The Imaginative Mathematics Of Bloch’s Unimaginable Mathematics Of Borges’s Library Of Babel

Curtis Tuckey

n his charming and thought-provoking little book, The Unimaginable IMathematics of Borges’s Library of Babel, William Goldbloom Bloch, a Berkeley-trained mathematician and professor of mathematics at Wheaton College, in Massachusetts, has taken Jorge Luis Borges’s seven-page Kaf- kaesque fan­tasy about a vast library containing all possible books and put together a small volume of mathematical variations, improvisations, and extrapolations. He is delighted to share his musings with anyone who cares to follow along, and his enthusiasm is infectious. He says,

for those of mathematical bent who’ve not read Borges, I hope this vol- ume inspires two things: a desire to explore more of Borges’s work. . . and, equally, a desire to learn more about the math tools I employ. . . [T]here are three main themes woven into this book. The first one digs into the Library, peels back layers uncovering nifty ideas, and then runs with them for a while. The second thread is found mostly in the “Math Aftermath” sections [in which] I develop the mathematics behind the ideas to a great- er degree. . . . The third focus is on literary aspects of the story and Borges. . . . [M]y purpose is to make explicit a number of mathematical ideas inher- ent in the story. . . I am a tour guide through a labyrinth . . . Besides enhanc- ing the story, the reader’s reward will be an exposure to some intriguing and entrancing mathematical ideas. [xiii-xiv]

There is little focus on “literary aspects” of the story, but once Bloch’s mind latches onto a “nifty idea” he lets his mathematical imagination run freely. Nifty ideas include: the number of books in the Library of Babel

