prior probability

Via Vatican News: “The upcoming Conclave starting on May 7 to elect the 267th Pope will be the 76th in the form we know today, which was established by Pope Gregory X in 1274, and the 26th held under the gaze of Michelangelo’s Last Judgment in the Sistine Chapel.” See also this 2013 report by Jason Horowitz, via The Washington Post.

Now that I have brought my 11-part series on “David Hume in Borges’ Library of Babel” to a close, what better way of marking the end of this intellectual escapade than with a melancholic piece of music performed by one of my favorite musicians of all time — the late great Argentine tango composer and virtuoso bandoneónist, Ástor Pantaleón Piazzolla (1921-1992):

I will conclude my series on “David Hume in the Library of Babel” by returning to to the question I posed in my previous post: Why hope? Why faith? Hope/faith/belief that the decoder book or master index will one day be found or rediscovered. Hope that the incomprehensible volumes of the infinite hexagons are full of meaning. Hope that the Universal Library contains some secret sequence, that it “is unlimited but periodic.” While others are driven to suicidal despair, our narrator remains full of hope.

A potential clue to this puzzle is the concept of the “infinite.” In all, our unnamed narrator employs this word eight times. (See Borges 1998/1941, Paras. 1, 2, 7, 14, & 15.) But is the Library really “infinite”? And if not, where are its outer limits? Alas, The Library of Babel is full of contradictions. On the one hand, we are told in the first and last paragraphs of the story that the Universal Library “is composed of an indefinite, perhaps infinite number of hexagonal galleries” and that it is “unlimited.” But at the same, we are also told in the middle of the story that the total number of tomes, though astronomically large, is not infinite: “the Library is ‘total’—perfect, complete, and whole—and … its bookshelves contain all possible combinations of the twenty-two orthographic symbols (a number which, though unimaginably vast, is not infinite) ….” How can an infinite Library not contain an infinite number of books? Or, what about our Universe? Is the Universe infinite? Alas, scientists still don’t have a definitive answer, for while the observable universe is finite, with a radius of about 46 billion light-years, the total universe could be larger, potentially infinite.

Perhaps there are some things we cannot assign a Humean probability value to, such as the infinite or the impossible. Consider once again the possibility of a decoder book or master index. If such a literary holy grail were to exist, will it ever be found? Or to put this question in Humean terms (cf. Hume 1748), is the probability p₁ of locating the Universal Library’s master index higher or lower than the probability p₂ that any of the miracles reported in the New Testament Gospels of the Bible really happened? Alas, given the astronomical size of the Universal Library and the historical distance between our day and Biblical times, how would we even begin to quantify such probability values? So, what is to be done? What happens when we cannot assign probability values to carry out Hume’s probabilistic test?

What if the decision to believe in God—or in any other impossibility or improbable possibility, I might add—is not so much a belief but a lifestyle: “a choice founded not on evidence but on the way we choose to live in the face of inadequate evidence.” (Tanya Luhrman, When God Talks Back, p. xiv) In other words, what if we turn Hume’s argument against miracles—and religious skepticism more generally—on its head? Or to quote Tanya Luhrman again: “If you could believe in God, why wouldn’t you?” (Ibid., p. xvi)

On this Luhrmannian view of Borges’ Library of Babel, our narrator has “out-Humed” David Hume. Despite the lack of any direct evidence that the randomly-arranged and composed books in the Universal Library have any meaning, the narrator still holds out hope that those incomprehensible and indecipherable volumes are susceptible to cryptographic methods or allegorical readings. How could it be otherwise, for without this hope, where would we find the requisite curiosity and motivation to make sense of the impossible?

