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 Deep Mathematical Analysis
In one of the most brilliant proofs in the history of mathematics, Cantor was able to show that the rational numbers—fractions made of integers as numerator and denominator—are of the same infinite “size” as the integers.
Cantor understood that the operation of exponentiation was the lowest (and the only one we know) arithmetical move that could take an infinite quantity from one level to a higher level of infinity. Exponentiation is essentially a move to the power set—the set of all subsets of a given set. This is one of the reasons why Bertrand Russell’s paradox is indeed a paradox: We cannot find a universal set because no set can contain its own power set! Let’s look at an example, the set containing only two distinct elements. Let’s call this set X and its elements a and b. Now the power set of the set X with only two elements, a and b, is the set that contains all subsets of X. It is, therefore, the set that contains: 0 (the empty set), a, b, and (a, b). These are all the subsets of X. We see that the power set is always larger than the set itself (because the set X itself contains only a and b). The power set has more elements, and the reason for this is that the power set has 2n elements, where n is the number of elements of the original set. So for any set, the power set associated with it is always larger than the set itself. If there were a set containing everything, its power set would still be larger, obviating the assumption that the original set contained everything. “Everything is not everything,” as the monk had told me.1
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 2n 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.
18
Just when I needed someone to clarify some of the ideas I’d been pursuing, I got a call from my friend Jacob Meskin, a Princeton-educated philosophy professor and expert on the religions of eastern and Southeast Asia. We hadn’t talked for many months and caught up on conversation about mutual friends and about philosophy. I tentatively explained to Jacob my view about zero, Shunyata, and the catuskoti. “That’s an interesting connection,” he said. “In fact, Nagarjuna does talk about the void as a key principle, and of course the emptiness, the empty set if you will, is a solution to the ‘four corners’—we don’t see too many other good solutions here.”
He chuckled. “Buddhism has a lot of numbers in it: the three marks of existence, the four truths, the eightfold path, the twelve-link chain of dependent co-origination, and so on. But tell me about the numbers. Why is the zero so important? I don’t really understand that.”
I explained to Jacob how the use of the place-holding zero is what allows the numerals to cycle; it is what enables us to use the same nine signs (plus the zero itself) again and again—for different purposes. For example, we can use the numeral 1 to mean the number one; but when we place a zero to its right, that same sign, 1, now means ten. A 4 alone means four, but when followed by two zeros it becomes four hundred. It means 4 hundreds, 0 tens, and 0 units. The existence of a place-holding zero is what gives meaning even to numbers that contain no zeros in them. The number 143, for example, could not be written this way if the number 140 didn’t exist, and this number needs the zero as a place-holder for the empty units. Without the zero, none of these numbers and manipulations of such numbers would be possible. “That’s interesting,” he said. “Nagarjuna actually talks about the void being movable from place to place—just like your place-holding zero. Perhaps he understood that, too. I like to think about this as the little plastic toy that children play with: a square with numbers that can be moved around, but the numbers can only be moved because there is an empty space for one missing little number-square. This missing piece allows us to move the numbers around one at a time until they are in numerical order. So, you see, the void is everywhere and it moves around; it can stand for one truth when you write a number a certain way—no tens, for example—and another kind of truth in another case, say when you have no thousands in a number!” Seeing Nagarjuna’s apparent view of the void, and perhaps the zero as well, as a dynamic, movable piece was certainly intriguing.
I mentioned my feeling that in India, math and sex and religion were intertwined. Jacob responded: “Forgive me, I am sure you’ve thought of this, but there is, it seems to me, a line of connection in this conversation here between numbers and . . . ahem . . . sex. It’s a bit weird, but here it is. Nagarjuna expresses it toward the end of chapter 24 of his Mula-madhyamaka-karikas (Fundamental middleway verses). In that chapter he imagines a critic attacking the view that he, Nagarjuna, has been presenting in the preceding 23 chapters. Nagarjuna imagines a critic accusing him of something like nihilism. This imaginary critic says, in effect, ‘Hey Nagarjuna, you’ve made Buddhism into the teaching of Shunyata. But that means that everything is empty. And isn’t that just like saying that nothing is really true? And doesn’t that mean that everything the Buddha said isn’t really true?’
“Nagarjuna’s answer is fascinating. He says that his critic has everything backwards and that it’s only because everything is in fact empty (Shunyata) that everything actually works, including the truths the Buddha came to state. Sort of like ‘without Shunyata, nothing works; with Shunyata, everything works.’” Then Jacob paused. “I will send you an Internet translation of some chapters from Nagarjuna’s MMK, and it includes chapter 24. I’d love to go over it with you if you’d like. But the bottom line is that if everything really did possess an eternal, unchanging character—an essence, sva-bhava in Sanskrit—then the basic claim of the Buddha, namely that everything arises only via a complex set of cooperating and conjoined factors, could not be true. The Buddha is insisting that everything is endlessly intertwined with a vast causal network of many other things, and so no single thing can ever truly be thought of as independent, as having its own essence. This is the fundamental Buddhist truth of what is called dependent co-origination.
