Imagine that I have a simple, black and white dot matrix display, with
a fixed resolution, attached to a computer.  The display can show an
A4 page of text or monochrome bitmaps.

I write a loop program that displays all possible combinations of
pixel, starting with a totally white page and ending with a totally
black one - a very large, but finite, number of iterations. Each
iteration of the loop shows a page on the display.

Within this set of pages should be every page of text that it is
possible to write, including the complete works of Shakespeare, my
biography from now until the day that I die, this question and all of
it's answers.  There should also be bitmaps of the faces of everyone
who has ever lived and will ever live and streetmaps of every town.

Does this prove that human knowledge is finite?

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Clarification of Question by blakkandekka-ga on 14 Dec 2005 15:37 PST

I've been thinking about this a little too since the question occurred
to me, so let's simplify it a little.

Instead of a big display let's imagine one of those one line LCD
alphanumeric displays that you see on cash registers or pagers.  Say a
10x5 grid and 60 characters.  That's 3000 dots which gives us 2^3000
combinations.  The program should give us every 60 character line that
it's possible to write using the latin alphabet (and probably quite a
bit of Kanji).  Thus do we have every line from every book that it's
possible to write in 2^3000 combinations?  Restricting the set to just
those combinations displaying latin characters should reduce the
numbers still further. A book of usable knowledge is unlikely to
repeat lines so the good stuff will likely be a smaller subset again.

A mathematical proof comprehendable by a numerical ignoramus such as
myself would be great if anybody can come up with one.

Phil

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Clarification of Question by blakkandekka-ga on 20 Dec 2005 05:02 PST

A further clarification.

Thanks for the enlightening comments everyone.

It?s true that in effect all communicable human knowledge may be
reducible to a set consisting of the digits 0 and 1 (information
theory experts may be able to help me out here).  That set will be an
integer number of digits and, as we all know, there are an infinite
number of integers.  More interestingly subsets of the infinite number
of integers are also infinite ? even and odd integers for example each
represent half the total possible set of integers but are both still
infinite.

Thus any subset of the integer total of binary digits can be infinite
but does not *have* to be.  What I?m after is whether the total of
linguistically expressible human knowledge is one of the finite
subsets or not.  I'm interested in the subset of that infinite stream
of zeros and ones that's *worth bothering* with.

To put it into the context of the Borges story (which was the thing
that first got me thinking about this) the Library of Babel is
infinite (I think I remember correctly) and contains all possible
books.  The question is whether all possible *readable* books are a
finite or infinite subset.  In the story there were only rumours of
readable pages in all of the billions of miles of volumes.

The computer monitor conundrum suggests that it?s finite.  I?m fairly
sure that that?s wrong, but I don?t know why.

Bear in mind when answering that it's possible to have a finite number
larger than the number of particles in the universe and that all human
knowledge is not containable within one puny brain.

The sun will burn out, the solar system disintergrate, and the
universe itself dissolve into mush long before a recognizable face or
page of text appears on the screen.

Does that add any useful perspective on your conundrum?

I suppose that if you accept the premise that all of human knowledge
can be reduced to material that can be displayed on your
fixed-resolution display, you've proven that human knowledge is
finite. But that seems to me like a mighty shaky premise.

Assuming a screen resolution of 1280 * 1024 which is fairly standard
today, you end up with a total of 1,310,720 pixels on your screen.
That means that you have a total of 2^1,310,720 different combinations
for your pixels to arrange themself into. To give you an idea of how
large that number is consider the following:

2^100,000 = 9.99 * 10^30,102. Just call it 1 * 10^30,103 for simplicity. 

Since 2^1,310,720 is approximately = (2^100,000)^13
                                   = (1 * 10^30,103)^13
                                   = 1 * 10^391,339

1 * 10^391,339 is a 1 followed by 391,339 zeros after it (and I
rounded down). It is impossible even imagine that number of images
without even displaying them.

