The Curious Infinite of Jorge Luis Borges

Previous to the Argentine writer Jorge Luis Borges, (1899-1986) many writers, philosophers, and mathematicians have for a long time played and juggle with the language trying to analyze and decipher the infinite.

Jorge Luis Borges, the Argentinian writer, author of the famous short story The Book of Sand.


In The Book of Sand, a translation from the Spanish original El Libro de Arena, Borges shows us an unexpected view of his understanding of the infinitude.

In this short story that bears the same as the title of the collection of short stories where it first appeared, Borges is visited by a Bible salesman that after a brief introduction offers him a special book: "I don't only sell Bibles. I can show you a holy book I came across on the outskirts of Bikaner. It may interest you."

What kind of book can a Bible salesman offer Borges—the writer—that could capture his attention? He was a professor of English literature and librarian with a huge collection of books of his property, so he was accustomed to seeing every kind of book. Thus, what was so interesting in a book that on its spine the title was the simple unexciting and dull phrase: Holy Writ - Bombay?

Borges opened the book at random just to find that the pages were worn, written in an unintelligible tightly printed text with poor typography. But this was not a surprise for him, the true surprise was in the numbering of the pages.

I noticed the one left-hand page bore the number (let us say) 40514 
and the facing right-hand page 999.

Was this a misprint? Was it a typographical error? There was only one way of finding the truth: look for the numbering in other pages. And that he did, but

"I turned the leaf; it was numbered with eight digits"

What should we expect after the number 999? The number 1,000, of course, but an eight-digit number is greater than 9,999,999; so the "error" persisted. In fact, no book on the earth is so big, so the first conclusion that comes to our mind is that from the very beginning of the book there was no intention in numbering it correctly.

He told me his book was called the Book of Sand, 
because neither the book nor the sand has any beginning or end.

We can reproduce this unordered numbering with any book from a bookcase: choose a book, erase or overwrite all the page numbers and substitute each one with a random one, no matter how big or small.

But there was an additional attribute of The Book of Sand that we cannot do with any book from any bookstand, bookshelf, or library: you see each page only once. This happened to Borges with one of the illustrations of the book:

It was at this point that the stranger said: 
"Look at the illustration closely. You will never see it again."

Borges' metaphor of choosing the sand to convey the idea of non-repeating events is perfect. Go to the beach —any sandbox is useful for our example— take a handful of sand, drop all of the grains and keep only one in your hands; the sentence applies perfectly: look at it closely because you will never see it again.

The book was so mysterious and weird that it even lacked the first page, not because there was no number 1 in the first page, but because in some inexplicably, or magical, puzzling and perplexing way there appeared more and more pages between the book cover and the "first page".

The stranger asked me to find the first page. 

I laid my left hand on the cover and, trying to put my thumb on the flyleaf, 
I opened the book. It was useless. 
Every time I tried, a number of pages came between the cover and my thumb. 
It was as if they kept growing from the book.

Trying to find the last page of the book was equally frustrating for Borges as baffling can turn to be for us to find the first and the last grain of sand on a beach.

Now find the last page.

Again I failed. In a voice that was not mine I 
barely managed to stammer: "This can't be".

A book with no first and last page is nothing more nor less than an infinite book. That was The Book of Sand: an infinite book. Somehow, the book was infinite in pages, but not infinite in weight, nor in volume. The book was not infinitely big, it was an ordinary book, but with the particularity that its pages were constantly appearing and disappearing, new pages substituting existing ones.

The strange salesman that visited Borges was aware of his astonishment with the bizarre book he was holding in his hands: the same thing happened to him—that's the reason why he called it a "devilish book". Thus, ceremoniously he told Borges:

If space is infinite, we may be at any point in space. 
If time is infinite, we may be at any point in time.

Borges' concept of infinitude is different

In mathematics, we usually associate the infinite with the sequence of the natural numbers: 1, 2, 3 ... We say that the natural numbers are infinite because they don't ever end. To every number no matter how big it is, we can always add 1 to find a bigger number. Thus, there is no way of reaching a limit, of reaching an end, there is no last natural number. The natural numbers are the best example of the most elementary idea of infinitude. But the sequence of natural numbers is far from Borges' idea of what is the infinite: we already saw that the Book of Sand had no first page; it had no page 1.

But equally important as not having a first page is the fact that the book's page numbering followed no ordered sequence; any number can follow any other number at any moment. The page numbers were random at its purest stage.

If as he says, when he first opened the book he saw the page number 40514 followed by the page number 999, at some other time the same number 40514 may be followed by, let's say, the number 23089.

Can we say then that the book's numbering is just a scrambled number sequence? No, we cannot compare the book's numbering with a scrambled number series because in this case or ordering we always have a first number: the first number we choose for the scrambled sequence. But Borges' book had no first page, so he is not writing about unordered sequences of natural numbers: his metaphor is something beyond that.

Borges' Book of Sand confronts us with a different concept of infinitude: an infinite too far beyond our mental conception, an infinite that avoids any ruling, an infinite that escapes any possible ordering or any possible prediction. To Borges, the infinite is the kingdom where the chaos reign; the infinite is the source of every possible finiteness.

Borges chooses a simple short story to convey his idea of the infinitude because for him the infinite is not only unreachable, but any part of it is also inconceivable. The simple random numbering of a small book like the Holy Writ, that can be held in our hands is enough to take us to the vertigo of the infinitude.

Borges and Cantor: two minds where the infiniteness meet

Georg Cantor, the founder of the Set Theory in mathematics, discovered many other types of infinities. In mathematics there are many manipulations that can be done with the infinite; the infinitude of the natural numbers is just the simplest of them. One of the many breakthroughs in this field came when the mathematician George Cantor (1845-1918) introduced a more complex notion of infinitude with what he called the transfinite numbers which are an infinite class of infinities. For Cantor, there is a ladder of infinitudes, where the infinitude of the natural numbers is just the simplest of all the infinities. For him, this ladder of ever-growing infinity has no end. To this obscure but interesting field of mathematics belongs the abstract field called transfinite arithmetic.

Borges had a literary mindset with deep interests in mathematics; that's the reason why he exposes so excellently—and in a very short story—how the infinite is beyond our comprehension. On the other hand, Cantor was the pure mathematician that worked on the abstract concepts of the theory and the cardinality of sets. From there he discovered that the infinites are infinites in themselves.

The first of the "infinitude of infinites" discovered by Cantor is the one called the Aleph. From there he also introduced the so-called the Continuum. The connection I try to establish between Borges and Cantor is that in The Book of Sand we can begin to understand what is the Continuum without recurring to deep mathematics.

Imagine that you open that "devilish book" and by some unknown power you can write the sequence of the page numbering as it appears page by page. Now close the book and reopen it again and repeat the process again and again. You will "finally" obtain all possible orderings of the natural numbers. I cannot show it here, but it is possible to prove, that your "list" of all possible orderings of the natural numbers is not countable, not even infinitely countable.

Thus opening an closing The Book of Sand is an act of delving into the Mathematical continuum.

No comments:

Post a Comment