References:
Cœdès, George, “A propos de l’origine des chiffres arabes,” Bulletin of the School of Oriental Studies, University of London, Vol. 6, No. 2, 1931, pp. 323–328.
Diller, Anthony, “New Zeros and Old Khmer,” The Mon-Khmer Studies Journal, Vol. 25, 1996, pp. 125–132.
Ifrah, Georges. The Universal History of Numbers. New York: Wiley, 2000.
26
I had been concentrating all along on the importance of zero as a place-holder within our number system, and on showing how our numbers work because we are able to insert a sign that says there are no tens or no hundreds or no thousands, and so on, in the representation of any number whatsoever using simply the ten numerals in our decimal number system. But what about the system as a whole?
Sitting at the departure lounge at the Phnom Penh airport waiting to board my flight back to Bangkok, I was pondering the rich history of zero the number—a concept I am convinced could only have arisen through a purely Eastern way of thinking (and, independently, by the Maya in the West).
Equally, I was thinking of the idea of infinity, also prevalent in Eastern thinking: the “endless sea,” Ananta the sea serpent, eternity, and innumerable other forms of extent that go beyond the simple numbers 1, 2, 3, and so on. But the development of our numbers in a purely mathematical setting, including both Eastern concepts of zero and infinity, was to take place in the West—or rather, both in the East and in the West (the rational, irrational, and complex numbers were explored theoretically in Europe between the fifteenth and nineteenth centuries).
We’ve seen that it is possible to define the numbers starting simply from the void, the empty set, and proceeding through the use of set membership: the set containing the empty set for 1, the set containing the empty set and the set containing the empty set for 2, and so on. But of course this is a sophisticated way of defining the numbers, using sets and the idea of set membership. In reality, numbers developed in a very different way.
The ancient Babylonians and Egyptians, thousands of years ago, learned to assign numbers to objects and thus to abstract the concept of number from the magnitude of the sets of things they observed. Perhaps the greatest intellectual discovery of early antiquity—at the dawn of civilization, really—took place when someone, or very likely several individuals at different places and times, could look at three stones on the ground, three cows in a meadow, three people walking on a path, three grains of wheat, three pyramids, three goats, three children, and understand that all of them had one and only one aspect in common: the quality of “being three.” Similarly, four could be defined as that aspect of many different things that are four, and so on. The numbers could grow and grow, and the magic of this understanding—that things that are of the same discrete magnitude are in some sense the same—was, and is, overwhelmingly powerful.
Soon enough, people of antiquity added words to their languages to represent these numbers. In fact, in India especially and in several other Asian countries under its influence, there were special words, nouns that everyone knew belonged to categories that were of universally accepted numbers, and these nouns became synonymous with the numbers. Here is an excellent example, from Cœdès’s seminal 1931 article, commenting on the stele of Changal (my translation from the French): “The year of the king of the çaka expressed in numbers is: the flavors, the organs of sense, and the Veda.”
Cœdès explained that there were six known flavors of food, five senses, and four Veda (the ancient collections of Hindu holy writings). Thus, this is a way of expressing the number 654 in words. This method was widespread in Cambodia, India, and other countries in south and Southeast Asia.
Next, Cœdès gave the example of an inscription from a place called Dinaya, discovered in 1923, in which the date çaka 682 is given as “the flavors, the Vasu, and the eyes.” Again he noted that flavors stood for 6 and explained that Vasu (deities attending to Vishnu, of which there were eight) stood for 8, and we know that a person has two eyes.
But Cœdès also discussed the problems that arose here. Across geographical areas and through time, there was not always complete agreement on which number was represented by which noun; sometimes ambiguities existed.1 This difficulty is similar to one we encounter today when we use the phonetic alphabet.
When asked to spell my name over the phone to someone whose English may not be perfect, or over a bad telephone connection, I often have to use this words-for-letters system. “Aczel,” I say, “A for apple, C for Charlie, Z for Zebra, E for Europe, L for Larry.” I use these words because they are the first ones that pop into my head, and I usually have to repeat this a couple of times.
But of course I am mostly wrong, as the accepted NATO Phonetic Alphabet is: Alfa, Bravo, Charlie, Delta, Echo, Foxtrot, Golf, Hotel, India, Juliet, Kilo, Lima, Mike, November, Oscar, Papa, Quebec, Romeo, Sierra, Tango, Uniform, Victor, Whiskey, Xray, Yankee, Zulu. But does anyone remember these?
By analogy, a system of numbers, which was prevalent in the East for centuries—in which one says “Veda” for four, “flavors” for six, and so on—certainly could not have been uniformly well understood by everyone. This was one important reason why written signs for numbers had to be invented.
Cœdès described the ancient Khmer number system, which was not decimal. Even today, despite the borrowing of numbers above 30, which are decimal, lower numbers in modern Khmer are not perfectly based on 10, but also on 5 and on 20. The Khmer, Cœdès noted, use many multiples of 20—as the French do only once, for 80 (quatre vingt, or four twenties) and numbers that go with it (for example, quatre vingt dix neuf, for 99). The Khmer use more of these multiples, clearly a vestige of a base-20 system due to our having ten fingers and ten toes, which persisted. This is reminiscent of the Maya number system, which was almost exclusively base 20 (with the exception of the partial calendrical use of base 18).
