Guillermo Martinez on Borges

 Guillermo Martínez, the brilliant Argentine mathematician and author (e.g. of the 'Oxford Murders' which was made into a film starring Elijah Wood) has written a marvelous book on 'Borges and Mathematics'. In this post, I extract and comment on a lecture he gave some fifteen years ago which is available here. 

the elements of mathematics that appear in the work of Borges are... 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, 'There is one concept that corrupts and deranges the others. I speak not of Evil, whose limited domain is Ethics; I refer to the Infinite'.

Sometimes, Leibniz's 'law of continuity' regarding generalizations from the finite to the infinite holds true- i.e. is useful. Sometimes, it is highly mischievous. 

Here the playful yet sharp linking of the Infinite with Evil

conventionally evil was associated with finitude or lack. The Gnostic element here is to suggest that the Demiurge may be part of an infinite regress of stupidity and evil.  

immediately removes infinity from the serene world of mathematics

not so serene- at least for Brouwer who said 'the (mathematical) construction is an art, its application to the world an evil parasite.' I should explain Brouwer felt that a vegetarian who did not also renounce the fruits of our carnivorous civilization is just a parasite. He has no virtue. What is properly constructed has integrity, but its use may be evil thus making the whole project parasitical. 

and sheds slightly menacing light on the elegant and almost technical discussion that follows. And when Borges goes on to say that the “numerous Hydra” is a foreshadowing or an emblem of geometrical progression, he is again playing the game of projecting monstrosity and “convenient horror” onto a precise mathematical concept.

I think he was also referencing John Wallis, the Cambridge Platonist, who saw 'the fourth dimension' as  'a monster in nature, and less possible than a Chimaera or Centaure'. Theosophists of the period were constantly quoting him.

 Doestoevsky, in 'Brothers Karamazov, suggests that God's Justice might be 'four dimensional' in which case Man, who is born Euclidean, must reject it or remain outside its Mercy. 

How much mathematics did Borges know? ...it is clear that Borges knew at least the topics contained in '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.

By then Gentzen calculi, based on conditional tautologies, had appeared. Borges may have heard of this. Apparently Gentzen was a Nazi and there were plenty of Nazi sympathizers in Argentina. 

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.

The problem here is that of 'existence proofs'. Do Brouwer 'fixed points' actually exist? American mathematicians were scandalized when Brouwer showed up there in the Sixties and appeared to be denying that they do. Errett Bishop reacted by publishing a paper on the 'debasement of meaning' and 'Schizophrenia in contemporary Mathematics'. To give an example one might say 'this is an existence proof' but actually it is merely a proof that if compactness obtains then such and such would exist. But in that case, what you are really saying is 'I have proved that if everything is as I want it to be such that x exists, then x exists.'  

The bigger problem with the enumerable was expressed by Brouwer's 1920 lecture 'Does Every Real Number Have a Decimal Expansion?' which he answered in the negative. We may say such and such real number must be 'sandwiched between two rational numbers or else that it is the limit of a sequence of rational numbers or finally that it is in the nature of a real number that it have a non-repeating decimal expansion. In other words, it looks like we have the thing pretty much cornered. On either side of it are ducks who quack- indeed we find quacking ducks no matter from which angle we approach it and moreover we know whatever it might be, it can certainly quack, but does this mean it necessarily is a duck? The problem here is for things to be meaningful, what is necessarily meaningful must be undefined or else we end up with the circular semantics of a quack. 

In their day-to-day work scrutinizing the universe of forms and numbers, mathematicians come across certain connections and patterns again and again, certain relationships that recur and that are always verified. By training and habit they are accustomed to thinking that if these relationships and patterns always hold true, then it must be for some discoverable reason. They believe that the universe of forms and numbers is arranged according to some external, Platonic order, and that this order ought to be deciphered. When they find the deep, and usually hidden, explanation, they exhibit it in what is called a demonstration or proof.