Variaciones Borges 30 » 2010 Curtis Tuckey 218 Awards for Professional and Scholarly Excellence (the PROSE Awards), and The Library ofB Melancholy (and unceasing), the librarians are forced to sleep standing up in tiny cu tiny in up standing sleep forcedto are librarians the unceasing), (and In “The Library of Babel” (1941), Borges described a mythical-religious­ a described Borges (1941), Babel” of Library “The In delighted by Bloch’s mathematical tour through the Borgesian labyrinth.Borgesian the through tour mathematical Bloch’s by delighted de­ filled galleries that have existed for all time. The ceilings of the galleries are on thechapters called “Information Theory” and“RealAnalysis.” a theme of Borges originally appeared in this journal (see “The Unimagi “The (see journal this in appearedoriginallyBorges of theme a and how large unimaginably it is; the possibility or impossibility of a cata alphabet—“all thatisabletobeexpressed, every in language.” the lettersof the of combinations possible all included are which among unreadable, but unsearchable,unclassifiable, and are which of all almost books, of collection enormous incomprehensibly an by surrounded are librarian,insufficient lightnormal is the a height of the hardlytallerthan mathemat the of some into furtherlooking worth Nonetheless,it think I Publishers American 2008 the for mathematics in mention honorable log of the collection; mathematical models of a book with infinitely many shafts furnish a constant source of mortal danger. Moreover, the librarians of properties surprising hexagonal galleries, and the for plans floor sible fantastical universe consisting of an impossibly vast honeycombbook- vastimpossiblyof an of universefantasticalconsisting tions of the twenty-threeletters.”on Indeed,variationsBloch’sthe several ofof tions ics behind Bloch’sconjectures behind conclusions,ics and emphasis particular with varia the contemplate shall you way this “In here:appropriate equally is nable”). paths through the labyrinth. Robert Burton’s exhortation in in exhortation Burton’s Robert labyrinth. the through paths pages;alternativeLibrary; universephysicalthe pos geometriesofthe for bicles, and the very low railings around the vast, bottomless,vast,ventilation the around railings low very the and bicles, servedly so.Reviewers,servedly professionalamateur,uniformlyhave and been The All gels, the faithful catalog of the Library, thousands and thousands of false of thousands and Library,thousands the of catalog gels,faithful the catalogs, the proof of the falsity of those false cata­ of the true catalog, the gnostic gospel of Basilides,commentarycatalog,uponof gospel the gnostictrue the the of that gospel, the commentary on the commentary upon that gospel, the gospel, that upon commentary the on commentary the gospel, that —the detailed history of the future, the autobiographies of the archan the ofautobiographies future, the the of history detailed —the nmgnbe ahmtc o Bre’ Lbay f Babel of Library Borges’s of Mathematics Unimaginable , quoted by Borges at the beginning of “The Library of Babel,” of Library “The of beginning the at Borges by quoted , abel logs, a proof of the falsity Anatomy of of Anatomy garnered ------(Gustav Fechner, Lewis Car Lewis Fechner, (Gustav The idea of a physical library of all possible books has a long history,but long a has books possible all of libraryphysical a of idea The uni­ ol b fud mn al osbe obntos f etr ad spaces and letters of combinations possible all among found be could me­ by created the calculating words (Llull, Pascal, Swift) to the idea that all possible works of literatureof possibleworks allwords (Llull,that idea Pascal,the to Swift) generatecouldchance(Leucippus,disorder fromorderwhether Democri each ream of paper were compressed to a thickness of one inch, the height f t icuin n lfo Fdmns eenal pplr mathematical popular perennially Fadiman’s Clifton in inclusion its of on 672folio pages (see Eco andKnuth) distanceexceed the wellforstackwouldthen-current estimates a such of encyclo an publishing of impracticality the discussed he Theremusic. of anthology, en­ an publish to age avail line and one-hundred lines to the page, bound into books one-thousand page,books the into to bound one-hundredlines and line seemingly unremarkable story, “The Universal Library” is perhaps more perhaps is Library” Universal “The story, unremarkable seemingly aiir o mrcn mathemati American to familiar the spacerequiredaccommodatethe to warehouses thesethe exhaustwould the to characters sixty printed wereletters twenty-three and three tween be­ of combinationspossible all if that determined PaulGuldin tronomer tus, Aristo­ tus, about arguments ancient from books possible all of library a of idea the if and sheet each onpermutationswere notes:musicalprinted two720 if in the seventeenth century, of methodical techniques for generating per generating for techniques methodical of century,seventeenth the in matician Marin Mersenne performed a similar calculation in in calculation similar a performed Mersenne Marin matician as­ and mathematician Swiss the 1622 In combinations. and mutations pedia of all 1,124,000,727,777,607,680,000 permutations of just twenty- just of permutations 1,124,000,727,777,607,680,000 all of pedia then side, a on feet 432 warehouses cubical in housed and long, pages between the earth and the stars. (On the other hand, Mersenne did man did Mersennehand, other the stars.(On the and earth the between verselle In his 1939 essay “The Total Library” Borges traced the evolution of evolution the traced Borges Library” Total“The essay 1939 his In ­able surface of the globe. In 1636 the Jesuit music theorist and mathe­ ­ interpola the story oftrue your death, the translation of every book every into language, havewrittenSaxon mythologyofthe the on not) did (but people, lost the books of Tacitus. ( , his monumental work on musical instruments and the theory the and ,instruments musicalmonumentalon work his tle, Cicero), through the conception that knowledge could be could knowledge that conception the through Cicero),tle, FantasiaMathemat ­ hncly obnn poete, oia pooiin, or propositions, logical properties, combining chanically size tions of every book in all books, the treatise that Bede could Bede that treatisebooks, the all in everybook of tions of such a collection had to wait for the develop­ the for wait to had collection a such of cyclopedia of all 40,320 permutations of eight notes,eight of permutations 40,320 all of cyclopedia Collected roll, Kurd Lasswitz, Theodor Wolff). BecauseWolff). Theodor Lasswitz, Kurd­roll, ica 7) (1958),Lasswitz’s relatively obscure and in ta Bre’ wrdfmu “Li world-famous Borges’s than ­cians Harmonie ment, - - - - - 219 The Imaginative Mathematics Curtis Tuckey 220 1,312,000 characters per book. Each character has twenty-five possibilities 10 Concern­ It is with the determination of this large number that William Goldbloom descrip­ can create using one-hundred characters (upper- characters one-hundred using create can characters printed on the spine, then each book will contain 80 characters wildernesses of books affirm, deny, and confuse everything like a deliri a like everything confuse and deny, affirm, books of wildernesses ah eaoa gley otis ie okhle o tit-w books of thirty-two bookshelves five contains gallery hexagonal each that says Borges precisely,Very spots. crucial in vague being by Library on each of four walls; each book contains four hundred ten pages; each pages; ten hundred four contains book each walls; four of each on ( ous god” oblivionsubalterna horror: vast,the contradictory Library, whose vertical his wife and his publisher, calmly calculates the number of books one books of number the calculates calmly publisher, his and wife his and all possibilities must be realized, so there are a total of 25 are Bloch begins hisimaginativeBloch begins book. “approximatelycontains eightybook each of line vaguely,then,each But his in books of number total the of determination a confounds Borges letter Lasswitz’swhereas hell.And librarian’s a of vision comicdarkly livereda fantastic. As he wrote in “The Total Library,” “I have tried to rescue from rescue to tried have “I TotalLibrary,”“The in wrote he fantastic.As the spines of thebooks.the spines twenty-two letters of the alphabet,” the comma, the period, and the space. magni­ per line times 40 lines per page times 410 pages per book, for a total of total a for book, per pages 410 times page per lines 40 times line per page, forty lines. There are exactly twenty-five orthographic symbols:“the pages per book, per page, 40 lines and 50 characters per line. (His answer: char special other and spaces, marks, punctuation by musings mathematical his in onegged handy, paper and pencil pipe, brary of Babel.” The professor, sitting in his armchair and smoking his smoking and armchair his in sittingprofessor, The Babel.” of brary black letters,” and there are an unspecified number of letters printed on printed letters of number unspecified an are there and letters,” black 2,000,000 exactly To the contrary, Borges was keen on emphasizing the mystical and the However, if we tidy up Borges’s description by assuming that there that assuming by description Borges’s up tidy we if However, ahr hn efrig cmuain n suyn te resultant the studying and computation a performing than Rather lc’ dsuso prles asizs bt ih upe prose. purpler with but Lasswitz’s, parallels discussion Bloch’s h tude as a mathematician or astronomer might, Borges instead de instead Borgesmight, astronomer or mathematician a as tude among uniform “anotherwise desert of theletter tion of the physical parameters of the books is everywhereis books precise,physicalthe parametersofthe of tion .) ing the number of books containing sixteen occurrences of the of occurrences sixteen containing books of number the ing eighty weignoreif characters and spacesand line per extra any Selected 216). and lower-case letters, lower-case ­and acters) if one has 500 has one if ­acters) g ,” Blochwrites, 1,312,000 books. - - “Informa “ own Catalog. Any other catalog is unthinkable descriptions. From this calculation Bloch concludes that “ that concludesdescriptions. Bloch calculationFrom this character string is defined to be the length of (one of) the shortest com­ shortest the of) (one of length the be to defined is string character contents of each andevery one of them. connection withthescience of information theory. written descriptions a and theme unifying a both provoking,lacks thought and written well output, or directions on how to locate that book in the Library of Babel (in of a book might be a computer program that has precisely that book as its able,” but it does show that if a catalog is a collection of descriptions of descriptions of collection a is catalog a if that show does it but able,” alog entry for each book: there are 25 are therebook: each for entry alog lc cnlds i iiil hpe wt a ucnt caatrsi theme: characteristic succinct, a with chapter initial his concludes Bloch Bloch notes that if one considers a library catalog to be a collection of de­ of collection a be to catalog library a considers one if that notes Bloch let us define a define us let string string of length 1,312,000, in the 25-character alphabet of the Library, and scriptions of books, and if one limits oneself to, say, half-page descriptions, result tiesBloch’s ideastogether andanswers hisconjectures. encod mystical a or position), and shelf, number, gallery floor, of terms encompasses length. notion finite This of library) the of setcharacter the brary, thedescriptions (more mostof than95%)will be atleast as long as abook. ing ing only understood by some oracle. The following information-theoretic in an important sense, but he draws it too quickly, and his chapter, though in the alphabet of the Library, then it is impossible to provide a unique cat­ books, then one can hardly do better than by simply listing the completethe listing simply by betterthan do hardlycan books,one then The books number in theLibrary, of thougheasily notated, is unimagin­ The information-theoretic concept of the of concept information-theoretic The “unthink is catalog other any that demonstrate quite not does This Theorem. Mathematically, let us identify a book in the Library with a character a with Library the in book a identify us Mathematically,let These books, droning wearily of wearily droning books, These than three timesourknown universe. (21) of instancessixteen scant the inflame the imagination or addle the senses, and yet if they were all col all were they if yet and senses, the addle or imagination the inflame lected into a subsection of the Library, they would occupy a space greaterspace a Library,occupywould theythe of subsection a into lected tion Theory” (Ca For any way of assigning unique descriptions to books in the Li the in books to descriptions unique assigning of way any For description of anykindwhatsoever of of a book to be any character string (again, in in (again, string character any be to book a of t a h l , are not typographicalto ,not phantasmagoriaare ogs of g 1,312,000 , with a little respite provided only by only provided respite little a with , , anylanguage. in Adescription ” (39). His conclusion is cor the Coll books but only 25 only but books Kolmogorov complexity Kolmogorov ection) The LibraryThe its is 1,600 able

