Finding Zero

by Amir D. Aczel

  Numbers, to Grothendieck, were everything. The magic of a number—this most powerful invention of the mind, or discovery of a preexisting truth—was astounding, and he could not stop thinking about numbers and how they came to be. On page 31 of Recoltes et Semailles, he writes that when he was small, he loved going to school. (In a French camp for “undesirables,” where he lived with his mother, going to school was a rare privilege.) At school, he wrote, “Il avait la magie des nombres” (There was the magic of numbers).

  But the world also has shapes and forms and geometry and measure, as the child knew, and while he was still a schoolboy he dreamed to complete the goal the ancient Greeks had pursued, of uniting numbers with shapes and geometry. On page 48, Grothendieck already writes about what he describes as: “Les épousailles du nombre et de la grandeur” (The wedding of number and size). It would be this idea—starting with the invention of numbers—that would lead him to his greatest achievements, including inventing revolutionary concepts such as the entities he called a motive, a sheaf, and the topos. All these abstract concepts derive from the basic idea of a number but extend it to immensely vast realms of highly theoretical mathematics.

  Algebra was linked with geometry through algebraic geometry—a field in which Grothendieck would leave his greatest mark. Geometry, the theory of shapes, is extended to topology, an area of mathematics concerned with notions of shape in a more abstract way: deformations of spaces by continuous functions and ideas of distance. It was in this area that Grothendieck defined sheaves and the topos.

  Prime numbers were important for Grothendieck’s work—as they are to most mathematicians since they are the building blocks of numbers (the nonprimes are made up of products of the prime numbers, hence the primes are elemental). At one time, Grothendieck was giving a lecture on some topic in which he used prime numbers as the skeleton on which to flesh out his general results. A member of the audience raised his hand and asked, “Can you please give us a concrete example?” Grothendieck said, “You mean an actual prime number?” The questioner said yes. Grothendieck was impatient to continue with his highly abstract derivation, and so, unthinkingly, he said, “Fine, take 57,” and went back to the board. Of course 57 is not a prime number, as it is equal to 19 times 3, so this number has become affectionately known as “Grothendieck’s prime.”

  Grothendieck, whose work in category theory and the topos would free us from the confines of the theory of sets, also understood that sets and their memberships were the most beautiful way to define numbers in the first place. This highly theoretical definition of what a number actually is uses the most powerful idea humans have ever come up with—that of a complete emptiness, the void, Shunyata. In mathematics, absolute nothingness is defined as the empty set.

  And it turns out that we can define the numbers—using the empty set—as follows: Zero is simply the empty set; we now define the number 1 as the set whose only member is the empty set. We can now define 2 as the set that contains two distinct elements: the empty set and the set containing the empty set. The number 3 will be a set that contains the empty set, the set containing the empty set, and the set comprising the empty set and the set containing the empty set. Continuing in this way, having started with sheer emptiness and the idea of a set, we can define all the natural numbers (the positive integers) all the way to infinity. As we see, each number is contained within the next-larger number as a series of Russian dolls each placed inside its larger mother. It was this derivation that I thought about when Jacob explained his Shunyata-womb idea. In a sense, the empty set here “gives birth” to all the numbers.

  From such concepts Grothendieck, the great master, was able to construct very complicated mathematics. But did he really know about the Eastern concept of nothingness—the Buddhist Shunyata? Well, for much of his life Grothendieck was indeed a Buddhist. And even when he wasn’t, he followed Buddhist ideas of peacefulness, charity to others, and dietary habits. He founded an antiwar survivalist group called Survivre Pour Vivre (Survive to live); his home was always open to people who were destitute and needed help; and he was active in many antiwar and environmental groups.

  During the great student demonstrations in Paris in 1968—the year he turned 40 and saw it as a milestone—Grothendieck decided to abandon mathematics (although he did produce some mathematical work over the years to come). I wondered if the zero and the Eastern void perhaps played important roles in the life of this leading mathematician. How much did Buddhism influence the thinking of Alexander Grothendieck? I didn’t know the answer.

