In his prologue to the Spanish-language edition of Mathematics and the Imagination, by Kasner and Newman, Borges says that mathematics, like music, can dispense with the universe. I want to thank you for dispensing with the universe—and with Argentina—in order to be here to listen to this talk this afternoon./

**The angle, the slope, and the interpretation; Thomas Mann and twelve-tone music; The game of interpretation as a balancing act**

Whenever one chooses an angle or a theme—and here we have chosen Borges and mathematics—that choice somehow distorts the phenomenon being studied. This is something well known to the physicists, of course. It also occurs when a critic tries to tackle a body of work from a particular angle: very soon he finds himself caught in the quicksands of interpretation. We must keep in mind that the game of interpretation is a balancing act that can become upset by too much or too little. If we take a very specialized and purely mathematical approach to Borges's writings we run the risk of remaining

*above*the text. Here “above” really means “outside” we force the text to imply things that the author neither said nor intended—an error of erudition. On the other hand, if we do not completely understand the mathematical elements that are present in Borges's work, we run the risk of remaining

*below*the text. Therefore I am going to attempt an exercise of equilibrium. I know that here in this room, there are people who know a lot of mathematics, but I am going to speak to those who only know how to count to ten. This is my personal challenge: everything that I say ought to be understandable by anyone who can count to ten.

*Doktor Faustus*recognizing Schönberg's intellectual authorship of the theory of twelve-tone music. Thomas Mann did so reluctantly because he thought that this musical theory had been transmuted into something else, something that could be molded literarily by him “in an ideal context, in order to associate it with a fictitious person” (his composer, Adrian Leverkühn). In the same way, the elements of mathematics that appear in the work of Borges are also molded and transmuted into “something else,” within literature, and we will try to recognize these elements without separating them from their context of literary intentions. For example, Borges begins his essay “Avatars of the Tortoise” by saying,

### What Borges knew of mathematics; Taking precautions with his library; Truth in mathematics and literature

*Mathematics and the Imagination,*and these topics are more than enough. This book contains a good sampling of what can be learned in a first course in algebra and analysis at a university. Such classes cover the logical paradoxes, the question of the diverse orders of infinity, some basic problems in topology, and the theory of probability. In his prologue to that book, Borges noted in passing that, according to Bertrand Russell, all of mathematics is perhaps nothing more than a vast tautology. With this observation Borges showed that he was also aware of what at least in those days was a crucial, controversial, and keenly debated topic in the foundations of mathematics: the question of

*what is true*versus

*what is demonstrable.*

*demonstration*or

*proof.*

*I did it*or

*I didn't do it.*There is a fact of the matter and they know what it is, but Justice can only come to this truth through indirect means: digital footprints, cigarette butts, and alibi-checking. Often, the justice system can prove neither the guilt of one nor the innocence of the other. Something similar occurs in archeology, where the notion of truth is provisional in nature: the ultimate truth remains out of range, as an unobtainable limit, being the unceasing compilation of the bones of the demonstrable.

*within mathematics*the two terms do indeed coincide; that mathematics is nothing more than a “vast tautology.” This is also related to

*Hilbert's Program,*the sweeping attempt by mathematicians to guarantee that any statement that can be proved true, by any means whatsoever, can also be demonstrated

*a posteriori*using a computational algorithm that can corroborate this truth in a mechanical way, without the aid of intelligence. The goal of Hilbert's Program was basically to reduce mathematics to those statements that can be proved by a computer.

*Incompleteness Theorem*—showed that this is not the way things are, and that mathematics resembles criminology in this respect: there are statements that are true but which remain, nevertheless, outside of the scope of formal theories. More precisely, in any formal theory that is complex enough to develop arithmetic, there are statements that the theory itself can neither affirm nor deny: they remain suspects for whom the theory can demonstrate neither guilt nor innocence. What I want to point out is that Borges envisioned the origin of this discussion. (It does not appear that he was aware of its outcome, however.)