unique ” (22). of a of ­rect - - - -

221 The Imaginative Mathematics Curtis Tuckey 222 “Rea “legitimately wonder if every integer might have a remarkably condensedremarkably havemighta integerevery if wonder “legitimately 1, and hence that for 1 25 questions athand. characters. In particular, for any kind of character-string representation for the numbers easily verify algebrausing that the product ( ( In a footnote (first included in the edition of 1944) attached to the last the to attached 1944) of edition the in included (first footnote a In m would be a much lengthier 1,440 digits long) and remarks that one may one that remarks and long) digits 1,440 lengthier much a be would word of the last sentenceBorges, “Thewordlast ofBabel,” ofthe Libraryof voicethe of in Library are incompressible incomprehensible). addition tobeing (in no a given character string is no shorter than the string itself, the string is string the itself, string the than shorter no is string character given a 0 through 0 shorter than a book is equal to (25 said to be to said By a similar argument one can conclude that for any assignment of descriptions to any assignmentto descriptionsany forof that concludecan argumentone similar a By words, most of them can be no more compressed in length than they are in ordinary in are they than length in compressed more no be can them of most words, form” (37). Bloch does not make a strong argument either way, and his way,and either argument strong a make not does Bloch (37). form” representation base in their than condensed more be hardly can descriptions numerical their output to programshortest the If output. its as string given the has that in condensed form, if we are restricted to restricted are we if form, condensed in definite a is question the to answer The herring. red a is digits” 100 in inconclusive discussion of median “the of the prime numbers expressible the entirethe Library. Thisdescriptionswould ofthe booksmore 95% that than implies ubr, uh s 2 as such numbers, havebook.longa as as least at be to versionA of argumentthe textany can found be in pressible (or very nearly so). However we might try to describe numbers describe to try mightso).Howeverwe nearly very (or pressible puter program(s), written in some fixed universal programminglanguage, base-25 place-holder notation. Better results are available, but these suffice for the for suffice these but available, are results Better notation. place-holder base-25 book on information theory. See, for example, LiandVitani 109. sum, the by given is characters) 1,312,000 (of book a than shorter descriptions of ber ber x distinct items, most of the descriptions will have to have at least as many as [log items,havewillas descriptions many the havedistinct ofas to most least at n 0

