In Luis Borges' "Library of Babel," the Universe
takes on the form of an immense library filled with books. The volumes
contain the sum total of all writings that could exist given 22 letters,
the space, the comma and the period. Each book consists of 410 pages
with 40 lines of 80 characters each on each page. The inhabitants of
the library have spent their existence attempting to decipher the mystery
of the universe and have arrived at the conclusion that every conceivable
work is represented without duplication. Having arrived at this conclusion,
generations spent their fruitless existence in search of that Text of
Vindication written especially for them.
Indeed Borges' Universe falls short of infinite inasmuch
as there is a precise and finite number of volumes which would meet
the criteria of containing all that could be written, without duplicating
any volume. None of Borges' beings saw fit to mention that figure or
even its magnitude in his essay (which was of course present in it's
entirety on one of the library shelves). In fact the exact figure was
also contained in a book or set of volumes although it was spelled out
long hand, i.e., "two thousand fifty six" instead of "2056."
Of course had someone found it they would have very likely confused
it with any of millions of other texts that spelled out magnificently
long numbers in near endless variety.
As to the question of the size of the Universe, it is
a relatively simple matter to calculate the number of volumes needed.
Before embarking on this exercise, perhaps it would be wise to answer
a few simple questions that may have already presented themselves to
the reader.
What about those works much shorter than 410 pages?
They are there. In fact, they are present more than once. Each exists
as a single book, the final pages consisting of a pattern of repeating
spaces filling out the remainder of the volume. And of course, each
short story is reproduced in numerous anthologies of collected stories
and in the context of other works as well. Thus we find "The Gift
of the Magi" among a collection of O'Henry as well as planted amidst
the adventures of Sherlock Holmes, between chapters 7 and 8 of Genesis
and once again as part of tale recited by Kubla Khan. Finally each short
story resides as the only intelligible passage, contained in the center
of the text and preceded and followed by absolute gibberish.
But what of the long-winded epics or of Will and Ariel
Durrant's "History of Civilization?" All present and accounted
for in multi-volume works. No guarantee that the texts would be adjacent
to one another on the shelves; but then, no one ever did decipher the
filing system (Dewey Decimal from Hell).
What about sentences that end in a question mark or
exclamation point or that need a quotation mark or an apostrophe? Well,
we could add these, each one increases the size of our library by several
sextillion fold; but then we'd admit that they aren't essential to communication
as they are readily inferred from context. And since the original concept
came from an Argentine author, we find ourselves limited to only 22
actual letters. Fortunately the variants of Chinese texts appear as
a translation or the thousands of characters would have a sizeable impact
and we'd find ourselves digging out the cinderblocks and planks for
a little extra capacity.
So, we accept the completeness of Borges' Universe;
but how big is it? With the three non-letters and 22 letters, we have
25 symbols to work with. The 410 pages of 40 lines of 80 characters
of text produce 1,312,000 locations for each of the 25 symbols. Thus
it is easy to calculate the number of volumes as 25^1,312,000.
This is indeed the exact figure but we round it off as we restate it
in the more familiar form: 1.959x10^1,834,097. This is considerably
more than a "google," 10^100, but very considerably less than
a "googleplex," 10^google.
What we need is a means of "getting at" something
this size. It vastly exceeds the number of atoms in the Milky Way; no
contest in fact. For that matter, it exceeds the number of electrons
in the known universe. But these are both trifling figures in comparison.
No we need to think of something truly grand.
Perhaps we can attempt to reconstruct the creation of
this Universe and let our perception of time guide us to some comprehension
of this number. We'll need a clock. Probably one that runs very slow.
It would be too easy to simply create one book per minute and have the
"minute" hand circle our imaginary clock for 1.959x10^1,834,097
minutes. No, we need a slow clock we can think about.
In addition to our "slow" clock, we need a
fast typist. In fact, the Supreme Typist has been clocked at a rate
of one book per microsecond; that's one million books per second!
Our clock'll need a large "face" so let's
start at the center of a sun just as it casts its first rays of light
into space. We'll hop aboard that first photon and ride it for a thousand
years. It passed the orbit of "Earth," 93 million miles away,
after a mere 8.3 minutes. It passed interstellar distances in less than
five years but we said we needed a BIG face so we ride on for another
995 years hurtling outward at 186,000 miles per second. When our millenium
is at last up (we'll let the Supreme Typist work while we build our
clock) there are 3.156x10^16 books completed (but we'll never
find them among all those empty shelves).
Well, this seems a suitable radius, one thousand light
years from our origin. We'll scribe a circle about the tiny sun in the
distance. Though our circumference is quite respectable we need more,
more immensity, so let's give our "clock face" some depth.
Instead of a simple, flat clock, we'll create a large water glass with
a depth ten times the diameter of the lip and fill it with water (far
more than we have in our own piddling universe).
Now we'll place a tiny snail on the lip of the glass
with instructions to crawl around the entire circumference and when
he returns to his starting point, he can stop and have a little drink.