For comparison, a googol (1*10^100) is larger than the number of atoms
in the universe. For a somewhat complicated explanation of this idea
go to http://van.hep.uiuc.edu/van/qa/section/Stuff_about_Space/The_Rest_of_the_Universe/982241844.htm

Still think that human knowledge is finite?

A very interesting question. Certainly all of recorded human knowledge
is finite, since there is a finite number of documents or images, each
of which can be represented in a finite number of digits. Let's ignore
the fact that there are analog artifacts, like photgraphic negatives
or analog tape recordings, since we can represent them in a finite
number of digits to any desired degree of accuracy.

The fact that the number of possible combinations on a video screen is
astronomical does not address the poster point, which is simply that
human knowledge is finite.

I would assert that the sum total of knowledge that has ever existed
in human consiousness is also finite, since there are a finite number
of neurons, a finite number of synapses, and for all practical
purposes, a finite number of strengths for each of those synpases, in
a finite number of brains that have ever been conscious. The
combinatorics here put the combinations of outputs from a video screen
utterly to shame, but the result is still finite.

You can attack my argument because of the analog nature of the
synapse, but I will still assert that the total information content of
a brain is finite. Various estimates place the storage capacity of the
brain between a few terabytes and 1000 terabytes. Not that large given
the fact that a terabyte of hard disk space is approaching $100 of
cost in the near future.

And eventually you would also get a black and white image of Marilyn
Monroe in the first Playboy.
You could stick to text and get to your goal faster, but your computer
and everything else will have burned up before you get much to read.

kottekoe-ga,

I think at some point 'astronomical' has to be considered infinite.
Also if you consider the fact that human knowledge continues to grow
at an astronomical rate, it is fair to say that it is infinite.

b&d-ga,

On top of everything else, I really don't see how your set-up creates
much in the way of human knowledge.

Yes, page 1 of Romeo and Juliet may be created by happenstance, and
page 2 created a million years later.  But there's nothing in your
system that would connect page 1 with page 2.

In other words, no Romeo and Juliet!  And not much in the way of
knowledge, methinks.

paf

I think that kottekoe-ga
" since there are a finite number of neurons, a finite number of synapses .."

underestimates capacity of human memory, which (some theories maintain) is
stored inside neuron nucleus - by molecular mechanism.

However, that is not main issue. Main problem is that you get lot of 
known and unknown facts, but also lot of nonsense:

You will get page saying that GWB was best president, worst president,
first president, second president ,...

 - as amount of human intelligible nonsense exceeds amount of true and
sensible statements (no pun here) - your collection of sense and
nonsense
will be useless.

So there.

also consider if I have an lcd screen that holds x amount of
characters 0-9 by combining different screens I can produce any
number, have I proven that the set of integers is finite... no, I have
however shown that any element in an infinite set can be produced.

For another perspective, the proton lifetime is thought to be around
10^34 years, but let's be generous and call it 10^50. And let's assume
your computer is a super-fast one that can display 10^50 images per
second (note that current computers do less than 10^10 operations per
second, and there are multiple operations for each image). Then after
10^51 years, 99.9% of the matter in the universe will have decayed,
and you will have displayed about 3 * 10^108 images. That's a long,
long way short of 10^391,339. By the time the last proton in the
universe decays, you will still have displayed less than 10^110
images. In terms of your display, you'll only get a chance to toggle
the first 366 pixels on the first line.

If the folks contributing to (and reading) this thread would probably
enjoy the short story "The Library of Babel", by Jorge Luis Borges. 
The full text (translated to English) of the story is available on the
web at <http://www.analitica.com/bitblioteca/jjborges/library_babel.asp>

Pafala-ga hit the nail on the head here.

even if the number of possible pages wasn't stupifyingly huge, as in
the case of the 60 character LCD display, it's the stringing together
of the 'pages' that represents the knowledge. 
"TWOHOUSEHOLDSBOTHALIKEINDIGNITYINFAIRVERONAWHEREWELAYOURSCENE" does
not represent the entire works of shakespere just as "TWO" does not
represent the above-mentioned line.  Your idea could be simplified
further by claiming that the letters A through Z can represent
anything in the latin language. Most human knowledge today is stored
as 1 and 0, it's the way the 1's and 0's are arranged that defines
them, though.

