The Unimaginable Mathematics of Borges' Library of Babel

by William Goldbloom Bloch

The Library of Babel is arguably Jorge Luis Borges' best known story--memorialized along with Borges on an Argentine postage stamp. Now, in The Unimaginable Mathematics of Borges' Library of Babel, William Goldbloom Bloch takes readers on a fascinating tour of the mathematical ideas hidden
within one of the classic works of modern literature.

Written in the vein of Douglas R. Hofstadter's Pulitzer Prize-winning Gödel, Escher, Bach, this original and imaginative book sheds light on one of Borges' most complex, richly layered works. Bloch begins each chapter with a mathematical idea--combinatorics, topology, geometry, information
theory--followed by examples and illustrations that put flesh on the theoretical bones. In this way, he provides many fascinating insights into Borges' Library. He explains, for instance, a straightforward way to calculate how many books are in the Library--an easily notated but literally
unimaginable number--and also shows that, if each book were the size of a grain of sand, the entire universe could only hold a fraction of the books in the Library. Indeed, if each book were the size of a proton, our universe would still not be big enough to hold anywhere near all the books.

Given Borges' well-known affection for mathematics, this exploration of the story through the eyes of a humanistic mathematician makes a unique and important contribution to the body of Borgesian criticism. Bloch not only illuminates one of the great short stories of modern literature but also
exposes the reader--including those more inclined to the literary world--to many intriguing and entrancing mathematical ideas.

Genres: Mathematics, Nonfiction, Science, Philosophy, Literary Criticism, Literature, Criticism

224 pages, Hardcover

First published August 25, 2008

About the author

Professor of Mathematics
Wheaton College

Reviews

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[Brixton · Rating 3 out of 5]

*2.5/5

As a book of mathematics, this is an absolutely wonderful book! Very fun—very easy to follow for aspiring mathematicians of all calibers! Five out of five!

As a book on the literature of Borges, this is a terrible book. Incredibly reductive in its approach to the importance of mathematics in Borges' work. Very easily a one out of five.

This book was not so much an exploration of the intersection between mathematics and literature as it was a sort of fun excuse to explore some wonderfully complex mathematics in a digestible way. If you are solely interested in learning some basic number theory or think you'd like to explore higher dimensional topology and geometry, but are afraid because you might not have any math background, this is a book for you! Bloch does a wonderful job laying out complex ideas in a very approachable way. He rarely makes any assumptions at all about what his readers might know about math, and he moves from there. If you fall into this category, please by all means, pick up this book.

However, that being said, I'm not sure Borges himself would necessarily find this book entirely applicable to his Library of Babel. This book fails to explore the sheer mystery and mysticism of mathematics which Borges so clearly wished to bring to the forefront with his fiction. A sensitive approach to Borges' mathematical literature is one which explores our many interpretations of infinities, which investigates the arbitrariness of our often too unwavering mathematical axioms, our seemingly inbred fascination with the notion of number and the like. A sensitive approach to Borges' mathematical literature means asking what unimaginable mathematics means for our psychology and our humanity, like any other sort of literature. Borges' mathematics cannot and must not get bogged down with questions of the 3-torus and incomprehensibly large integers. As someone who enjoys math for math's sake, this was still a very pleasant read, but I felt as though I was betraying Borges' intentions as I read.

Read this book. I will recommend this book. This is what imaginative mathematics can and should look like, and this is by all means a great gateway into it. But please, if you pick up this book, read it with a grain of salt, and do not allow this to inform your understanding of Borges' infatuation with mathematics.

[Jonathan · Rating 5 out of 5]

This book is *awesome*! Why? The author takes a short story, "The Library of Babel," by Jorge Luis Borges and analyzes it along many different mathematical dimensions. The results are stimulating and accessible to non-mathematicians. The story is about a library, which is composed of:

An indefinite number of ... hexagonal galleries. In the center of each gallery is a ventilation shaft, bounded by a low railing. ... Twenty bookshelves ... line four of the hexagon's six sides... One of the hexagon's free sides opens onto a narrow sort of vestibule, which in turn opens to another gallery, identical to the first -- identical in fact to all.

