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.