But, as Godel understood, these proofs depend on the existence of an 'Absolute Proof'. If even 'natural proofs' are not available, where is this supposed to come from?  

Thus there are two moments in mathematics, as in art: a moment that we can call illumination or inspiration—a solitary and even “elitist” moment in which the mathematician glimpses, in an elusive Platonic world, a result that he considers to be true; and a second, let's say, democratic, moment, in which he has to convince his community of peers of its truth. In exactly the same way, an artist will have fragments of a vision and then at a later time execute that vision in the writing of a book, the painting of a picture, or some other creative activity. In that sense, the creative processes are very similar. What is the difference? That in mathematics there are formal protocols under which the truth that the mathematician wishes to communicate can be demonstrated step by step from principles and “ground rules” that all mathematicians agree on.

Sadly, it turns out, these are always arbitrary and if looked at closely, it will be found nobody actually agrees to that crazy nonsense.  

The demonstration of the value of an esthetic work is not so straightforward, however.

It is equally arbitrary. But for any practical purpose there is always a good enough 'witness'.  

An esthetic work is always subject to criteria of authority, to fashion, to culture, and to the personal and ultimate criterion—often perfectly capricious—of taste.

So there is still the problem that 'naturality' is far to seek. I think this is because everything is 'co-evolved' such that there is no objective function to optimize. This is because the fitness landscape does not directly feature. Thus there are problems of concurrency, computability and complexity. 

Mathematicians believed for centuries that in their domain these two concepts—truth and demonstrability—were basically equivalent: that if something were true then the reasoning behind it could be shown with a logical demonstration, a proof.

Some did. Some didn't. A useful lemma might be a gift-horse one should not look in the mouth.  

On the other hand, judges in a court of law, for example, have always known that truth is not the same as demonstrability.

Generally, there is a 'reasonable doubt' test- though it may favor the prosecutor.  

Let's suppose that there has been a crime committed in a locked room with only two possible suspects. Both of the suspects know the whole truth about the crime: 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.

But the justice system might be able to stipulate what 'clinching evidence' might be. In some jurisdictions, the case can be dismissed because of lack of evidence at the current time but 'without prejudice'- i.e. the case may be revived at a later point.  

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.

Unless it isn't at all. Some new scientific technology may show all previous 'bones of the demonstrable' were irrelevant. The issue can be decided once and for all. Thus various theories about the origins of races based on philological analysis had to yield the stage to DNA based studies. This too may change.  

Thus we see that in fields other than mathematics, truth does not necessarily coincide with demonstrability.

Truth is sublatable as is demonstrability. Is there a necessary connection between them? Must there be either a proof or disproof of the Reimann hypothesis? Even if we say yes, would we also say the same about the Continuum hypothesis.  

Interestingly, Martinez thinks Borges did not know Godel's theorem. Would this matter? Surely, Borges would have thought- as most of us do- that somethings which are true can't be proven while some proofs are simply wrong?

... my aim is to connect the mathematical elements with stylistic elements in Borges. I am trying an approach that is stylistic rather than thematic. Some of the stories and essays where mathematical ideas loom most conspicuously are

Tlon? It is the one closest to a recently closed question in math- viz. the Hilbert program to resolve the foundational crisis. So long as there is an undefined term in a first order language- e.g. membership in ZFC- consistency is not possible for anything rich enough to do Arithmetic. But maybe no language about a language should want any such thing. Mathematics might be not just more of a free creation of the mind, it might be such utter anarchy or antagonomia as frees the mind of itself thus defeating in advance madness or such logicism- i.e. Platonic realism- as is in the world, but only as Hell. But that's cool. If there must be a Hell, let it be in commuting distance. Otherwise, one begins to feel foolish for having turned one's life into a suburban villa of the stockbroker type. 

these: “The Disk,” “The Book of Sand,” “The Library of Babel,” “The Lottery of Babylon,” “On Rigor in Science,” “An Examination of the Work of Herbert Quain,” and “Argumentum ornithologicum”;

that last is interesting. It goes as follows ' I close my eyes and see a flock of birds. The vision lasts a second or perhaps less; I don’t know how many birds I saw. 

