We avoided these delicacies and shared a meal of curried rice with vegetables—the curry was better than Indian curry. And we talked about the project. “What would you do if K-127 doesn’t exist anymore?” Debra asked cautiously when we finished our dessert of crème brûlée.
I thought for a moment, and then said, “I will find it . . . even if I have to spend the rest of my life here in Southeast Asia.”
“So I guess we may have to take Miriam out of school and move here?” she said with a smile. “And how many more bribes do you think you might need to pay to do this?” We both laughed. But I knew she would stand behind me. Somehow—perhaps for no objective reason—we both had that confidence that I would succeed in finding K-127, even if it was now no more than a heap of broken pieces of stone. Debra knew how much this quest meant to me, and I was touched by her deep support. But neither of us knew how ugly it would get toward the end.
However, what we came to Laos for were knowledgeable monks serving Buddha in ancient temples, the Wats of Southeast Asia. We made our way to the oldest and architecturally most impressive temple: Wat Xieng Thong, a steeply pitched wooden pagoda with elaborate glass mosaics and gilded decorations built in 1565. We strolled through the wide grounds to the temple overlooking the Mekong on one side and the town on the other. The monks were walking the grounds or chatting in small groups. While Debra was taking pictures of the old temple, I approached one of the monks and explained what I was looking for.
He led me to the most learned scholar in this temple, who was sitting meditating by an impressive Buddha image inside the old Wat. I took off my shoes, came in, and sat on a wooden bench in a corner, waiting. When the monk finished his meditation, I introduced myself and asked him my question: “What is the meaning of the Buddhist void, the Shunyata?” He looked at me and thought for a moment, then replied, “Everything is not everything.”
This was a considered answer from someone who knew a lot, and I understood that I shouldn’t take it lightly. He was not searching for the right expression or confused about the use of English words. He meant exactly what he said: “Everything is not everything.” I had to ponder this curious answer for a while. But I knew what he meant. Perhaps to an Eastern mind, “everything is not everything” might be intuitive and obvious in some sense. A Westerner has to think about it, and then it becomes clear and reveals its great depth and meaning.
To explain what “everything is not everything” means, I need to appeal to the work of the great English philosopher and mathematician Bertrand Russell. Russell proved in the early twentieth century that there is no such thing as a universal set, a set that contains everything inside it, leaving absolutely nothing outside. There is no container in which the entire universe or set of universes all exist with nothing left on the outside: There must always be something remaining outside of any kind of enclosure.
This mathematical idea has profound implications for the structure of the universe: The universe, whatever it is, cannot be all there is. Russell proved this surprising mathematical finding by an ingenious argument. He said, “Let’s consider sets that contain themselves and sets that do not contain themselves.” For example, the set of all dogs does not contain itself as a member, simply because it is not a dog.
But the set of all things that are not dogs does contain itself as a member. Why? Because it is not a dog, and hence it belongs with the collection of all things that are not dogs. Then Russell asked himself, What about the set of all sets that do not contain themselves? Does this set contain itself as an element? If it does, then by definition it cannot contain itself, and if it doesn’t, then it does contain itself.
Russell used this paradox to expose some of the problems with the then-emerging theory of sets. We now know that the theory of sets does not agree well with the basic Eastern logic of Nagarjuna and the tetralemma, and we’ve seen how Linton, using Grothendieck’s work—which was based on categories, rather than sets—was able to circumvent the problem. “Everything is not everything”—there is always something that lies outside of what you may think covers all creation. It could be a thought, or a kind of void, or a divine aspect. Nothing contains everything inside it. I found this idea profound. But he went on.
“Here,” said the monk, motioning for me to come closer and offering me a tiny stool about 12 inches tall and made of an embroidered seat and four little wooden legs. I sat down beside him. “When we meditate,” he said, “we count.” He looked at me intently. “We close our eyes and are aware only of where we are at the moment, and of nothing else. We count breathing in, 1; and we count breathing out, 2; and we go on this way. When we stop counting, that is the void, the number zero, the emptiness.” Here it was, I thought: the Shunyata and the number zero all in one.
I was beginning to understand what I had come here for. Here was the intellectual source of the number zero. It came from Buddhist meditation. Only this deep introspection could equate absolute nothingness with a number that had not existed until the emergence of this idea.
The monk continued. “We are born, we grow and develop, we become a quantity. Then we die, and this quantity becomes zero. This is the secret to meditation, and to existence.” I sat there for a while on the tiny uncomfortable chair, contemplating what the wise monk had given me. Then I thanked him and left.
Crossing the square inside the temple grounds I ran into a crowd of European tourists speaking loudly in French, Italian, and German. In their midst was a tall Caucasian man dressed in a yellow robe, with a long white beard and hair bound in a ponytail. He was hard not to notice. I walked over to him and began a casual conversation about the temple we both had just exited. Eventually, I got around to the question that I wanted to ask this Western man who’d adopted the dress of the East: “What does the Buddhist void mean to you?”