### Elements of mathematics in the works of Borges

*Borges and Science*(Eudeba). This book contains essays on Borges and mathematics, on Borges and scientific investigation, on the subject of memory, on Borges and physics. My favorite section was “Borges and Biology.” After some evasiveness, the author of that section writes, almost apologetically, that after reading the complete works of Borges he is forced to conclude that there is no connection between Borges and biology. None! The man was terrified to have discovered something in this world—biology—that Borges had not touched.

*Argumentum ornithologicum*”; the essays “The Perpetual Race of Achilles and the Tortoise” together with “Avatars of the Tortoise,” “The Analytical Language of John Wilkins,” “The Doctrine of Cycles,” “Pascal” together with “Pascal's Sphere,” and so on. Some of these even contain small mathematical lessons. And though the topics considered are quite diverse, I see three recurring themes. Furthermore, these three themes all come together in one story, “The Aleph.” I propose that we begin our study there.

### Cantor's Infinity

*En Soph,*the unlimited and pure deity. It is also said that it has the form of a man who points to the heavens and to the earth, indicating that the earth below is the mirror and map of the heavens above. For

*Mengenlehre*it is the symbol of the transfinite numbers, in which the whole is not necessarily greater than each of its parts.

*Mengenlehre*is the German word for the theory of quantities. The symbol for aleph appears thus:

*the whole is not necessarily greater than each of its parts,*a notion that contradicts the Aristotelian postulate that a whole must be greater than any its parts. This is one of the mathematical concepts that really fascinated Borges. I would like to give a short explanation of how this idea of infinity arises in mathematics.

*something.*Indeed, he can be reasonably certain that

*he and I have the same number of cards.*This he knows even though he does not know what this number is.

*A*and

*B*have the same number of elements if and only if there is a perfect one-to-one correspondence between them.” This assertion is very easy to prove. But what happens when we jump to an infinite context? Of the two concepts of “having the same number of elements” that we have discussed for finite collections—(

*i*) counting both collections and seeing if the count is the same, and (

*ii*) establishing a perfect, one-to-one correspondence between the two collections—one of them no longer makes sense when we pass from the finite to the infinite. What could the first notion mean for an infinite collection if the process of counting it cannot come to an end? In the context of infinite collections the notion of counting no longer makes sense. But the second notion survives: for infinite collections it is still possible to establish perfect one-to-one correspondences, just as we did between people and chairs.

*There is a part that is equivalent to the whole.*This is the kind of paradox that amazed Borges: in the mathematics of the infinite, the whole is not necessarily greater than any of its parts. There are proper parts that are as great as the whole. There are parts that are equivalent to the whole.

### Recursive objects

*recursive*objects. Borges's Aleph, the little sphere that encompasses every image in the universe, is a recursive object in this way, albeit a fictional one. When Borges says that the application of the name “Aleph” to this sphere is not accidental and immediately calls attention to the connection with this property of infinite sets—that a part can equivalent to the whole—he is inserting his conception into an environment that makes it plausible. This is the technique that he explains in his essay “Narrative Art and Magic,” at the point where he discusses the narrative difficulty in making a centaur believable. Just as in the case of infinity, where a part can be equivalent to the whole, it is conceivable that there is an element of the universe that encompasses the data or information content or knowledge of everything.

*opposite*property. What would

*anti-recurive*objects be like? They would be objects in which each part is essential and no part can be used as a replacement for the whole thing. Finite sets are examples of anti-recursive objects because no proper subset of a finite set is equivalent to the whole set. Jigsaw puzzles are also examples because, if they are good ones, no two pieces will be alike. From an existential point of view, human beings are anti-recursive. There is an intimidating phrase that is due not to Sartre but to Hegel: “Man is no more than the sum of his actions.” It does not matter how flawless a man's conduct has been during each day of every year of his life: there is always time to commit some final act that contradicts, ruins, and destroys everything that has happened up to that moment. Or to take the literary turn given by Thomas Mann in

*The Holy Sinner,*his book based on the life of St. Gregory: no matter how incestuous and sinful a man has been throughout his entire life, he can always confess his sins and become Pope.

