I have often pondered over the roles of knowledge or experience, on the one hand, and imagination or intuition, on the other, in the process of discovery. I believe that there is a certain fundamental conflict between the two, and knowledge, by advocating caution, tends to inhibit the flight of imagination. Therefore, a certain naivete, unburdened by conventional wisdom, can sometimes be a positive asset.
—HARISH-CHANDRA, Indian mathematician (1923–1983)
Gangadhar, his friend from school, was briefly a master of Hindi literature at the municipal school and is now an academician at the Amravati Heritage Library, and a poet in his spare time. He is the only person to whom Abdul Karim can confide his secret passion.
In time, he too becomes intrigued with the idea of infinity. While Abdul Karim pores over Cantor and Riemann, and tries to make meaning from the Prime Number theorem, Gangadhar raids the library and brings forth treasures. Every week, when Abdul Karim walks the two miles to Gangadhar’s house, where he is led by the servant to the comfortable drawing room with its gracious, if aging mahogany furniture, the two men share what they’ve learned over cups of cardamom tea and a chess game. Gangadhar cannot understand higher mathematics but he can sympathize with the frustrations of the knowledge-seeker, and he has known what it is like to chip away at the wall of ignorance and burst into the light of understanding. He digs out quotes from Aryabhata and Al-Khwarizmi, and tells his friend such things as:
“Did you know, Abdul, that the Greeks and Romans did not like the idea of infinity? Aristotle argued against it, and proposed a finite universe. Of the yunaanis, only Archimedes dared to attempt to scale that peak. He came up with the notion that different infinite quantities could be compared, that one infinite could be greater or smaller than another infinite . . .”
And on another occasion:
“The French mathematician, Jacques Hadamard . . . He was the one who proved the Prime Number theorem that has you in such ecstasies . . . he says there are four stages to mathematical discovery. Not very different from the experience of the artist or poet, if you think about it. The first is to study and be familiar with what is known. The next is to let these ideas turn in your mind, as the earth regenerates by lying fallow between plantings. Then—with luck—there is the flash of insight, the illuminating moment when you discover something new and feel in your bones that it must be true. The final stage is to verify—to subject that epiphany to the rigors of mathematical proof . . .”
Abdul Karim feels that if he can simply go through Hadamard’s first two stages, perhaps Allah will reward him with a flash of insight. And perhaps not. If he had hopes of being another Ramanujan, those hopes are gone now. But no true Lover has ever turned from the threshold of the Beloved’s house, even knowing he will not be admitted through the doors.
“What worries me,” he confides to Gangadhar during one of these discussions, “what has always worried me, is Gödel’s Incompleteness Theorem. According to Gödel, there can be statements in mathematics that are not provable. He showed that the Continuum Hypothesis of Cantor was one of these statements. Poor Cantor, he lost his sanity trying to prove something that cannot be proved or disproved! What if all our unproven ideas on prime numbers, on infinity, are statements like that? If they can’t be tested against the constraints of mathematical logic, how will we ever know if they are true?”
This bothers him very much. He pores over the proof of Gödel’s theorem, seeking to understand it, to get around it. Gangadhar encourages him:
“You know, in the old tales, every great treasure is guarded by a proportionally great monster. Perhaps Gödel’s theorem is the djinn that guards the truth you seek. Maybe instead of slaying it, you have to, you know, befriend it . . .”
Through his own studies, through discussions with Gangadhar, Abdul Karim begins to feel again that his true companions are Archimedes, Al-Khwarizmi. Khayyam, Aryabhata, Bhaskar. Riemann, Cantor, Gauss, Ramanujan, Hardy.
They are the masters, before whom he is as a humble student, an apprentice following their footprints up the mountainside. The going is rough. He is getting old, after all. He gives himself up to dreams of mathematics, rousing himself only to look after the needs of his mother, who is growing more and more frail.
After a while, even Gangadhar admonishes him.
“A man cannot live like this, so obsessed. Will you let yourself go the way of Cantor and Gödel? Guard your sanity, my friend. You have a duty to your mother, to society.”