“I’m not a Buddhist,” he answered. “I am a Hindu. I am from Bezier, in France, but have lived for 41 years in Chennai—that’s Madras.”
“Yes, I know it’s the old Madras,” I said. “So what are you doing in a Buddhist temple?”
“Just visiting,” he laughed. “I live here now, temporarily. I am Jean-Marc,” he said, smiling.
Debra saw me speaking to him and waited. I asked Jean-Marc about the Hindu gods and their meaning. “I believe,” he answered, “that God is not in heaven. You see, Shiva is in me and in you.”
“Then we are all the destroyers of worlds?” I asked. At that moment the crowd of visitors surrounding us noticed this unusual man wearing a robe, which was somewhat different from those of the Buddhists: darker and a tad greenish. They flocked to him with questions. He didn’t finish answering me. I walked over to Debra and told her about our conversation.
“Maybe you’ll see him again,” she said. “He seems to know interesting things.” We walked back into town and had drinks at a French cafe overlooking the river.
The next day, I ran into Jean-Marc in a local shop as we were walking down the main street. He was there with his young Indian male companion; they were purchasing two colorful Buddha images. “Wrong religion, isn’t it?” I asked him. He laughed. I went back to our interrupted conversation from the day before, about Shiva being in all of us. “Well,” he said, “Shiva is indeed everywhere; he is in all of us, whoever we may be.” I thought of the one man I knew who had justly—and tragically—applied this idea to himself: Robert Oppenheimer. At dawn on July 16, 1945, just as the first atomic bomb exploded a few miles away from him and other scientists in the New Mexico desert, Oppenheimer ruefully recalled a verse referring to Shiva in the Indian epic the Bhagavad Gita: “I am become death, the destroyer of worlds.”
“So you are interested in Buddhism,” I said, pointing to the two Buddha statuettes in his hand, one painted red and the other green.
“Yes, absolutely,” he said. “There are interrelations among the religions of the East, as you can see, for example, at Angkor
Wat. The temple started as a Hindu shrine to Vishnu—you know about the remains of a tenth-century statue of Vishnu they had just discovered at the top?—and now it is an active Buddhist temple. So, there you have it. And I’m sure you’ve seen images of Shiva’s mount, the big bird Garuda, everywhere in these Buddhist countries of Southeast Asia.”
I nodded, then said, “Well, I want to ask you about the catuskoti, the tetralemma . . . I’ve been reading Nagarjuna.”
“You don’t need Nagarjuna to understand the tetralemma,” he said. “That weird-looking logic—from a Western point of view—is as old as Buddhism itself. Nagarjuna is just one of its later interpreters. You should study its earliest manifestations in Buddhism. As a mathematician, you will probably want a philosophical analysis . . . maybe it will answer both of your questions. Why don’t you both come to my place?” he offered.
We thanked him for the offer and walked with him and his friend down the quiet street toward the Nam Khan, the Mekong tributary. We reached the riverbank and descended down to the sandy shore of the river, where a precarious-looking bamboo bridge stretched to the other bank above fast-churning waters. We walked carefully, holding both bamboo railings until we reached the other side. Then we climbed up a steep, dusty path to a hut perched at the top of the hill. Our host opened the door for us. We sat down in his small living room, and he offered us some tea. Then he turned to the bookshelf behind him.
He chose one volume, opened it, and read aloud to us: “In early Buddhist logic, it was standard to assume that for any state of affairs there were four possibilities: that it held, that it did not, both, or neither. This is the catuskoti (or tetralemma). Classical logicians have had a hard time making sense of this, but it makes perfectly good sense in the semantics of various paraconsistent logics, such as First Degree Entailment. Matters are more complicated for later Buddhist thinkers, such as Nagarjuna, who appear to suggest that none of these options, or more than one, may hold.”1
As he read, he sat lotus-position across from us on the mat, playing with his long gray beard and stopping every few minutes to sip hibiscus tea. His eyes suddenly closed in meditation. After some time he opened them again and continued: “Within Buddhist thought, the structure of argumentation that seems most resistant to our attempts at a formalization is undoubtedly the catuskoti or tetralemma.”2
The description showed how the catuskoti appeared very early in Buddhist thought, as early as the sixth century BCE when the Buddha, the Indian prince Siddhartha Gothama, lived. Jean-Marc read to us what happened when the Buddha was asked about one of the most profound metaphysical issues:
How is it Gothama? Does Gothama believe that the saint exists after death and that this view alone is true and every other false?
Nay, Vacca. I do not hold that the saint exists after death and that this view alone is true and every other false.
How is it Gothama? Does Gothama believe that the saint does not exist after death and that this view alone is true and every other false?
Nay, Vacca. I do not hold that the saint does not exist after death and that this view alone is true and every other false.
How is it Gothama? Does Gothama believe that the saint both exists and does not exist after death and that this view alone is true and every other false?