We don't see a number of birds anymore than we see a certain number of grains of sand in a pile we heap up on the beach. On the other hand, suppose any 'intuition' (immediate apprehension) is 'constructed' in a mathematical sense then, for a point on a plane, by a property of the (Smith-Volterra-) Cantor set or Denjoy–Riesz theorem, it has a single coordinate. 

Were they a definite or an indefinite number?

We can think of a number as an interval. 

 This problem involves the question of the existence of God. If God exists, the number is definite,

Only if God is a Hilbertian Finitist. but that won't keep him safe from self-contradiction

 because how many birds I saw is known to God. If God does not exist, the number is indefinite,

The God who created birds knows the number of birds must always be indefinite because of death and birth and the problem of deciding just when a particular bird dies or is born. 

 because nobody was able to take count. In this case, I saw fewer than ten birds (let’s say) and more than one; but I did not see nine, eight, seven, six, five, four, three, or two birds. I saw a number between ten and one, but not nine, eight, seven, six, five, etc. That number, as a whole number, is inconceivable; ergo, God exists.

So, if there is a sorites problem, God exists. This is like saying if masturbation exists, God must exist because only an omniscient being could punish wankers for each and every one of their horrible offences. 

Taken together with Tlon, on the other hand, Borges's birds suggest that creation is not construction. The world does not embody mathematics though, no doubt, it arises sooner or later for utilitarian reasons. What is strange is that its foundations are in every incompossible heaven though its theorems are to be found here down below. 

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.

Martinez is both a literary master as well as a productive mathematician. Perhaps, in informal discussion he would point to current mathematical theorems and conjectures which he finds foreshadowed in not just Borges's work but other great literature- even that of the ancients. 

I am going to talk about these three recurring themes in reverse order. The first theme is the infinite, or, more accurately, the infinities. Toward the end of “The Aleph,” Borges wrote: 'I want to add two observations: one on the nature of the Aleph, the other on its name.

When we think of Beatrice Viterbo we think of the Universal Set designated as V.  Martinez goes on to give a good account of Cantor's work. However, for ordinary people, there was already the notion of 'uncountable infinity'- indeed it is a dogma of the Jains- and this is basic to Zeno, Parmenides's pupil. 

 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.

Indeed. It may be less. There could be a sort of negative synergy or 'cancelling out'. A group may be less effective than any one of its members.  

There are proper parts that are as great as the whole. There are parts that are equivalent to the whole.

And the whole may be quite horrible though all its parts are marvelous.  

Recursive objects This particular property of infinity can be abstracted and applied to other situations in which a part of an object encompasses the data or information content of its entirety.

Though 'infinity' only arises as something useful by Leibniz's law of continuity. Well read chaps like Borges- back in those days- would actually read Liebniz and Spinoza and so forth. Hegel too was big at the time. He had his own 'bad Infinite'.  

We will call such objects recursive objects.

Sadly, as Hegel realized, everything caught in the web of predication is recursive of itself and everything else. In India, the Jains make a big deal of this but the notion is not absent in any mystical literature.  

Borges's Aleph, the little sphere that encompasses every image in the universe, is a recursive object in this way, albeit a fictional one.

Because everything is a mirror of everything. What is interesting is that Borges decides it is a false Aleph. This is like Razborov-Rudich. But the idea is very old. The Chinese unicorn can only be encountered if you don't know it is a unicorn. But, it turns out that knowing a thing is only possible if the things isn't known at all. Adam eats the apple of knowledge and only succeeds in bringing death into the world.  

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.

