In Folha, Viana details Borges’ fascination with mathematics
Reproduction of Marcelo Viana’s column in Folha.
I was already doing my doctorate at IMPA when I discovered the Argentine writer Jorge Luís Borges and the fascination with mathematics that permeates his work. There are more than 180 explicit references to mathematical ideas, especially those that deal with the concept of infinity. Not the mere infinity of mathematicians: in Borges it takes on a cosmological character. “There is a concept that is the corruptor and deceiver of all the others. I’m not talking about evil, whose limited empire is ethics: I’m talking about infinity,” he wrote in “Avatars of the Tortoise”. Adding: “Five or seven years of metaphysical, theological and mathematical learning would (perhaps) enable me to plan your story decently. Needless to say, life forbids me this hope, and even this adverb.”
More than the formal concept of infinity, what interests Borges is the perplexity of the finite mind in the face of what it cannot fully grasp. In “The Aleph”, the narrator comes across a point in space that contains all points. To observe it is to see the totality of the universe, from all angles, at the same time. He tells us that he sees the sea and the land, dawn and sunset, letters and numbers, everything. But in doing so he is restricted to listing visions one after the other: the actual infinity, which coexists in the Aleph, can only be expressed by means of a potential infinity, built up gradually in the narration.
There is also the question of the whole and the parts. Aristotle said that the whole is necessarily greater than any of the parts. But Cantor postulated that two sets have the same number of elements if there is some one-to-one correspondence between them. To check that passengers and seats are equal in number, the crew doesn’t need to count one or the other: it’s enough that every passenger sits down and every seat is occupied. But in the realm of infinity, Cantor’s postulate has counterintuitive consequences. To each integer we can associate its double, an even number, and this is a one-to-one correspondence. Therefore, there are as many integers as there are even numbers, even though the latter are only a part of the former!
In “Book of Sand”, Borges adds to the mystery. At one point, the narrator explains that he was given a challenge: “He told me it was called the Book of Sand because, like sand, it has no beginning or end, and he told me to open it to the first page. I took the book in my left hand and opened it, my thumb and forefinger almost touching. But it was impossible: no matter how hard I tried, there were always several pages between the cover and my hand.”
It’s not just that the pages of the Book of Sand are infinite in number: between any two there are always many more! Borges wonders, perplexed, how to number the pages of such a book. But mathematicians know the answer: fractions. The front cover corresponds to zero, the back cover to one, and each page of the book corresponds to a fraction between these two numbers (2/9, 1/2, 3/5 etc). After all, between any two fractions there are always many others. And the challenge is impossible: the Book of Sand has no first page, just as there is no first fraction after zero.
But that’s not the end of the story. Cantor also proved that there are as many fractions as there are whole numbers, even though fractions are only a part of whole numbers. This means that, despite all its peculiarities, the Book of Sand has no more or fewer pages than any other infinite book!
To read it in full, visit Folha’s website.