Abdul Karim cannot make Gangadhar understand. His mind sings with mathematics.
The limit of a function f(N) as N goes to infinity . . .
So many questions he asks himself begin like this. The function f(N) may be the prime counting function, or the number of nested dolls of matter, or the extent of the universe. It may be abstract, like a parameter in a mathematical space, or earthy, like the branching of wrinkles in the face of his mother, growing older and older in the paved courtyard of his house, under the litchi trees. Older and older, without quite dying, as though she were determined to live Zeno’s paradox.
He loves his mother the way he loves the litchi tree; for being there, for making him what he is, for giving him shelter and succor.
The limit . . . as N goes to infinity . . .
So begin many theorems of calculus. Abdul Karim wonders what kind of calculus governs his mother’s slow arc into dying. What if life did not require a minimum threshold of conditions—what if death were merely a limit of some function f(N) as N goes to infinity?
A world in which human life is but a pawn
A world filled with death-worshipers,
Where death is cheaper than life . . .
That world is not my world . . .
—SAHIR LUDHIANVI, Indian poet (1921–1980)
While Abdul Karim dabbles in the mathematics of the infinite, as so many deluded fools and geniuses have done, the world changes.
He is vaguely aware that there are things going on in the world—that people live and die, that there are political upheavals, that this is the hottest summer yet and already a thousand people have died of the heat wave in Northern India. He knows that Death also stands at his mother’s shoulder, waiting, and he does what he can for her. Although he has not always observed the five daily prayers, he does the namaz now, with her. She has already started becoming the citizen of another country—she lives in little leaps and bends of time long gone, calling for Ayesha one moment, and for her long-dead husband the next. Conversations from her lost girlhood emerge from her trembling mouth. In her few moments of clarity she calls upon Allah to take her away.
Dutiful as he is to his mother, Abdul Karim is relieved to be able to get away once a week for a chess game and conversation with Gangadhar. He has a neighbor’s aunt look in on his mother during that time. Heaving a sigh or two, he makes his way through the familiar lanes of his childhood, his shoes scuffing up dust under the ancient jamun trees that he once climbed as a child. He greets his neighbors: old Ameen Khan Sahib sitting on his charpai, wheezing over his hookah, the Ali twins, madcap boys chasing a bicycle tire with a stick, Imran at the paan shop. He crosses, with some trepidation, the increasingly congested market road, past the faded awnings of Munshilal and Sons, past a rickshaw stand into another quiet lane, this one shaded with jacaranda trees. Gangadhar’s house is a modest white bungalow, stained an indeterminate gray from many monsoons. The creak of the wooden gate in the compound wall is as familiar a greeting as Gangadhar’s welcome.
But the day comes when there is no chess game at Gangadhar’s house.
The servant boy—not Gangadhar—ushers him into the familiar room. Sitting down in his usual chair, Abdul Karim notices that the chess board has not been laid out. Sounds come from the inner rooms of the house: women’s voices, heavy objects being dragged across the floor.
An elderly man comes into the room and stops short as though surprised to see Abdul Karim. He looks vaguely familiar—then Abdul remembers that he is some
relative of Gangadhar’s wife—an uncle, perhaps—and he lives on the other side of the city. They have met once or twice at some family celebration.
“What are you doing here?” the man says, without any of the usual courtesies. He is white-haired but of vigorous build.
Puzzled and a little affronted, Abdul Karim says:
“I am here for my chess game with Gangadhar. Is he not at home?”
“There will be no chess game today. Haven’t you people done enough harm? Are you here to mock us in our sorrow? Well, let me tell you . . .”
“What happened?” Abdul Karim’s indignation is dissolving in a wave of apprehension. “What are you talking about? Is Gangadhar all right?”
“Perhaps you don’t know,” says the man, his tone mocking. “Some of your people burned a bus on Paharia road yesterday evening. There were ten people on it, all Hindus, coming back from a family ceremony at a temple. They all perished horribly. Word has it that you people did it. Didn’t even let the children get off the bus. Now the whole town is in turmoil. Who knows what might happen? Gangadhar and I are taking his family to a safer part of town.”