: most character strings, made from a fixed set of characters,incom of are set fixed a from strings,made character :most

− 1)/( − + 25 The theorem is easily demonstrated by a counting argument. For each natural num n n rltd usin Boh osdr cmat oain fr large for notations compact considers Bloch question, related a In ,25 arethere l A 1

x + 25 na − 1).Taking− m – m incompressible 2 l

+ ··· + 25 ysis” (Model 1 , most of them of most n

descriptions of length ofdescriptions x not equal to 1, the sum 1,312,000−1 x = 25 and 25 = 4,781 n. wih f rte i odnr dcml notation decimal ordinary in written if (which . The theorem says that most of the books in the in books the of most that saystheorem The. 1

. Even if one has not seen a geometric sum before one can

will have to have at least [log least at have to havewill ing InfiniteBooks) n 1,312,000 =1,312,000, we see that the number of descriptions of number the =1,312,000,that seewe

− 1)/24, or just a bit more than 4% of the size of n . From this one can determine that the num the that determine .can Fromone this x

0 x + 0 + x 1 x

+ 1

n + x 2 x symbols, then almost all of all almost then symbols,

+ ··· + 2

+ ··· + x n 25 x -1 n

m )( -1 is equal to the fraction ] characters—in other characters—in ] x − 1) telescopes to 25

x m n ]

− - - - - “having measure zero.” This doesn’t help in the least in elucidating what elucidating in least the in help doesn’t Thiszero.” measure “having “ A what is meant by the phrase ‘infinitely thin pages.’ Three different inter different Three pages.’ thin ‘infinitely phrase the by meant is what wouldbe. Having first of which does not have infinitely thin leaves) Bloch concludes that concludes Bloch leaves) thin infinitely have not does which of first Kindle-like,carry,could one volume one in Here me.” with “go for Latin es the issue when Bloch tries to explain how thick a book of such pages such of book a thick how explain to tries Bloch when issue the es are rather uncontroversially infinitely thin already. And it actually confus actually it already. And thin uncontroversiallyratherinfinitely are are theproblems withBloch’s analysis. all possible books for ready reference atanytime! then by necessity the is[book] infinitely thin itself leaves have no thickness, Bloch takes several pages to explain the point- the explain to pages several takes Bloch thickness, no haveleaves set definition of definition set mistaken.and result,misleading turns surprising by is it but Briefly, here sonal friendof his: the editor of the manuscript, relates the following observation of a per a of observation following the relates manuscript, the of editor the tors) models infinitely thin leaves as rectangular sections of a Euclidean a of sections rectangular as leaves thin infinitely models tors) it means for a leaf to be “infinitely thin”—planes have thin”—planes “infinitely forbe means to it leaf a zero thickness and the of interval unit the in ing parallel to the made up of a countably infinite collection of six-by-nine inch “leaves” ly inch six-by-nine of collection infinite countably a of up made being as “book” a such imagine canleaves. example,oneFor such many plane, and models an infinite book as a box-like collection of infinitely of collection box-like a as book infinite an models and plane, (the interpretationsthree his presenting After(46). details” the in parate dis spirit,but in similar Books three to lead thin’ ‘infinitely ofpretations Regardless of which interpretation we assume, if the pages are ‘infinitely thin,’ ‘infinitely are pages the if assume, we interpretation which of Regardless vademecum In his second interpretation, Bloch (as have many other commenta other many have (as Bloch interpretation, second his In Bloch says, “The mathematical analysis of [such a book] hinges on hinges book] a [such of analysis mathematical “The says, Bloch size, printed in nine- or ten-point type, that would consist of an infinite an of consist would that type, ten-point or nine- in printed size, common the of volume, single a is required is that all speaking, strictly ent page would open into other, similar pages; the inconceivablemiddle pages;the similar other, into open would page ent pointless: is Library vast the that observed has Toledo de Álvarez Letizia of planes.) Using that silken vademecum would not be easy: each appar each easy: be not would vademecum silken that Using planes.) of number of infinitely thin leaves. (In the early seventeenth century, Cava century, seventeenth early the leaves. (In thin infinitely of number lieri said that all solid bodies are the superposition of an infinite number infinite an of superposition the are bodies solid all that said lieri page would have ( no“back.” is a book for ready reference, or a pocket handbook, from the measure zero measure xy defined -plane -plane and projecting rigidly from each rational point “infinitely thin” to mean “having measure mean to thin” zero”“infinitely z -axis. Uncontent with merely saying that the that saying merely with Uncontent-axis. and then and Collected defines 118) ” (54). This is a spectacularly “infinitely thin” to mean to thin” “infinitely ------223 The Imaginative Mathematics Curtis Tuckey 224 2 Weierstrass Theorem, not proved until the late nineteenth century, that in a book with book a in century,that nineteenth Weierstrasslate the Theorem,proveduntil not convergence” having the property that between that leaf of convergence and any other convergenceany of and leaf that between that propertyconvergence” the having of convergence. (Kline 349, emphasis).(Kline my Cavalieri’s standardsupportedthe not intuitiveby pictureis ways,Thisseeit,it.” to conclusionopen able alone be is let wewouldnot finity of pages densely packedinto a bookin thickness, inch of one or any other thickness.other real-world(Forcomparison,a the by havemeasurezero,of be would,to itselfbook the he concludes that since a countably infinite collection of such leaves would said that “a line is made up of points as a string is of beads, a plane is made up of lines lines of up made is beads, plane of a is string a as points of up made is line “a that said Perhaps one reason for the confusion is that, unlike the pages of an actual as a cloth is of threads, and a solid is made up of areasplane He thickness. nonzero of nevertheless but thin infinitely are which indivisibles,” allel mous principle and many other important theorems by imagining geometrical solids geometrical imagining by theorems important other many and principle mous model of Cartesian space, however. It is a consequence of what is known as the Bolzano- the infinitesimals of analysisRobinsonian (nonstandard analysis), buthe structure.) physical a of volume spatial the of 0.00000000000001% than compo the of volumes the of sum the mean not thickness.do Wetimes respond to spatial extension. It is certainly possible to have a countable in ­ infini to be to mis­ nent electrons, neutrons, and protons, which science tells us make up less cor not does “measure” structures geometrical for because misleading if our model has a leaf sewn in at each rational point in its spine,its in then eachrational at sewn leaf point a in has model our if of “leaf one least at be must there covers, finite between bound pages many infinitely leaf, there is always yet another leaf. If having one such frightful leaf were not enough, not were leaf frightful such one having leaf.If another yetalways is leaf,there book, sections planar cannot bestacked one atopthenext. extendedbook,heightstructure:the volumeoftimes the width mean we be infinitely thin. “In other words,” he says,“if we looked at the Book side between any given two, but instead consisting of infinitely many “equidistant and par and “equidistant many infinitely of consisting instead two,givenbut any between TheBonaventuramathematicianItalian Cavalieri eponyarrivedhis ­ (1598–1647) at takenly assumes that an infinite sum of infinitesimals must itself be itself must infinitesimals of sum infinite an that assumes takenly In his third interpretation Bloch defines “infinitely thin” in terms of terms in thin” “infinitely defines Bloch interpretation third his In book like book tesimal. Hewrites: other infinitesimal, so the book is, again, infinitely thin: never to be seen, be to thin: never is,infinitely again, book the infinitesimal, so other as this one, adding the infinite number of infinitesimals produces yet an yet produces infinitesimals of number infinite the one, adding this as Book by adding together all of the infinitesimals. For a summation such summation a infinitesimals. For the of all together adding by the Book of thickness the compute we analysis, nonstandard of rules the By never tobefound, never tobeopened. (54) : not consisting of plane regions of zero thickness, with another plane plane another with thickness, zero of regions plane of consisting not : by his definition his by as a book ispages made up of volume 2

every of a physicala of leaf is a leaf a is leaf ,have to ” ------A possible source for the error is perhaps that in 3 The OtherChapters The thickness of such a book is given by the infinite sum of the thick of sum infinite the by given is book a such of thickness The discussed above, we have the following situation: a countably infinite book with a leaf a with book infinite countably a situation:following havethe above,discussedwe centerishexagonwhosecircumference anyand is unattainable construction worthy of Borges) I note that although Bloch (and several other commenseveralother (and Bloch although that note Borges)I of worthy construction ( In Chapter 4, Bloch uses the physical universe of the Library as a foil for foil a as Library the of universe physical the uses Bloch 4, Chapter In dimensional analogs of spheres and toruses. (Incidentally, these topics topics these (Incidentally, toruses. and spheres of analogs dimensional definitely notlimits of finite partialsums. with a long history and many interpretations—including a vision of infi of vision a interpretations—including many and history long a with of finite partial sums.offinite partial nonstandardinfinite sumsof The analysis most are no are also well represented in representedin well also are popular of exposition) book’s the of quarter full (a discussion lengthy a at each rational number strictly between 0 and 1 will satisfy (b) but not (a), while the while (a), not but (b) satisfy will 1 and 0 between strictly number rational each at ” Sand’ of Book ‘The story, short Borges’s in described book the to structure similar a mentary on afootnote. for this chapter,this for Borges’s “ statementthat to take to topics topics from elementary topology: Möbius bands, Klein bottles, and higher there is no first page or last page. Comparing the Book of Sand with the two models two the with Sand of Book the page. Comparing last or page first no is there thicknesswaya such in every (a) that page nexta pagehas previousa and page, (b) and book The described. have commentators) other the of any (or Bloch that models the of is ‘The Babel’ footnote Library final of the proposedin book believe“the tators) that not infinite: 1. is sum that leaves,and the of nesses of consisting book ideal an consider and mal, hard to construct a Robinsonian model of the book described in “The Book of Sand,” of Book “The in described book the of model Robinsonian a construct to hard hyperfinite book with “Thein Book of has Sand” literally infinitely many pagesinto bound a book of ordinary but I shall leave that for a later commentary on a footnote to a commentary on a com a on commentary a to footnote a on commentary later a for that leaveshall I but The Unimaginable The infinite sums, and instead one must make do with limits of sequencesof limits make sums,with must infinite instead do one and In this footnote to a commentary on a commentary on a footnote (a delightfully (a footnote a on commentary a on commentary a to footnote this In This is not the case. In Robinsonian analysis it is perfectly reasonableperfectly is it analysis Robinsoniancase. In the not Thisis N to be an infinite hypernatural number, so that 1/ that so number, hypernatural infinite an be to

131), this latter book is manifestly different in structure from any of any from structure different in manifestly is latterbook 131),this 1/N +. . . +1/N=1 N leaves ofthickness 1/ N terms Fantasia Mathematica Fantasia

exactly N will satisfy (a) but not (b).not but very(a) willnot satisfy is It the Librarythe sphereexact whosea is 3 equal to 1. Not infinitesimal, Not 1. to equal N standard leaves of thickness 1/ thickness of leaves .) The main inspiration inspiration main The .) analysis ,” is a metaphora is ,” N is infinitesi is there

are N - - - - - . 225 The Imaginative Mathematics Curtis Tuckey 226 “despite not offering much of a framework forbecause the it proof” would 4 (and not entirely successfully) when he edited the original story of 1941 of story original the edited he whensuccessfully) entirely not (and At any given time there will be a collection of galleries that have been visited at that time arquitectura either chosen gallery, let ushave two librarians say “Imay besometime,”leave thatgallery by opposite doors, visit a new gallery on that floor in each time unit, and never turn back. never turn unit, and time each in floor that on gallery new a oppositedoors,visit plane,in the and galleries in ordersthe hallway or linearly path long,that one winding damning. As with the description of the parameters of the books (charac books the ofparameters the of description the with damning.As all of Borges’s requirements at the same time, but this in itself is hardly is itself in this but time, same the at requirementsBorges’s of all Library.the for able housing Perhaps following suggestiona of Einstein’s But it is nonetheless fun to think about what other spaces might be availmightspacesbe other what about think to nonethelessfun is it But and another collection of galleries that are adjacent to (i.e., sharing a wall with) at least at with) wall a (i.e., sharing adjacentto are that galleriescollectionof another and essentially is therecase, this accessible.In mutually are galleries the of all that assume layout of the Library is a mixture of specificity and confounding vague confounding and specificity of mixture a is Library the of layout some mathematics into his discussion, when he finally does come to a to come does finally he when discussion, his into mathematics some 3-dimensional the is Library the houses spacethat the for model sonable for the reader. The proof is not quite small enough to fit into this margin, this into fit to reader.enough smallthe quite for The not proof is original, and interesting is discussion Bloch’s 1958. in edition new a for satisfies that layout a with up come not does Babel.He of Library the for rea a that chapter lengthy this in proposes Bloch universe, own our for ters per line, lines per page, pages per book), Borges’s description of the of description Borges’sbook), per page,pages per lines line, per ters requirements to the letter either, but Toca’s elegant architectural render architectural elegant Toca’s but either, letter the to requirements ings are especially compelling. And although Bloch would like to inject inject to like would Bloch although And compelling. especially are ings misstates the hypotheses in a way that makes the conclusion false, and false, conclusion the makes that way a in hypotheses the misstates mathematical conjecture (his “Conjecture of Extreme Disconsolation”) he JorgeTocaarquitecturade nio “La “La Luis articleBorges” in his also (see ness, and Borges himself adjusted the architecture subtly but significantly nite Euclidean space; see Borges’s essay “Pascal’s Sphere” ( have to be wall-sharing galleries arbitrarily large finite walking distances apart. Let us Let apart. distances walking finite large arbitrarily galleries wall-sharing be to have particular, there is a unique path between any two given galleries. From some arbitrarily biblioteca de Babel” for more references) and Cristina Grau in in Grau Cristina references)and more for Babel” de biblioteca Anto of contributions substantial the account into take not does he but boundary of the4-dimensional sphere. but asketch of aproofinto a easilyfits footnote. be “too messy to offer up in these pages,” he leaves the proof as an exercise Bloch’s Conjecture is not quite true, as stated: he should probably have said that said have probably should he stated: as true, quite not is Conjecture Bloch’s In ChapterIn Five, “Graph Theory,” considersBloch possible floor plans there are some galleries that are completely inaccessible to others to inaccessible completely are that galleries some are there (1989). Neither Toca nor Grau manage to satisfy all of Borges’s 4 Selected or Borges y la y Borges 351-53). there will there ------20) perd n h sm ya as year same the in appeared infinito” el (2008) y Borges “Letizia, essay lengthy Castellón’s Alberto (1999). in articles several and Martínez, Guillermo by (2003) considercollectionany of hexagons necessarily(not having connecting vestibules) that Borges’sLibraryBabel, of of the of show to is trick The visited. been yet not have which but galleries, visited the of one can only beinterpreted byagodlike external observer” (125). vincing. In Chapter 7 he draws a superficial (though poetic, and perhaps poetic, and (though superficial a draws he 7 Chapter In vincing. on bookshelves.on calleddiscussionbe While nightmarishhis to is enough books of .. . permutations all of set the of permutations all of set the of appeared too late toget Bloch’s attention (andvice versa). every­ neglects but mathematics, and Borges about English in said have Borgesian) analogy between Turing machines and the human (or librar (or human the and machines Turing between analogy Borgesian) Borge­ adjacent to already-visited galleries but which have not as yet been visited.each(From been yet as havenot which but galleriesalready-visited to adjacent many unvisited adjacent galleries on its perimeter that the librarians will necessarily will librarians the that perimeter its on galleries adjacent unvisited many By our assumption that all of the galleries are mutually accessible by some path, each of Hence the (unique) path from path (unique) the Hence thing thing written in Spanish, including the very relevant ian) condition, saying, “The librarian’s life and the Library together em together Library the and life librarian’s “The saying,condition, ian) find the first adjacent, unvisited gallery in each direction.) From time From direction.) each in gallery unvisited adjacent, first the find form azigzag of length those proofthe conjecturethe of that,is time, that 2 at least thereat will be time time:some i.e.,be will there travelingfuturethe ofsome other at librariansor hexagons one these by visited be will take a long time to reach all of them in their continuing journey. One way to do this is to so be will theresize, certain a reached has galleries visited of collection the when that the librarians (it does not matter which one) at least Thenone.) either pick then length of thezigzag we started with, tobeaslarge aswe like. least body a Turing machine, running an unimaginable program whose output permutations with space all available filling involvesThisBabel. of brary by one of the librarians (it does not matter which one) at time In Chapter 8 he gives a thoughtful review of some of what other critics Li­ the Pattern”for “Grand a of idea his discusses Bloch 6,Chapter In n 2n more time units until all of those adjacent, unvisited galleries will be visited.Of be adjacent,will galleries those unvisited of all until units time more n sian, there is too much hand-waving for it to be mathematically con hexagons in the zigzag, look along the lines both north and south until you until south and north both lines the along lookzigzag, the in hexagons galleries,let B be the be n : and makes many of the same points, but perhapspoints, samebut the of makesmany and B has a face-sharing galleryface-sharing a has last A to one that is eventually visited. (If there are two “lasts” “lasts” two are thereeventuallyvisited. (If is that one t n at which allwhich at B must be longerthan be must The Unimaginable Mathematics of of Mathematics Unimaginable The n of them havethem visited.beenof Thekey to n steps to get to A that had already been visited been already had that n .Finally, maytakewe Borges y la matemática t n , and it has taken one of Borges y la ciencia la y Borges B n starting at time galleriesare that t n it will take at take will it n ,the t n - - -

