Since its early development, regardless of having started as a sequence of bodily gestures or vocal sounds, language might have shown to its inventors its inherent combinatorial nature. Given the relative limitation of the available repertoire of signs, shortened still further by the selection of just a few among all the possible ones, it may have not passed unnoticed to their users that virtually all of them could be combined with each other in any type of sequence, many without even being assigned a meaning. The recourse to writing, while still in its early pictographic form, seems to evidence this perception, mainly after the continual stylization it underwent, ending up in the syllabic and alphabetical systems of today: the fact that a varied palette of sounds could be represented by several combinations of written signs is probably witness to our having already had the intuition that either in speech or their written format the elements of language could be permuted to provide all that can be said, as well as, certainly, much more.
The storyteller and essayist J. L. Borges goes even further, supposedly beyond language (if something in such confines is there), as to finding vestiges of that hunch in the philosophical mullings of Aristotle about the cosmology of the atomists, in which reality itself was set as a phenomenon for pure combinatorics to explain. One can say, however, that we have seemingly reached this comprehension much before the atomism, when the world was first seen as a variegated melange of a handful of primary elements, or still earlier, when Thales said that all comes out of permuted water transmutations: as it indeed seems, it is rather our minds that somehow are set up to make out of all that goes through them a result of the nearly exhaustive operation of coupling one or a bunch of basic units whatsoever, the most conspicuous example of this being the act of numbering. However, what might have caught the attention of Borges was Aristotle’s use of alphabet characters to illustrate how atoms differentiate from each other in form and reciprocal positioning, and, of course, his astounding observation that both tragedy and comedy end up being composed indifferently with the very same twenty-four letters.
But, according to the Argentinian, the concept of what he called a ‘total library’, the collection of all possible character combinations – texts – dwelling somewhere in the realm of ideas, would still take about two millennia and a half to become clear, this time in the mind of the German philosopher Gustav Theodor Fechner, its formal inventor – if ideas, and this one, in particular, are things liable to be invented. Its first exposition came out at the beginning of the XXth century, in a short story by the mathematician and writer Kurd Lasswitz, ‘The Universal Library‘, describing a mathematical way to obtain the collection of every book or single text that already exists, is still to be written, or even will never find its author, containing as well an all-astounding amount of gibberish, i.e., totally meaningless and unpronounceable strings of symbols coming along as a penalty in the process of exhausting their very permutations to assure that each and all of the meaningful, speakable portion of whatever can be read be obtained.
As for the set of symbols, Lasswitz conceived of adopting a generous one – containing around one hundred of them (among which the space between words) – that would enable generating, besides the fictional, the scientific literature, all displayed in books with a fixed number of pages, each of these with a fixed number of lines that won’t exceed a certain amount of symbols each. Needless to say, the letters are treated therein as numbers (from the zero, which is the space between words, to ninety nine, corresponding to any character that happens to be the last of that set), every book thus consisting in a figure of about one million digits in a number system of base one hundred, and so the volumes housed by the Library will correspond, in total, to the appalling quantity indicated by the number one followed by one million and two zeroes, compared to which – like most of the recent commentators of the Library never forget to add – the number of atoms in the known Universe looks trifling, once it is believed to not exceed a modest two-digit figure followed by no more than eighty zeros.
About forty years later, Borges would propose a version of this monster, building it up from a much smaller set, of twenty-five symbols printed to books not as bulky as those proposed by Lasswitz and counting no more than three thousand and two hundred characters per page. He did that in ‘The Library of Babel’, the short story that turned the concept of ‘total library’ into one of the last hundred years’ myths. Despite counting only one-quarter of the symbols in Lasswitz’s Library, this new collection would contain books in a quantity that is neither much less gargantuan nor believed to be less complete than the older one, since the numbers and other special characters Borges left out can be or, rather, are written therein in full as names or expressions, all utilizing only the lower-case letters, the space, the comma and the period – for, yes, whereas inconceivable in life’s practical terms, books exhibiting written-in-full formulas from mathematics or logic are not only possible, but exist indeed in such a library. Hence, since both collections are believed to contain all that is possible to write, and since much of what in Lasswitz’s version is expressed with a single character will take a minimum of two from Borges’ much smaller set of symbols, at least two questions are risen, i.e., one about the minimum limit of characters and books necessary for such a project being considered complete, and another about whether and how it’s completeness would be exceeded, thus containing duplicates of the works. In short: what would be the limits, regarding the size of the volumes and the number of characters in their pages, below which it is not certain that the collection will contain all that is possible to be written and, conversely, whether any of those setups – if there is one such – would allow for repeated books?
