Finding Zero (14 page)

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.

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
é
pousailles 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.