(urth) Sigh

[Adam Thornton]

On 01/11/2011 11:19 PM, Jeff Wilson wrote:
> On 1/11/2011 10:12 PM, Adam Thornton wrote:
>> On 01/11/2011 09:04 PM, Jeff Wilson wrote:
>>> Borges' Library? Do you mean the Aleph? How is either one incorporated
>>> into the context of the Sun cycle?
>> Ultan's library occupies some space between (in a mythical sense)
>> Borges' Library of Babel and Borges' National Library.
>
> That's fine, but we need a literal antecedent for the 
> see-around-corners thing to be a natural feature of space-time.
> na
Does the supposition in "The Library of Babel" that, if you were 
immortal and went in any direction long enough, you'd eventually 
encounter the same books in the same disorder, which would thereby 
become The Order, count?

I've been doing a little meditative combinatorics.  I arbitrarily 
assigned the length of the printing on a book's spine (which the 
Librarian does not tell us) to be one 80-character line.  With that 
assumption (and ignoring the Librarian's unsupported supposition that 
the titles are meaningless), we find that there are approximately 
1.3383768226039 x 10 ^ 1834209 books in the Library.  (I've used the 
freely available "hypercalc" program, 
http://mrob.com/pub/perl/hypercalc.txt, in this meditation.)

But of course the supposition that you'd come, when you began seeing 
books you had already encountered, upon the same Library in the same 
order, is delusionally optimistic.  There are, of course, the factorial 
of the number of books number of orders of each book in the library.  
That turns out to be roughly 10 ^ ( 2.4548624015714 x 10 ^ 1834215 ) 
libraries before one must, perforce, begin seeing the books in an order 
in which one has before seen them.  It is *this* collection, of course, 
that must contain the wonders the Librarian speaks of, such as the true 
catalogue and all the false catalogues, because a single book is clearly 
far too small to contain such a thing.

But of course that, in turn, assumes that all the copies of the library 
are complete.  If we then postulate that any iteration of the library 
may or may not have a copy of a particular book then the number of 
(complete and partial) libraries our immortal traveler must traipse 
through before encountering repetition becomes the power set of the 
number of complete libraries, or 2 ^ ( 10 ^ ( 2.4548624015714 x 10 ^ 
1834215 ) ) .  This, rightly, seems like a big number.

But, in fact, to an arbitrarily good approximation, *all* integers are 
bigger than that.

Indeed, if we adopt even a naive encoding for the books of the story, 
"01" for the first character through "25" for the last, and simply 
endeavor to transcribe the 
set-of-all-books-in-all-libraries-in-each-possible-order as an integer 
encoded this way...then there is probability zero that this number does 
not appear somewhere in the decimal representation of pi.  So all our 
wandering is for naught: we could just sit crosslegged and contemplate, 
very closely, a ring.

Adam