These are just some problems of conceiving a ‘total library’: how would one know that it is ‘total’ and not incomplete or repetitive, or still both? As a mathematician, Lasswitz roughly assumes that his proposed setup is sufficient to generate all that’s possible to write in any conceivable variations, whereas Borges, from the heights of his everlasting irony, seems never striving to spur any certainty about this point, indeed claiming that “some” believed the collection to be infinite, thus containing – a necessary outcome of that statement he didn’t care to mention – uncountable exact replicas of the finite combinations implied by the conditions he established for each book and the set of letters. However, in considering a volume to consist of a vast number, the issue about the repetition of texts seems partially answered, as most sequences, regardless of their lengths, will indeed appear several times throughout the collection. Nevertheless, this fact ends up raising a good deal of additional problems, like the possible existence of numbers providing, each, a single work (say, the Iliad) in its entirety, or like the amount and the lengths of identical text blocks present in several books, seemingly probing the limits of what is deemed plagiarism. More generally, however, some even see that most, if not all, of what is tied to the conception of authorship had already crumbled by the mere consideration of the library’s potential existence, a point that, being thought of past certain depth or from a different perspective, will reveal itself moot.
The thought that any given text – or any construct – can be obtained out of merely the permutations of its constituents seems obvious, having even been foreseen long ago by someone like Cicero, as told by Borges in ‘The Total Library’, the essay alluded above, predating about two years the short story ‘The Library of Babel’. But Cicero did seemingly consider obtaining the texts by randomly selecting the characters, like many after him, up to Carrol and Huxley, did, and what is special about Lasswitz’s take is the proposal of an order for the combinations, to which the sequence of Natural numbers comes in handy, that is, given a set of numbers large enough, it is assumed that within such a limit all the combinations of what their digits stand for – in the case, the written characters – will be provided. To some extent, this seems guaranteed if one thinks of the possible combinations of not only the basic characters, but still of the words they form. However, it might be not the case for bigger word blocks, like those corresponding to lengthy works, which will certainly appear torn apart, separated by enormous amounts of coherent or incoherent material, hadn’t each of them corresponded in its entirety to a distinct number, which certainly they all do, some certainly exceeding in quantity of digits what each book in the library was determined to contain. An obvious and easy – yet also cumbersome – way to assess the eventuality that a large meaningful text exists uncut somewhere inside Lasswitz’s or Borges’ proposal is to count its characters to see if they fit what is allowed for a single volume, although it is also guaranteed, whatever be the case, that several of its parts will also appear scattered throughout other volumes. This shows that, despite displayed in the numerical order, the random combinations of words and characters that result from it don’t easily provide a means to find coherent texts, a fact showing the disparity between the ways words and numbers are organized to provide their proper, specific meanings.
Up to now there’s no algorithm to find, out of nothing, the number to which an already written text corresponds, not to mention others yet to be created: this means that, if a work is to be found in Borges’ or Lasswitz’s collection, it must be known in its entirety beforehand. At least from this standpoint, authorship stands unharmed by the harrowing and ideal presence of the Total Library. However, one can ask what should be the utility of a set of objects, despite it being complete, if it is impossible or, at least, highly improbable to find something worthwhile within it. An answer to this should consider that, other than allowing for browsing at leisure through their books, no libraries are useful if one has no idea of what to search for into them, although this is, by the way, what a catalog usually helps to accomplish quite well, yet not in a Universal Library. To start with, its catalogs will be books of a certain class – of the catalogs – in the collection, and, like any stochastic combination of symbols, whether or not senseless, they have, first, to be found and, second, as shown above, one would have to know them in advance as well or be willing to chance spending several times the age some attribute to the Universe in the perusal of the books, it being also by no means guaranteed that the first-found catalog will be either complete or even authentic: like any potential riches, those ones stashed in the Total Library must be harvested, mined, first – preferably, using the habitual method by whose means words are organized.
