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While taking a class on infinity at Stanford in the late 1980s, Ravi Kapoor discovers that he is confronting the same mathematical and philosophical dilemmas that his mathematician grandfather had faced many decades earlier--and that had landed him in jail. Charged under an obscure blasphemy law in a small New Jersey town in 1919, Vijay Sahni is challenged by a skeptical judge to defend his belief that the certainty of mathematics can be extended to all human knowledge--including religion. Together, the two men ...
While taking a class on infinity at Stanford in the late 1980s, Ravi Kapoor discovers that he is confronting the same mathematical and philosophical dilemmas that his mathematician grandfather had faced many decades earlier--and that had landed him in jail. Charged under an obscure blasphemy law in a small New Jersey town in 1919, Vijay Sahni is challenged by a skeptical judge to defend his belief that the certainty of mathematics can be extended to all human knowledge--including religion. Together, the two men discover the power--and the fallibility--of what has long been considered the pinnacle of human certainty, Euclidean geometry.
As grandfather and grandson struggle with the question of whether there can ever be absolute certainty in mathematics or life, they are forced to reconsider their fundamental beliefs and choices. Their stories hinge on their explorations of parallel developments in the study of geometry and infinity--and the mathematics throughout is as rigorous and fascinating as the narrative and characters are compelling and complex.
Moving and enlightening, A Certain Ambiguity is a story about what it means to face the extent--and the limits--of human knowledge.
I punched in the number 342 without thinking about it. It was the same number I had entered 25 years ago when the calculator was brand new.
"Want to see some number magic?" my grandfather had asked as he watched me push the buttons more or less randomly. I was sitting in his room completely taken by his birthday present, if not quite sure what to do with it. He put his notebook down, temporarily giving up on the math problem that had resisted solution since morning.
"Yes, Bauji!" I had rushed over to him.
"Enter any three-digit number in your calculator and do not let me see it." Thatis when I had first entered 342, the same three digits I entered now. "OK. Now enter the same number again, so you have a six-digit number," he had said. I punched in 342 again, so now I had 342342 entered in my calculator. "Now, I do not know the number you have in there, Ravi, but I do know that it is evenly divisible by 13."
By "evenly divisible" he meant that there would be no remainder. For example, 9 is evenly divisible by 3 but not by 4.
Bauji's claim seemed fantastic to me. How could he know that my number, randomly chosen and completely unknown to him, would be evenly divisible by 13? But it was! I divided 342342 by 13 and I got 26334 exactly, with no remainder.
"You're right," I said, amazed.
He wasn't finished, though. "Now, Ravi, I also know that whatever number you got after you divided by 13 is further divisible by 11." He was right once again. 26334 divided by 11 was 2394. Why was this working? "Take the number you got and divide by 7. Not only will it divide evenly, but you will be surprised at what you get." He had begun his pacing and I knew that he was as excited as I was.
I divided 2394 by 7 and I got 342! "Oh! Oh! It's the number I started with! Bauji, how did this happen?"
My grandfather just sat there, grinning at the completeness of my astonishment. "You will just have to figure that one out Ravi," he said, walking out to check the state of his tomato plants, the newest additions to his vegetable garden in the backyard. He seemed to be the only person who could grow tomatoes in New Delhi's dry summer heat.
The first thing I did was to check the divisions by hand. My hypothesis was that Bauji had rigged the new calculator somehow. But no, the numbers worked out exactly the same way when I did the long divisions by hand. Next, I decided to try this with some other three-digit numbers. The same thing worked every time. Whatever the repeated number, I could divide it evenly by 13, 11, and 7, and each time I got back to the number I started with. A few minutes of checking and rechecking convinced me that this property was true of any three-digit number. I tried doing the same thing with four-digit numbers, and it did not work any more. Neither did it work for two-digit numbers. What was going on?
I tried reversing the order. Instead of dividing first by 13, then 11, and then 7, I divided the six-digit number first by 7, then 11, and then 13. It made no difference at all. After dividing by each of those three numbers I would get back my original three-digit number. Why was this happening?
I wasn't getting anywhere and it was getting to be dinnertime. Ma had already called me twice. I knew that risking a third "I'll be right there" would be unwise, and so I put my notebook away and headed to the kitchen. I stopped thinking about the problem. And then mysteriously, out of nowhere, just when I was wondering if Ma would let me have ice cream, a new idea occurred to me. Even now, with more experience in such things, I cannot quite explain the inception of the moment of insight-the Aha moment-when out of nowhere a new idea comes, and chaos is replaced by understanding.