Abdul Karim’s eyes are wide with shock. He can find no words.
“All these hundreds of years we Hindus have tolerated you people. Even though you Muslims raided and pillaged us over the centuries, we let you build your mosques, worship your God. And this is how you pay us!”
In one instant Abdul Karim has become “you people.” He wants to say that he did not lift an arm to hurt those who perished on the bus. His were not the hands that set the fire. But no words come out.
“Can you imagine it, Master Sahib? Can you see the flames? Hear their screams? Those people will never go home . . .”
“I can imagine it,” Abdul Karim says, grimly now. He rises to his feet, but just then Gangadhar enters the room. He has surely heard part of the conversation because he puts his hands on Abdul Karim’s shoulders, gently, recognizing him as the other man has not done. This is Abdul Karim, his friend, whose sister, all those years ago, never came home.
Gangadhar turns to his wife’s uncle.
“Uncle, please. Abdul Karim is not like those miscreants. A kinder man I have never known! And as yet it is not known who the ruffians are, although the whole town is filled with rumors. Abdul, please sit down! This is a measure of the times we live in, that we can say such things to each other. Alas! Kalyug is indeed upon us.”
Abdul Karim sits down, but he is shaking. All thoughts of mathematics have vanished from his mind. He is filled with disgust and revulsion for the barbarians who committed this atrocity, for human beings in general. What a degraded species we are! To take the name of Ram or Allah, or Jesus, and to burn and destroy under one aegis or another—that is what our history has been.
The uncle, shaking his head, has left the room. Gangadhar is talking history to Abdul, apologizing for his uncle.
“. . . a matter of political manipulation,” he says. “The British colonialists looked for our weakness, exploited it, set us against each other. Opening the door to hell is easy enough—but closing it is hard. All those years, before British rule, we lived in relative peace. Why is it that we cannot close that door they opened? After all, what religion tells us to slay our neighbor?”
“Does it matter?” Abdul Karim says bitterly. “We humans are a depraved species, my friend. My fellow Muslims address every prayer to Allah, the Merciful and Compassionate. You Hindus, with your ‘Isha Vasyam Idam Sarvam’—the divine pervades all. The Christians talk on about turning the other cheek. And yet each of them has hands that are stained in blood. We pervert everything—we take the words of peace spoken by prophets and holy men and turn them into weapons with which to kill each other!”
He is shaking so hard that he can barely speak.
“It is in mathematics . . . only in mathematics that I see Allah . . .”
“Quiet now,” Gangadhar says. He calls for the servant to bring some water for the master sahib. Abdul Karim drinks and wipes his mouth. The suitcases are being brought out from inside the house. There is a taxi in front.
“Listen, my friend,” Gangadhar says, “you must look to your safety. Go home now and lock your doors, and look after your mother. I am sending my family away and I will join them in a day or so. When this madness has passed I will come and look for you!”
Abdul Karim goes home. So far everything looks normal—the wind is blowing litter along in the streets, the paan shop is open, people throng the bus-stop. Then he notices that there aren’t any children, even though the summer holidays are going on.
The vegetable market is very busy. People are buying up everything like crazy. He buys a few potatoes, onions and a large gourd, and goes home. He locks the door. His mother, no longer up to cooking meals, watches as he cooks. After they eat and he has her tucked into bed, he goes to his study and opens a book on mathematics.
One day passes, perhaps two—he does not keep track. He remembers to take care of his mother but often forgets to eat. His mother lives, more and more, in that other world. His sisters and brother call from other towns, anxious about the reports of escalating violence; he tells them not to worry. When things are back to normal they will come and see him and their mother.
How marvelous, the Universal Mystery
That only a true Lover can comprehend!
—BULLEH SHAH, eighteenth century Punjabi Sufi poet
Logic merely sanctions the conquests of the intuition.
—JACQUES HADAMARD, French mathematician (1865–1963)
One morning he emerges from the darkness of his study into the sunny courtyard. Around him the old city writhes and burns, but Abdul Karim sees and hears nothing but mathematics. He sits in his old cane chair, picks up a stick lying on the ground and begins to draw mathematical symbols in the dust.
