1 Introduction

1.1 The Library of Babel as a self-contained backup memory

THE LIBRARY, as Borges described, has many books.¹

Each book has 410 pages, each containing 40 lines, each containing 80 characters. Therefore, each book contains:

\[ 410 \frac{\text{pages}}{\text{book}} \times 40 \frac{\text{lines}}{\text{page}} \times 80 \frac{\text{characters}}{\text{line}} = 1312000 \frac{\text{characters}}{\text{book}} \]{#eq:charsperbook}

It’s easy to see that lots of things can be said with this many characters. The Gutenberg Project edition of «Alice’s Adventures in Wonderland» is only about 160-thousand characters, and that includes extraneous text (like Project Gutenberg’s own license, indices, etc.). The numbers closely match rules of thumb for printed text, so it’s reasonable to think of a single tome of THE LIBRARY as being approximately as long as a 410-pages book in regular type.²

In Borges’ research paper / fictional account, there are people who destroy books left and right, but the narrator isn’t fazed: after all, if the axioms about THE LIBRARY³ are true,⁴ one must remember two things:⁵

One: The Library is so enormous that all reduction of human origin is infinitesimal. Another: each tome is unique, irreplaceable, but (since the Library is total) there are always several hundreds of thousands of imperfect facsimiles: of works that differ by no more than a single letter or a comma.

Just how many imperfect copies are there, given the working theory of THE LIBRARY?

Let’s imagine any one book that will be called the Original. It’s possible that the last character of the last line of the last page is a period. A «typo» is, then, having any other character in that particular place. There are, then 24 «incorrect» characters for this particular place and therefore, 24 books that differ from the Original only in the last character.

This argument can be extended to and and all of the other 1,311,999 «places» in the book. In other words, there are 24 «incorrect» options for each and every one of the 1,312,000 characters in the book. So, there are 24 almost-perfect books that differ only on the first character, 24 more that differ only on the second, 24 more for the third… Then, the total number of books that differ from the Original in one and only one character, is:⁶

\[ 24^{1312000} \approx 1.4096... \times 10^{1810837} \approx 10^{10^{6.257}} = T_1 \]{#eq:almostperfect}

Let’s call this number \(T_1\), because it will indicate how many books have exactly 1 typo (compared to the Original). This mighty number, written complete, is about 1,810,000 digits long! In other words, imagine a book from THE LIBRARY that contains only the number 9 repeated in each and every one of its pages.⁷ That mighty number is still way, way smaller than the number of nearly-perfect copies of the Original. How much smaller? Roughly speaking, adding a single digit to a number generally means it grows by a single order of magnitude. The number of copies is 500 thousand orders of magnitude larger than the largest number that can fit inside a book from THE LIBRARY. In comparison, the known universe’s diameter⁸ is larger than a Planck unit of length⁹ by only 50 orders of magnitude, give or take.

But wait! This calculation is inspired by the fact that there are nearly perfect copies of the Original, and whose mistakes are so small as to make them trivial. One typo in a 410-page book is certainly not enough to make it unreadable.

What about two typos? Arguably, having two typos in a book is also not enough to render the book indistinguishable from the Original.¹⁰ How many two-typo books are there? What’s the value of \(T_2\)?

For every 1-typo book, there will be \(T_1\) books that differ from it in only one character, but only one of them will be the Original. Then, the number of 2-typo books will be:

\[ T_2 = T_1 \times (T_1 - 1) = T_1^2 - T_1 \]{#eq:2typos}

Without even reaching for a calculator, I can say that in the above expression, the \(T_1^2\) dominates. And how much! It’s approximately \(1.98 \times 10^{3621674}\). Subtracting \(T_1\) is nothing. Given these magnitudes, it may be easier to remember that \(T_2 \approx T_1^2\)

The number of 3-typo books is again calculated by reasoning that it comes from making almost all changes to a 2-typo book, except for the one that would «reduce» it to a 1-typo book. There are \(T_1 - 1\) such changes, therefore…

\[ T_3 = (T_1 - 1) \times T_2 = (T_1 - 1) \times (T_1^2 - T_1) = T_1 \left(T_1 - 1\right)^2 \]{#eq:3typos}

Again, the number with the largest exponent dominates all others (\(T_1^3\) in this case). The exact value for an \(n\)-typo book can be expressed like so:

\[ T_n = (T_1 - 1) \times T_{n-1} = T_1 \left(T_1 - 1\right)^{n - 1} \]{#eq:ntypos}