The solution is to describe it naturalistically. However, Borges refers to a 'law of sympathy' ultimately founded on the fact that all things mirror all things in the web of predication or the monadology of existence. Magic, then, is more real than our determination to exile ourselves from it so as to maintain the illusion of our own reality.  

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.

I think Borges was abreast of the mystical currents of his age. Everything encodes all information about everything else.  But 'self-information' is 'surprisal'. We just don't want to surprise ourselves too much and are thus content to remain solitary prisoners of our own dream of a world. 

There are other recursive objects that Borges played with in his works. For example, the maps in “Rigor in Science,” where the map of a single province occupies an entire city, and “in whose abandoned parts, in the deserts, lived animals and beggars.”

This another way of saying to know everything about something is to know everything. Again a Jain dogma. But what is dogma in one religion is heresy- and thus sexy- in another. For Indians, karmic metempsychosis is dogma. In the West it is heresy though it has a lively literary tradition of its own. But, by the time Borges began to write, the Irish were playing with the notion that 'tuirgen' or 'investigative birth-seeking' causes all souls to live the lives of all souls.  

And from the point of view of biology, human beings are recursive objects. A single human cell is enough to make a clone. Certain mosaics are clearly recursive objects: in particular, those in which the design inherent in the first few tiles is repeated throughout.

The mystics went further. If you knew everything about the mosquito that just bit you, you would know everything about everything. Perhaps that is why you kill the mosquito and think no more about it.  

Now consider objects that have the opposite property. What would anti-recurive objects be like?

It would be anti-reflexive. This is generally the case for intensional or epistemic objects.  

They would be objects in which each part is essential and no part can be used as a replacement for the whole thing.

In which case the object must have a well ordered 'extension'- i.e. every member of the set must be known and distinguishable. The problem is, we know of no such objects. One might say, my car is anti-recursive, but is it really? On the one hand, taking out some of the parts might not degrade performance too much. On the other, changing the initial or 'boundary' conditions greatly alters the outcome. Your car gets you to where you want to go because you drive it. If I do, it will end up wrapped around a tree.  

Finite sets are examples of anti-recursive objects because no proper subset of a finite set is equivalent to the whole set.

Only if none of their members are epistemic, impredicative, or intensional in a certain sense.  

Jigsaw puzzles are also examples because, if they are good ones, no two pieces will be alike.

but they contain information. Some pieces must be from the 'borders'.  

From an existential point of view, human beings are anti-recursive.

No. From a phenomenological point of view, this may be the case because of some illicit assumption of an epistemological or ontological type.  As far as bare existence goes, humans are highly recursive. It takes two of them to make another human being. 

There is an intimidating phrase that is due not to Sartre but to Hegel: “Man is no more than the sum of his actions.”

Sartre said “Man is nothing else but what he purposes, he exists only in so far as he realizes himself, he is therefore nothing else but the sum of his actions, nothing else but what his life is.” It seems the French can be more verbose than the Germans. 

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 not. Generally not.  

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.

I can't. Is it due I iz bleck? I suppose one could mention Hinton's 'alterable past' or Wilde saying the repentant sinner can do what the God of Aristotle could not- viz. change the past. 

Infinity and the Book of Sand What I have said up to this point about the infinite would be enough to clarify this small fragment. I am going to extend the discussion a little further in order to explain the relationship between “The Library of Babel” and “The Book of Sand.” We have just recently seen that there are “as many” natural numbers as even numbers. But what happens if we consider fractions? Fractions are very important in Borges's thinking. Why? Let us recall that fractions (also called 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:


3/5, 5/4, 7/6, 7/16, ....

What property of these numbers did Borges use in his stories? 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.
Because of this, there can be no first number greater than zero: between any positive number and zero there is always yet another.

