The Unimaginable Mathematics of Borges' Library of Babel
William Goldbloom Bloch
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"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.
log(n), that we looked at 43 44 S u n i m a g i n a b l e m at h e m at i c s earlier, and indeed, after multiplication by a constant, they are the same function.) We are interested in knowing approximately the number of primes expressible in 100 digits, so we compute (10100 ) for a good estimate: 10100 ≈ 10100 1098 1098 10100 = = ≈ ≈ 1097 . 100 · ln (10) ln (10) 2.3 ln 10100 three Real Analysis The Book of Sand To see a World in a Grain of Sand And a Heaven in a Wild Flower Hold
of number, which is inﬁnitely small and yet not equal to zero . . . The real line is a subset of the hyperreal line; that is, each real number belongs to the set of hyperreal numbers. Surrounding each real number r , we introduce a collection of hyperreal numbers inﬁnitely close to r . The hyperreal numbers inﬁnitely close to zero are called inﬁnitesimals. The reciprocals of nonzero inﬁnitesimals are inﬁnite hyperreal numbers. The collection of all hyperreal numbers satisﬁes the same algebraic
Extending this reasoning, for each and every hexagon on our ﬂoor, the hexagons above and below it are exact clones. That means that the labyrinthine paths we ran are precisely the same above and below—the stairs and shafts dictate this. Each ﬂoor plan is inevitably, invariably, precisely the same as every other ﬂoor plan. There is no advantage gained by taking a set of stairs up or down. 99 100 S u n i m a g i n a b l e m at h e m at i c s figure 54. The spiral staircases and the air shaft
agenda—what else could explain such an egregious omission? Blockhead’s analysis triumphantly concludes with the unique insight that this interesting homme de lettres employs these strategies to avoid the literary trap of being read as merely a colonist writing in and about a colony. It’s conceivable that some of Blockhead’s remarks may be of independent interest, and some may even apply to Borges—after all, there are some real political and historic parallels between Argentina/Spain and South
problems of analysis into geometric form. Unhappily, our senses can not carry us very far, and they desert us when we wish to soar beyond the classical three dimensions. Does this mean, beyond the restricted domain wherein they seem to wish to imprison us, we should rely only on pure analysis and that all geometry of more than three dimensions is vain and objectless? [. . . ] We may also make an analysis situs of more than three dimensions. The importance of analysis situs is enormous and can not