  20

  When I came back to my hotel and turned on my computer, there was a much-anticipated message from Rotanak Yang (Andy Brouwer’s friend). He told me that his father, who spoke no English, was the director of an institution called Angkor Conservation, where many Cambodian inscriptions, statuary, and other artifacts have been taken over the decades to protect and conserve them. K-127 might have been placed there. But Angkor Conservation had been looted in 1990 by the Khmer Rouge in the last recrudescence of their violence in Cambodia, and many of the pieces kept there had been destroyed or plundered. So it was unclear whether K-127 would still be there, if indeed it had ever been brought to this repository. And then he wrote, “But neither I nor my father can help you any further. In order for us to give you any more information, you need to contact the Cambodian Ministry of Culture and Fine Arts to obtain permission to receive our information.”

  I closed my PC and sighed. Here I was, in Bangkok, waiting to go to Cambodia to look for the missing inscription, and now I needed to deal with a bureaucracy I didn’t know or understand. I thought about this new hurdle and sent a message back to Rotanak: “Would you please give me some idea as to where I should start this request? Do you know anyone at the Ministry of Culture and Fine Arts who I could contact?” I sent the message and went into town to visit Buddhist temples for inspiration.

  Next to my hotel was one of the stops for the boats that ferry tourists and Thais up and down the Chao Phraya. I caught one of these packed boats going north and disembarked close to the Royal Palace. The palace was closed—as it was a holiday, the king’s birthday—and an armed guard chased me away when I tried to walk in through one of its gates. I crossed the street and within a block found the largest of the temple complexes in Bangkok, Wat Pho. This was the home of the famous golden reclining Buddha. I admired the 150-foot-long gold statue of the Buddha lying on his side, supporting his head with his hand. A sign inside the temple read, “Beware Non-Thai Pickpocket Gangs.” I instinctively touched my pocket—my wallet was still there.

  Outside, as I crossed the crowded street heading in the direction of the boat landing, a middle-aged man rushed toward me. He opened a color-picture brochure displaying photos of a naked woman. “Young girl,” he said, “young girl.” I literally pushed him away. This was the scourge of Southeast Asia. Ever since the Vietnam War, when the US military would bring its war-tired soldiers for rest and recreation in Bangkok, the people here found a lucrative trade in selling their women and girls to Westerners. But the improving economy has made the phenomenon less prevalent in recent years.

  A few months earlier, at an international conference in Mexico where I had been invited to speak, I met Nicholas Kristof of the New York Times. When I told him that I was headed to Cambodia, he said that he had just returned from there. “When I was there, I bought from a brothel two Cambodian girls who had been forced into prostitution, and set them free and sent them into a program that would train them to live on their own,” he said. I was glad the world had people like him. He probably could have also bought the freedom of the young girl this pimp was peddling.

  When I returned to the Shangri-La, there was an answer from Rotanak. “Try to contact H. E. Hab Touch,” he wrote. “I don’t have a phone number or e-mail address, but maybe you can find it.” I didn’t know what H. E. stood for, but I spent some time on the computer looking
for Hab Touch. I recognized the name as that of one of the former directors of the Cambodian National Museum in Phnom Penh, which was a good sign. This man must know much about antiquities, I felt, and I hoped he would support my quest. I located his address and wrote him an e-mail message requesting his help in my search. But for a while there was no response.

  After a few days, to my delight, Hab Touch answered my e-mail. He told me he and his people would look into it and try to find more information about the inscription’s whereabouts. Mr. Hab (in Cambodia the surname comes first) then indeed spent a significant amount of time trying to help me find the inscription. He finally determined that on November 22, 1969, the artifact was sent to the place called Angkor Conservation (where Rotanak’s father worked) in Siem Reap, home of Angkor Wat and a thousand other smaller temples in the jungles and fields of western Cambodia. What happened to it afterward, no one knew. He suggested that I also contact the director of the local museum. It turned out that Chamroeun Chhan, whom my friend the art dealer in Bangkok had suggested I contact, was the director of that local museum in Siem Reap, and I was glad that two leads now pointed me to him. I would try to contact him later, but my greatest need now was to obtain information about K-127 from Angkor Conservation.