Borges, being a bookish cove, would have been aware of the argument for the eternity of the Universe attributed to Aristotle. The idea here is that anything that is eternal is necessary. If the present form of the world always was and always will be, it is necessary and no other form is possible. The Book of Sand, whose cover gives its name as 'Holy Writ' and its place of publication as Bombay, was given by an Indian untouchable to a Scotsman who exchanges it for some cash and a blackletter Wycliffe bible. For Catholics, Lollard literacy opened the door to the horrors of the Reformation. Clearly, there is some atrocious reversal of 'natural' hierarchy here. The heathen pariah has unleashed an unholy terror upon his masters. India, at about this time, was gaining independence. The world was being turned topsy turvy. Instead of Genesis, everything is always in medias res which wouldn't matter if the thing was entertaining. But it isn't. It is a nightmare from which you can't wake up- probably because you are already dead. 

This is exactly the property that Borges borrowed in “The Book of Sand.”

I think Borges shows awareness of an argument against every real number having a decimal expansion 

Remember the moment in the story when Borges (as a character) is challenged to open the Book of Sand to its first page.

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

Borges might have read Samuel Butler, author of Hubidras, who wrote ' I leave to my said children a great chest full of broken promises and cracked oaths; likewise a vast cargo of ropes made with sand'. 

He asked me to find the first page.

I lay my left hand on the cover and opened the book, with my thumb almost touching my index finger. All was useless: there were always several pages interposed between the cover and my hand.

Borges may have been known that the Big Bang Theory was first proposed by a Roman Catholic priest in 1927. If the Universe doesn't have a beginning, how can it have a Creator? 

The front cover of the Book of Sand corresponds to zero, the back cover corresponds to the number one, and the pages in between correspond to the fractions between zero and one. Among the fractions there is no first number after zero or last number before 1. Whatever number I choose, there are always others in between. In this situation it is tempting to conjecture that the infinity of the fractions is tighter, denser, or richer than the infinity of the natural numbers. The second surprise that awaits us is that this is not the case: there are “as many” rational numbers as natural numbers. How can this be?

The cardinality is different but, it turns out there is no cardinality for the total number of cardinalities.  Martinez then gives a good account of Cantor's discoveries. However, because 'diagonal lemmas' establish 'self-reference' and this has always been known to give rise to paradoxes, Borges would have been aware the various philosophical questions this method begs. 

Moreover, this enumeration gives a 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.

Martinez is assuming a bookbinder- i.e. that the book is 'constructed'. If there is no creator how can there be a non-arbitrary curator or compiler? Borges's bible-seller says 'None is the first page, none the last. I don't know why they're numbered in this arbitrary way. Perhaps to suggest that the terms of an infinite series admit any number.' 

We don't know that they are numbered. All we can say that each page known to us has some numbers on it. But they may not be segments from 'normal' numbers- i.e. they are too random to permit any association with a real number and hence can't be partially ordered. 

To see why, think of random segments from the decimal expansion of a transcendental number. We don't know their order and have no way of doing so if the underlying real is 'normal'. 

Infinity and the Library of Babel
...what kind of infinity corresponds to the collection of all the various books that can be written in our universal alphabet, if we admit words of any finite length and allow books to be of any finite number of pages?

Cantor's Diagonal Argument can be used to show that this collection of books 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.

I came to a similar conclusion. There is a well ordering.  

How is this important in understanding Borges's story? In a note at the end of the story, Borges wrote that a lady-friend of his had observed that the entire construction of the Library of Babel was superfluous or excessive (he used the word 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.

I admit that this is a very mathematical way of looking at things. “The Library of Babel,” is meaningful on many levels, and I am not saying that this work of literature reduces to mere mathematics. But at the end of the story Borges arrived at the idea that all of the books can be united in a single, infinite volume.

or just an infinite binary string 

This closing footnote contains the germ of the idea that foreshadows and culminates in “The Book of Sand.” I want to draw attention to this way of thinking about Borges's stories and essays in order to abstract a key idea that is repeated or duplicated elsewhere. It is our first example of a literary “operation” that is reminiscent of mathematical methods. We will study this topic more thoroughly later.

