Finding Zero
Page 4
Anything is either true,
Or not true,
Or both true and not true,
Or neither true nor not true.
This is Lord Buddha’s teaching.2
Upon reading these words, I exclaimed, “God, what strange logic!” Raju laughed and explained that it came from the prominent second-century CE Buddhist philosopher and teacher Nagarjuna. He looked at me and said, “Spend some time in the East and you will understand it.”
I stared long and hard at Nagarjuna’s statement, and considered. True, not true, both true and untrue, neither true nor untrue?—What is this set of options? They made no sense at all. This certainly was a bizarre way of thinking about reality. Raju quickly explained, “The key to everything here is called Shunya.” He smiled widely, and when he saw that I did not understand what Shunya meant, he continued. “Shunya means zero in our language. But it also relates to the Buddhist philosophical concept of the void, which is called Shunyata. You see, zero, the number, and the Buddhist emptiness—the goal of meditation and an ideal striven for on the road to Nirvana, or enlightenment—are one and the same. Emptiness is a deep philosophical concept, and from it we get our zero.”
Professor C. K. Raju in Shimla, the foothills of the Himalayas, India.
Then Raju picked up his worn leather bag full of papers, shook my hand a few times, and said, “But you will figure it all out.” He flashed his big, toothy smile again and disappeared. He had to rush, he had explained; he was leaving for Malaysia, where he had a visiting professorship at the university. The Malaysians, apparently, were interested in work that would bring scientific recognition to Asia and were willing to pay for it; Raju was regularly presenting the research he was producing in Malaysia at international conferences in Germany.
The Germans, too, were involved in projects on the history of Asian science, and for them Raju was a gold mine, someone who prodigiously produced papers on this topic. I was glad for the idea about Shunya, the zero, being derived from the Buddhist void, Shunyata. I also had a vague feeling that the different kind of logic—true, untrue, both true and untrue, and neither true nor untrue—somehow was related to the concept of the void, and hence to the Eastern zero.
5
I began my search in south Asia working backward in time, starting with the tenth century CE. The day after my memorable meeting with Raju, I took a yellow-green tuk-tuk—the ubiquitous three-wheeled, motorcycle-powered vehicle that can navigate even the narrowest streets of the most crowded of Asian cities—through Delhi’s early-morning mist to the airport. There I boarded a Kingfisher Airlines flight to Khajuraho, a complex of Hindu and Jain temples situated in what was once a dense forest in the state of Madhya Pradesh. After two hours of flight, the Kingfisher stopped at the holy city of Varanasi on the Ganges, where we stayed on the ground for about 20 minutes, and then took off again for Khajuraho, less than an hour away. As we were about to land, I glimpsed stone temples below in the remains of a lush tropical jungle.
A few decades ago, a Japanese mathematician named Takao Hayashi had photographed some mysterious numbers here, but he had no information on the name of the temple where the ancient inscription could be found. I had to discover it for myself.
Khajuraho has a tiny airstrip, and a hut serves as a terminal. I got a cab, a rickety old Ford with no upholstery or much of an interior, smelling of sweat and rotted vegetables. I took it up the straight, dusty road through dried-up, hardscrabble fields to one of just a handful of hotels, a Best Western—the nicest place in this tiny town, it turned out. The young man at the desk was not as interested in assigning me my room as in selling me tourist trinkets in his side business. I turned him down and asked for my room key—I would have no time for shopping, I explained. After checking in, I left the hotel and in about 15 minutes of walking on deserted roads managed to find my way to the temples. A man with apparent signs of leprosy sat on the ground outside the entrance to the fenced compound. I paid the admission fee and entered. Most tourists come here to gawk at the graphic erotic statues and friezes that adorn these unusual temples.
In 1838, a British military officer, Captain T. S. Burt, was exploring the jungles of Madhya Pradesh some 400 miles southeast of New Delhi with his company of Bengal Engineers when he and his men came upon a group of ancient temples that had been reclaimed by the jungle. What they saw stunned Burt. In his logbook, he noted that these temples were among the finest he had ever seen, but he was also at a loss on how to describe the nature of the erotic art he saw at Khajuraho. About 10 percent of all the magnificent, eleventh-century stone statuary here depicted sexual situations, some of which seem startlingly bold even today.
In the West, we don’t see sexual imagery in public locations—certainly not in places of worship. But the statues Burt saw, of men and women engaged in a variety of sexual positions, many quite acrobatic and imaginative, were on the outside walls as well as the insides of Hindu and Jain temples built a thousand years ago. Burt wrote in his diary, “I found seven Hindoo temples, most beautifully and exquisitely carved as to workmanship, but the sculptor had at times allowed his subject to grow a little warmer than there was absolute necessity for his doing.”1 There were once 85 temples at Khajuraho, and 20 of them survive to this day; they are both Hindu and Jain.
