Year’s Best SF 15 Read online

  In mathematics there is the sequence of natural numbers, walking like small, determined soldiers into infinity. But there are less obvious infinities as well, as Abdul Karim knows. Draw a straight line, mark zero on one end and the number one at the other. How many numbers between zero and one? If you start counting now, you’ll still be counting when the universe ends, and you’ll be nowhere near one. In your journey from one end to the other you’ll encounter the rational numbers and the irrational numbers, most notably the transcendentals. The transcendental numbers are the most intriguing—you can’t generate them from integers by division, or by solving simple equations. Yet in the simple number line there are nearly impenetrable thickets of them; they are the densest, most numerous of all numbers. It is only when you take certain ratios like the circumference of a circle to its diameter, or add an infinite number of terms in a series, or negotiate the countless steps of infinite continued fractions, do these transcendental numbers emerge. The most famous of these is, of course, pi, 3.14159…, where there is an infinity of non-repeating numbers after the decimal point. The transcendentals! Theirs is a universe richer in infinities than we can imagine.

  In finiteness—in that little stick of a number line—there is infinity. What a deep and beautiful concept, thinks Abdul Karim! Perhaps there are infinities in us too, universes of them.

  The prime numbers are another category that capture his imagination. The atoms of integer arithmetic, the select few that generate all other integers, as the letters of an alphabet generate all words. There are an infinite number of primes, as befits what he thinks of as God’s alphabet…

  How ineffably mysterious the primes are! They seem to occur at random in the sequence of numbers: 2, 3, 5, 7, 11…There is no way to predict the next number in the sequence without actually testing it. No formula that generates all the primes. And yet, there is a mysterious regularity in these numbers that has eluded the greatest mathematicians of the world. Glimpsed by Riemann, but as yet unproven, there are hints of order so deep, so profound, that it is as yet beyond us.

  To look for infinity in an apparently finite world—what nobler occupation for a human being, and one like Abdul Karim, in particular?

  As a child he questioned the elders at the mosque: What does it mean to say that Allah is simultaneously one, and infinite? When he was older he read the philosophies of Al Kindi and Al Ghazali, Ibn Sina and Iqbal, but his restless mind found no answers. For much of his life he has been convinced that mathematics, not the quarrels of philosophers, is the key to the deepest mysteries.

  He wonders whether the farishte that have kept him company all his life know the answer to what he seeks. Sometimes, when he sees one at the edge of his vision, he asks a question into the silence. Without turning around.

  Is the Riemann Hypothesis true?

  Silence.

  Are prime numbers the key to understanding infinity?

  Silence.

  Is there a connection between transcendental numbers and the primes?

  There has never been an answer.

  But sometimes, a hint, a whisper of a voice that speaks in his mind. Abdul Karim does not know whether his mind is playing tricks upon him or not, because he cannot make out what the voice is saying. He sighs and buries himself in his studies.

  He reads about prime numbers in Nature. He learns that the distribution of energy level spacings of excited uranium nuclei seem to match the distribution of spacings between prime numbers. Feverishly he turns the pages of the article, studies the graphs, tries to understand. How strange that Allah has left a hint in the depths of atomic nuclei! He is barely familiar with modern physics—he raids the library to learn about the structure of atoms.

  His imagination ranges far. Meditating on his readings, he grows suspicious now that perhaps matter is infinitely divisible. He is beset by the notion that maybe there is no such thing as an elementary particle. Take a quark and it’s full of preons. Perhaps preons themselves are full of smaller and smaller things. There is no limit to this increasingly fine graininess of matter.

  How much more palatable this is than the thought that the process stops somewhere, that at some point there is a pre-preon, for example, that is composed of nothing else but itself. How fractally sound, how beautiful if matter is a matter of infinitely nested boxes.

  There is a symmetry in it that pleases him. After all, there is infinity in the very large too. Our universe, ever expanding, apparently without limit.

