The Musical Brain: And Other Stories

The Infinite

AS A KID I PLAYED some extremely strange games. They sound made up when I explain them, and I did, in fact, make them up myself, but many years ago, when I was still in the process of becoming the self that I am today. I made them up, or else my friends at the time did: it comes to the same thing because those kids contributed to the accumulation that resulted in me. The reason I’ve set out to describe the games and give a written account of them is that people have told me on more than one occasion that they really should be recorded, so that if I die tomorrow, the ideas won’t be lost forever. I’m not so sure of their uniqueness. Children are always coming up with the craziest things, but the repertoire is not infinite. Relying on my intuition and the law of probabilities, I’d be prepared to bet that the same or similar ideas have occurred to other children, at some point, somewhere. If that is the case, and a copy of this publication falls into the hands of a reader who was one of those children, these descriptions will serve as a reminder, and perhaps a resurrection, of a forgotten past. It will, I think, be necessary to go into some fairly complicated details, and this may lead to excessive technicality, but I’m undertaking this task in the hope of discovering what my childhood had in common with other distant, unknown childhoods, and since the shared element is bound to be something small, a fine point, and I don’t know which small thing it is, which detail in particular, I have no choice but to set them all out. There’s also a more practical reason, which relates to comprehensibility: even the most insignificant details are important for the complete explanation of mechanisms that might, at first glance, seem absurd. One has to work through the list of senseless oddities so as not to miss the one that has the magic power to make sense of everything.

I will begin with a mathematical or pseudomathematical game for two players, which consisted simply of naming a bigger number than the one just named by your opponent. If one player said “four” the other had to say “five” (or more: he could also say “a thousand”) in order to stay ahead, and so it went. Basically, that was the essence of the game; as you can see, it was extremely simple. Obviously, given the ordered sequence of numbers, to win you had to avoid the mistake of naming a number smaller than the one that had just been named. . But it’s also obvious that victory by default would be accidental in such a game and would not affect its essence. The winner was, essentially, the one who came up with a number so big that the other player couldn’t find a bigger one. We respected this principle: we never made mistakes, and if one of us had slipped up, the other would have been more than prepared to ignore it and keep going. So it’s hard to imagine how the game could ever have played itself out fully. There seems to be a contradiction in the fundamental idea. But I think all the difficulty springs from adopting an adult perspective, trying to understand the theory of the game and reconstruct a session of play. For us it wasn’t hard to understand; on the contrary, it was almost too easy (that’s why we complicated it a bit). The difficulties, which in any case we found amusing and absorbing, were on another level, as I will try to show. The game itself seemed perfectly natural to us.

Before getting down to details, however, some clarifications are necessary. First: age. We would have been ten and eleven years old (or eleven and twelve: Omar was a year older than me; we were at primary school but in the final grades). Which is to say that we were no longer little children learning how to count, fascinated and amazed by the miracle of arithmetic. Not at all. Also, back then, thirty-five years ago, learning was no game: it was straight down to business; not a minute was wasted. Even in our semirural school (School Number 2 in Coronel Pringles: it still exists), the academic level was remarkably high; these days it would seem too much to ask. And all the children, though most of them came from farms and had illiterate parents, kept up with the pace, no two ways about it. The “hump” was sixth grade, and many stopped there, but if you were in that class you marched with the rest, and it was no dawdle.

The characters: Omar and myself. I never played this game with other people. I can’t remember if I ever tried, but if I did, it didn’t work. It was the kind of game that has to find its players, and does so only by a modest miracle. It had found the two of us, and we had adapted so well to its intricate, crystalline recesses that we had become a part of it, and it a part of us, and everyone else was necessarily excluded. Not so much because we would have had to explain the rules, or allow for idiosyncrasies (it was a mathematical game), but because the two of us had already played so much — all afternoon, hundreds of times — and we couldn’t start over; other players could, but not Omar and I.

Omar Berruet was not my oldest friend; his family had moved to the neighborhood a couple of years earlier, from Greater Buenos Aires (Berazategui), but his parents were from Pringles. His mother and mine had been childhood friends; one of his father’s sisters lived around the corner and had two sons, the Moraña boys, whom I’d known for much longer; the older one was in my grade all through primary school. The Berruets rented the house next door to ours. Omar was an only child, a year older than me, so we weren’t in the same class at school, but being neighbors we got to be friends. We’d spend the whole day together. He was tall and thin with straight blond hair, pale-skinned and lymphatic, unlike me in every way: the attraction of opposites brought us together. I suspect that I tended to boss him around and subject him to my erratic and fanciful moods.

He was happy to go along with my whims, but he also had a hidden strength that a number of painful experiences taught me to respect. Omar wasn’t lacking in intelligence, but when it came to demonstrating it, he was, again, my opposite: while I was all boasting, noise, and display, he responded quietly, with irony and realism. (This is as good a place as any to mention that he stayed in Pringles, became a bank teller, and had eight children, one of whom died.)

And finally, the scene. Back then, the town of Coronel Pringles was more or less like it is today, but a bit smaller, not so built up, with more dirt roads. Calle Alvear, where we lived, was the last paved road; another hundred yards and there were vacant lots (whole empty blocks), farmhouses, the country. On our block there were five houses, all on the same side: Uruñuela’s place on the corner, the house where my aunts Alicia and María lived, our place, Gonzalo Barba’s house (he was my dad’s nephew and business partner), and the Berruets. On the other corner: my dad’s business Aira & Barba, with its yard and offices. The houses rented by Gonzalo and the Berruets belonged to Padelli, and their backyards adjoined his place, which was just around the corner. On the other side of the street, behind a long wall, was the land belonging to the corner houses, Astutti’s on the left and Perrier’s on the right. The most interesting things in those wild tracts were, in Astutti’s yard, a supermodern mobile home that the owner’s brother (I think) was building or cobbling together (this hobby outlasted my childhood), and in Perrier’s yard, a tree, which was in fact a pair of twin trees, with intertwined branches, a gigantic conifer, the biggest tree in Pringles, as high as a ten-story building and perfectly conical in shape.

Nothing ever happened in the street: a car went by every half hour. We had vast amounts of free time: we went to school in the mornings, and the afternoons lasted entire lifetimes. We didn’t have extracurricular activities the way kids do today; there was no television; the doors of our houses stood open. To play the number game we climbed into the cabin of the little red truck that belonged to Omar’s father and was almost always parked just outside the front door. .

Right. Now, the game.

Who came up with it? It must have been one of us. I can’t imagine us taking it from somewhere else, ready-made. Thinking back, I’ve always seen the game as a blend of invention and practice. Or rather, I see the practice of it as permanent invention, without any kind of prior idea. And if I try to work out which of us was behind it, I have to conclude that I was the inventor. There’s something about the thrust of it, a kind of fantasy or exuberance, something elusive but utterly typical of me as I was at that age. Omar was at the opposite extreme. But, strangely, those vertiginous tunnels could be entered from the opposite extreme as well.

There were no rules. Although we spent our lives inventing rules for all our games, as kids always do, this game had none, perhaps because we realized that they were inadequate, bound to fall short, or just too easy to make up.

Now that I think of it, there was a rule, but it was transient and could be revoked at our convenience. We applied it once and forgot it the next time, but for some reason it has remained in my memory, and it must have remained in the game as well. It was pretty inoffensive: all it did was specify that the biggest possible number, the upper limit, would be eight. Not the number eight itself, but any number containing eight: eight tenths, eight hundred thousand, eight billion. It was really an extra accelerator (as if we needed one!) to take the game to another level.

It’s not that there were levels in the game, or series within the series, or if there were such things, we didn’t bother with them. But there were differences in speed, alternations between “step by step” and “leap,” and we could take them to extremes that are not to be found among the mobile, spatiotemporal sculptures of physical reality. These differences were always rushes, even our lapses into the hyperslow. But it never got out of control; even the all-encompassing acceleration was a kind of slowness. Which meant that within the game’s austere monomania, we could use speed to keep changing the subject of the conversation (since subjects are speeds).

“Three.”

“A hundred.”

“A hundred and one.”

“A hundred and one point zero one.”

“Eight hundred and ninety-nine thousand nine hundred and ninety-nine.”

“Four million.”

“Four million and one.”

“Four million and two.”

“Four million and three.”

“Four million and four.”

“Four million and four point four four four.”

“Four million and four point four four.”

“Four million and four point four.”

“Four million and four point three.”

“Four million and four point one.”

“Half a trillion.”

We never bothered to find out what a trillion was (or a quadrillion, a quintillion, a sextillion, although we used the terms). Whatever it was, we stuck with it.

“Half a trillion and one.”

“A trillion.”

“Eight trillions.”

“Eight trillions and eight.”

We did the same with “billion,” although in that case we knew that it meant a thousand millions. So if a million was “one,” a billion was a thousand of those “ones.” But we never went as far as counting how many zeros it contained and using that to calculate (there should be nine, I think). It would have been tedious, a drag, no fun. And we were playing a game. We were impatient, like all kids, and we had invented a game ideally suited to impatience: the leaping game. Although we spent hours and whole afternoons sitting still in the cabin of the little red truck that belonged to Omar’s dad, we were exercising our impatience. Otherwise, it would have been a sort of numerological craftwork, and I would describe our game as art, not craft.

We didn’t even know if a billion was bigger than a trillion. What did it matter? It was better not to know. We both hid our ignorance, and never put each other to the test. And in spite of this, the game remained very easy to play.

We were attracted by big numbers, inevitably: it followed from the nature of the game. They were the gravitational force accelerating our fall. But at the same time we held them in contempt, as indicated by the fact that we didn’t bother to find out exactly how big they were. Numbers were one thing and big numbers were another: with numbers we were in the domain of intuition (eight could be eight things or eight points; the same with eighty, or even eight hundred million); but when it came to really big numbers we were thinking blind; the game became purely verbal, a matter of combining words, not numbers.

“A billion.”

“A trillion billions.”

“Half a billion trillion billions.”

“A billion billion trillion billion trillions.”

It’s true that numbers reappeared on the far side of these accumulations.

“A billion billions.”

“A billion billions and six.”

“Six billion billions and six point zero zero zero zero zero zero six.”

These were luxuries, embellishments that we allowed ourselves, as if to stave off a boredom that we didn’t feel and couldn’t have felt, but could nevertheless imagine. On the other hand, we both agreed not to accept things like “six billion six billions”: that wasn’t a number but a multiplication. We had more than enough to do with numbers pure and simple. Why make life complicated?

I don’t know how long this game lasted. Months, years. It never bored us, never ceased to surprise and stimulate us. It was one of the high points of our childhood, and when we finally stopped, it wasn’t because we’d exhausted the game, or tired of it, but because we had grown up and gone our separate ways. I should add that we didn’t play it all the time, and it wasn’t our only game. Not at all. We had dozens of different games, some more extravagant and fantastic than others. I have resolved to describe them one by one, and this is the one I happened to begin with, but I wouldn’t want the rather artificial way in which I’ve isolated the number game to give a false impression. We weren’t a pair of obsessives permanently shut up in the cabin of an old truck spouting numbers. A new fantasy would excite us and we could forget about the numbers for weeks at a time. Then we’d start over, exactly like before. . On reflection, the way I’ve presented the game in isolation is not so artificial after all, because various features did set it apart: its immutable simplicity, its naturalness, its secrecy. I think we kept it secret, but not for any special reason, not because it was a secret: just because we forgot to tell anyone, or the opportunity never arose.

The game was very simple and austere, and that’s why it was inexhaustible. By definition, it couldn’t be boring. And anyway, how could we have been bored? It was pure freedom. In the playing, the game revealed itself as part of life, and life was vast, elastic, endless. We knew that prior to any experience. We were austere, like our parents, the neighborhood, the town, and life in Pringles. Today it’s almost impossible to imagine just how simple that life was. Having lived it myself doesn’t help. I’m trying to imagine it, to give some form to that idea of simplicity, putting memories aside, avoiding them as much as possible.

Sometimes, in the plenitude that followed an especially satisfactory session of play, we did something that seemed to depart from simplicity, but in fact confirmed it. We played the same game as a joke, out of pure exuberance, as if we hadn’t understood, as if we were savages, or stupid.

“One.”

“Zero.”

“Minus a thousand.”

“Zero point zero nine nine nine.”

“Minus three.”

“A hundred and fifteen.”

“A million billion quadrillions.”

“Two.”

“Two.”

This didn’t last very long, because it was too dizzying, too horizontal. A minute of it gave us a totally different perspective on what we’d been doing for hours before, as if we’d jumped down off a horse, descended from the world of mental numbers to that of real numbers, to the earth where the numbers lived. If we had known what surrealism was, we would have cried: Surrealism is so beautiful! It changes everything! Then we went back to the normal game like someone going back to sleep, back to efficiency and representation.

All the same, a certain nostalgia crept in, a vague feeling of dissatisfaction. It didn’t happen at a particular moment, after a day or a month or a year. . I’m not writing a chronological history of this game, from invention and development to decadence and neglect. I couldn’t, because it didn’t happen like that. The successiveness of this narration is an unavoidable defect; I don’t see how I could avoid it while still giving an account of the game. The dissatisfaction had to do with the difference between numbers and words. We had made the very austere decision to limit ourselves to real, “classical” numbers. Positive or negative, but everyday numbers, of the kind that are used for counting things. And numbers are not words. Words are used to name numbers, but they’re not the same.

This, of course, had been a choice, a pact that we renewed each time we started playing, and we didn’t complain. The game made thought mobile and porous, loosened it like a kind of relaxing yoga, allowing us to see the kingdom of the sayable in all its amplitude while preventing us from entering it. Words were more than numbers; they were everything. Numbers were a little subset of the universe of words, a marginal, faraway planetary system where it was always night. We hid there, sheltered from the excesses of the unknown, and tended our garden.

From our hiding place we could see words as we’d never seen them before. We’d distanced ourselves from them so that we could see how beautiful, funny, and amazingly effective they were. Words were magical jewels with unlimited powers, and all we had to do, we felt, was reach out and take them. But that feeling was an effect of the distance, and if we crossed the gap, the game dissolved like a mirage. We knew that, and yet some strange perversion, or the lure of danger, sustained our crazy longing to try. .

We were testing the power of words every day. I never missed an opportunity: I’d see one coming, feel that I was grasping the mirage, taking control of its unerring death ray, and

I wouldn’t rest until I’d fired it. My favorite victim, needless to say, was Omar:

“Let’s play who can tell the biggest lie.”

Omar shrugged:

“I just saw Miguel go past on his bike.”

“No, not like that. . Let’s pretend we’re two fishermen and we’re lying about what we’ve caught. The one who says the biggest lie wins.”

I emphasized “biggest,” to suggest that it had something to do with the number game. Omar, who could be diabolically clever when he wanted, made it hard for me:

“I caught a whale.”

“Listen, Omar. Let’s make it simpler. The only thing you can say is the weight of the fish, its length in yards, or its age. And let’s set some upper limits: eight tons, eighty yards, and eight hundred years. No! Let’s make it really simple! Just the age. Let’s suppose fish go on growing until they die. So by saying the age you’re saying the length, the width, the weight, and all that. And let’s suppose they can live any number of years but the highest number we can say is eight hundred. You start.”

Omar would have had to be really stupid not to realize by this stage that I had something up my sleeve, something very specific. And he wasn’t stupid at all; he was very intelligent. He had to be, supremely: he was the measure of my intelligence. In the end, he resigned himself:

“I caught an eight-hundred-year-old fish.”

“I caught its grandfather.”

Omar clicked his tongue with infinite scorn. I wasn’t especially proud of the idea myself: it was an unfortunate attempt to play a practical joke on my friend by recycling a gag I’d read in a magazine, which must have gone something like this: “Two fishermen, inveterate liars, are talking about the day’s catch: ‘I caught a marlin this big.’ ‘Yeah, yeah, that was the newborn baby. I caught its mother.’ ” What a flat joke! I worked so hard to set it up, and for such a paltry result! What did I ever see in it? Nothing but the power of the word. The joke contained, in a nutshell, both our number game (the lying fishermen could go on increasing the dimensions of the fish ad libitum) and that which transcended it: a word (like “mom,” or “dad,” or “grandfather”) triumphed over the whole series of numbers by placing itself on a different level.

So that’s what I was referring to. That was the game’s limit, its splendor and its misery.

Until we discovered the existence of that word. This, I repeat, did not occur at a particular moment in the game’s history. It happened at the beginning — it was the beginning.

The word was “infinity.” Logical, isn’t it? Perhaps even blindingly obvious? In fact, it has been a strain for me to call it the “number game,” when it was really the “infinity game,” which is how I’ve always thought of it. If I had to transcribe the archetypal session, the original, the matrix, it would be simply this:

“One.”

“Infinity.”

Everything else sprang from that. How could it have been otherwise? Why would we have denied ourselves that leap when every other kind of leap was allowed? In fact, it was the other way around: all the leaps that we allowed ourselves were based on the leap into the heterogeneous world of words.

From this point on, we can, I think, begin to glimpse an answer to the question that has been building subliminally since I began to describe the game: when did the sessions come to an end? Who was the winner? It’s not enough to say: Never, no one. I’ve given the impression that neither of us ever fell into the traps that we were continually laying for each other. That’s true in the abstract, in the myth that was ritually expressed by the various series, but it can’t always have been the case in the actual playing of the game. To be honest, I can’t remember.

I feel I can remember it all, as if I were hallucinating (otherwise, I wouldn’t be writing this), but I have to admit that there are things I don’t remember. And if I were to be absolutely frank, I would have to say I don’t remember anything. An escalation, once again. But there’s no contradiction. In fact, the only thing I remember with the real microscopic clarity you need in order to write is the forgetting.

So:

“Infinity.”

Infinity is the limit of all numbers, the invisible limit. As I said, with the big numbers we were thinking blind, beyond intuition; but infinity is the transition to the blindness of blindness, something like the negation of negation. And that’s where the real visibility of my forgotten memory begins. Do I actually know what infinity means? It’s all I can know, but I can’t know it.

There’s something wonderfully practical about leaping to the infinite, the sooner the better. It thwarts every kind of patience. There’s no point waiting for it. I loved it blindly. It was the sunny day of our childhood. That’s why we never wondered what it meant, not once. Because it was the infinite, the leap had already happened.

Our refusal to think it through had a number of consequences. We knew that it didn’t make sense to talk about “half an infinity,” because in the realm of the infinite the parts are equal to the whole (half of infinity, the series of even numbers, say, is just as infinite as the other half, or the whole). But, returning surreptitiously to healthy common sense, we accepted that two infinities were bigger than one.

“Two infinities.”

“Two hundred and thirty million infinities.”

“Seven quintillion infinities.”

“Seven thousand billion billion quintillion infinities.”

“A hundred thousand billion billion trillion quintillion infinities.”

And so we continued until the word made its triumphant return:

“Infinity infinities.”

This formula could, in turn, be included in a series of the same kind:

“Ten billion infinity infinities.”

“Eight thousand billion trillion quadrillion quintillion infinity infinities.”

We didn’t pronounce these words, of course. I should make it clear that in general we didn’t actually articulate all the little series that I’ve been transcribing here; neither these particular ones nor others of the same kind. I’ve set it out in this long-winded way to make myself clear, but it wasn’t our intention to labor the obvious; on the contrary. All these series, and in fact all the series that might have occurred to us, were virtual. It would have been boring to say them. We weren’t prepared to waste our precious childhood hours on bureaucratic tasks like that; and, above all, it would have been pointless, because each term was surpassed and annihilated by the next. Numbers have that banal quality, like examples: they’re interchangeable. What matters is something else. Stripping away all the stupid and bothersome foliage of examples, what we should have said was:

“A number.”

“A number bigger than that.”

“A number bigger than that.”

“A number bigger than that.”

Although, of course, if we’d done that, it wouldn’t have been a game.

The word returned once more:

“Infinity infinity infinities.”

Only one number was bigger than that:

“Infinity infinity infinity infinities.”