As observed, despite being read also as a string of numerals, the ways a text and a number are organized are quite different, so that, if any combination of a basic set of characters, regardless of it containing symbols for the numerals, always makes sense when interpreted as a number, not every numeral organization makes for a comprehensible text. Also, whenever written in full, with words, numbers will always use more characters than in their standard numeral form. This isn’t valid of all the numbers that consist in the books of the Library: once each of them contains, according to Lasswitz’s setup, exactly one million digits, the first book in the numerical order, with blank pages only (one million spaces), will be named with the four only characters of the word ‘zero’, while the second, with only – say – a dot (period) at the last position in the last page, needs only the three characters of ‘one’, and so on. However, naming numbers will become a problem when the characters involved exceed the quantity of one million (the total a book can contain), making it impossible for at least the biggest, last numbers – but also countless smaller ones – in that series to fit into a single volume. As the stochastic permutation it is, the Library surely has lots of books with large or small numbers written in full, many of these incomplete, due to evident reasons. This is surely the case of the catalogs whose strategy is bad enough as to list the volumes according to the numbers that their strings of symbols consist in.
In reality, even a true and complete catalog of the Universal Library, one that would correctly and succinctly list all the books by the numbers of their shelves, will exceed, of necessity, the number of characters allowed in each book, and much before having listed just a very small fraction of the collection. So, once the size of books is thus limited in the Library, a usable catalog displaying in full their places must spread over a large number of volumes that will be very hard to find if a catalog of the catalog isn’t at hand first. This constraint, however, is one that believably only a book with an unpronounceable number of pages would circumvent, had it been allowed in the Library’s setup. Nevertheless, Borges took care of surreptitiously letting slip into his Babelian stock of books a chimera of a like-kind, although having reserved the description of all its frightening details for another short story, more than a quarter of a century ahead. In The Library of Babel, he alludes to this oddity as “the formula and perfect compendium of all the rest”, later denominating it “The Book of Sand”.
If there is an appropriate numerical correspondent to an aberration like this, it should possibly be the set referred to by Cantor’s theories, the one dwelling between zero and one – or any two contiguous whole numbers – and whose elements never follow one another closely, there being infinities hiding betwixt every pair of them, no matter how near to each other they appear to be: including both covers, there’s no page of this compendium, all of them equally and inconceivably thin, from which countless other can’t be separated, it being impossible to find a complete text if it exceeds what fits into two open pages, since those apparently contiguous to both will certainly show a different work, and probably in a language that only God knows which – incidentally, this entity’s description appears as the last note to the Library of Babel story, out of the observation, attributed to Letizia Álvarez de Toledo, that the collection would be useless, had the Book of Sand existed. Borges most certainly intended to convey the analogy with Cantor’s work, yet it crumbles at the observation that it would require, to start with, that the text in each pair of open pages had been infinite like most decimals in the set of the Real numbers.
Given the monolithic setup that a Universal Library requires, which comprises books with a fixed number of pages, each containing a limited amount of characters, all resulting in a finite – albeit astronomical – collection (that in Borges’ story is believed by “some” to be infinite), the Book of Sand is indeed another disobedience Borges purposefully committed as regards the starting conditions he appeared willed to impose upon his very plot, possibly to endorse two of his intuitions: one, appearing at the start of the story, is about the hexagonal rooms in the Library being infinite in number; the other, coming at the end, is where he considers every combination of symbols to be somehow meaningful in at least one of the “Library’s secret tongues”. In sum, the Book of Sand looks to be that presage inherited from some that Cantor himself has had, about something lurking from within each pair of any contiguous whole numbers, which he believed to amount to a quantity inconceivably bigger than that of all these ones together and that, despite inhabiting also every minimal slice of space between any two seemingly close infinitesimal elements of the Real number series, is just the one and same infinity that only with the idea of ‘God’ it shares its disquieting ubiquity.