  He turns to the work of Georg Cantor, who had the audacity to formalize the mathematical study of infinity. Abdul Karim painstakingly goes over the mathematics, drawing his finger under every line, every equation in the yellowing textbook, scribbling frantically with his pencil. Cantor is the one who discovered that certain infinite sets are more infinite than others—that there are tiers and strata of infinity. Look at the integers, 1, 2, 3, 4…Infinite, but of a lower order of infinity than the real numbers like 1.67, 2.93, etc. Let us say the set of integers is of order Aleph-null, the set of real numbers of order Aleph-One, like the hierarchical ranks of a king’s courtiers. The question that plagued Cantor and eventually cost him his life and sanity was the Continuum Hypothesis, which states that there is no infinite set of numbers with order between Aleph-Null and Aleph-One. In other words, Aleph-One succeeds Aleph-Null; there is no intermediate rank. But Cantor could not prove this.

  He developed the mathematics of infinite sets. Infinity plus infinity equals infinity. Infinity minus infinity equals infinity. But the Continuum Hypothesis remained beyond his reach.

  Abdul Karim thinks of Cantor as a cartographer in a bizarre new world. Here the cliffs of infinity reach endlessly toward the sky, and Cantor is a tiny figure lost in the grandeur, a fly on a precipice. And yet, what boldness! What spirit! To have the gall to actually classify infinity…

  His explorations take him to an article on the mathematicians of ancient India. They had specific words for large numbers. One purvi, a unit of time, is seven hundred and fifty-six thousand billion years. One sirsaprahelika is eight point four million Purvis raised to the twenty-eighth power. What did they see that caused them to play with such large numbers? What vistas were revealed before them? What wonderful arrogance possessed them that they, puny things, could dream so large?

  He mentions this once to his friend, a Hindu called Gangadhar, who lives not far away. Gangadhar’s hands pause over the chessboard (their weekly game is in progress) and he intones a verse from the Vedas:

  From the Infinite, take the Infinite, and lo! Infinity remains…

  Abdul Karim is astounded. That his ancestors could anticipate Georg Cantor by four millennia!

  That fondness for science…that affability and condescension which God shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on calculating by al-jabr and al-muqabala, confining it to what is easiest and most useful in arithmetic.

  —Al Khwarizmi, eighth century Arab mathematician

  Mathematics came to the boy almost as naturally as breathing. He made a clean sweep of the exams in the little municipal school. The neighborhood was provincial, dominated by small tradesmen, minor government officials and the like, and their children seemed to have inherited or acquired their plodding practicality. Nobody understood that strangely clever Muslim boy, except for a Hindu classmate, Gangadhar, who was a well-liked, outgoing fellow. Although Gangadhar played gulli-danda on the streets and could run faster than anybody, he had a passion for literature, especially poetry—a pursuit perhaps as impractical as pure mathematics. The two were drawn together and spent many hours sitting on the compound wall at the back of the school, eating stolen jamuns from the trees overhead and talking about subjects ranging from Urdu poetry and Sanskrit verse to whether mathematics pervaded everything, including human emotions. They felt very grown-up and mature for their stations. Gangadhar was the one who, s
hyly, and with many giggles, first introduced Kalidasa’s erotic poetry to Abdul Karim. At that time girls were a mystery to them both: although they shared classrooms it seemed to them that girls (a completely different species from their sisters, of course) were strange, graceful, alien creatures from another world. Kalidasa’s lyrical descriptions of breasts and hips evoked in them unarticulated longings.

  They had the occasional fight, as friends do. The first serious one happened when there were some Hindu-Muslim tensions in the city just before the elections. Gangadhar came to Abdul in the school playground and knocked him flat.

  “You’re a bloodthirsty Muslim!” he said, almost as though he had just realized it.

  “You’re a hell-bound kafir!”

  They punched each other, wrestled the other to the ground. Finally, with cut lips and bruises, they stared fiercely at each other and staggered away. The next day they played gulli-danda in the street on opposite sides for the first time.