We can see that the pattern holds: in general the number of books with \(n\) typos can be approximated by \(T_1^n\) and so it grows exponentially. Five typos scattered regularly in a 410-page book are hardly enough to render a book unreadable—or unrecognizable—but the total number of books with only 5 typos easily dwarfs the number of all information humanity has ever produced.¹¹

How many typos are needed to render a text completely unreadable? That question will remain open for now, because a complete answer will require refinement of ideas, concepts and tools beyond the scope of this small fragment. It’s obvious, for instance, that if I were to insert a typo in 50% of all characters in «Hamlet» it would most likely become unreadable, but just using that number and calling it a day seems cheap. For once, it’s not constructive and for twice, it’s completely arbitrary. At 0% typos the Original is readable and at 100% typos it’s unrecognizable. It stands to reason that the «turning point» must be somewhere in between, but this is not a continuous problem, since there’s no way to have, say half a typo. Even defining what that «turning point» is will prove to be very much non-trivial. Therefore, the question of how many typos are needed to render a text unrecognizable from the Original will be left open for future curious minds.

1.2 The Library of Babel as repository of tangential truths

But, just how many are true typos?

There exists a book in THE LIBRARY that is a perfect copy of «THE TRAGEDY OF HAMLET, PRINCE OF DENMARK» followed by nothing but blank characters.¹² There exists a book that is almost the same except for the fact that every occurrence of the string HAMLET_¹³ has instead the string OPHELIA. The play reads a bit weird, but it’s understandable nonetheless. Does this book qualify as a typo-riddled version of «Hamlet» or is it a new book in and of itself? I have no good answer.

However, this small exercise speaks to us of another kind of books existing in THE LIBRARY: books¹⁴ that are very related to other books through similar concepts. The book I mentioned in the above paragraph is an example of such: a book discussing the tragedy of Ophelia. But there’s more to it.

These books might discuss things that are false, or have never been imagined by any human, ever, but are hinted at from our existing books in this Earth. For instance, there’s the Hundred Years’ War, but there are books on the Ninety-nine Years’ War, the Ninety-eight Years’ War… None of those conflicts have actually happened on this Earth¹⁵ but they are written down in a book somewhere in THE LIBRARY in quite the detail.

Another example: the Clay Institute of Mathematics has offered a prize for whoever proves or disproves the Hodge conjecture, but nothing is said of the Aaron conjecture, the Aaronson Conjecture, the Smith Conjecture, the Jones Conjecture or the Williams Conjecture. Maybe THE LIBRARY even contains proof of all of them.

These I will call Tangential Truths: the ideas that don’t exist, but are hinted at through texts and ideas that do exist on this Earth. The Tangential Truths are those that are revealed to us not directly,¹⁶ but only through preexisting texts.

The Tangential Truths are not the same as truths that are yet unknown, like a mathematical proof or scientific result. The difference is that the latter are susceptible to be found through other means, and not as a literary epiphany from the Platonic Ideal that is THE LIBRARY.¹⁷ The Tangential Truths are what come to mind from imagination, trying to substitute a single word in a title and thinking what that could be. A single, non-ambiguous definition of Tangential Truths will be impossible beyond mere demonstration.

1.3 The Book of Books

This Book of Books is merely one in myriads, but it contains a few of these Tangential Truths. The Book of Books is not mine, for I cannot claim to be a Creator that large. Instead, I’m merely a prophet, a head that rearranges characters and words and puts down in paper what he finds. Others might be there, doing the same.

This Book of Books is one of many Books of Books, all describing books of Tangential Truths that are in THE LIBRARY. There exists a book that correctly catalogues all of these Books of Books, but it’s impossible for us to even understand what it is, for it would require an even greater Epiphany. It’s difficult to even comprehend the Tangential Truths, comprehending the Tangential Truths derived from they would require a demiurgic act beyond the abilities of mortal men and women.

Through abstraction of Truths and their Tangential Truths, one becomes closer and closer to the Creator Himself. I don’t know if such a thing should even be attempted. After all, before THE LIBRARY there was a THE TOWER, and It was doomed to failure by decree from Above. THE LIBRARY exists and alone should suffice, lest we once again reach what should not be reached..

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