All temples in this area—“a stone’s throw away from each other,” as Burt wrote—have statues adorning their walls. These sculptures depict scenes from everyday life as well as images of deities. But the erotic scenes strongly dominate these temples because they are so explicit and so unexpected. These are almost life-size statues and friezes carved in gray, yellow, or reddish stone showing people in every imaginable sex act. At one temple, high above the observer, a man and a woman are supported by two seminude women; he is held upward, facing away from the viewer; below him and facing us is the naked woman, her head on the floor; her legs are spread above, and the two of them are attached in their genital region. At another temple, at eye level, there are statues of several elephants led by people, and on the right, unexpectedly, is a statue of a man mounting a woman from behind—oblivious of the elephants. At another temple, an entire panel depicts men and women engaged in several different positions of oral sex, as well as a combined oral and vaginal sex act by one woman with two men.
To date, no good explanation has been proposed for this unusual art. Tour guides lecture about the Kama Sutra to naive tourists who venture here, and magazine articles and tour books suggest that some of the sexually explicit statues represent the Hindu god Shiva and his Shakti, Parvati. The dreaded destroyer of worlds, if this is true, is only interested in his consort’s body; and anyway, many of the temples are not Hindu but Jain. Some of the more scholarly sources conjecture that the imagery may have served as fertility symbols. But nobody really knows the answer.
As inexplicable as the erotic art of Khajuraho was, so was the discovery more than a century ago of a piece of complex mathematics in this location. I had gleaned hints of it in an old book on the history of mathematics. David Eugene Smith says, “[A magic square] appears in a Jaina inscription in the ancient town of Khajuraho, India, where various ruins bear records of the Chandel dynasty (870–1200).”2 I had originally thought that Hayashi might have been led here by this reference; instead, it turned out that he had read the very first historical description of this curious mathematical piece, written much earlier. Hayashi had possessed the actual announcement of the discovery of this ancient magic square in the notes made by the prominent British archaeologist Sir Alexander Cunningham, who in the 1860s found the mathematical inscription by the entrance to a Jain temple.3
I spent several hours visiting all the temples of the compound, but nowhere found an inscription. Where was Hayashi’s magic square? I asked every tour guide I met—nothing. Then a French tourist who had overheard me ask his guide said, “I think I may have seen some ancient numbers by the door of o
ne of the temples in the Eastern Group.” This was all the way across town, in a more deserted and rarely visited set of temples.
Takao Hayashi in the former summer palace of the Maharaja of Mysore (with his son Makoto) in 1983 while researching the history of mathematics in India.
I left the fenced-off compound and walked for half an hour, passing stray cattle, which roam freely in all Indian towns, feeding on garbage, and finally found the ancient temples composing the Eastern Group. The edifices here were mostly Jain rather than Hindu. I went from one temple to the next, followed by stray dogs and children dressed in rags asking for money. Otherwise, the grounds were eerily deserted. A wind blew in from the fields, kicking up swirls of dust. I saw nothing. Then I came to the Parsvanatha temple, a Jain temple built in the mid-tenth century CE. The doorway was framed with erotic statues. A man, or perhaps a god, stared amorously into the eyes of the woman or goddess in his arms, her head turned up to him. His left hand fondled her ample breast.
And there on the right, inside the doorway, I finally found what I had come here for—the numerals Hayashi had seen 40 years earlier but could not remember exactly where. It was a magic square, with Hindu numerals (some of which are like our own and some so different they are not recognizable by a nonexpert), inscribed on the door of this thousand-year-old temple. This magic square was a four-row by four-column square with the following digits (here transcribed using our modern numerals):
Now notice some amazing facts: The sum of every horizontal row is 34; so is the sum of every vertical column, the sums of the two diagonals, the sums of all two-by-two squares at the four corners of the larger square, and also the sum of the central two-by-two square. The temple bearing this curious inscription is definitively dated by another inscription to 954 CE. So as early as the mid-tenth-century, the people who built and worshipped here understood how to construct such sophisticated magic squares. The Khajuraho magic square is one of the oldest four-by-four squares (earlier three-by-three squares are known in China and Persia).
The magic square at the entrance of the tenth-century Parsvanatha temple at Khajuraho.
With its magic square and numerals found on site, Khajuraho provides the greatest example of Hindu numerals from as early as the tenth century. The numbers found at Khajuraho and at other early temples in India suggest that numbers here may have originated in connection with religious needs and practices. For example, ancient Indian documents called the Vedas—dating from as early as the second millennium BCE—specify the sizes of temples and the numbers of animals to be sacrificed; all are represented numerically, and it is perhaps for this reason that the earliest Hindu numerals appear in ancient temples.
The decorated façade of one of the ancient temples of Khajuraho.
Also, these numbers have been identified in one of the most curious early documents ever discovered. Interestingly, in 1514, the German artist Albrecht Dürer, who was fascinated by numbers, made a celebrated engraving called “Melancholia,” which featured at its upper right corner a four-by-four magic square:
This, like the Khajuraho magic square constructed almost six centuries earlier, is a “normal” magic square, meaning that all numbers from 1 to 16 must appear on it, and its sums are all 34. But while the Khajuraho magic square is surrounded by smiling naked or seminude figures engaged in carnal pleasures, Dürer’s magic square is placed next to a melancholic, solitary, and fully clothed female figure. This is one more example of the differences I perceived in logic—and outlook—between East and West.