  Then they ran into each other in the school library. Abdul Karim tensed, ready to hit back if Gangadhar hit him. Gangadhar looked as if he was thinking about it for a moment, but then, somewhat embarrassedly, he held out a book.

  “New book…on mathematics. Thought you’d want to see it…”

  After that they were sitting on the wall again, as usual.

  Their friendship had even survived the great riots four years later, when the city became a charnel house—buildings and bodies burned, and unspeakable atrocities were committed by both Hindus and Muslims. Some political leader of one side or another had made a provocative proclamation that he could not even remember, and tempers had been inflamed. There was an incident—a fight at a bus-stop, accusations of police brutality against the Muslim side, and things had spiraled out of control. Abdul’s elder sister Ayesha had been at the market with a cousin when the worst of the violence broke out. They had been separated in the stampede; the cousin had come back, bloodied but alive, and nobody had ever seen Ayesha again.

  The family never recovered. Abdul’s mother went through the motions of living but her heart wasn’t in it. His father lost weight, became a shrunken mockery of his old, vigorous self—he would die only a few years later. As for Abdul—the news reports about atrocities fed his nightmares and in his dreams he saw his sister bludgeoned, raped, torn to pieces again and again and again. When the city calmed down, he spent his days roaming the streets of the market, hoping for a sign of Ayesha—a body even—torn between hope and feverish rage.

  Their father stopped seeing his Hindu friends. The only reason Abdul did not follow suit was because Gangadhar’s people had sheltered a Muslim family during the carnage, and had turned off a mob of enraged Hindus.

  Over time the wound—if it did not quite heal—became bearable enough that he could start living again. He threw himself into his beloved mathematics, isolating himself from everyone but his family and Gangadhar. The world had wronged him. He did not owe it anything.

  Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world.

  —The Mathematician Bhaskara, commenting on the

  6th century Indian mathematician Aryabhata,

  a hundred years later

  Abdul Karim was the first in his family to go to college. By a stroke of great luck, Gangadhar went to the same regional institution, majoring in Hindi literature while Abdul Karim buried himself in mathematical arcana. Abdul’s father had become reconciled to his son’s obsession and obvious talent. Abdul Karim himself, glowing with praise from his teachers, wanted to follow in the footsteps of Ramanujan. Just as the goddess Namakkal had appeared to that untutored genius in his dreams, writing mathematical formulas on his tongue (or so Ramanujan had said), Abdul Karim wondered if the farishte had been sent by Allah so that he, too, might be blessed with mathematical insight.

  During that time an event occurred that convinced him of this.

  Abdul was in the college library, working on a problem in differential geometry, when he sensed a farishta at the edge of his field of vision. As he had done countless times before, he turned his head slowly, expecting the vision to vanish.

  Instead he saw a dark shadow standing in front of the long bookcase. It was vaguely human-shaped. It turned slowly, revealing itself to be thin as paper—but as it turned it seemed to acquire thickness, hints of features over its dark, slender form. And then it seemed to Abdul that a door opened in the air, just a crack, and he had a vision of an unutterably strange world beyond. The shadow stood at the door, beckoning with one arm, but Abdul Karim sat still, frozen with wonder. Before he could rouse himself and get up, the door and the shadow both rotated swiftly and vanished, and he was left staring at the stack of books on the shelf.

  After this he was convinced of his destiny. He dreamed obsessively of the strange world he had glimpsed; every time he sensed a farishta he turned his head slowly toward it—and every time it vanished. He told himself it was just a matter of time before one of them came, remained, and perhaps—wonder of wonders—took him to that other world.

  Then his father died unexpectedly. That was the end of Abdul Karim’s career as a mathematician. He had to return home to take care of his mother, his two remaining sisters and a brother. The only thing he was qualified for was teaching. Ultimately he would find a job at the same municipal school from which he had graduated.