What's the implication of this? The author, William Goldbloom Bloch, takes us through combinatorics, information theory, real analysis, topology and cosmology (a tough chapter), geometry and graph theory, and more combinatorics. He explains things in an easy to understand way, and then if you want more, has "Math Aftermath" sections that get more in depth.

In the end, he gives a long list of suggested readings. I have started a list here http://amzn.com/w/236UZ2PNNR3W2. What math books do you all suggest?

[Kane Faucher · Rating 4 out of 5]

The author does a very good job of explaining some of the mathematics that are implied from Borges' short story, and thus uses this as a generative point. The section on the manifold may be a bit confusing for many (although the author does give his frequent "advisories"). Perhaps a little too much time is spent in the beginning making qualifying comments / apologies when they may not be necessary. In my view, there were a few other avenues the author could have explored, but that is hardly considered a failing.

[Christopher · Rating 4 out of 5]

Of all the short stories in Argentinian writer Jorge Luis Borges' masterpiece collection FICCIONES, "The Library of Babel" is one of the most peculiar. This weary narration by an aged caretaker of a library of seemingly infinite expanse involves several exotic mathematical principles, yet ones fairly easily graspable by the layman. The mathematician Bloch has written a fine book about all the thought-provoking concepts in Borges' story.

The complete text of "The Library of Babel" is included here, so if you like the intersection of maths and literature, you have all you need here to explore Borges' vision. Still, I'd recommend neophytes read this story first in FICCIONES, as there you'll also find some other enjoyable and influential short stories.

Each chapter discusses the relevant concepts in accessible prose, followed by a "Math Aftermath" for those who want to see rigorous figures and calculations. First we have combinatorics, namely how to calculate the number of possible books in the library. Bloch A remarkable conclusion is drawn, perhaps unrealized by Borges himself. If the library contained every possible book, even if only a single copy of each, then its contents would still be exponentially too large to fit in our universe. The second chapter concerns information theory, namely the (im)possibility of creating a catalogue for the Library.

In Chapter 3, Bloch discusses real analysis, with the springboard being Borges' footnote that instead of an infinite library, one could conceive of a single book of infinitely thin pages. A trip through non-standard analysis reveals a complication that Borges evidently didn't realize.

The fourth chapter discusses topology. The idea of the Library as a Pascal sphere is well-known to Borges fans, but Bloch also describes how a 4-dimensional sphere could meet Borges' description of an infinite but periodic universe. This is the most challenging of all the chapters, especially the Math Aftermath which talks about klein bottles and the like. You'll find this chapter much easier if you've read Edwin Abbott's FLATLAND.

Chapter 5, devoted to Geometry and Graph Theory, examines the honeycomb layout of the Library and possible paths through it, presenting multiple possible interpretations of Borges' text that have quite different ramifications for the inhabitants. The following chapter introduces more combinatorics to ponder how the disorder of the Library might be the Grand Order.

So as you can see, Borges' little story, that many people have no doubt read, thought "How cute", and moved on straightaway, touches on an immense amount of mathematical concepts. The final chapter is dedicated to informed speculation on just how much of the mathematical ramifications of the text Borges was conscious of.

My maths skills have seriously atrophied since I left school, but this was a friendly, approachable text, a catalyst for the all too rare utterance "Who knew maths could be fun!"

My only complaint is that Bloch occasionally goes off on flights of fancy that depart far from Borges' work, when a discussion rooted in the text is already more than enough to satisfy or overwhelm the layman. Also, there is a chapter dedicated to critics that he doesn't like, where he suggests that people stop looking at the text from certain literary criticism perspectives instead of venerating its mathematics.