“Now, here comes the intriguing connection to sex. Shunyata would seem to be the fundamental openness of reality, its receptivity, the yielding framework through which and within which change and fluctuation and movement become possible. It is as if we are saying, à la Nagarjuna, that it is only because of zero (Shunyata) that there can be variation in intensity (number). Without emptiness there could be no movement; without zero, there could be no numbers. Does one dare to hazard the (by now obvious) surmise that zero is the (in a sense!) principle of the womb, the vagina, and that the numbers, that is to say numerical quantities as opposed to zero, are the principle of the phallus? Are enumeration, measurement, even the ticking off of a Geiger counter or digital display perhaps, an echo of . . . sexual intercourse, where numbers move back and forth in a field opened up to their waxing and waning only by the blessing of a receptive, enveloping vacuity ready to receive them?”
Jacob’s theory was fascinating and intriguing, and I looked forward to pursuing these ideas further.
19
To while away my time while waiting for information on the Khmer inscription with the first zero, I went to Jim Thompson’s house, one of my favorite sites in Bangkok. Jim Thompson, born in 1906, was an American businessman, graduate of Princeton, and CIA operative during World War II. He then abandoned a successful business career in New York to come to live in Thailand.
Here, he made one of the most remarkable contributions a foreigner has ever made to the country. He single-handedly revived a dying cottage industry: silk production. Within a few years, his vision and business acumen turned Thailand into a major world producer of silk and silk products. He did it by providing incentives to small manufacturers all over Thailand, mom-and-pop businesses, to weave silk, and he arranged for its sale on fair terms to export companies.
Thompson became a prominent expat living in Bangkok, and it is likely that he knew another leading expat: George Cœdès. There were many functions in which leaders in the close-knit expatriate community of the city met and interacted. But we have no clear evidence that they did indeed meet. Thompson was divorced when he came to Thailand, and here he knew many women in the European and American community; several of them became intimate friends, although he did not have a long-term romantic attachment, as far as we know.
Thompson built a house—actually a series of several connected houses—in the heart of the city by a canal. These buildings were designed in the typical style of the Thai countryside: made from wood, elevated on stilts to prevent flooding from overflowing rivers or canals, and painted dark red. He was also an avid collector of Asian art, and so today his houses, still containing his impressive collection of fine Asian art treasures, function as a museum.
In 1967, when he was 61 years old, Thompson took a trip to neighboring Malaysia with three friends, a couple and a woman friend of his. They went to a forest recreation area called the Cameron Highlands, where they stayed in a lodge. In the late afternoon, Thompson told his friends he was going for a walk and left the compound to follow a hiking trail. He was never seen again.
Within hours of his disappearance, a large search party was organized, including hundreds of police and other public safety personnel scouring the area in search of Thompson. The entire mountainous region was searched methodically for several weeks, as he was a prominent foreign missing person. But to this day, not a single credible clue has surfaced about Jim Thompson’s fate. His disappearance is one of the greatest mysteries of this kind.
I came to Jim Thompson’s house to ponder disappearances. A somewhat similar story is that of the brilliant Italian theoretical physicist Ettore Majorana, who had worked with Enrico Fermi in Rome. In 1938, Majorana took a ferry from Sicily to Naples, where he was living at that time, and disappeared without leaving a trace. As in the case of Thompson, conjectures and theories abound about what might have happened to him. One hypothesis was that he did make it to shore but left the ferry unseen and then went into a monastery to hide from the world, perhaps sensing that a terrible war was about to erupt and that his and Fermi’s work in physics might be used to make a doomsday weapon.
I also thought about yet another vanished person, one whose work is so close to my topic: Alexander Grothendieck. We have good indications that he is, indeed, alive. Majorana and Thompson never left behind evidence that they were still
living—but who knows, maybe either or both lived for at least some time after their disappearances.
We know that Grothendieck is still alive because he does send communiqués from time to time. The last one was in 2011, when he sent a letter from his hiding place, addressed to a Paris mathematician, in which he demanded that all his published and unpublished works be immediately pulled from any kind of circulation, private or public or anything in-between. Surprisingly, his colleagues agreed to this demand, even though it meant that the mathematics world would lose access to his work. Within days, most of his publications—even copies existing in cyberspace—were removed from circulation. Fortunately for me, I had already secured a copy of Grothendieck’s most bizarre, and gargantuan (stretching over 929 pages), mathematical-autobiographical screed titled Recoltes et Semailles (Reapings and sowings). This rambling document, written in French and circulated among his friends in manuscript form in 1986, is a mixture of mathematics, biographical descriptions, and thoughts about the universe. He had hoped to get it published but had not succeeded in doing so. In the meantime, the manuscript had achieved great success among mathematicians, although most of its copies, paper or electronic, had by now been destroyed in compliance with his request.
Fittingly, I now sat under a banyan tree in the garden of another missing person and read from my copy of Grothendieck’s book. Grothendieck describes how he had been fascinated with the idea of numbers since he was a very young child still living in Hamburg, cared for by a foster family as his anarchist parents, Hanka Grothendieck and Sacha Schapiro (they never married; Grothendieck uses his mother’s last name), were fighting with the Republicans on the losing side in the Spanish Civil War of 1936. When the Republicans were defeated by Franco’s fascists and routed out of Spain, the couple recrossed the Pyrenees into France but were immediately caught by the French police. They eventually ended up in detention camps. Alexander would join his mother to spend the Second World War in a wartime camp, while his father was sent to his death at Auschwitz.
Finding Zero Page 12