  I therefore returned to Rotanak, but he insisted that I get formal permission to be given any more information on artifacts at Angkor Conservation. So again I wrote to Hab Touch, and in a few days the desired permission arrived. I was cleared to travel to Siem Reap to visit Angkor Conservation to look for the lost artifact. I could not believe my search was about to begin in earnest, and with a specific destination. I took note of what I now knew: K-127 was discovered in 1891 at Sambor on Mekong. By 1931, George Cœdès had translated it and realized that it contained the oldest extant zero and published his finding. The stele was taken to the national museum in Phnom Penh. In November 1969, it was moved to Angkor Conservation in Siem Reap. In 1990, a thousand artifacts from Angkor Conservation were destroyed or stolen by the Khmer Rouge. Now I had official permission to search for it at its last-known home. I packed carefully, planning for what could be a difficult search for something that might or might not be found. This quest could take a long time.

  21

  I was now headed to Siem Reap to see if K-127 still existed and could be located. Most important in this search, Hab Touch had told me on the phone that he would get me in touch with his people at Angkor Conservation, to see if they could locate this elusive artifact.

  With these hopeful signs, I took a long taxi ride to the Don Meuang airport, the secondary Bangkok airport, and boarded an Air Asia flight to Siem Reap. Air Asia provided a comfortable flight in a two-engine turboprop plane, which can use far smaller runways than jets do. It took off after a couple of minutes on the runway, but the airline provided no food or drink—you had to pay even for a glass of water. On arrival, I went through the lengthy visa application process, had my picture taken, paid the fee (I knew they only accepted US dollars and I was prepared), and after an hour of immigration bureaucracy I emerged to find a cab to take me to my hotel.

  The Angkor Miracle Resort Hotel, which caters mostly to Chinese visitors, turned out to be comfortable and surprisingly quiet given the number of tour buses that arrive each day. It was January—high season here—and tourists mobbed the city. When I asked for a taxi, I got the only driver in town who knew absolutely no English or French: not hello, not yes or no. The reason for this was that he was not really a cab driver. The hotel’s concierge had called the last available taxi driver in town for me. But even this man was now too busy driving tourists around; so he volunteered his father to drive me. I had never before had to rely on someone who could not even guess what yes or no meant (and head motions don’t work here, since in Asia they often mean different—even opposite—things from what they mean in our culture).

  Angkor Conservation, a small, specialized foundation dedicated to the conservation of Cambodian antiquities, is not indicated on any tourist map. So I knew it would be hard to find it under any circumstances, and nearly impossible with a driver who couldn’t communicate with me. Fortunately, the concierge at the hotel thought to use his iPhone to find Angkor Conservation. He circled its location on the map he handed me. It was situated by the Siem Reap River, away from the main part of town, and the map showed that the general place he had circled was near an Italian restaurant named Ciao.

  I knew this would be hard. I showed the cab driver the map and he grumbled something in Khmer. After a few moments of talking past each other, he decided to move and drove somewhere—I wasn’t sure he understood anything. Siem Reap is still a traditional Southeast Asian town, in which cars are relatively rare and most locals use bicycles or, if they are wealthier, motorcycles. Transportation for hire is usually by tuk-tuk. We managed to maneuver our way through a series of traffic jams in which motorcycles and tuk-tuks competed for road space and ignored the legal direction of traffic.

  We made our way through the downtown area and headed north on Charles de Gaulle Boulevard, in the direction of the Angkor Wat complex. After passing the lush tropical gardens belonging to a new hotel, we turned right on a narrow road, once paved but now gravelly and lined with rickety tables set up by scores of vendors hawking fruits and vegetables. We continued toward the river. Then the driver turned right into a dirt road and we bumped along for some time, finally stopping by a small farm with chickens running around in the dusty yard, some of them scattering at the approach of our car. There was nobody around. We both got out and stood there, scratching our heads and staring blankly at a map that seemed to say nothing to either of us.