### Infinity and the Book of Sand

*rational numbers*) are obtained by dividing integers. Fractions may be thought of as pairs of integers, with one integer in the numerator and another (which cannot be zero) in the denominator:

*That for any two fractions there is always another one between them.*Between 0 and 1 we find 1/2, between 0 and 1/2 we find 1/4, between 0 and 1/4 we find 1/8, and so on. Any number can be divided in half.

*first*number greater than zero: between any positive number and zero there is always yet another. This is exactly the property that Borges borrowed in “The Book of Sand.” Remember the moment in the story when Borges (as a character) is challenged to open the Book of Sand to its first page.

^{1}

*Cantor's Diagonal Argument.*The enumeration proceeds as follows:

To the fraction 1/1 we assign the number 1. |

To the fraction 1/2 we assign the number 2. |

To the fraction 2/1 we assign the number 3. |

To the fraction 1/3 we assign the number 4. |

We skip 2/2 because it is already counted (1/1=2/2). |

To the fraction 3/1 we assign the number 5. |

To the fraction 1/4 we assign the number 6. |

And so on. |

*consecutive*ordering to the positive fractions. This ordering is of course different from the way that the fractions lie along the number line, but it might provide an explanation for the unusual page-numbering in the Book of Sand. (This is something that Borges might not have known.) The page numbering seems mysterious to the Borges character in the story, but in principle there is no mystery. There is no contradiction between the fact that for any two leaves of the Book of Sand there is always another between them, and that each page can be assigned a unique page number: the same skillful bookbinder who could stitch those infinitely many pages into the Book of Sand could perfectly well number each page while doing so.

### Infinity and the Library of Babel

*A,*27 is

*B,*the number 526 might be a Chinese ideogram, and so on.

*any*finite length and allow books to be of

*any*finite number of pages?

*is enumerable as well.*The idea is to display all the books that consist of a single page in the first row, all the two-page books in the second row, all the three-page books in the third row, and so on. We then enumerate the books by following Cantor's diagonal path. Since every book in the Library of Babel is also included somewhere on our bookshelves, we conclude that the collection of books in the Library of Babel must also be enumerable.

*useless*) because all the books of the Library of Babel could fit into a

*single*volume of infinitely many, infinitely thin, pages—“a silken vademecum in which each page unfolds into other pages.” The book formed by piecing together all the various books of the Library of Babel into a single volume, one after another, would not be longer than Cantor's diagonal path.

### The sphere with center everywhere and circumference nowhere

*vidita,*” as Bioy Casares would say—effectively count all the numbers. But he has a way of generating them in thought, and in this way can attain numbers as large as necessary. From the ten digits of decimal notation he can reach numbers as large as he likes. However bound to his earthly situation, he can still extend his arm to the sky. That is the objective and the difficulty of counting.

*whose center is everywhere and circumference nowhere.*This occurs in “Pascal's Sphere” and elsewhere. Borges warns his reader: “Not in vain do I recall those inconceivable analogies.” It is a very precise analogy that adds plausibility to the little sphere that he describes in “The Aleph.” In order to understand the geometric idea of such a sphere, something that might seem to be a play on words, we shall first ponder it in the plane, and instead of spheres we shall consider circles. Consider an ever-expanding circle: if it continues to grow indefinitely then it will eventually encompass any given point in the plane. The location of its center is not really important and it could be anywhere.

*infinitely expanding*one, where these words have a dynamic sense.” In other words, we can replace the plane with a circle that grows and grows, because each point in the plane is eventually encompassed by such a circle. Now, in this indefinitely expanding circle, the circumference is lost at infinity. We cannot delimit any circumference. This, I think, is the idea that he is referring to. In making the jump to the infinite, the entire plane can be thought of as a circle with center at any point and circumference nowhere.

