Finding Zero (13 page)

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.

He also understood something counterintuitive: There are various
levels
of infinity, meaning that not all infinite quantities are equally big. Though infinite, some numbers can be larger, in some sense, than other infinite numbers. And Cantor was able to prove mathematically that, while both sets are certainly infinite, the set of irrational numbers is of a
higher order of infinity
than the set of all the integers. That is, there are more irrational numbers than there are integers.

Cantor's concept of infinity was controversial within the mathematical community of his time, and his frustration at the reception
to his work, as well as the difficulties he faced in developing his theories of the infinite, contributed to years of mental instability.
Cantor suffered bouts of depression throughout his life, underwent lengthy hospitalizations, and died in a mental institution in Halle in 1918. He explained infinity to the world. Cantor showed that the continuum of numbers between any two numbers on the number line must have 2
n
elements if there are
n
integers (
n
is infinite here). We know what “grows exponentially” means—so you can see that raising a number, here 2, to a power that is
infinite
creates an exponentially larger infinity. Cantor thus showed that the order of infinity of the real numbers (meaning rational numbers and irrational numbers, like pi or
e
, the base of the natural logarithms) is higher than the order of infinity of the rational numbers alone, or the integers.

In any case, it is possible, based on what Jean-Marc had said to me, that the Jains of ancient India understood that exponentiation raises the level of infinity when an infinite number is exponentiated, and creates a really large number when the number being exponentiated is large but still finite.

“You see that the ancient Indians understood infinity almost as well as Cantor did in the late 1800s,” Jean-Marc said.

“So, let me see,” I said. “Zero comes from the Shunyata through the logic of the catuskoti; and infinity comes from Hindu, Jain, and perhaps also Buddhist mathematical and philosophical considerations that go back 2,000 years.”

“Sounds reasonable to me,” Jean-Marc answered, seeming to be distracted by something. He rubbed his forehead and brushed back his long, curly gray hair. Then, as if an afterthought: “But,
tell me, then. Do numbers really exist?” He looked at me triumphantly, like a chess player delivering checkmate.

“That's the biggest problem in all of the philosophy of mathematics,” I said.

“Yes, indeed,” he answered.

“Numbers are our greatest invention, and zero is the capstone of the whole system,” I said. “But whether they exist outside our minds, outside their role as a construct that helps us understand the world around us, is a big open problem. I've interviewed many mathematicians about their views.”

“And what do they say?” he asked.

“The majority are Platonists: They believe that there is a Platonic milieu in which numbers exist, quite independently of people or animals or any physical reality. But others are divided on this question. What do you think?”

“As a Hindu,” he said, “I certainly believe in an immanent, divine reality. As I've told you, I believe that Shiva is in me and in you and in everything and everybody. If we aren't here, Shiva still is—and so are numbers and mathematical and all spiritual essentials. There is a reality that goes beyond people, and it includes numbers.”

I was impressed with his erudition, his Eastern kind of Platonism. It was definitely time for me to continue my search for the first-known tangible evidence in Asia of the discovery of the idea of a zero—either invented or deduced from a latent reality.

I thanked Jean-Marc for the meal and the fascinating discussion and went down the hill to the bamboo bridge. I paid the bridge-keeper a dollar toll to cross it back to Luang Prabang, and walked through town to meet Debra at our cafe.

We had dessert together. Outside Paris, Luang Prabang is perhaps the best place in the world for delicious French pastries. We shared a
tarte aux pommes,
and I had a smoked tea while Debra drank a cappuccino. We watched the sun set over the Mekong—glorious shades of red and orange through a filmy white mist over the river. I told her about my conversation with Jean-Marc. “He sounds a bit like Roger Penrose,” she said, referring to a book we had both read, Penrose's
The Road to Reality,
which discusses the issue of whether numbers were invented or discovered. Then we looked at the pictures she had taken during the day and the ones she had just taken of the spectacular sunset.

We walked together to our secluded hotel on the hill overlooking the town. I had found what I had come to this enchanted town to discover: the source of the zero and the source of infinity, embedded in the millennia-old wisdoms of Buddhists and Hindus and Jains. I now badly needed specific information about K-127 and was eager to fly to Cambodia to look for the Khmer zero that George Cœdès had studied 80 years ago. I hoped the inscription had survived the vicissitudes of time and the ravages of the Khmer Rouge in Laos's neighbor to the southeast.

The next day, we packed our bags and took a cab to the small airport, still nearly empty despite the growing demand from tourists. People were talking about a new airport to be built, and the construction of a high-speed rail line from China in the north. Once these two projects were completed, the town of Luang Prabang would be full of Chinese and other tourists. We knew prices would go up, high-rise hotels would be built, and the peaceful atmosphere would likely change.

I was a little worried that I might be asked to pay a penalty again, this time in order to leave the country on an “improper” passport, but to my relief my passport was stamped and we were allowed to board our flight. We returned to Bangkok and Debra flew home. Our mini second honeymoon in the oriental gem of Luang Prabang was over too quickly. I remained in Bangkok and awaited information on the whereabouts of K-127.