Borges and the Infinite: How Mathematics Shaped a Literary Genius

A mathematician deep into doctoral research at Brazil's Institute of Pure Mathematics stumbled onto something unexpected: the Argentine writer Jorge Luís Borges had woven mathematics throughout his entire body of work, over 180 explicit references scattered across his stories and essays. But this wasn't the mathematics of textbooks. Borges was after something else—the vertigo of a finite mind confronting what it cannot fully grasp.

In an essay called "Avatares da Tartaruga," Borges wrote that infinity was "the corruptor and deceiver of all other concepts." He wasn't interested in the formal definitions mathematicians use. What seized him was the perplexity itself—the way infinity breaks the mind's ability to comprehend anything whole. He acknowledged this plainly: five or seven years of study in metaphysics, theology, and mathematics might have equipped him to write its history properly. But life forbade it. Even the word "perhaps" felt too certain.

Consider "The Aleph," where the narrator discovers a single point in space containing every other point simultaneously. To observe it is to witness the entire universe from all angles at once—the sea and land, dawn and dusk, letters and numbers, everything. Yet the narrator can only describe what he saw by listing the visions one after another. The infinite that coexists in the Aleph can only be expressed through a potential infinity, built gradually across the page. The actual infinity collapses into sequential time.

Then there is the question of wholes and parts. Aristotle held that a whole must always be larger than any of its pieces. But Georg Cantor proved something stranger: two sets contain the same number of elements if a one-to-one correspondence exists between them. You need not count passengers and airplane seats to know they match—just watch everyone sit down and every seat fill. But in the realm of infinity, Cantor's principle produces results that defy intuition. For every integer, you can pair it with its double, a even number. This creates a perfect one-to-one match. Therefore, there are as many integers as there are even numbers, even though the even numbers are only a part of the integers.

Borges deepened this mystery in "The Book of Sand." The narrator receives a book with a peculiar property: it has no beginning and no end, like sand itself. When he tries to open it to the first page, holding it in his left hand with thumb and forefinger nearly touching, it proves impossible. No matter how hard he tries, multiple pages always exist between the cover and his hand. The book's pages are not merely infinite in number—between any two pages, infinitely many others always exist.

Borges asks, bewildered, how one could number such pages. But mathematicians know the answer: fractions. The front cover is zero, the back cover is one, and each page corresponds to a fraction between them—two-ninths, one-half, three-fifths, and so on. Between any two fractions, infinitely many others always lie. The challenge becomes impossible: the Book of Sand has no first page, just as no first fraction exists after zero.

Yet Cantor proved something more: there are as many fractions as integers, even though the integers are only a part of the fractions. This means that despite all its peculiarities, the Book of Sand contains neither more nor fewer pages than any other infinite book.

Why do mathematicians care about Borges? The answer may be that he worked the same way they do. Whether his subject was mathematical or not, he began with concrete examples and moved toward general truths. "The general can be more intense than the concrete," he wrote in "A History of Eternity." That bridge between the particular and the universal, between what we can hold in our hands and what we can only think about, is where both literature and mathematics live.