  Some ten minutes later, someone emerged from a shack some distance away and approached us. The driver and the man who came over started an animated discussion apparently about where we were and where we were going; both men soon became very excited, raising their hands up in the air, pointing first in one direction, then in another, their voices rising. Finally, the driver returned to the car and slammed the driver’s door. I tried to say, “This can’t be right. This is not the place . . .” But he understood nothing, and seemed to care even less. He put the car in reverse and drove backward on the narrow dirt path until we came back to the road. There he just stopped and started talking fast in Khmer, looking at me expectantly.

  This is crazy, I thought. I picked up the map. I couldn’t make out anything—and there was no Ciao restaurant anywhere, either. After trying to find the place while driving slowly back along the road for another half an hour, up and down and scrutinizing both sides, I decided it was time to give up. “Hotel!” I said. He understood. “Hotel,” he said, smiling for the first time since I hired him, “hotel,” and he made the way back double-time, traffic and all.

  It was late afternoon, and as I got out of the cab at the Angkor Miracle Resort Hotel’s driveway, I noticed five tuk-tuks parked at the entrance, their drivers lazing in the sun by their vehicles. One of them, a boy with a soft, round face who looked 16 or even younger, noticed me and ran over. “Sir,” he said, “please, please hire me; I am a good driver, and I need the money. Please.” The tuk-tuk behind him had in big letters on its side “Mr. Bee.” I dislike tuk-tuks: I find them unsafe since the bed behind the motorcycle engine has no protection at all, generally being made of wood, and the ride is bumpy.

  He looked up at me. He was small and thin, his eyes bright and pleading. “Sure, Mr. Bee,” I said. “But the place I am looking for is closed by now.” He looked very disappointed, and his head dropped somewhat; he started to turn to walk away. “Tomorrow,” I said. “I promise, really!”—I knew that these tuk-tuk drivers hear tomorrow and know there likely will be nothing: The customer will find something else to do, or another driver. “Be here at eight in the morning, please, and I will hire you for a full day—I promise.”

  Mr. Bee smiled. “I’ll be here,” he said. I gave him $2 to show my good faith.

  T
he next morning before eight, Mr. Bee was there. He saw me from afar as I approached the hotel door and ran over. “Good morning, sir,” he said.

  “Good to see you, Mr. Bee,” I answered. Then I showed him the map. “Can you read it?”

  “Yes, I can,” he said confidently and we looked it over together. He smiled and said, “I’ll take you there,” and he put on his helmet—few other tuk-tuk drivers use helmets, and I took this as a sign of his being very careful; he would prove himself to be careful, intelligent, and considerate. He helped me into the tuk-tuk’s cab. It became clear very quickly that not only did he speak close to perfect English, but this young man, a boy, really, was very bright—far cleverer than the driver from the day before.

  “First, let’s go to the Siem Reap Museum,” I said. “And then we’ll try to find what I need.” We drove through heavy traffic until we reached the museum. I asked Mr. Bee to wait for a little while outside and went in. “I need to see Mr. Chamroeun Chhan,” I said, “the director.” The woman at the desk asked what it was about. “Mr. Hab Touch sent me to speak with the director about an inscription I am looking for.” She picked up the phone and spoke in Khmer. The only words I understood were “His Excellency Hab Touch.” And then it dawned on me: I had been dealing all along with perhaps the highest official responsible for antiquities in all Cambodia—someone accorded such a lofty honorific. And I realized how ignorant I had been in not understanding what H. E. meant in Rotanak’s e-mail message about “H. E. Hab Touch.” I was embarrassed that I had not used the proper salutations when communicating with someone who had been so helpful in my quest. Surely His Excellency had many more important things to do than help a random academic looking for an obscure inscription.