Authors: D B Hartwell
, Indian mathematician (1887–1920)
bdul Karim is his name. He is a small, thin man, precise to the point of affectation in his appearance and manner. He walks very straight; there is gray in his hair and in his short, pointed beard. When he goes out of the house to buy vegetables, people on the street greet him respectfully. “Salaam, Master sahib,” they say, or “Namaste, Master Sahib,” according to the religion of the speaker. They know him as the mathematics master at the municipal school. He has been there so long that he sees the faces of his former students everywhere: the autorickshaw driver Ramdas who refuses to charge him, the man who sells paan from a shack at the street corner, with whom he has an account, who never reminds him when his payment is late—his name is Imran and he goes to the mosque far more regularly than Abdul Karim.
They all know him, the kindly mathematics master, but he has his secrets. They know he lives in the old yellow house, where the plaster is flaking off in chunks to reveal the underlying brick. The windows of the house are hung with faded curtains that flutter tremulously in the breeze, giving passersby an occasional glimpse of his genteel poverty—the threadbare covers on the sofa, the wooden furniture as gaunt and lean and resigned as the rest of the house, waiting to fall into dust. The house is built in the old-fashioned way about a courtyard, which is paved with brick except for a circular omission where a great litchi tree grows. There is a high wall around the courtyard, and one door in it that leads to the patch of wilderness that was once a vegetable garden. But the hands that tended it—his mother’s hands—are no longer able to do more than hold a mouthful of rice between the tips of the fingers, tremblingly conveyed to the mouth. The mother sits nodding in the sun in the courtyard while the son goes about the house, dusting and cleaning as fastidiously as a woman. The master has two sons—one is in distant America, married to a gori bibi, a white woman—how unimaginable! He never comes home and writes only a few times a year. The wife writes cheery letters in English that the master reads carefully with finger under each word. She talks about his grandsons, about baseball (a form of cricket, apparently), about their plans to visit, which never materialize. Her letters are as incomprehensible to him as the thought that there might be aliens on Mars, but he senses a kindness, a reaching out, among the foreign words. His mother has refused to have anything to do with that woman.
The other son has gone into business in Mumbai. He comes home rarely, but when he does he brings with him expensive things—a television set, an air-conditioner. The TV is draped reverently with an embroidered white cloth and dusted every day but the master can’t bring himself to turn it on. There is too much trouble in the world. The air-conditioner gives him asthma so he never turns it on, even in the searing heat of summer. His son is a mystery to him—his mother dotes on the boy but the master can’t help fearing that this young man has become a stranger, that he is involved in some shady business. The son always has a cell phone with him and is always calling nameless friends in Mumbai, bursting into cheery laughter, dropping his voice to a whisper, walking up and down the pathetically clean drawing-room as he speaks. Although he would never admit it to anybody other than Allah, Abdul Karim has the distinct impression that his son is waiting for him to die. He is always relieved when his son leaves.
Still, these are domestic worries. What father does not worry about his children? Nobody would be particularly surprised to know that the quiet, kindly master of mathematics shares them also. What they don’t know is that he has a secret, an obsession, a passion that makes him different from them all. It is because of this, perhaps, that he seems always to be looking at something just beyond their field of vision, that he seems a little lost in the cruel, mundane world in which they live.
He wants to see infinity.
It is not strange for a mathematics master to be obsessed with numbers. But for Abdul Karim, numbers are the stepping stones, rungs in the ladder that will take him (Inshallah!) from the prosaic ugliness of the world to infinity.
When he was a child he used to see things from the corners of his eyes. Shapes moving at the very edge of his field of vision. Haven’t we all felt that there was someone to our left or right, darting away when we turned our heads? In his childhood he had thought they were farishte, angelic beings keeping a watch over him. And he had felt secure, loved, nurtured by a great, benign, invisible presence.
One day he asked his mother:
“Why don’t the farishte stay and talk to me? Why do they run away when I turn my head?”
Inexplicably to the child he had been, this innocent question led to visits to the Hakim. Abdul Karim had always been frightened of the Hakim’s shop, the walls of which were lined from top to bottom with old clocks. The clocks ticked and hummed and whirred while tea came in chipped glasses and there were questions about spirits and possessions, and bitter herbs were dispensed in antique bottles that looked as though they contained djinns. An amulet was given to the boy to wear around his neck; there were verses from the Qur’an he was to recite every day. The boy he had been sat at the edge of the worn velvet seat and trembled; after two weeks of treatment, when his mother asked him about the farishte, he had said:
That was a lie.
My theory stands as firm as a rock; every arrow directed against it will quickly return to the archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things
, German mathematician (1845–1918)
In a finite world, Abdul Karim ponders infinity. He has met infinities of various kinds in mathematics. If mathematics is the language of Nature, then it follows that there are infinities in the physical world around us as well. They confound us because we are such limited things. Our lives, our science, our religions are all smaller than the cosmos. Is the cosmos infinite? Perhaps. As far as we are concerned, it might as well be.
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?
Are prime numbers the key to understanding infinity?
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
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
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
, 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 grownup and mature for their stations. Gangadhar was the one who, shyly, 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.