What is the key difference between the two texts? In the first there is a community of librarians. In the second there is an atrocious book which nobody will ever read precisely because it has been placed in an actual Argentinian library. We may feel, if we are in Babel's library, that we could get lucky and find the book Borges would have written had he been us. But of the book traded by the illiterate untouchable for the Christian bible, we feel only horror.  

The sphere with center everywhere and circumference nowhere
We now consider the second element of mathematics in “The Aleph.” It shows up when Borges is about to describe the Aleph, and wonders “how to convey to others the infinite Aleph, that my fearful memory scarcely embraces?”

I have something more to say about the symbol for aleph. The figure of a man with one arm touching the earth and the other pointing to the sky seems particularly fitting because, in a way, the operation of counting is the human attempt at attaining infinity. That is to say, a human being cannot, in his finite life—in his “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.

The problem here is that we can't get to the number we can easily specify- e.g. how many more pizzas will I eat in my life? The intension is not connected by any means known to us to the correct extension. True, a Doctor may say to me 'Vivek, I guarantee to you, that if eat even one more pizza you will die.' But if I stop eating pizzas I will still die sooner or later. I want to know the day I can eat a pizza because I'm going to be hit by a bus.  

Borges wrote something similar when he asked himself “how to convey to others the infinite Aleph, that my fearful memory scarcely embraces? Mystics, in a similar hypnotic state, are lavish with emblems: to signify divinity a Persian says of a bird that it is in some way all birds; Alanus de Insulis spoke of a sphere whose center is everywhere and circumference nowhere.” A little farther down he says, “the central problem—the enumeration of an infinite set—is unsolvable.”

This is the case unless certain 'intensions' have 'natural' extensions or there is some outside 'witness' or 'oracle' or editor who can remove repetitions.  

Borges attempts to describe the Aleph, but it is infinite, and it is impossible to run through an infinite description in writing because writing is sequential and language is “successive.”

Language is intensional and thus can give us a good enough 'intuition'. Then, if a 'witness' comes along, we're off to the races. Borges succeeds as a writer because the witness is the darkness in our own wretched hearts.  

Since he cannot give a complete description of the Aleph, in its place he has to provide a sufficiently convincing idea or example, and it is his well-known enumeration of images that follows. We will have more to say on this later.

Borges was a poet- better yet at least some of the books he valued were ones ordinary people have read.  

The second recurring theme is the sphere 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.”

I suppose, even back then, people had some vague idea that Einstein's universe might be bounded but eternal or else unbounded but subject to collapse. 

It is a very precise analogy that adds plausibility to the little sphere that he describes in “The Aleph.”

It is just a 'naked' monad- i.e. something simple or 'atomic' in the sense that it can't be decomposed any further. But if one such exists, it must contain all information just as if we knew one 'atomic proposition' we could deduce all others. There would be an algorithmic way to crank out all knowledge. These are very old ideas. If you know the secret name of God, you are God. If you know yourself, you know everything. In Islam, Allah enables Alexander to invent the mirror. If he can conquer what he sees in the mirror, he conquers the entire Universe. I don't know how much the Mathematical allusions in Borges adds to their message. What is certain is that it conveyed the sort of thrill that Science Fiction does when we are young and imaginative.  

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.

This assumes there is plane or, more generally, that there is a hypokeimenon- i.e. a material substratum or undergirding for reality. It is something that can't be a predicate but which permits other things to have predicates. For Pascal- but not Parmenides who taught that everything that can be thought of or named (i.e. every 'intension') must be (i.e. have a well defined extension)- the nightmare is that no such undergirding exists- at least as far as we are concerned. God may have a geometry of His own, but we don't inhabit it. 

In the essay, “Pascal's Sphere,” at the point where he wants to make this image a bit more precise, Borges writes, “Calogero and Mondolfo reason that Pascal intuited an infinite sphere, or rather an 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.