My first real Aha moment was at the dinner table, two days after my 12th birthday. The idea that loosened the knot was the realization that division was the reverse of multiplication, a fact I had long known, but never applied in quite the fashion this problem demanded. Instead of dividing by 13, 11, and 7 one at a time, why not multiply them together and then divide by the product all at once? Would this approach even lead to the same answer? I thought it would, and a confirming example showed this to be the case. In my head I divided the number 24 first by 2 and then by 3. I got 4 as the answer. Next I divided 24 by 6 (which is 2 x 3) and got 4 as well. So, it should work to divide the six-digit number by the product 13 x 11 x 7. I did some more examples on a paper towel just to be sure. It appeared that I might be onto something.
Ma noticed that I was completely distracted. "Ravi, what's going on? Why aren't you eating?" But I hardly heard her. I had to find out what 13 x 11 x 7 was; perhaps that would lead me to understand why Bauji's magic worked.
"I'll be right back, Ma," I said, getting up quickly before she could react.
"No, sir, you won't. Sit here and finish your dinner." She looked like she meant it. But my grandfather must have known what I was going through.
"Its okay Anita. Let him go." He must have been convincing enough, for I saw unwilling permission in my mother's eyes.
I ran up the stairs two at a time and fired up the calculator. 13 x 11 x 7 was ... 1001.
I knew this was terribly significant, though as yet I was not quite sure why. I tried dividing 342342 by 1001. As I expected, I got 342. But wait a minute. That must mean that the reverse is true as well. So if I multiply 342 by 1001 I should get 342342. Of course! 342 x (1001) = 342 x (1000+ 1)= 342,000 + 342 = 342342. So, taking a three-digit number and repeating it was just like multiplying it by 1001. And if you multiplied it by 1001, you could divide the six-digit number by 1001 to get the original three-digit number. What had confused me was dividing by 13, 11, and 7-but by dividing by those numbers I was in effect dividing by 1001. How simple! How could I have not seen it?
"Bauji! Bauji! I've got it!"
And he too bounded up the stairs two at a time, just as I had a few minutes before. "Tell me," he gasped. He was out of breath, but not overly so- not bad for 85.
"When you asked me to repeat the three-digit number, you were actually having me multiply it by 1001. And then you made me divide it by 1001, except you did it in three stages. So of course I ended up with the same number I started with!"
He looked at me and smiled. "Good work," he said ruffling my hair, his most characteristic gesture of affection. "I'll give you another one to think about tomorrow." When I told him I wanted another one right then, he laughed. "Looks like you'll be the next mathematician in the family. We might have to send you to The Institute of Advanced Studies! Maybe we could collaborate on some research."
Sitting in his lap, surrounded by his books and papers, I could not imagine a better fate.
* * *
The next evening Bauji died. I remember going to his room, calculator in hand, ready for my next puzzle, but as I neared the door I heard my mother's voice-really a whisper-coming from inside the room. The door was ajar, about half open. I could hear an urgent pleading in her tone even though her sounds did not seem to have the rhythm of words. From the hallway I saw that she sat on the floor, cross-legged, near my grandfather's desk. She had Bauji's-her father's-head in her lap and she was massaging his forehead, beseeching him. Even though I had never seen a dead person before, and even though I was not standing near him, I could tell that Bauji was gone. His posture had an unalterable finality that sleep lacks. For the longest time I couldn't move. I stood in the doorway of his room which suddenly seemed extraordinary in its sameness: his desk was piled high with its usual mountain of mathematics books; three of them were open. On the far wall his books were spilling out of the two large bookcases; many were on the floor. On the wall near me were his music records and cassettes, mostly instrumental jazz. The tape-recorder was set on auto-replay and was then softly playing Louis Armstrong's trumpet rendition of "Summertime."
I walked across the room to Ma unsure what to say or do. After many minutes she seemed to understand that she was no longer alone. For a second I saw her face collapse with grief. Her eyes closed tightly on themselves and tears flowed from the corners. Then she quickly pulled me towards her so I wouldn't see her cry. From over her shoulder I could see Bauji's last expression. He wore the face of happy surprise, as if he had at last glimpsed the solution to a difficult mathematical problem, and the answer was not at all the one he had expected.
As my mother hugged me more tightly the calculator slipped from my hands and fell to the floor near Bauji's hands. The "I" of Texas Instruments fell out and could not be reattached despite my, and then my father's, best attempts.
Many years later my mother told me that Bauji had decided that I had a mathematician's mind and he had wanted to push me to excel in the subject. The calculator was to have been a catalyst in my development as a mathematician. "I'm going to use it to get Ravi passionate about mathematics," Bauji had said.
* * *
Although Bauji was secular, he participated in religious functions with some regularity. "Religion is about community," he would announce after each such function, "and everyone needs community." I once heard an uncle refer to him as an "atheist with goodwill towards God." I didn't know quite what he meant, but it somehow seemed to fit Bauji.