There is a farishta standing at the edge of his vision.
He turns slowly. The dark shadow stays there, waits. This time Abdul Karim is quick on his feet, despite a sudden twinge of pain in one knee. He walks toward the door, the beckoning arm, and steps through.
For a moment he is violently disoriented—it occurs to him that he has spun through a different dimension into this hidden world. Then the darkness before his eyes dissipates, and he beholds wonders.
All is hushed. He is looking at a vast sweep of land and sky unlike anything he has ever seen. Dark, pyramidal shapes stud the landscape, great monuments to something beyond his understanding. There is a vast, polyhedral object suspended in a pale orange sky that has no sun. Only a diffuse luminescence pervades this sky. He looks at his feet, still in his familiar, worn sandals, and sees all around, in the sand, little fish-like creatures wriggling and spawning. Some of the sand has worked its way between his toes, and it feels warm and rubbery, not like sand at all. He takes a deep breath and smells something strange, like burnt rubber mixed with his own sweat. The shadow stands by his side, looking solid at last, almost human but for the absence of neck and the profusion of limbs—their number seems to vary with time—at the moment Abdul Karim counts five.
The dark orifice in the head opens and closes, but no sound comes out. Instead Abdul feels as though a thought has been placed in his mind, a package that he will open later.
He walks with the shadow across the sands to the edge of a quiet sea. The water, if that is what it is, is foaming and bubbling gently, and within its depths he sees ghostly shapes moving, and the hints of complex structure far below. Arabesques form in the depths, break up, and form again. He licks his dry lips, tastes metal and salt.
He looks at his companion, who bids him pause. A door opens. They step through into another universe.
It is different, this one. It is all air and light, the whole space hung with great, translucent webbing. Each strand in the web is a hollow tube within which liquid creatures flow. Smaller, solid beings float in the emptiness between the web strands.
Speechless, he stretches out his hand toward a web-stra
nd. Its delicacy reminds him of the filigreed silver anklets his wife used to wear. To his complete surprise a tiny being floating within the strand stops. It is like a plump, watery comma, translucent and without any features he can recognize, and yet he has the notion that he is being looked at, examined, and that at the other end is also wonder.
The web-strand touches him, and he feels its cool, alien smoothness on a fingertip.
A door opens. They step through.
It is dizzying, this wild ride. Sometimes he gets flashes of his own world, scenes of trees and streets, and distant blue hills. There are indications that these flashes are at different points in time—at one point he sees a vast army of soldiers, their plumed helmets catching the sunlight, and thinks he must be in the time of the Roman Empire. Another time he thinks he is back home, because he sees before him his own courtyard. But there is an old man sitting in his cane chair, drawing patterns in the dust with a stick. A shadow falls across the ground. Someone he cannot see is stealing up behind the old man. Is that a knife agleam in the stranger’s hand? What is this he is seeing? He tries to call out, but no sound emerges. The scene blurs—a door opens, and they step through.
Abdul Karim is trembling. Has he just witnessed his own death?
He remembers that Archimedes died that way—he had been drawing circles, engrossed with a problem in geometry, when a barbarian of a soldier came up behind him and killed him.
But there is no time to ponder. He is lost in a merry-go-round of universes, each different and strange. The shadow gives him a glimpse of so many, Abdul Karim has long lost count. He puts thoughts of Death away from him and loses himself in wonder.
His companion opens door after door. The face, featureless except for the orifice that opens and shuts, gives no hint of what the shadow is thinking. Abdul Karim wants to ask: who are you? Why are you doing this? He knows, of course, the old story of how the angel Gabriel came to the Prophet Mohammad one night and took him on a celestial journey, a grand tour of the heavens. But the shadow does not look like an angel; it has no face, no wings, its gender is indeterminate. And in any case, why should the angel Gabriel concern himself with a humble mathematics master in a provincial town, a person of no consequence in the world?
21st Century Science Fiction: The New Science Fiction Writers of the New Century Page 3