I'm not sure about this. I have a vague notion that Guido Calogero wrote something about Gorgias the Nihilist or 'paradoxologist' who gave an Eleatic type argument for non-existence! In this case, the true horror is that though we are as self-centered as fuck our 'extension' is empty! Indeed, we don't exist precisely because of our desperate 'conatus' or determination to continuously remain just as fucking stupid and lazy and selfish as we already are. But only because we don't actually exist. In their heart of hearts, nobody doesn't know this. The trouble is that we defend ourselves against non-existence by falling in love- i.e. subscribing to a Religion whose God isn't just fallible, this is a God who quickly comes to loathe us and to find the heart in which we have erected his Throne to be a fucking gas-chamber. We kill what we love so as to exist- as nothing at all. 

A similar construction is valid for three-dimensions: a globe that grows indefinitely will eventually encompass any given point in space.

Space doesn't exist. God may have such a notion but for us it is an intension without an extension.  

In this way, the universe can be thought of as an indefinitely expanding sphere.

But we can't think- we are incapable of the thought- of what undergirds the thought outlined above.  

This, by the way, is the conception of the universe in contemporary physics: the universe was a little sphere of infinitesimal magnitude and infinitely concentrated mass that once upon a time—in the Big Bang—suddenly expanded in all directions. Why is this “inconceivable analogy” interesting?

It is like Edgar Alan Poe's anticipation of 'Olber's paradox'. The nightmare here is of being shut up in a light-cone which gradually gets cut off from everything else. The Universe itself is burying us alive! 

Because the Aleph is a little sphere. If the universe is viewed as a great big sphere, then the idea that every vision of the universe can be reproduced in a little sphere at the foot of the stairs in some basement is much more believable. Simply through contraction every point in the big sphere of the universe can be translated into the small sphere of the Aleph.

Why then does Borges decide it was a false Aleph? He says the entire universe is contained within the pillar of a certain mosque in Cairo. Borges ends his tale thus 'Does this Aleph exist in the heart of a stone?

If there are two Alephs, indiscernibly identical in terms of information that can be gleaned from them, nevertheless they are distinct if one Aleph actually is itself that which it conveys information about. It is the 'true' Aleph. Why might it exist 'in the heart of a stone'? The stony-hearted feel no love, no longing, no 'regret of Heraclitus'. If all information can incarnate anywhere it must be in the 'heart of a stone'. 

 Did I see it there in the cellar when I saw all things, and have I now forgotten it? 

Borges knew from his reading of TS Eliot that Memory is Love- at least in Sanskrit. Did he also know that an epithet of Shiva, the Hindu God, is 'smarahara'-destroyer of Love, destroyer of Memory- but this is a necessary destruction so Love, and 'Smriti'- i.e. Religion- can survive its own Apocalypse or rending of the veil. 

Our minds are porous and forgetfulness seeps in; I myself am distorting and losing, under the wearing away of the years, the face of Beatriz.' 

As the Prophet said 'everything is going to destruction save the face of God.' What does the lover's loss of face matter? Beatrice's beauty burgeons elsewhere. I suppose a mathematician might mention Grothendieck's God- the dreamer who dreams us and our dreams. Perhaps we do meet our beloved in dreams but forget doing so when we wake. Or perhaps, mathematics is a labyrinth in which everything exists because everything is itself that labyrinth. 

Russell's Paradox
The third paradox is what I call the “paradox of magnification.” (The technical term in logic is 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.”

Magnification, or the postulation of very vast objects, gives rise to curious paradoxes, and Borges was certainly aware of the most famous one, due to Bertrand Russell. Russell's paradox—which shook the foundations of mathematics and toppled the “naive” theory of sets—shows that one cannot postulate the existence of a set that contains all other sets; that is to say, one cannot postulate an Aleph of sets.