Act III, scene iii of David Hume in the Library of Babel

Below are the last two paragraphs of Jorge Luis Borges’ short story “The Library of Babel.” Here, the Argentine author (pictured above) concludes his story by contrasting “the present condition of humanity” (which teeters at the verge of extinction!) with the “elegant hope” that the books in the Universal Library have meaning:

Methodical composition distracts me from the present condition of humanity. The certainty that everything has already been written annuls us, or renders us phantasmal. I know districts in which the young people prostrate themselves before books and like savages kiss their pages, though they cannot read a letter. Epidemics, heretical discords, pilgrimages that inevitably degenerate into brigandage have decimated the population. I believe I mentioned the suicides, which are more and more frequent every year. I am perhaps misled by old age and fear, but I suspect that the human species—the only species—teeters at the verge of extinction, yet that the Library—enlightened, solitary, infinite, perfectly unmoving, armed with precious volumes, pointless, incorruptible, and secret—will endure.

I have just written the word “infinite.” I have not included that adjective out of mere rhetorical habit; I hereby state that it is not illogical to think that the world is infinite. Those who believe it to have limits hypothesize that in some remote place or places the corridors and staircases and hexagons may, inconceivably, end—which is absurd. And yet those who picture the world as unlimited forget that the number of possible books is not. I will be bold enough to suggest this solution to the ancient problem: The Library is unlimited but periodic. If an eternal traveler should journey in any direction, he would find after untold centuries that the same volumes are repeated in the same disorder—which, repeated, becomes order: the Order. My solitude is cheered by that elegant hope. [Borges 1998/1941, Paras. 14-15; emphasis in the original; footnote omitted]

The final word of the final sentence of the final paragraph of the story is the word “hope.” (Borges 1998/1941, Para. 15) But why is the narrator of the story so hopeful, especially considering the penultimate paragraph of the story, which paints a bleak and foreboding picture of “the present condition of humanity.” (Ibid., Para. 14) Mankind is on the verge of intellectual and physical extinction: illiterate and ignorant youth blindly worship books they cannot read. Others ignite brain epidemics, commit heresies, or embark on “pilgrimages that inevitably degenerate into brigandage,” all of which have “decimated” the library’s frail and teetering population. Still others resort to suicide. Although the doom and gloom and foreboding of many of the previous paragraphs all come to a head here, what is most surprising to me is that this dark picture still contains a ray of hope: the Universal Library will endure, even if humanity does not.

So, why is the narrator so hopeful? I will consider this question and conclude my series on “David Hume in the Library of Babel” in my next post.

Act III, scene ii of David Hume in the Library of Babel

Pictured above is a sample page from a random book in the imaginary world of Jorge Luis Borges’ short story Library of Babel. (Words that are English-sounding are highlighted for reference, though these are no more frequent than random chance predicts.) Since the text of every volume in the Universal Library is randomly generated, and since a decoder book has yet to be found, how can we find any meaning in these lines? Are they just meaningless gibberish (i.e. “non-sense”), or do they contain deep secrets waiting to be discovered? To this end, the antepenultimate paragraph of Borges’ short story begins thus:

Infidels claim that the rule in the Library is not “sense,” but “non-sense,” and that “rationality” (even humble, pure coherence) is an almost miraculous exception. They speak, I know, of “the feverish Library, whose random volumes constantly threaten to transmogrify into others, so that they affirm all things, deny all things, and confound and confuse all things, like some mad and hallucinating deity.” Those words, which not only proclaim disorder but exemplify it as well, prove, as all can see, the infidels’ deplorable taste and desperate ignorance. For while the Library contains all verbal structures, all the variations allowed by the twenty-five orthographic symbols, it includes not a single absolute piece of nonsense. [parenthetical in the original]

The reference to infidels in this passage is quite curious, for as it happens, David Hume was commonly referred to as “the Great Infidel” during his lifetime for his skeptical religious views. (See generally Rasmussen 2017.) By way of example, consider Hume’s hyper-rational argument against miracles. (Hume 1748) For Hume, “no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavours to establish ….” (Ibid., Para. 13) To put Hume’s argument in the context of Borges’ Library of Babel, which possibility is more likely: (A) that the pages pictured on pages 14 and 19 above, for example, which we know were randomly generated, is just meaningless gibberish—let’s call this the non-sense hypothesis—or (B) that these pages have some deeper secret meaning?

