any human writing, which can be defined as a finite
string of encoded numbers, already exists in pi?
I am not trained in math, so please excuse (or correct) any mistakes
on my part. But I noticed that the comment section above has what I
think are some pretty convincing answers to the above questions.
Specifically,
1. Sounds like this can be done trivially but is not in any way
practical. For example, this comment:
----
Anonymous Coward @ 2006-05-25 22:51:37
Finding long continuous sequences in PI that exactly match the byte
values in, say, a 6 MB MP3 is intractable. In fact, for anything but
trivial sequences (such as your INFO example), the problem is
intractable.
You could provide a bunch of offsets, however, say one for every
longword in the data. But this definitely isn't compression; the bits
representing the offsets will very likely be as large, if not larger,
than the original data.
You are essentially describing a one-time pad, except that it won't
work for encryption purposes because everyone knows the codebook.
----
2. I think not. Most of the comments at this link that respond in the
negative gave answers I found more convincing than those who answered
in the affirmative. For the negative:
----
Dan @ 2006-05-27 05:40:21
First of all, copyrighted materials are not 'plaigarized from Pi'
because any given number of the size of, say, a 250kb picture has
certainly not been discovered yet. So, unless you can prove that the
creators of the material had the knowledge that the paticular string
of digits in question already existed within Pi, they have done
nothing wrong.
Furthermore, suppose I took all of the bits of an mp3 and mixed them
up so that the file could no longer be used to hear the song. If you
downloaded this new file, would you be infringing on a copyright? No,
because the owner of the song does not lose any money by you having it
- you still have to purchase a CD if you actually want to listen to
the song. It is only when you acquire the information to de-scramble
the file does the copyright holder lose business. (If this were not
true, then most novels wouldn't be legal - I'm sure you often can
rearrange the letters of one novel to make another, albeit shorter,
copyrighted story).
By this logic, simply downloading Pi would not be illegal - only when
you acquire the offset(s) would you be breaking the law. And judging
by how difficult it is even to find a 1000 digit number in Pi, I
suspect that most offsets would be extremely large. This means that it
would not be reasonable to argue that you stumbled upon that
information by chance - you had to have knowingly and proactively
searched for it. Thus, you had acted upon an intent to acquire
copyrighted material illegally, and could be prosecuted.
----
3. It sounds like pi itself cannot appear "in" pi - aside from the
degenerate case one wry commenter presented ("at offset 0"), but that
doesn't mean that a given string of finite length cannot be found. I
found this comment to be really good at pointing out why a previous
comment was wrong on this and other points.
I also think that it doesn't make sense to claim that one cannot
copyright works because their encoded versions exist in pi. After all,
until one has the entire work in hand, one doesn't know the starting
offset of the pi-encoded version. Getting the work "in hand" - that
is, writing it or stealing it - is the whole point, after which
finding it in pi - even if it takes an astounding amount of time to
find it - is relatively beside the point.
Also, I wonder if one can simply take the length of the given work,
then calculate how many digits of pi one would need to raise the
probability of finding the string to something reasonable (> 0.5). All
you can say is, yes, chances are that given a search of these n digits
of pi, you are more likely than not to find the work. But that does
nothing to negate the hard work of, say, Shakespeare, of kindly
preparing the actual string that you need to feed to the search
function.
----
truth machine @ 2006-05-27 13:05:29
"We would need a very very very long binary expansion of pi and some
kick-ass interface for searching for a file"
No, we would only need to be able calculate it -- and, as is noted
above, the hexadecimal expansion of pi has the advantage of being
calculable from any starting point, so no significant amount of
processing power is required. OTOH, downloading the start and end
positions of an arbitrary item would generally require longer than the
life of the universe and more storage than there is matter.
"This thought pops up with astounding regularity on the net. Without
fail, the same logical error is made every time"
The thought pops up frequently because there's a nearly inexhaustible
supply of sloppy thinkers -- and the same logical error is made every
time because anyone recognizing the error immediately dismisses the
thought.
"An infinite series of numbers is not an exhaustive set of numbers.
Another way to say that is just because there are an infinite number
of digits in Pi does not mean that any particular combination of
numbers must exist. You might be able to find your phone number or
social security number quite easily, but good luck finding any series
of numbers that represents a program, music, or anything else. Not
that it is impossible, it is just very very improbable."
Too bad you have no idea what you're talking about. The digits of pi
aren't just "an infinite series of numbers", they are the decimal
expansion of a transcendental number. While it's not proven that every
finite string of digits occurs within the decimal expansion of pi, it
is very very likely, not "very very improbable", that any specific
finite string does appear. The vast majority of mathematicians
believes this to be the case, as there is absolutely no reason to
expect otherwise, and it is known that the first 30 million digits of
pi are very uniformly distributed.
"To further explain why Pi is not an exhaustive set, let's assume that
all number sequences exist in Pi. That statement means that I should
be able to find a series of 100 fives, or 1000 fives, or a billion
billion billion fives, and even an infinite number of fives. Any
number of fives (or any other number) is a valid series of numbers, so
it would have to be found if our assumption were true."
This does not "further explain why Pi is not an exhaustive set" -- it
is to be expected that all of those different sequences occurs at some
point in the *infinite* string of digits of the decimal expansion of
pi. If pi contains all strings of digits, then they all occur; if it
does not, then one or more may not occur. Rather than "explain why Pi
is not an exhaustive set", you have offered a simple case of petitio
principii.
To see why it is almost certainly true that all these sequences occur,
imagine breaking up the decimal expansion of pi into blocks of a
billion billion billion digits. There 10^27 different such blocks, but
an infinity of such blocks in the expansion, and we have no reason to
expect any of these blocks to occur any more than any other; we
certainly don't have any reason to expect any of them to never occur
-- and that's the case for the block of all 5's. So not only should we
expect such a sequence to occur, but we should expect it to occur
infinitely many times.
"It also would mean that I should be able to find all the numbers for
e (Euler's number) as well. Since e is a non-repeating, non-
terminating number like Pi, its number sequence cannot be found within
Pi."
Um, do you know of any images, music, etc. of infinite length? e and
other infinite sequences are irrelevant. But we should expect any
finite subsequence of the decimal expansion of e to occur within the
decimal expansion of pi. Certainly you have offered no reason not to
expect that, and there is every reason to expect that, as the digits
of pi show no statistical patterns and there is no known reason why
there would be any.
----
4. I don't think there are serious implications for creativity if any
given finite string can be found in pi. Given what I think about item
3 above, I still think the whole point is knowing what string to look
for, which presupposes someone's having independently created or
"discovered" it. Regarding free will, however, I'm less certain. It
would seem strange if it has been proven that any given finite string
will appear in pi given enough digits, and it does seem a lot more
uncanny than arguing that the keys of a keyboard can be worked into
any written work. It's the consecutive nature of the pi strings that
seems unsettling. But perhaps there already is a philosophical answer
to this question?
Thanks for any feedback or advice on the above.
Shepherdmoon
.
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