I had a similar idea a few years ago (odds of faces I know being
randomly generated on a monochrome 640x480 monitor) and I know it's a
fun concept, but reality (math in particular) has a way of shattering
these thoughts.  After realizing the odds against my mothers face
spontaneously generating itself I became quite interested in (as
jparaig-ga pointed out) the science of behemouth numbers.  The fact
that math can handle numbers larger than the number of atoms (or
electrons, or quarks) in the universe is incredible when you give it
some thought.  You may still enjoy the idea that human knowledge,
being capable of understanding ideas larger than the universe itself,
will never be able to measure itself.

B&D:

To respond to your second clarification: All possible readable books
are finite. To be readable, a book must be short enough to read in a
single person's lifetime. Thus it must contain a finite number of
bits. There are a finite (though ridiculously large) number of
combinations of this finite number of bits (2^(# of bits), only a
small fraction of which are readable (but still a ridiculously large
nubmer).

Some of the commenters made a good point that I neglected to point out
in my first comment. The information content of a random sequence of
video screens is exactly zero. In fact this is almost a defining
concept in information theory. No information is carried by a random
sequence of bits, even though any english sentence, or for that
matter, any book, is encoded somewhere within that sequence. My main
intention was to discuss the more interesting aspect of your question,
i.e. whether all of human knowledge is finite, which I still maintain
is true.

Your hypothetical scenario proves 1 thing: randomness CAN equal truth.
 By this, I mean that of all the random screens, every now and then,
one would appear that a human could recognize.  This is what I'm
referring to as 'truth'.  However, this would vary human to human.  We
have not all been exposed to the same things that live as memories in
our minds.

Consider your question again, with only a 2x2 pixel display.  Only 4
pixels.  They can be black or white, in any and all patterns.  There
are only 16 possible solutions, yet none of them will result in a
recognizable letter, face, image, or Shakespearean play.  This is
because the 'box' has been made too small.

As the box gets larger, to the point where it could display much
larger, recognizable things, the pool of recognizable results will
grow as well.  However... the box can always be larger.  This proves
that human knowledge is INFINITE.

With each different size of the 'box', the resolution changes, and
more variances in shading, etc. can be created.  However, at any time,
if the box was just a little bigger (maybe just 1 row and 1 column
bigger), that would create more possibilities that did not previously
exist in the smaller box, but still could have existed in the human
mind.

BOOYAH!  :)

It does not prove that knowledge is finite.  Since the image size is
finite, what it can even display is finite.  Since infinite things
exist, such as the universe or the number Pi, it would take an
infinite amount of time to know these things.  To display Pi, the
screen would infinitely need to change in order to display the
continuing number. As far as books go, it is possible to write a book
which never ends, with an infinite ending.

Further, the screen wouldn't even do a good job of helping someone
learn everything. Aside from the size limitations, it couldn't even
display faces properly.  Two faces could be identical except in areas
such as brightness, or color, though the screen couldn't show the
differences.  Nor could the screen teach some one the way a orange
sounds like when opened, the way it smells, or what it feels like, or
something else that is totally unknown.

I think I'm repeating thoughts already expressed here but, well, I can, so I will.

If the books in the library are of finite length or, in other words,
do not contain infinite subject matter, then the number of books must
be finite.

Does this mean that human knowledge is finite? No, it simply means
that the human knowledge that can be contained in such a storage
system must be finite.

If there are things which are infinite in quantity, infinite in size,
infinitely divisible etc. (basically anything mathematical) then one
can obviously create an infinite number of true statements about them.
Also, I can create an infinite number of (admittedly trivial) true
statements just by having each new statement refer to the previous
one.

And, even though the concern here was linguistically expressible
knowledge, let's not forget about non-linguistically expressible
knowledge.

I hope this wasn't totally useless.

Can a book describe what it's like to read itself?

gotcha!