For his part, Borges’ narrator rejects the non-sense hypothesis out of hand. For him (or her!), the Universal Library “includes not a single absolute piece of nonsense.” But this belief poses a Humean puzzle. If our narrator were a true Humean, his degree of belief about any disputed fact (e.g. about whether any given line of any given page of any given volume in the Universal Library is “non-sense” or not) should correspond to the amount of relevant evidence available to him, or in the immortal words of David Hume: “A wise man, therefore, proportions his belief to the evidence.” (Hume 1748, Para. 4) Does this puzzle have a solution?

Act III, scene i of David Hume in the Library of Babel

The last part of Jorge Luis Borges’ short story “The Library of Babel” presents a tantalizing possibility: the existence of a literary holy grail, a master index “that is the cipher and perfect compendium of all other books”:

We also have knowledge of another superstition from that period: belief in what was termed the Book-Man. On some shelf in some hexagon, it was argued, there must exist a book that is the cipher and perfect compendium of all other books, and some librarian must have examined that book; this librarian is analogous to a god. In the language of this zone there are still vestiges of the sect that worshiped that distant librarian. Many have gone in search of Him. For a hundred years, men beat every possible path—and every path in vain. How was one to locate the idolized secret hexagon that sheltered Him? Someone proposed searching by regression: To locate book A, first consult book B, which tells where book A can be found; to locate book B, first consult book C, and so on, to infinity. . . . It is in ventures such as these that I have squandered and spent my years. I cannot think it unlikely that there is such a total book on some shelf in the universe. I pray to the unknown gods that some man—even a single man, tens of centuries ago—has perused and read that book. If the honor and wisdom and joy of such a reading are not to be my own, then let them be for others. Let heaven exist, though my own place be in hell. Let me be tortured and battered and annihilated, but let there be one instant, one creature, wherein thy enormous Library may find its justification. [ellipsis in the original; footnote omitted]

Imagine if such a book really existed. This master tome would allow us to decode all the other books in the Universal Library and thus unlock the secrets and solve the mysteries contained therein. But does this miracle book really exist, and if so, will it ever be found? Or to put this question in Humean terms, is the probability p₁ of locating the Universal Library’s master index higher or lower than the probability p₂ that any of the miracles reported in the New Testament Gospels of the Bible really happened? [1] Alas, given the astronomical size of the Universal Library (see, e.g., William Goldbloom Bloch, The Unimaginable Mathematics of Borges’ Library of Babel, 2008) and the historical distance between our day and Biblical times, how would we even begin to quantify these Humean probability values? To be continued …

Act II, scene iii of David Hume in the Library of Babel

My previous post was about Borges’ paradox: although the Universal Library contains all possible books, including those books that have not yet been written, the probability of finding any particular book is close to zero.

This cruel paradox, in turn, gives rise to three different literary factions in the imaginary world of The Library of Babel. One is the “inquisitors” or “official searchers” whose job it is to comb through the hexagons full of books with random combinations of letters and symbols in order to find those books that might have some meaning (Para. 9). But according to our unnamed narrator, the inquisitors have given up even the pretense of searching for the truth. Instead, they limit their investigations to detecting “disgraceful or dishonorable words” (ibid.).

Another faction consists of a now-defunct “blasphemous sect” whose members abandoned the search for those “precious books” altogether and instead tried to generate or recreate those mysterious texts for themselves using elaborate games of chance (Para. 10). But by trying to create their own holy grail, so to speak, these literary heretics were “feebly mimicking the divine disorder” (ibid.), i.e. committing the sin of blasphemy. Yet another faction was the so-called “Purifiers” (Para. 11). The Purifiers’ reaction to this general despair was “hygienic, ascetic rage” and “holy zeal.” In short, they “condemn[ed] entire walls of books,” thus causing the “senseless loss of millions of volumes” (ibid.).