  On the train home, he saw a woman. The train was stopped on a bridge. Below him was the sleepy curve of a small river, gold in the early morning light, mists rising faintly off it, and on the shore a woman with a clay water pot. She had taken a dip in the river—her pale, ragged sari clung wetly to her as she picked up the pot and set it on her hip and began to climb the bank. In the light of dawn she was luminous, an apparition in the mist, the curve of the pot against the curve of her hip. Their eyes met from a distance—he imagined what he thought she saw, the silent train, a young man with a sparse beard looking at her as though she was the first woman in the world. Her own eyes gazed at him fearlessly as though she were a goddess looking into his soul. For a moment there were no barriers between them, no boundaries of gender, religion, caste or class. Then she turned and vanished behind a stand of shisham trees.

  He wasn’t sure if she had really been there in the half-light or whether he had conjured her up, but for a long time she represented something elemental to him. Sometimes he thought of her as Woman, sometimes as a river.

  He got home in time for the funeral. His job kept him busy, and kept the moneylender from their door. With the stubborn optimism of the young, he kept hoping that one day his fortunes would change, that he would go back to college and complete his degree. In the meantime, he knew his mother wanted to find him a bride…

  Abdul Karim got married, had children. Slowly, over the years of managing rowdy classrooms, tutoring students in the afternoons and saving, paisa by paisa, from his meager salary for his sisters’ weddings and other expenses, Abdul Karim lost touch with that youthful, fiery talent he had once had, and with it the ambition to scale the heights to which Ramanujan, Cantor and Riemann had climbed. Things came more slowly to him now. An intellect burdened by years of worry wears out. When his wife died and his children grew up and went away, his steadily decreasing needs finally caught up with his meager income, and he found for the first time that he could think about mathematics again. He no longer hoped to dazzle the world of mathematics with some new insight, such as a proof of Riemann’s hypothesis. Those dreams were gone. All he could hope for was to be illumined by the efforts of those who had gone before him, and to re-live, vicariously, the joys of insight. It was a cruel trick of Time, that when he had the leisure he had lost the ability, but that is no bar to true obsession. Now, in the autumn of his life it was as though Spring had come again, bringing with it his old love.

  In this world, brought to its knees by hunger an
d thirst

  Love is not the only reality, there are other Truths…

  —Sahir Ludhianvi, Indian poet (1921–1980)

  There are times when Abdul Karim tires of his mathematical obsessions. After all, he is old. Sitting in the courtyard with his notebook, pencil and books of mathematics for so many hours at a stretch can take its toll. He gets up, aching all over, sees to his mother’s needs and goes out to the graveyard where his wife is buried.

  His wife Zainab had been a plump, fair-skinned woman, hardly able to read or write, who moved about the house with indolent grace, her good-natured laugh ringing out in the courtyard as she chattered with the washerwoman. She had loved to eat—he still remembered the delicate tips of her plump fingers, how they would curl around a piece of lamb, scooping up with it a few grains of saffron rice, the morsel conveyed reverently to her mouth. Her girth gave an impression of strength, but ultimately she had not been able to hold out against her mother-in-law. The laughter in her eyes faded gradually as her two boys grew out of babyhood, coddled and put to bed by the grandmother in her own corner of the women’s quarters. Abdul Karim himself had been unaware of the silent war between his wife and mother—he had been young and obsessed with teaching mathematics to his recalcitrant students. He had noticed how the grandmother always seemed to be holding the younger son, crooning to him, and how the elder boy followed his mother around, but he did not see in this any connection to his wife’s growing pallor. One night he had requested her to come to him and massage his feet—their euphemism for sex—and he had waited for her to come to him from the women’s quarters, impatient for the comfort of her plump nakedness, her soft, silken breasts. When she came at last she had knelt at the foot of the bed, her chest heaving with muffled sobs, her hands covering her face. As he took her in his arms, wondering what could have ruffled her calm good nature, she had collapsed completely against him. No comfort he could offer would make her tell what it was that was breaking her heart. At last she begged him, between great, shuddering breaths, that all she wanted in the world was another baby.