The Khajuraho inscription not only shows that the tenth-century Indians were adept at this kind of magic-square arithmetic, but it also showcases the numerals they used at that time (shown in the picture of the Parsvanatha magic square) and the correspondence to our numerals. Which numbers are the same and which different? How did the Hindu numerals become our own? And how did they change?
6
There are different kinds of logic—not just the Western, “linear” kind of logic. Though—very broadly speaking—in the West religion often seems to be antithetical to sex, in the East religion and sex are part of one grand celebration of life and all its pleasures. Mathematics is linked to both sex and religion, as the placement of the Khajuraho magic square and the erotic imagery in religious temples suggests. In fact, sex embodies perhaps the greatest mystery of life, and mathematics—the abstraction of logically based processes we perceive in nature around us—is arguably the greatest intellectual mystery.
I wondered whether the ancient Jains and Hindus who built these temples pondered such deep mysteries. How could they not have, given the evidence they left us? Why are our lives so deeply ruled by sex? And why is the universe fundamentally ruled by mathematics? What is the secret to desire? And why do numbers behave in such curious ways, as evident in a magic square and in the remarkable way arithmetic works? These may well have been some of the questions that Eastern peoples asked themselves in antiquity, and their answers might have led them to belief, and hence to the establishment of their religions and the invention of their gods and the construction of places of worship, where they placed symbols of their greatest mysteries: sex and mathematics. This, at least, was my conjecture. Was I on the right path?
The erotic statues of Khajuraho reminded me of the previous time I had seen ancient erotic art. It was in Pompeii, during another cruise I took aboard my father’s ship when I was 14. Laci couldn’t accompany me this time because my father had asked him to supervise the loading of specially roasted Italian coffee. The company insisted on serving its passengers only the highest-quality Italian coffee, so this commodity was always loaded onto the ship at an Italian port. My mother and sister went shopping, and my father stayed aboard. So the wife of the chief engineer, Ruth Chet, an attractive and sophisticated 32-year-old, accompanied me on the visit to Pompeii.
We arrived at the archaeological site and visited the antiquities, and then entered the special exhibits area, where erotic statues and frescoes found in the ruined city were on display. But the Italians had a strange, sexist rule in those days: Men of any age could visit the exhibit, but no women. I was naturally curious about this art and went in. My young age was no issue, but Mrs. Chet was barred, and despite her loud protestations, pleading, begging, and threatening, the guard would not let her pass.
In the exhibit hall I saw a statue of a small man with a giant erect penis in his hands, reaching almost to his neck, and couples on beds copulating in various positions. The women’s breasts were often covered with strapless bras. These statues and frescoes were all pre-Christian, as Pompeii was destroyed in 79 CE, although the covered breasts might demonstrate a degree of modesty even in sexual situations. Once Christianity was adopted in the West, the use of erotic imagery declined drastically. This is in contrast with what was happening at the same time in India. As a 14-year-old, I was deeply curious about the subject and, naturally, also very embarrassed by it. And Ruth Chet, being barred from the display, took her frustrations out on me. As soon as we returned to the ship, she ran to my father. “Your son has a dirty mind!” she cried. “He went into the pornographic exhibit—and they wouldn’t allow me to go in.” My father laughed.
At 14, I was intensely shy looking at the Roman art. But now, at Khajuraho, I was a mature adult on a mission to discover ancient numerals. The mysterious suggestive statues of Khajuraho were in a sense like the mathematical objects that hid in their midst. Thinking about the similarities and the contrasts between the two assemblages of sensual art, I came to the conclusion that Eastern peoples of the tenth century had no hang-ups about sex and sexuality. The freedom exhibited by the Khajuraho statuary evidences such openness and sheer excitement about life and its pleasures that I felt certain it pointed to a fundamental difference between East and West. I wondered whether this disparity of views was somehow connected with the fact that Eastern logic is different from the usual Western way of thinking, a
nd whether both relate somehow to the ability to abstract numerals out of the void and thus create a number system so powerful that it would one day take over the world. In the East, sex and logic and math seemed to be related.
We tend to think that our Western logic is the only valid kind of logic. A few years ago, I became frustrated that my sister, Ilana, who had been diagnosed with breast cancer, was not making what I believed to be logical decisions. She shunned Western medicine in favor of Chinese qigong as her only treatment for the disease. In desperation, trying to understand how anyone could be so “illogical,” I bought the book Logic for Dummies by Mark Zegarelli. He says that Aristotle was the true founder of classical logic. I read,
For example, here’s Aristotle’s most famous syllogism:
Premises:All men are mortal
Socrates is a man
Conclusion:Socrates is mortal.1