[Celebrilomiel · Rating 3 out of 5]

Fascinating, accessible, and thought-provoking.

In the first-person-singular sections that bookend the first-person-plural mathematical chapters, Bloch's prose had such a friendly and humorous tone that I liked him right away and went from merely intellectually open to fully invested and eager to see what he had to say on approaching Borges' short story through the lens of mathematics. He was like a best friend who delights you with his presence as well as his turn of phrase and makes you instantly comfortable.

It seems that many people have a habit of skipping the front- and back-matter of a book, but I make a point of reading cover to cover, and this book proved why I do so: I enjoyed the acknowledgements more than any other I've ever read, and I'm glad I read the glossary, because he slipped in several sly quips. The nine central chapters on mathematics (and, later, literature analysis) did not carry the tongue-in-cheek style that the enclosing chapters did, since they were focused on the math and consequently were more straightforward, but his wit peeped through in a phrase here and a wry allusion there.

The chapters on mathematics were at times mind-boggling, as eponymously promised, but they were always understandable. Personally, I found the chapters on combinatorics, information theory, and graph theory the most interesting. The chapter on the topology of the library was intriguing, but the fourth dimension was a bit hard to wrap my mind around: it all made sense, the way he explained it, but trying to imagine it was… Well. You try to imagine the fourth dimension as physical space. The notion of the mirror-reversal in the 3–Klein bottle Library was fascinating, though — it made me want to write fiction with recursive multidimensional travel through mirrors.

Once I've studied higher mathematics and have an increased technical appreciation for the concepts propounded within this book, I think I shall come back to it and enjoy it all over again.

[Parker · Rating 4 out of 5]

A clear and straightforward explanation (for laypeople) of the complex and high-level math implicated in Borges' classic short story, "The Library of Babel." Though the story is just a few pages long--a fabulous translation, much better than the one I had previously read, is included in the beginning of this text--it raises questions from all different fields of mathematics, relating to the vastly huge number of books, the possible shape of the Library, and the textual combinations that might appear therein. It's a nice and deep look into details that might be overlooked but deserve to be examined. In a later chapter, too, he provides some personal insights, which were really great: the comparison between Borges' librarians and the Turing machine is inspired and wholly correct, and I never would have thought of it.

In addition to lucidly treating concepts that might otherwise baffle, Bloch has clearly done a tremendous amount of research in preparation, and it shows. He addressing what seems to be the entirety of the criticism that's been written, and he writes of his experiences researching in Argentina. On top of it all, he provides an e-mail address at the end of the book, promising a more tailored reading list for those interested in the story!

"The Library of Babel" is really a beautiful work of fiction and a provocative thought experiment, and Bloch has given it here the treatment it merits.

[to'c · Rating 5 out of 5]

One would have to write like Borges himself to praise this book adequately. I only grant five stars to books that expand my mind in some great and meaningful way and Mr. Bloch certainly succeeded in doing that. In many ways Borges' "Library of Babel" is one of his simplest and most accessible stories. The Universe as a gigantic library. Yet it is full of too much depth to be contained in that very library. Mr. Block now informs us that there is far more depth to be found than you may have imagined. And he does so in a charming and very accessible manner. From simple combinatorics to Turing machines, each turn of each page yielded even more surprises. This is one of those books that causes me to dig deeper and follow some of the leads it contains. Even tho' I've finished reading the book it will be with me for a long, long time.

Interest in either Borges or math is not required.

[Craig · Rating 3 out of 5]

Quite accessible, but it jumps from concept to concept without really digging deep into anything.

[Jim Razinha · Rating 4 out of 5]

I like math, and I’ve learned in the past fifty years that I usually like math writings. There are exceptions, of course*. I’m squaring the circle here with this one. In a book group, someone recommended Michael Graziano’s The Divine Farce (and I don’t, now that I’ve read it) for reasons I don’t remember. Many commenters said that the novella, A Short Stay in Hell by Stephen Peck which is based on Borges’ Library short story, was better (it was, but then it had to be). I shelved them both TBR and then I came across this book, which looked to be an interesting analysis/hyperextension of the Library. So, I did read both the Farce, and the Short Stay, along with the Library, and then this. How Mr. Bloch wrote 200+ pages on a 8-10 page short story, I had no idea, but now I know.