. 227 The Imaginative Mathematics Curtis Tuckey 228 Although he was not a mathe­ a not was he Although Postscrip In his review of review his In cians), which he found inscribed in one of those books.thoseof concludes,one He in cians), “It inscribed found which he chief editor of the Book-of-the-Month Club, he published Borges’s published he Club,Book-of-the-Month the of editor chief the of Eco.thinking WhenUmberto was tell) quite cannot whom one might write such a book for, the late Clifton Fadiman comes Fadiman Clifton late the for, book a such write might one whom ematics. Intheintroduction to (139). .” . infinity. of emblem an and universe a also is Babel of Library in Library” “Universal Lasswitz’s a magician with his props, but the magic is more than legerdemain.His than more is magic the but props, his with magician a Fadiman rec rinths Bloch prefaced his book with a question, “Who is the intended audience intended the is question,“Who a with book his prefaced Bloch mathemata mathematicis scribuntur for this work?”, and his own initial response (whether tongue in cheek I cheek in tongue (whether response initial own his work?”,and this for the National Library of Argentina, and reminds us of Copernicus’s dictum, tations in the mathematical books in Borges’s personal library, now part of tember 24, Manguel 2008)Alberto wrote (italics added), immedi­ is myhopethatthisbook beliesthatsentiment.” mentioning the Library of Babel: “Borges plays with ideas,” he said,“like he ideas,” Babel:“Borgesofplayswith Library the mentioning At the end of the book, Bloch talks about the wear patterns and anno­ patternsand wear the about talks book,Bloch the of end the At (which includes “The Library of Babel”) in 1964. Twentyin lateryears Babel”) “The of Library includes (which Though I confess that my mathematical illiteracy mathematical my that confess I Though I still know no real mathematics, who has received a con ­ receiveda has who about Moron,”Mathematical a of “Meditations calledessay an where,in something of the way the mathematical mind seems to work, and a fair a and work, to seems mind mathematical the way the of something amount about the lives of the great mathemati great the of lives the about amount follow many of his formulas and graphs, the lucidity of Mr. Bloch’s arguBloch’sMr. of lucidity graphs,the and formulas his of many follow the pleasures of reading about mathematics, pleasures open to anyone to open pleasures mathematics, about reading of pleasures the ing todoalittleing mental work. (xiv, myemphasis) ments enlightened menevertheless . . . ately to my mind. At Willy Ley’s suggestion Fadiman included included Fadiman suggestion Ley’s Willy At mind. my to ately ­ommended