This is why this Book must be the single one of its kind. Another reason for its uniqueness, according to the short story it stars, is the apparent absence of gibberish in it, not to mention the basic set of symbols it uses, seemingly encompassing all those that already exist or (a mulling of mine) even that are possible. The first of these reasons may suggest that, since the infinite between zero and one would necessarily contain infinitely many sequences expected to correspond to meaningless texts, there couldn’t be any analogy, apart from the inexhaustible quantity of pages found between any two of them, linking those numbers and the Book, although it also hints at Borges’ intuition that every combination of characters would, in fact, mean something in any given so-called “secret tongue” of the Library. As to the second reason, in believing that all the existing written characters – rather than all the possible ones – already consist in a finite, albeit monstrous set, and, consequently, that the base of the number system that provides all their combinations is certainly inconceivably large, whereas each page is allowed to contain just some of them, one should consider the following: either a pair of open pages or one of them is a number with a relatively modest amount of digits or it is the Book itself a single infinitely large numeral; but if it is a page or pair of pages that correspond to a number (a finite one), then the book wouldn’t probably have infinitely many pages (each containing a finite amount of characters), unless it comprises infinitely many repetitions of them as well, all consisting of the numbers (finite ones) that exclusively furnish coherent texts (for, again, there’s apparently no garbage in the Book); finally, by the way, being capable of providing only coherent texts could also be the case of the infinite numeral that as a whole the Book would hypothetically consist in. These considerations won’t hold if all combinations of digits happen to ultimately issue texts coherent in any of the “secret tongues” of the Borgesian Library, although it remains in question which very large set of very large numerals or which specific infinite sequence of digits could stand for exclusively coherent combinations of characters in at least these non-secret languages we are used to.
Algorithms – and mathematics in general – might have a lot to develop before stumbling upon the heuristic for the numerical representation of meaningful texts and the largest sequences of numerals that provide them, for as surely as, e.g., the Iliad corresponds to a number, there must exist – likely infinitely – many of such numerical sequences that account, each, for all that has already been and is still to be written, displayed in every conceivable order, including the rigorously chronological! But, even so, whereas finding in these collections what has been written may not be a major problem at all for a well-built algorithm, identifying what truly will be produced is impossible, because many strings of numerals, even those purged from all garbage, will surely show, for example, texts in countless variations and attributed to anyone except their true future authors, not to mention that most of them won’t ever be written by anybody, yet will be surely attributed to somebody that either has existed, will exist or not. (Note: we may have started talking here of one or more number figures corresponding to another of Borges’ infinite books, this time that one described in ‘The Garden of Forking Paths’, the story quoted by Hugh Everett III in the epigraph of his doctoral thesis, which is about the countless many bifurcations the flow of time would make if, at the quantum level, each particle takes all its possible paths instead of this very single one we are witnesses to. There might or must be in the minimum one sequence of numerals that encompasses all the existing or even possible writings in each and all of those parallel times, on which we could have a glimpse, were we lucky enough to stumble upon it.)
Regardless of any apparent uselessness, the search for such a chronology (which certainly corresponds to a single, finite and enormously large sequence of digits hidden within uncountably many spots inside the so-called ‘potential infinity’) can likely shed priceless lights on the path to a mathematical description of the structure of language, thus revealing number relations that wouldn’t surely be found through other means, a task that, although destined to never being completed, can’t be even cogitated without computers. By the way, it is with the help of them, computers, that now we can have a look on ‘The Library of Babel’ – besides the pure imagining at the short story’s reading – in two ways, one of them, a fanciful (yet also monotonous, in consonance with the story’s scenario) three-dimensional virtual tour built upon a previous, text-only rendition whose creator does his best at adapting as economically and as functionally as possible the constraints that the collection’s original setup still imposes on the current technological limitations. As a result, one can locate a handful of versions of any text that doesn’t exceed what a single book page can comprise, i.e., the permutation of a couple of thousands and a few hundred characters chosen from a set of twenty-something of them, roughly close to how Borges envisaged the collection, and, as a consequence of this, the hope of finding a large work that is complete and soundly organized inside a single one of the volumes wouldn’t be likely fulfilled without some inhuman amount of work, naturally, if and only if this variation of Borges’ starting setup allows for such a possibility. But even if just some fully displayed works are available, although finding them be a matter for living countless lives, there is still a lot to do or to play with this ‘little’ marvel lost in the mass of information – most of it sheer gibberish – of an ever-growing Internet.