Nay, Vacca. I do not hold that the saint both exists and does not exist after death and that this view alone is true and every other false.
How is it Gothama? Does Gothama believe that the saint neither exists nor does not exist after death and that this view alone is true and every other false?
Nay, Vacca. I do not hold that the saint neither exists nor does not exist after death and that this view alone is true and every other false.3
The only way to solve the conundrum, according to the article Jean-Marc was reading to us, was to conclude that the last two possibilities, both true and untrue and neither true nor untrue, “are empty.”4 Jean-Marc looked up in triumph, and I knew that my hunch had turned out to be right: The catuskoti, or tetralemma, collapses. Once we insist on four corners, these corners vanish, leaving us with the empty set: the void, Shunyata, or simply zero.
The connection I had been seeking for so long between the unusual Eastern logic of the catuskoti and the void, leading to zero, was now clear. The only mathematical solution to the logical paradox of the four possibilities of the catuskoti was the mathematical empty set: the great void, utter nothingness, the ultimate zero.
“So there you have it,” Jean-Marc said. “The tetralemma leads directly to the Shunyata.” We all looked at him, and he continued. “Buddhism emphasizes the void—something you in the West do not have. If you want, this may be what you are looking for—the source of the zero in the East is as old as the Buddha himself, 1,600 years old.”
We sat there quietly for a while and then I said, “Thank you. Maybe now you can tell me about infinity?”
He laughed and said, “Ah, that’s too big—maybe for another day?”
“May I come to see you tomorrow?” I asked.
“With pleasure,” he replied, and shook our hands. Debra and I stood up, and his Indian friend took us down the hill to the bamboo bridge.
17
Debra planned to devote the next day to taking photographs. We agreed to meet in the late afternoon at the same cafe we had enjoyed the previous day, with its view of the Mekong. In the meantime, I went to see Jean-Marc in his small hilltop home to learn more about his view of the Eastern infinity. I felt he would likely know much about the concept, since Hinduism was comfortable with the infinite. He was in a jovial mood and offered me a bowl of green curry vegetables and rice. We sat down at his table and ate.
When we were finished, he said, “So you want to know about the infinite in Eastern philosophy?” I said that I did because I believed that both zero and infinity—the extremes of our modern number system—had to have come from the East. “The Buddha himself was a mathematician, you know,” Jean-Marc said. “In early books about him, such as the Lalita Vistara, he is described as being excellent in ‘numeracy’ and able to use his ability with numbers to try to win the attention of Princess Gopa. Numbers, including very large numbers and their limit of infinity, appear in that text already. Then, of course, we have in Hinduism many references to infinity: infinite time, infinite space, and so on. It is far more widespread in Indian philosophy of that time than it is in the West. In the West, you only have some vague notions of God being infinite—whatever that means. But you should definitely look at Jainism, a religion that began early as well. The Jains, in particular, were interested in very large numbers.” He walked over to his bookshelf and pulled out a volume and leafed through it. Then he said that infinite quantities are mentioned in a Jain text called the Anuyoga-vara sutra (Doors of Inquiry), written two millennia ago. The infinite quantities there are derived through an operation called “multiple multiplication,” which might have meant exponentiation. If so, it would imply that the Jains who lived two thousand years ago understood something very deep about infinity.
“This is stunning,” I said.
He smiled. “Yes, the ancient Indians understood infinity at least 1,800 years before mathematicians in the West did.”
“So you know about Cantor’s work?” I asked, surprised.
“Yes, of course. I studied philosophy for many years, including the philosophy of mathematics.”
What was so surprising—and something I had not realized before—was that what he told me based on the Jain text provided some proof of a real mathematical understanding of infinity so early, and so long before a great genius in Germany, the tormented mathematician Georg Cantor, was able to explain the same concepts.
Cantor was a mathematician at the University of Halle in eastern Germany in the late 1800s, where he single-handedly developed the mathematics of infinity. He had been a student at the University of
Berlin, one of the most important universities in Europe at that time, studying under a mathematics giant, Karl Weierstrass, who contributed hugely to our understanding of the real numbers: the numbers on the real number line, which include both rational numbers (integers or quotients of integers) and irrational numbers (numbers, such as pi, that cannot be expressed as quotients of integers). Weierstrass, together with another German mathematician named Richard Dedekind, understood that irrational numbers had infinite, nonrepeating decimal expansions: things like 0.1428452396 . . . , as compared with 0.48484848 . . . Decimals that repeat, such as the latter example, can be proven to always equal a rational number, meaning that they can always be written as ratios of two integers—In this case, the number 16/33.
Nonrepeating decimals—the best example is pi = 3.14159265359 . . . —are never rational, meaning they cannot ever be written as a ratio of two whole numbers. Cantor extended this entire study to a profound and new understanding of actual infinity. He understood that the decimal expansion of an irrational number is nonrepeating and infinite.
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