Now, what about David Hume? What if Hume found himself in Borges’ Universal Library? Which of these three factions would he belong to? Wouldn’t Hume reject the excessive formality of the inquisitors, the pointless games of chance of defunct blasphemous sects, and the “hygienic, ascetic rage” and “holy zeal” of the Purifiers? Assuming, then, Hume that would rejects all three factions, what would Hume’s response to Borges’ paradox be? Stay tuned. I will turn to the last part of Borges’ “Library of Babel” in my next post …

Act II, scene ii of David Hume in the Library of Babel

In my previous post, I surveyed the abstract axioms of Borges’ imaginary library. The next paragraph of the story (Para. 8) contains a paradox: the Universal Library contains all possible books, but the probability of finding any particular book is close to zero:

When it was announced that the Library contained all books, the first reaction was unbounded joy. All men felt themselves the possessors of an intact and secret treasure. There was no personal problem, no world problem, whose eloquent solution did not exist—somewhere in some hexagon. The universe was justified; the universe suddenly became congruent with the unlimited width and breadth of humankind’s hope. At that period there was much talk of The Vindications—books of apologiæ and prophecies that would vindicate for all time the actions of every person in the universe and that held wondrous arcana for men’s futures. Thousands of greedy individuals abandoned their sweet native hexagons and rushed downstairs, upstairs, spurred by the vain desire to find their Vindication. These pilgrims squabbled in the narrow corridors, muttered dark imprecations, strangled one another on the divine staircases, threw deceiving volumes down ventilation shafts, were themselves hurled to their deaths by men of distant regions. Others went insane. . . . The Vindications do exist (I have seen two of them, which refer to persons in the future, persons perhaps not imaginary), but those who went in quest of them failed to recall that the chance of a man’s finding his own Vindication, or some perfidious version of his own, can be calculated to be zero. [ellipsis and emphasis in the original]

On the one hand, the Universal Library must contain so-called books of “Vindication” — i.e. “books of apologiæ and prophecies that would vindicate for all time the actions of every person in the universe and that held wondrous arcana for men’s futures” — but at the same time, given the unimaginable astronomical scale of the library, the probability of finding any of these magical books is zero for all practical purposes.

In other words, even though individualized Vindication books must exist in theory — after all, Borges’ Universal Library is a “total” library, since it contains all possible books (past, present, and future) that could be written — in the absence of a catalog or index of all the books in the Library of Babel, the probability of any given person finding his own specific Vindication book during his own lifetime is virtually nil.

Moreover, this demoralizing paradox will give rise to several different factions and a wide range of responses. In short, the search for truth will be transformed into empty inquiries and formalistic rituals; hope and optimism will mutate into blasphemous pursuits and destructive pogroms. I will survey these responses in my next post. To be continued …

Act II, scene i of “David Hume in the Library of Babel”

In my previous three posts (see here, here, and here), we revisited the first three paragraphs of Jorge Luis Borges’ The Library of Babel. Today, I will turn to the middle part of Borges’ story, or “Act II” of my retelling of this tale (paragraphs 4 to 11).

In summary, our Humean narrator begins Act II by describing the fundamental “axioms” or basic laws, so to speak, that govern the meta-logic of the Universal Library:

1st: “The Library has existed ab æternitate.” (Para. 4, emphasis in the original)

2nd: “There are twenty-five orthographic symbols.” (Para. 5, emphasis in the original)

3rd: “In all the Library, there are no two identical books.” (Para. 7, emphasis in the original)

From these premises, our unnamed narrator concludes that “the Library is ‘total’—perfect, complete, and whole—and that its bookshelves contain all possible combinations of the twenty-two orthographic symbols (a number which, though unimaginably vast, is not infinite)—that is, all that is able to be expressed, in every language.” (Para. 7, emphasis added, parenthetical in the original) But given that the books in the Universal Library consist of a random distribution of 25 symbols, how does one find meaning in any of these enigmatic texts? Stay tuned: I will proceed to Act II of “David Hume in the Library of Babel” in my next post …

I will resume my series on “David Hume in the Library of Babel” tomorrow. In the meantime, below is one of my all-time favorite 80s hit songs as well as two bonus links: