* * * *
“Just by changing our path slightly so that we always stay below the line p=q, we'll be able to enumerate all the rational numbers between 0 and 1. Since there's a one-to-one mapping, or a bijection, between the naturals and the positive rationals, and a bijection between the naturals and the rationals between 0 and 1, we know that all three sets are the same size, or have the same cardinality. The cardinal number of the set of all natural numbers is called aleph-null, after the Hebrew letter aleph.”
“Aleph-null confounds our intuitions. You can see that all the rational numbers between 0 and 1 take up half of the plane of all rational numbers in the picture above, with all the other rational numbers in the other half, and yet one half is not bigger than the other, or the whole plane. Divide infinity in half, and you still have infinity. Turn the number line into a plane, multiply infinity by infinity, and you end up still with the same size of infinity.
* * * *
* * * *
“It seems to say that a part can be just as large as the whole. And it is possible to map the whole infinite line of rational numbers into the seemingly finite segment between 0 and 1. In every grain of sand is the universe.”
* * * *
* * * *
One of the few memories that David had of his father was a trip they had all taken to Myrtle Beach. David could not even be sure that it really happened; he was so young back then.
He remembered digging in the sand with a plastic shovel—red, yellow? Well, in this particular moment the shovel was blue, like the blazer the lady lawyer was wearing. Betty was sunbathing to the side, and his father was helping him by moving the sand that he dug up into a plastic bucket.
The sun was hot but not unpleasant. The voices of the people on the beach faded into an indistinct murmur. One shovelful.
He was mesmerized by the smooth, hypnotic way sand moved: solid particles that flowed like a liquid, falling, sliding, tumbling from the blue plastic shovel into the bucket. Two shovelfuls.
The particles were so fine, like flour, like salt. He wondered how many grains of sand had tumbled from his shovel between the time he began this thought and now, right now. Three shovelfuls. If he stared really hard, could he see the individual grains? Four shovelfuls. He held his breath.
“You're counting?” His father asked.
He nodded. The sounds and sights of the world flooded back into his awareness. He gasped, like a swimmer coming up for air.
“It will take a long time to count all the sand on this beach.”
“How long?”
“Longer than it took you to count the triangles on my towel,” Betty said. He felt her hand, cool and smooth, lightly caressing his back. He relaxed his back. It was a nice feeling.
His father looked at him, and he stared back. It was an intense stare that others would have found off-putting, but his father smiled. “It would take until infinity, David.”
“What's infinity?”
“It's beyond the time that you and I have. Let me tell you something the Chinese philosopher Zhuangzi once said: if a man can live a hundred years, that's a pretty long life. But a life is filled with sickness, death, sorrow, and loss, so that in a month of days you might have just four or five when you laughed out loud. Space and time are infinite, but our lives are finite. To experience the infinite with the finite, we should just count those transcendental moments, those moments of joy.”
Betty's hands continued to stroke his back, and he saw that his father was no longer looking at him, but at his mother.
This is one of those moments, he decided.
* * * *
“You keep on fucking around with those numbers and books, and you'll end up like those criminals on Wall Street,” Jack said. “Nobody in this country wants to work honestly with his hands any more. That's why the Chinese are eating our lunch.”
David took his books and notes and retreated to the bedroom that he shared with the baby. She was taking a nap, and David stared at her face, so peaceful, oblivious to the sounds of the TV blasting from the living room.
Perhaps the world did not make sense because he was not counting properly. Maybe he was out of sync with the world.
David sat down at the desk. He drew a vertical line down the page, labeled the bottom 0 and the top 1. Then he tried to map out the sequence of rational numbers between 0 and 1 along the zigzagging path of the Cantor pairing function traveling in the Cartesian plane that Ms. Wu had drawn on the board at school. He drew a short horizontal line for each rational in the sequence. Gradually, he filled the page.
* * * *
The lines accumulated one after another, reaching high on each zig up the vertical axis, before methodically stepping down and filling in the blank space left behind on each falling zag to the horizontal axis.
There are an infinite number of moments in a finite life. Who says that you must stay in the present, and experience them in order?
The past was not the past. The same moments would be experienced again and again, and each time something new would be added. Given enough time, the blanks would be filled with the rational. The lines would complete a picture. The world made sense. All you had to do was wait.
* * * *
* * * *
Parts of our brain, consisting of regions in the frontal, parietal, and medial temporal lobes, are active only when we are not presently engaged in a cognitive task. When we are computing the sum of 12,391,424 and 38,234,231, figuring out how to get from home to the next job interview, or reading the latest mutual fund prospectus, these regions of the brain go dark on an MRI brain scan. But when we are not actively thinking about anything, the dark network of the brain lights up.
A much younger David flipped the page. Betty was out with Jack, and he was locked alone in the house. She had left him with a warning that he shouldn't answer the phone or the door or let anyone know that he was home. He did not find this strange. For all he knew, this was how all eight-year-old boys spent their evenings when their mothers went on dates. He enjoyed the company of the boxes of books left behind by his father far more than the company of other people, or Jack.
He did not like the novels much, but forced himself to plod through them slowly, reading them as textbooks about social and emotional rules that he could not fathom. He preferred the books about math, with their beautiful equations, fantastic graphs, and strange symbols that he could not pronounce.
And then there were the books about science, which he devoured the way other children read fairytales. Like this one:
It turns out that the dark network is where our species practices its most amazing power, a power more uniquely human than language, than math, than our ability to go to war and compose poetry. The dark network is where we engage in time travel.
Jack had begun to come over more frequently and sometimes stayed nights. David cataloged and enumerated the changes in his mother carefully, analyzing them as clues to what he could not intuit: the way his mother giggled, like a young girl in movies; the dresses that he had never seen her wear; the way more and more of Jack's things accumulated in their apartment.
The brain's perception of time offers up one mystery after another. There is no easy answer to how the brain perceives the passage of time, the steady conversion of the future to the present, and the present to the past. Is there a cluster of neurons that pulse steadily, like a metronome or the clock signal in a modern integrated circuit? Or is it the analog delay of activation potentials cascading across the neurons that tells us time is passing? Or perhaps time is measured by the chemical diffusion of neurotransmitters, and maybe that explains why time slows down when we are under the influence of drugs like cocaine, which boosts dopamine.
The key rattled in the door, and Betty and Jack stumbled in. David paused in his reading and looked up. A momentary cool breeze was followed by the smell of cigarettes, sweat, and alcohol filling the hot, stale air of the apartment. Jack plopped down on the couch and flicked on the TV. Be
tty came back from the kitchen with a half-filled glass, and as she approached Jack, she laughed, lost her balance, and tumbled into Jack's lap. The drink, miraculously, stayed unspilled. She kicked off her heels and put an arm around his neck.
“The boy is leaving books everywhere,” Jack said. He surveyed the piles all over the floor. “You can hardly walk around without kicking over a stack. What's with all the books anyway? I never see you read anything.”
In any event, research seems to suggest that we do not so much live in the present as an illusion of the present. Although your eyes may perceive your foot striking the ground a fraction of a second before the nerve impulse carrying the sensation travels from your foot to your brain, you do not perceive a delay. The brain sits in the cranium in darkness, and signals from around the body are integrated into a sense of now only after the slowest signal has arrived, which suggests that our conscious awareness of the present is delayed, a bit like a “live” broadcast. We may be like train passengers in rear-facing seats, always perceiving the “present” only when it has become the recent past.
“His father was a big reader,” Betty said. “He did real well in school. Got into UVA.”
Realizing that she was killing the mood, Betty stopped. She tried to kiss Jack.
“Then he got into you,” Jack said, moving his lips away from hers. A nasty tone had crept into his voice. He caressed her breasts through her dress. Betty blushed and reached up to stop him. Jack slapped her hands aside and laughed.
“Stay still. I'm showing the boy what you can't teach him.”
David turned his eyes away. He was not good at reading faces and could not explain what he saw in his mother's face at that moment. He felt it was like looking at her when she was undressed.
Not only is our sense of the present illusory, but we do not even spend most of our time in it. The dark network is where the brain takes its trips down memory lane and simulates the future. We relive our experiences to draw out lessons and play out possibilities to plan for what is to come. We imagine ourselves in other times, and in the process we live out many lifetimes in one.
“We need to clean this pigsty up,” Jack said. “Too much stuff you don't need any more.”
Unlike a computer, which can retrieve data from long-term memory without alteration and process it in short-term memory, the brain's memories, patterns of activation potentials, are processed in-place, and are thus altered each time we remember. We cannot step into the same place in Heraclitus's river twice not only because we cannot physically go back in time, but because even our memory of each moment remains ever changing.
“The boy sits there and reads all day. It's not natural. Look at him. Not a peep out of him all this time we've been back. He's creeping me out. Hey, I'm talking to you!”
He threw the remote at David. It thudded against his chest and clattered to the ground. David flinched and looked up. Their eyes met. After a moment, Jack swore, and he began to push Betty off.
Jack he found most baffling of all. He could not figure out the rules needed to predict his outbursts.
In the end Betty coaxed Jack into coming with her into the bedroom. David was left alone in the living room. Slowly, he uncurled himself, ignored the pain, and cradled the book in his lap.
The dark network is the default mode for our brain. It is the state that our brain drifts toward whenever we are not occupied by some pressing concern in the present. Whenever we are not thinking about anything in particular, we drift in time, cast off from the anchor of the present to wander over the infinite paths of our lives, those taken, untaken, and yet to be mapped.
The brain's capacity to manipulate time remains mostly unexplored. If simultaneity of the senses is largely an illusion, could our sense of the linearity of experience similarly be constructed? We seem to be skipping over the river of time, aware of the present only from time to time by an act of will. If trauma or disease affected the relevant regions of the brain, could we cut experience into ever-finer slices, experience them out of order, or stay forever away from the present, lost in time?
The next day, Jack and Betty packed up all the books and brought them to the dumpster.
“You can't read those books anyway,” Betty said, trying to comfort David. “I don't even understand them. We need to move on with our lives.”
* * * *
“You might think,” Ms. Wu said, “based on what we studied last time, that all infinities are aleph-null, but this is not true. The countably infinite is only the smallest of the infinities.
“The set of all real numbers, for example, is not countably infinite. It is far bigger. Cantor found a way to prove this.
“Suppose that the real numbers are countably infinite. Then there must be a bijection from the naturals to the reals. The reals must be capable of being counted. Since every real number can be written as an endless sequence of decimal digits—just pad out the end with repeating 0's if they don't go on endlessly—we can imagine that the enumeration will look something like this:
“Remember, this is supposed to be an enumeration of all the real numbers. But we can easily construct a new real number that cannot be on this list. Just take the first digit in the first number on the list and write down a new digit that's different. And take the second digit in the second number on the list and write down a new digit that's different. Continue this diagonal movement down the list.
“When you put the new digits together, you have a new real number. But this is a real number that does not exist on the list anywhere. It differs from the first number on the list in the first digit, from the second number on the list in the second digit, from the third number on the list in the third digit, and so on.
* * * *
* * * *
“You can construct an infinite many such real numbers that cannot be found on the list just by drawing new diagonals and flipping to new digits. There is no bijection from the naturals to the reals. No matter how you try to arrange the reals, more of them will slip through your fingers. The real numbers are infinite, but it's a much bigger kind of infinity than aleph-null. There are so many more real numbers than the natural numbers that the real numbers cannot be counted. We call the cardinality of this uncountable infinity beth-one.”
* * * *
“But even beth-one is still only a very small transfinite number. There are many more numbers that are much bigger, a true infinity of infinities. We'll get to those in the next few days. When Cantor first wrote about their existence, some theologians were deeply threatened by his work. They thought Cantor was challenging the absolute infinity and transcendence of God.
* * * *
* * * *
“But even just knowing beth-one is bigger than aleph-null allows you to see some wonderful things. For example, we know that the rational numbers are countable and have cardinality aleph-null. But the real numbers are the union of the set of rationals and the set of irrationals, and we know that the real numbers have cardinality beth-one.”
* * * *
* * * *
“Therefore, the set of irrational numbers must have cardinality greater than aleph-null, since we know that doubling aleph-null only gets you to aleph-null, not beth-one. In fact, we have proven that there must be uncountably many—or beth-one—irrationals.
“In other words, there are many, many more irrational numbers than rational numbers. Almost all real numbers are irrational. And by a similar argument, you can prove that almost all irrational numbers are transcendental and not algebraic, not findable as the root of a polynomial with integer coefficients. Even though so few transcendentals seem to touch our daily life—like pi and e—they make up most of the number line. Most of the math you've been studying in school all these years has been focused on just tiny slices of the continuum.”
* * * *
* * * *
The textbook from Ms. Wu included poetry quotations at the beginning of the chapter. David did not usually like poetry, which seemed to be comp
osed of the same language beneath the language that he was not attuned to, with metaphors and figures that confused him. But these were different. These seemed to say how he felt.
* * * *
Ah, awful weight! Infinity
Pressed down upon the finite Me!
—Edna St. Vincent Millay, Renascence
* * * *
I am large, I contain multitudes.
—Walt Whitman, Song of Myself
* * * *
The lines he had been drawing would never complete the number line; he understood that now. The irrational space between them was infinite. The picture would never cohere and make sense. Life could not be reduced to its rational moments.
But the rational moments were not worth counting. There was nothing wrong with him. He finally understood. Isn't it wonderful to know, to really know, that the irrational is the rule, and not the exception, and to know further that most of that is transcendental, even if we are aware of so few of them? Life did not make sense. It did not need to. Why were the theologians afraid of Cantor? This was a truth to be celebrated. It is only the transcendentally happy moments that should be counted.
A scream from Betty on the other side of the bedroom door interrupted him, followed by the cries of the baby. David was amazed that such a small body could be the source of such loud cries, the full-throated demand for justice, for sense, so fearless and sad at the same time. When the baby stopped to take a breath, he could hear Betty's muffled voice pleading indistinctly. Then came the sound of plates smashing on the floor.
He opened the door.
He could tell that Jack was only a little drunk. He stood steady on his feet. Betty's long, smooth hair, of which she was very proud, was wrapped around Jack's hand and held in a fist. She was on her knees, her hands pulling at his hand holding her hair. She had put the baby down on the couch, where she flailed her limbs and her face was turning red from crying and the lack of air.
Maybe Jack was fired from his job again. Maybe he got into an argument with the Vietnamese grocers down the block. Maybe he didn't like Betty's dress when he got home. Maybe the baby was crying when he didn't want to hear her.
Asimov's SF, December 2011 Page 12