I mean: that was the smallest bigger number, not the only one, because the series of infinities could be extended indefinitely. And so we ended up repeating the word over and over in a typically childish way, at the top of our voices, as if it were a tongue-twister.

“Infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinities.”

There was, believe it or not, an even bigger number: the number that one of us would say next. It was pure virtuality, the state in relation to which the game deployed all its marvelous possibilities.

Amazingly, given our greediness, it never occurred to us to add the name of a thing to the numbers. Bare like that, the numbers were nothing, and we wanted everything. There’s no real contradiction between the two half-wild children I’ve been describing, in a society that seems archaic and primitive today, and the fact that we were greedy. We wanted everything, including Rolls-Royces and objects that would have been no use to us, like diamonds and subatomic particle accelerators. We wanted them so badly! With an almost anguished longing. But there’s no contradiction. The supernatural frugality of our parents’ lives had apparently achieved its goal, and perhaps that goal was us. They were still using the furniture they’d bought when they got married; the rent was fixed; cars lasted forever; and the mania for household appliances would take decades to reach Pringles. .

What’s more, we always had enough money to buy the few things on sale that interested us: picture cards, comic books, marbles, chewing gum. I don’t know where we got it from, but it never ran out. And yet we were insatiable, greedy, supremely avid. We wanted a schooner with a solid-gold figurehead and silken sails, and in our fantasies about discovering a treasure — doubloons and ingots and emeralds — we weren’t so rash as to spend it at once on this or that; we converted it into cash, placed the sum in a bank and, as the compound interest mounted, bought ourselves Easter Island statues, the Taj Mahal, racing cars, and slaves. Even then we weren’t satisfied. We wanted the philosopher’s stone or, better, Aladdin’s lamp. We weren’t deterred by the fate of Midas: we were planning to wear gloves.

The numbers were numbers and nothing more. Especially the big numbers. Eight could still be eight cars; one for each day of the week, and one extra with swamper tires for rainy days. But a billion? An infinity? Infinity infinities? That could only be money. Why we never talked about this is a mystery to me. Maybe it went without saying.

The tree, a giant dark-green triangle hiding half the sky, kept watch over the little red truck, with the two of us inside, tireless and happy. The day was a stillness of sunlight.

Among the many daydreams prompted by the natural world, an especially frequent variety explores the perfection of the mechanisms by means of which living beings function. Gills, for example. A fish, as it swims, lets water pass through what I presume is a sort of hydrodynamic valve, and extracts from that water the oxygen it needs. How it does this doesn’t matter. Somehow. To simplify and conceptualize, as I did in the two previous sentences, it’s relatively straightforward: you can imagine an apparatus, an alembic, in which water is broken down and oxygen retained while the hydrogen is allowed to escape. Daydreaming retains something, too, and lets something else escape. What it retains in this case is the size of the fish: some fish are tiny, no bigger than a match, and in a fish so small the apparatus becomes a marvel. . Or does it? To put it together and take it apart, we’d have to use magnifying glasses and microscopes, screwdrivers and tweezers and tiny hammers the size of needle points; it would be a feat of patience and dexterity. A feat that might be pulled off once, at a very optimistic estimate; but there are billions of those fish in the sea. . At this point we should bow to the evidence and admit that the reasoning behind the daydream contains an error. Two errors, actually. The first is having overlooked the difference between doing something and finding it done. No one has ever set about making gills for little fish. They are ready-made. Constructivism is an empty illusion. The second has to do with size. Here the error lies in taking our human size as a fixed standard. In fact, the demiurge chooses a scale appropriate to each case, or rather he chose it at the outset, in the process of creating all the sizes. It’s a fluid, elastic studio, where it’s always a pleasure and a joy to work, in comfortable conditions, by hand. I think that’s why concepts are so attractive, why humans cling to them so stubbornly, from childhood on, scorning all reality checks. It’s examples that are cumbersome and unwieldy; for them, we’re never well proportioned, we’re always giants or dwarfs.

Daydreams are always about concepts, not examples. I wouldn’t want anything I’ve written to be taken as an example.

MARCH 21, 1993