You can, by calling it a proper class. I once heard of an Argentine general who banned set theory because it was 'collectivist'.  I suppose he would have been cool with classes because, after all, there has to be an upper class- right? 

I wonder whether there was something, back in the Thirties, like what we call a 'Reflection principle' which holds that it is possible to find sets that, with respect to any given property, resemble the class of all sets. Godel said 'All the principles for setting up the axioms of set theory should be reducible to Ackermann's principle: The Absolute is unknowable. The strength of this principle increases as we get stronger and stronger systems of set theory. The other principles are only heuristic principles. Hence, the central principle is the reflection principle, which presumably will be understood better as our experience increases. Meanwhile, it helps to separate out more specific principles which either give some additional information or are not yet seen clearly to be derivable from the reflection principle as we understand it now.' 

I don't suppose it matters whether Borges came across some such essay or article. My own suspicion is that he was a Pragmatist. Why know the Absolute save if would be fun to gain Biblical knowledge of it? Scratch that. Why get jiggy with the Absolute if the lads at the pub don't believe your boasting about it? 

Why are mathematicians interested in Borges?

I think the answer is that they suspect their own integrity is orthogonal to existence. They are like Brouwer's vegetarian condemned to be an evil parasite if he uses or is of use to what is of this carnivorous world. 

The three elements that we have just examined appear time and again in Borges's works, molded in literary forms in various ways. In the essay, “Cartesianism as Rhetoric (or, Why are Scientists Interested in Borges?)” in 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.

The problem here is that Borges is clearly signaling that his style is baroquely parodic, exhausting its own possibilities in advance, because truth be told, there are no paradoxes. There is only scrutiny blinded by its own object.  

In a little article that I wrote on the same topic, “Borges and Three Paradoxes of Mathematics,” I point out the elements of Borges's style that have affinity with the mathematical esthetic. Here is my principal thesis:

I said before that traces of mathematics abound in the work of Borges. Even in the passages that have nothing to do with mathematics, there is something in his writing, an element of style, that is particularly pleasing to the mathematical esthetic.

I suppose genuine mathematicians are stimulated by what Grothenieck calls Yoga- i.e. ideas which can unite disparate subjects under the rubric of greater generality. Borges was scrupulous in presenting only such material. This meant he was relatively poor even during his most productive years.  

I think that a clue to this element is expressed, inadvertantly, is this extraordinary passage from 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.”

In English, gaucho is more concrete and  'individual' then 'farmhand' and 'pampa' is more concrete and individual than 'rounded plain'. Perhaps, an Argentine would think the reverse. Still the 'participation' or 'methexy' associated with gaucho or pampa has an element of oikeisosis or 'natural' appropriation. 

When Borges writes, he typically accumulates examples, analogies, related stories, and variations on what he wants to tell.

He is uniting disparate subjects in the light of a more general, perhaps universal, principle which he can convey in a superbly compact and lapidary manner.  

In this way the thrust of the story that unfolds is at once particular and general, and his passages give the impression that his particular examples are self-supporting references to universal forms. Mathematicians proceed in the same way. When they study an example, a particular case, they examine it with the hope of discovering a stronger and more general property that they can abstract into a theorem. Mathematicians like to think that Borges writes exactly as they would if faced with the challenge: with a proud Platonism,

surely not. There is no ever present danger of 'modal collapse' in his oeuvre precisely because he is so scrupulous in guarding against 'the contamination of reality by the dream' or simulacrum.  

as if there existed a heaven of perfect fictions and a notion of truth for literature.

There is a 'witness' for it- though that witness may lie in the heart of a stone.  

This summarizes, in some way, what I think about the articulation of mathematical thought in the style of Borges.

I would say Borges is like Voevodsky. He shows, for any useful purpose, there are always good enough 'univalent foundations' of a categorical type. Sadly, as Socrates explained long ago, categories are like the oars on a galley which are only resorted to when there is no wind to belly out the sails.