There are fun musings here, backed up with equally fun maths. *And there are some that I am not fond of, topology being a big one. I never got into the subject as much as Martin Gardner in Mathematical Games columns, and I still don’t. But that does t take away from what Bloch has done here.

Infinity, infinite series, infinitesimal elements, geometry of Euclidean and non-Euclidean natures, an aficionado should find something to like here.

On notes, in his Preface, Bloch says
“As a reader, when I encounter an endnote, I’m compelled almost against my will to flip to the back of the book to learn what the endnote says.”
{The note says, “Did you look?}

I always look.

He continues,

“As I writer, I find that despite my best efforts to incorporate them into the body of the book, my work includes diverting digressions, fine points of mathematics that might interest only specialists, and citations to other works. All of these are consigned to the endnotes.”

As a reader, I value those almost as much as the text.

I liked this, but perversely, I found the most value in the epigrams Bloch chose. And now I have even more on my TBR List.

[Yumeko (blushes) · Rating 4 out of 5]

To be fair my interest started dying out around page 100 in this 160 page book. In my defence , I never really cared much for Borges' book.

Regardless, I would think twice before discounting the brilliance of this book.
He points out sentences of interest from the story, develops mathematical concepts and constructs the formulae, all done very accessibly, leading to conclusions that were staggering to say least, also recognizing multiple interpretations of the text.
To tease, one was:
"The librarian’s life and the Library together embody a Turing machine, running an unimaginable program whose output can only be interpreted by a godlike external observer"
And
"The Library is its own catalogue. Any other catalogue is unthinkable."
And it's just, grand as fuck man. The guy even made a small murder mystery to prove a point.
The concept of mathematics in literary analysis is quite new to me, and this was a wonderful introduction to it.
Every chapter starts with quotes, the cover is pretty, and I quite like how the author talks, there's no reason not to like it🥀.

[Olivia Case · Rating 5 out of 5]

Pacing: ⭐️⭐️⭐️⭐️⭐️
Plot: ⭐️⭐️
Style: ⭐️⭐️⭐️⭐️⭐️
Setting: ⭐️⭐️⭐️⭐️⭐️
Character: ⭐️⭐️
Theme: ⭐️⭐️⭐️⭐️⭐️

[Therese Arkenberg · Rating 5 out of 5]

Written by a talented creative writer, Borges fan, and mathematician--and any one of those on its own could be a full recommendation--this book delves into the mind-blowing implications of a short story that is already staggering in its implications of scale.
If you have the slightest interest in mathematics, Borges, or glimpsing infinity, I'd suggest you track this one down through a library system at once. I don't think there's another book like it in the world. It's such an unusual that I can't in good conscience run about accosting strangers and shouting "BUY IT!!", but it would look excellent on a shelf in your office, den, or waiting room.

Cross-posted from my longer review at Story Addict.

[Luke Duncan · Rating 4 out of 5]

I picked up this book while working my way through Borges Labyrinths because 1) I've really been enjoying Borges and 2) this books was cited as being as compelling as GEB.

While it didn't quite live up to the GEB plug, it was pretty darn good. As someone with a kinda of mathy background in a technical field, I thought the first few chapters were a little weak but a nice warm up. I especially liked the chapter on Topology, which had the most in depth coverage.

I learned a few new things, got a good book recommendation from the topology chapters references, and had fun with an easy read over a couple of days. I also found myself thinking "man I need to give this to a younger me! I'd totally have gone into math instead."

[Gill · Rating 4 out of 5]

“Bloch is not only a mathematician, he's also a critic who has reviewed a lot of the literature on the Library of Babel and the life of Borges. So there is a lot in the book besides working out some of the mathematical implications of Borges' inspirations.