^{2}And, as I said before, mathematics slips into Borges's writings within a context of philosophical and literary references: the idea that the universe is a sphere is connected to a whole tradition of mysticism, religion, and Kabbalah. These other connotations are explained in more detail in “Pascal's Sphere.”

### Russell's Paradox

*self reference,*but this has a different meaning in literature and I don't want to mix up the two concepts.) The paradox appears when Borges gives the partial enumeration of the images of the Aleph. But it also occurs in other stories, where Borges constructs worlds that are so very vast and space-filling that they end up including themselves—or even their readers—within their scopes. In “The Aleph” this can be seen here: “I saw the circulation of my dark blood, I saw the workings of love and the modification of death. I saw in the Aleph the world and in the world once more the Aleph, and in the Aleph the world. I saw my face and my guts, I saw your face I was dizzy and I cried.”

*concepts.*The set of all concepts is indeed a concept. In other words, although such sets are rarer than ordinary collections of things, the idea that a set could be an element of itself seems to fit the conception of “set.” Furthermore, if I postulate the set of all sets, then in order for it to be a set, it would have to be an element of itself.

*X*= {

*A*:

*A*is a set and

*A*is not an element of

*A*}

*X*will contain the set of all natural numbers, the set of all trees, the set of people in this room, and so on. Now we may ask ourselves: is

*X*an element of

*X*? If

*X*were an element of itself then it would have to satisfy the definition written above. In other words, if

*X*belongs to

*X*, then

*X*is not an element of

*X*. But this is absurd. Does that imply then that

*X*is not an element of itself? If

*X*were not an element of itself, then by the definition it would have to be an element of

*X*. In other words, if

*X*were not an element of

*X*, then

*X*would have to belong to

*X*. But this is also absurd. Here we have a set that is in Neverland, a set that neither is nor is not an element of itself.

### Why are mathematicians interested in Borges?

*Borges and Science,*the author, Lucila Pagliai, asks why Borges's stories and essays are so dear to scientific investigators, philosophers, and mathematicians. She comes to the conclusion that there is an

*essentially essayistic matrix*in the work of Borges, especially in his mature work, and I think she has a point. Borges is a writer who procedes from a single principle—“in the beginning was the idea,”and conceptualizes his stories as incarnations or avatars of abstractions. There are also fragments of logical arguments in many of his stories. The kind of essayistic matrix that Pagliai refers to is, undoubtedly, one of the elements of Borges's style that bear a certain similarity to scientific thought.

^{3}

*The History of Eternity*: “I don't believe in bidding farewell to Platonism (which seems ever cold) without communicating the following observation, with the hope that it will be carried forward and further justified:

*the general can be more intense than the concrete.*There is no shortage of illustrations. As a boy, summering in the north of the province of Buenos Aires, I was fascinated by the rounded plain and the men who drank mate in the kitchen. But my delight was tremendous when I found out that the plain was ‘pampa’ and those men were ‘gauchos.’ The general...trumps individual details.”

*claim,*a statement that is affirmed in anticipation of being proved at some later point. In the next talk I will try to establish this claim, and will discuss some of Borges's nonmathematical stories and essays in this light. I thank you for having been here today. See you next week.

^{1}The negative fractions are not listed here, but if I can enumerate the positive fractions then it is an easy matter to enumerate the negative ones, and it is furthermore an easy matter to enumerate all of them, positive and negative, into a single list. I am speaking rather informally here; mathematicians will have to forgive me a few imprecisions.

^{2}From

*The Cricket,*by Argentine poet Conrado Nalé Roxlo. “Is this blue sky porcelain? Is a golden cup shinbone? Or is it that in my new condition, as a cricket, I am seeing everything as a cricket this morning?”

^{3}Another excellent essay in

*Borges and Science,*“Indications,” by Humberto Alagia, called my attention to the fragment of “The History of Eternity” that I cited in this passage.