The Unimaginable Mathematics Unimaginable The Labyrinths ventional secondary-school education, and is will is education,and secondary-schoolventional matician, Fadiman was not afraid of math of afraid not was Fadiman matician, Fantasia Mathematica but I know what mathematics is about, and (mathematics is written for mathemati­ in his in atsa Mathematica Fantasia Lifetime Reading Plan in the in made it difficult for me to me for difficult it made cians. I have written else writtenhave I ­cians. , hewrote, New York Sun York New n 98 ad as and 1958, in type , specifically of person of (Sep Laby ------When asked in 1983 whether there was a kind of kinship between poets between kinship of kind a was there whether 1983 in asked When In his preface Bloch went on to say that the potential audience for his book ue b te hlrns eeiin okhp ( Workshop Television Children’s the by duced calideas. After all, exponentials elementaryand combinatorics are already Library of Babel. possess who fantasy fans fearlessnesscertain a and mathemati fiction for of pants, shirts, and sweaters. From this calculation it is only a few steps few a only is it calculation this sweaters.From and shirts,pants, of how many dif many how and math­ Borgesaficionados should include mathematics-friendly Borges aficionados, together with together aficionados, Borges mathematics-friendly include should orh t sxhgae colhlrn Dezl ap, n mnt and minute a in Zappa, Dweezil schoolchildren, sixth-grade to fourth- twenty-one seconds, used combinatorics (“combina- combinatorics seconds,used twenty-one top in magnitude, not conception, to computing the number of books in the in books of number conception,magnitude,the not computing in to ics of grade-school mathematics. An example: in a 1996 TV spot pro spot TV 1996 a in example: An mathematics. grade-school of ­ics And I think there should be a kinship. And I suppose there is. There is There is. there suppose I And kinship. a be should there think I And (with whom I share nothing but the girth and the beard),the and girth the but sharewhomI (with nothing was delightedI My mathematics is very slight. very is mathematics My Mr. Bloch notes his in preface that the ideal reader of his book is Umberto say. ( cians, and poets and philosophers, who are a measure of poets, I should I poets,of measure a are whophilosophers, and poets and cians, knhp ewe al hns epcal bten ot ad mathemati and poets between especially things, all between kinship a by the book’sand instructed wit and wisdom, and grateful for the guided Eco. Unworthy as I am to aspire to the condition of the great polymath great the of condition the to aspire to am I as Unworthy Eco. Babel anditsmirror, ouruniverse, are sodelicately balanced. (24) tour throughtour mathematicalthe foundations on which Librarythe both of ematicians, Borges himself replied, Borges, thePoet ferent outfits he could put together from a certain number certain a from together put could he outfits ­ferent to beto 83, myemphasis) .latterthis In class young wouldI science- include But I have read and reread Bertrand Russell.Bertrand reread and read have I But Sesame Street Sesame what? Oracle Corporation ”) to calculate to ”) Curtis Tuckey , ie at aimed ), - - - 229 The Imaginative Mathematics Curtis Tuckey 230 ---. “Labiblioteca deBabel: unamodesta propuesta.” ---. ---. ---. ---. Works cited Toca Fernández, Antonio. “Laarquitectura deJorge Luis Borges.” Grau, Cristina. Costa, Renéde. Cortínez, Carlos ed. Castellón Serrano, Alberto. “Letizia, Borges yelinfinito.” Hayes, Brian. “Books-A-Million.” Kline, Morris. Eco, Umberto. Fadiman, Clifton, ed. Borges, Jorge Luis. Bloch, William Goldbloom. “The Unimaginable: Catalogues andthe Selected Non-fictions Labyrinths: Selected Stories andOtherWritings The Unimaginable Borges’s Mathematics of Library Babel of Lifetime Reading Plan Academia deExtremadura delas Letras ylas Artes 23-40. Tiempo Jill Levine andEliotWeinberger. New York: Penguin, 1999. Month Club, 1964. Oxford UP, 2008. Oxford: Blackwell, 1995. de Bellas Artes New York: Oxford UP, 1972. americanscientist.org/bookshelf/pub/books-a-million>. Penguin, 1998. Book of Sand‘The in Library of Babel’.” ter, 1958. 3.24(2009): 77-80. Mathematical Thought from AncienttoModern Times The Search for thePerfect Language Borges yla arquitectura El humorinBorges 7(1982). Collected Fictions Borges, thePoet. Fantasia Mathematica . Ed. EliotWeinberger, trans. EstherAllen, Suzanne . 3 . rd ed. New York: HarperPerennial, 1988. . Detroit: Wayne State UP, 2000. American Scientist . Trans. Andrew Hurley. New York: Fayetteville: Uof Arkansas P, 1986. . Madrid: Cátedra, 1989. . New York: Simon andSchus Variaciones Borges . New York: Book-of-the- . Trans. James Fentress. 97.1 (2009).

231 The Imaginative Mathematics