If you got as far as calculus in your math studies then you can probably follow most of the math without too much trouble. If you are a fan of Borges, there is a lot here about his math background and interests that you probably didn't know and which affected other works of his too.

If you are neither, pass this by. Go read a lot of Borges and if you like him, then come back and read this.

[Joegjoeg · Rating 4 out of 5]

Great book about the maths behind trying to make a library of every book that could ever be written. My favorite part of the book is in the preface when the authors asks "who is the intended audience for this work" and answers his question "Of the more than 6 billion people who are not Umberto Eco, I imagine that those who'd find this work appealing are..."

[Jafar · Rating 3 out of 5]

The Library of Babel is the best example of how Borges brilliantly weaved mathematical ideas into his fiction. This book explores some of the mathematical implications of this ingenious short story. Bloch has a great eye for spotting hidden idea in the story and then expanding upon them. The best was using the Library as a universal Turing machine.

[Joseph · Rating 4 out of 5]

Pretty neat little book. Uses Borges' Library of Babel as a springboard for introducing some interesting and relevant mathematical concepts in a more or less intuitive way. Though there's definitely some parts that require more concentration, this should be more or less accessible to anybody who's passed high school.

[Luke Crawford · Rating 5 out of 5]

Really quite good. Math for literature types, sort of. Explained to me, among other things, what a manifold is.

"The librarian’s life and the Library together embody a Turing machine, running an unimaginable program whose output can only be interpreted by a godlike external observer. A user. A reader."

[Eoin · Rating 4 out of 5]

Ok, so I'm an easy mark for this book. I was concerned that this would be insufficiently mathy or literary or both, but it is a careful balance of readablity and high concept math. What the title promises, the book delivers.

[Nick Douglas · Rating 5 out of 5]

Answered several questions I had about the math implied by Borges's story, and introduced many other fun implications and counterfactuals. Almost entirely accessible without a post-high-school math background. And clever without getting too cute.

[Stephen · Rating 5 out of 5]

Fantastic, non-trivial analysis of Borges' story, comes complete with topology, combinatorics out the wazoo, with just a hint of non-linear dynamics.

[Jon Gauthier · Rating 4 out of 5]

Very accessible; written in a lively and exciting tone that I wish were more common in this genre!

[Pierre_menard · Rating 4 out of 5]

Great hybrid of a literary math book. Wish there had been more math! It also doesn't discuss Quine's short essay on the story which would have been a good addition to the information theory chapter. But it's clear the author gets Borges and it's a great eye-opening companion to the story.

[Steve Gross · Rating 3 out of 5]

You will only enjoy this if you are a fan of both Borges and mathematics.

[James · Rating 5 out of 5]

For those who want to dive deeper into the implications of Borges' famous library

[Braden Thompson · Rating 3 out of 5]

Can you believe i didnt pick this up on my own volition

[Sevak · Rating 3 out of 5]

There are certainly interesting points in this book, but some parts can get really boring for people who know basic math.

[William Schram · Rating 4 out of 5]

This book applies ideas in pure mathematics to Jorge Luis Borges' short story, The Library of Babel. Given the description of the Library and the things mentioned in the story, we can come to a number of surprising conclusions about the Library itself. For one thing, it would not be possible to contain the Library in our Universe. The topology of the Library is also something of a sticking point for some since it is supposed to be infinite. Anyway, I thought this book was a charming introduction to the mathematics involved and to the works of Jorge Luis Borges.

[Austin Savill · Rating 4 out of 5]

This book is well written and shows many aspects of infinity as well as explaining much of what infinity means and how we can view or believe some things to be infinite. It explains the passage very well and analyzes many of the arguments of how Borgues conveys this library. I enjoyed it thoroughly but don't think many others would unless they enjoy math or explanations of philosophical concepts.