So when he died, no one was quite sure what type of ceremony was required. He would be cremated, that much was clear. All Hindus are cremated, and while Bauji never referred to himself as a Hindu, he never repudiated the affiliation either. But the family divided on the extent of the rites that should accompany the sacrament. Bauji's sister insisted upon a full recitation from the scripture, sprinkling of holy Ganga-jal, spreading of gold dust, application of sandalwood paste, the presence of six Brahmans, and the lighting of the pyre by the eldest male descendant.
My mother disagreed. "Bauji liked simplicity, he would not have wanted all this," she said. After much wrangling (that got her crying) Ma prevailed on every point except the lighting of the pyre by the eldest male descendant.
"There is no other option. It has to be a male descendent or his soul will not be properly liberated. It says so in the Vedas," insisted my great aunt, and on this one issue she refused to give in. And since Bauji had no sons and I was the only grandchild, the eldest male descendant they were talking about was me.
When I reached out with the kindling to light the funeral pyre, my hand started to shake. The shaking was strong, and it grew stronger when I unexpectedly started to shiver. I was embarrassed: I had wanted to present an image of dignified grief to my relatives-so becoming in a 12-year-old faced with the passing of his grandfather, whom he so adored-but it wasn't working. I had the odd sensation of watching myself from outside my own body, as if I were floating around the entire ceremony, watching it from above. I could see the boy in the middle who now (as a final embarrassment) appeared to be crying, who was losing the battle to still his hands, which were stubbornly refusing to obey his commands. Then I saw my father hold the boy from behind and steady his wrists. At his touch, I came back into myself. The flame caught. The pyre hissed.
* * *
A few days after all the relatives left I got in the habit of going to Bauji's room every afternoon after school. I would lie down on the floor in the center of the room and imagine he was still sitting on his desk by the window. When he was wrestling with a problem he would sit there with his eyes shut and his body perfectly still. He would stay like that for a long time and then, every once in a while, he would sit up very straight and furiously start writing in his notebook. If he liked what he wrote, he would jump up, as if released by a spring, and pace with great energy and intensity, muttering to himself, or sometimes to me, "Could this be it? Could this be it?" Sometimes he would end these walkabouts with a loud "Ha!" and take me in his arms and throw me high up, nearly to the ceiling, and then catch me under my armpits as I came down. "I see it now, Ravi! I see it!" Now and then he would challenge me to have a go at a mathematical question, and, if I succeeded, he and I would do a joyful postmortem on the insight that cracked the case.
Bauji saw grace in mathematics, and sometimes I could see glimpses of what he saw. Now, without him, there was only the monotonous drone of doing what needed to be done.
I did try to read his mathematics books, but their pages seemed cold and lifeless. The symbols spread themselves on page after page without reason or beauty. I looked at his handwritten notebooks and they too had an alien feel, except for one page whose margin contained the notation "Show this to Ravi" next to some ominous-looking calculations. For two days I tried and failed to decipher what he might have wanted to show me. I could only tell that it seemed to have something to do with prime numbers and infinity.
After I was beaten by his math, I took to going to his room and listening to his jazz records. At first they, too, seemed without order, like the mathematical symbols in his books. There was none of the predictable, repeating structure of the music I was accustomed to; instead there were notes that seemed floated on the spur of the moment, without a planned arrangement to guide them.
Then suddenly one day, I got it. I was listening to Charlie Parker (Crazeology) and I had a musical Aha moment. I realized that in most of these records there was, in fact, an underlying structure which allowed for inspired improvisation: the ensemble would first play the tune from beginning to end, with the melody played by the horns, and the harmony played by the rhythm section-the piano, bass, and drums. Then, as the tune went on, the rhythm section would continue to play the harmony while each horn improvised a solo. The soloist would select notes available within the harmonic structure while incorporating the soul of the original melody, but with his notes he would create something new each time he played. And it was all done with a casual, understated coolness. Within two weeks I was hooked.
So I spent the afternoons and evenings listening to Parker, Armstrong, Ellington, and then Goodman and Getz. For two months my parents let me be. They must have heard the music coming from Bauji's room-the music that everybody except Bauji thought was strange-yet they did not ask me about it.
It was not until a week before my final examinations that my mother came to institute some course correction. I was sprawled on the floor, my eyes shut and my feet keeping pace with the changing moods of "West End Blues." She tapped me on the shoulder and asked if she could turn off the record player. "We need to talk," she said. I could tell she was picking her words carefully. She told me she knew how much I missed Bauji and she knew I was listening to his music to "stay connected with him." I wasn't sure of this-I thought I was listening to his music because I liked it-but I did not think it best to volunteer this information. "Ravi, it's time to move on. Bauji wanted you to follow a path in life and you can't get on that path if all you do is listen to this ... music." (Continues...)
Excerpted from A Certain Ambiguity by Gaurav Suri Hartosh Singh Ba
Copyright © 2007 by Princeton University. Excerpted by permission.
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