The snail sets off at a snail's pace covering only a single meter each
hour. Worlds come into existence and expire at a frantic pace while
the snail covers less than one second of arc. Since the lip is 5.94x10^20
meters long, each full revolution takes a corresponding number of hours.
As the snail completes his first lap, the Supreme Typist has put another
2.14x10^30 books on the shelf (we still can't find them;
spread too thin).
After this first trip it becomes apparent that the snail
is a lot faster than we originally thought and he seems to be holding
up reasonably well (obviously did a good job of pacing himself) so we'll
change the rules and ask him to make another 9,999 trips before letting
him have that drink. As he wheels in after his 10,000th circuit, he
pauses and sips from the edge of the cup. We need to conserve water
as our shelves don't seem to be filling up very fast, so a single tiny
drop (about 1/100th of a teaspoon) is all we can afford.
Another 10,000 trips around and another drop of water.
So it goes and goes and goes. The water level doesn't SEEM to be dropping
very fast but then again the book shelves aren't filling up either.
After a time of truly cosmic period, 10^67 years, the water level has
dropped but a single inch. Still the shelves remain empty.
The millenia drag by. It now occurs to us that the
water is dropping way TOO fast, for after another 10^88 years
there remains only a small puddle (relatively speaking). Our Supreme
Typist, realizing that she has barely begun her task, casually gets
up from her typewriter, stretches, and refills the glass, flushing the
snail up to the top where he dutifully climbs onto the lip and begins
the arduous task of once again draining the tumbler.
By the time he drains it a second time, the Supreme
Typist has doubled the number of volumes but the total number is only
6.3x10^101. Ten more times the glass is refilled but alas
the number of volumes remains a paltry 3.2x10^102 and the
shelves look as empty as they did when we began. In fact, so few books
have been completed that the first 409 pages of every book are identical
as are 39 of the 40 lines on the last page. Only the very last line
is differentiated in the volumes that have been produced over the last
zillions of years at a rate of one million per second. The Supreme Typist,
intent on her task, now completely ignores the snail which spends the
next several eons draining the little puddle in the bottom of the tumbler
where he waits and eventually dies of dehydration.
With all this we are still far short of expressing
the enormity of Borges' Universe. I utterly despair of ever communicating
the magnitude of 10^1,834,097, (I figure any of you can grasp
the 1.959 part). I've been fast, one million books per second; I've
been long, one thousand light years; I've been voluminous, 31 billion
cubic light years; I've been slow, one meter per hour; I've been cruel,
"make that ten thousand laps before you get a drink of water;"
I've been stingy, "only a drop now, save some for later;"
and I've been tricky, "here let's fill that glass up." Still,
I haven't even come close, not even close to coming close. A quantitative
approach is not in the offing. Let's instead peruse this magnificent
library and try to feel of its immensity by grasping the surprise of
finding un-thought-of volumes.
We've noted that it contains every short story, essay,
treatise, epic, novel and novella. It also contains every textbook:
on subjects familiar - 4th grade geography, Spanish grammar, spelling
and biology; as well as some subjects of which 20th century Earth is
not yet acquainted - symonetics, Claussian geometry and basic quantum
cell-set theory. Oh sure, the math texts are a pain to read, what with
every number written out long-hand, but they are still there, every
one we've ever seen and imagined as well as the greater number which
we have not.
We have every historical account from the broad summaries
used in college courses, to the million volume detailed history of the
period from 1000-3000 A.D. But then again, if you really want detail,
we have the complete, verbatim transcript of every word, spoken by every
man, woman and child living in Germany during World War II. Of course,
we have that same record for every country and every period of civilization
but then it gets a little tiresome. More interesting is the complete
set containing the words which these people WISH they'd said but weren't
clever enough to have thought of until long after the occasion passed.
Moving to the "literature section" (wherever
that is), we find of course the complete works of Tolkien, Vonnegut,
deMaupassant, Asimov, Grey, London, and Borges, not to mention the future
works of Axataxes, Cho, Orowitz and Hisa-Fontina.
We are pleased but not surprised to find Hemingway's
"Moby Dick." We are however surprised to find on nearby shelves
the 1,312,000 editions each with a single typographical error and the
over 1.7 trillion unique copies which contain two and only two typos.
More interesting is the novel which faithfully refers throughout to
the great whale "Moby Duck." The version which we finally
check out tells the tale of the hunter and seaman, Captain A. Happenin,
and his nemesis, Moby Richard.
As if we needed any more convincing of the library's
completeness or scope we could peruse our own history as well as our
own personal future in any degree of excruciating detail we choose,
including what our first words will be when we awaken on July 5, 2016,
as well as what we probably should have said then, the day before, the
day before that and two or three decades earlier for that matter.
No, it's all here, every bit of it, every single solitary
bit of it, the past, the present (this text) and the future, the correct
future, the future in which our death is erroneously predicted as one
all too critical minute later than it actually occurs and the future
that might of been had we chosen differently. Nothing is missing.
So we can't grasp 10^1,834,097. Still we
ought to shudder at the secrets it contains.