Among these amusements, one can find anyone’s texts (like this very one here) or conversations through the Net (even those loaded with transcription errors made by an AI in still its infancy), although this might not be an activity one would endure long toying with. A video has suggested something possibly funnier, which consists in converting an image into a text, hiding somewhere inside it a message, locating this new text at the Library’s site and sharing its address, a trick that indeed looks quite useful in times of hardened censorship, had the author been careful enough to take effective measures to prevent the message and the ‘encoding’ method from being easily uncovered, like by sharing, instead, the image whose code has been edited to contain the location of the message in the Library.
But there are other – and probably countless – possibilities of having fun with the program, like testing some limits, contradictions or paradoxes the Library surely provides. One that easily comes to mind and likely deserves deep testing can be stated as ‘no text can contain its Library address’, and can be step-by-step described as follows: find the text location and add – paste – it to the text, then find the address of this new text and, if wanted, add this new address to it as well, and so forever or until no characters can fit in the page anymore. For obvious reasons (since at least according to its location each text can be deemed unique), none of the listed locations would be reached again as the list increases, but it remains still in question whether there is in the least one writing that contains its real location. (One should consider that in the Library a text can normally be found in several places, in one of them it may stay alone in a page, in another, surrounded by either other meaningful or meaningless texts.) Another game has to do with considering texts as numbers: select a coherent text that fits exactly the number of characters permitted in a page, consider it as a number in the Library’s twenty-some base system (by assigning a numeral value – like the zero to space – to each of the characters), and consider adding to, subtracting from, multiplying or dividing by it any numbers corresponding to other coherent texts, provided the results don’t exceed what a page is allowed to display. Would that provide other coherent texts? What would be the relative locations of texts – numbers – differing in one unit? Which of them would be the next page to each other? What about numbers differing in ten, one hundred, one million units?
A sure source of fun the site can offer dwells on meditating on the many aspects or meanings of its presence on the Internet. Some say – often precluding Lasswitz and Fechner – that Borges prefigured the World Wide Web or, in the least, that there are some astounding, direct relations between it and his story, which some ostensibly dismiss. However, one should consider that, as already pointed out, all that can be written with such an alphabetical set, including all the transcriptions made to it from whatever is written in every distinct language ever invented by the man, will be found through the site’s algorithm, which means that, as one of Internet’s elements, the Library encompasses itself, which is kind of an obvious take, although it is surely not that it is also bigger than the Web, for this one doesn’t contain, for instance, at least those writings I still keep on real paper and have never bothered to type on a computer and upload (the Internet is still far from harboring all we have produced in those more than half a dozen millennia we’ve been building cultures around the world). Thus, proportionally speaking, that site would be analogous to the Book of Sand, while the Library itself is what the Web would stand for, although in a still – and perhaps perpetual – growing state that believably will never manage to exceed what the site itself contains.
Notes:
1 – Not satisfied with bringing such a monster to its electronic life, Jonathan Basile gifted his work’s viewers with another amusing monstrosity, this time, one coming from the realm of images, which he called ‘universal slide show‘. In it, one can find every picture that fits into the set up dimensions and doesn’t need more than its palette of 4096 colors. Needless to say, yet saying it anyway – as a tribute to those fond of abusing such type of statements: each of the show’s images might have many descriptions in the Library itself, not to mention that every thinkable scene, including those that have actually happened and will surely take place, can be found in it, together with every frame of their videotapes, had or not all of them been filmed in reality, and from every possible angle. Again, that won’t be found for free, unless – as it happens to the texts – each image had been taken or produced already. And, again, as it happens to the collection of books, this one of images is a part of the Internet, and yet, in this case as well, it surely encompasses much more than the rest of the Web would ever dream of being able to.
2 – The idea of infinity, nicknamed ‘actual’, visited in this text isn’t something this author is very fond of, a matter already dealt with in another text. Calling it a quantity something that is defined as lacking a limit isn’t the finest example of coherence philosophy and science in general have been capable of providing in the few recent centuries, but here the advice made famous by Parmenides – and by Berkeley, later on – was heeded, according to which one may speak like the many, provided one thinks like the few.
Philosophy - a musing 