In antiquity, Cœdès explained, the Khmer possessed only the numbers: 1, 2, 3, 4, 5, 10, 20, and several multiples of 20. This was all they had as far as numbers they understood. At some point, they borrowed the Sanskrit word chata for 100. With these numbers they expressed all numerical information.2 All this was, of course, before the maturing of their numbers and the invention of zero (or its importation from India or some other place) as attested by inscription K-127.
What all this taught me was that fingers and toes are really important. If we had not had them, or had different quantities of them, maybe we would view numbers in a totally different way. If some day we were to meet aliens with only two fingers on each hand and two toes on each foot, their number system might well be binary, allowing them to communicate with the innards of their computers more directly than we do: Our numbers always have to be “translated” into a binary (0 and 1 only) code for a computer to understand them.
On the other hand, with two fingers per hand and two toes per foot, maybe their number system would be octal (based on 8). It was fun to speculate on such things, and it kept me entertained as I waited to hear about the fate of my precious find. In Bangkok, it helped me relieve the immense tension of waiting for news about the fate of K-127 and whether Hab Touch would follow through on his promise.
George Cœdès returned to his native France some years after French colonial rule in Indochina ended, as these new nations grappled with questions of democracy, parliaments, monarchy, and Communism. In Paris, he had a prestigious academic position and continued to write papers and books about Southeast Asia. He was highly decorated, having been awarded the rank of commander in Thailand’s Order of the White Elephant, as well as France’s prestigious Legion of Honor. He died in Paris in October 1969—a month before K-127 was brought to Angkor Conservation. Cœdès had several children, and one of them became the admiral of the Cambodian fleet. This fact made me feel a kind of nautical kinship with this great man.
27
On April 9, 201
3, I finally got the e-mail message I had been waiting for:
Dear Professor Amir,
I apologize for having taken so long to write to you. It was a great pleasure to meeting with you in Phnom Penh and delighted to hear about the history of Zero. Thank you for your research article on Khmer Zero, which is now the earliest Zero in the world civilization. I have shared this exciting news with my colleagues and hope this inscription will be on display in the National Museum in Phnom Penh. I look forward to meeting with you again and please let me know if I can be of assistance to you with this important research.
With best wishes,
Touch
I was elated. I couldn’t believe a successful conclusion was finally in sight. Could it be that my odyssey was now over? Following e-mails reassured me that what I had hoped for was going to take place at last. His Excellency Hab Touch would arrange for the priceless K-127 to be taken out of the hands of Lorella Pellegrino and placed in the Cambodian National Museum in Phnom Penh, where it once was, and where it belongs. From then on, scholars, mathematicians, historians of science, and the people of Cambodia and elsewhere would be able to see the very first zero of our numbers ever discovered—a find that changed our view of history, the one artifact in the history of science that proved definitively that the zero came from the East.
Debra met me again in Bangkok, and we flew to Paris together a week later, switching planes in Bahrain. At our small hotel on the Left Bank, I used the Internet and wrote a short article about the rediscovery of K-127 for the Huffington Post. It was published within hours. After I sent the link to Hab Touch, he responded that he was delighted that people would now be learning about “Khmer Zero,” as he called it. “Let the discussion begin!” he wrote me.
His country could benefit from displaying and explaining its antiquities, and I hoped he would succeed in his efforts to repatriate to Cambodia many statues that had been looted during the Khmer Rouge era and sold to museums around the world. The Tribune had an article about the New York Metropolitan Museum of Art agreeing to return two such statues, and other museums around the world were considering doing the same. I knew that this was the result of Hab Touch’s negotiations with museums, and I felt good that my own work had contributed in small measure to this larger effort.
Debra left to return to Boston and I stayed in France for a few more days. I had one last thing to do before this big adventure was over. After accompanying her to the counter for her transatlantic crossing, I walked over to another part of Terminal 2 at Charles de Gaulle Airport and boarded a domestic French flight to the south.
28
After landing in Toulouse, in southwest France, I walked over to the desk of a car rental agency and collected the keys to an Alfa Romeo. I headed due south, to the high Pyrenees.
The Alfa took the twists and turns in the steep mountain road beautifully. It was exhilarating to drive it uphill through so many quick, sharp turns. After two hours of climbing, I made it to the top, way above the tree line, having just crossed the border to the independent mountaintop principality of Andorra. I enjoyed a strong espresso at nearly 9,000 feet, was buoyed by the breathtaking view from the summit, and then headed back down somewhat, recrossing the French border. Two road turns below it, I found what I had come for.
I stopped by the gate of an alpine villa built of wood, the outside panels carved in the ornate designs one often sees in the Austrian Alps.1 I knocked on the door, and an attractive woman in her fifties opened it. She was wearing a long blue dress with a wide décolletage that revealed generous cleavage. “Oh, he’s been waiting for you all morning,” she said with a smile. “Let me get him . . . Laci!” she called.
He came down the stairs. At 88, he looked fit and healthy. “So good to see you!” he said, giving me a big hug. “It’s been so many years . . . what, 40 or so?”
I smiled and said, “Yes, yes, a very long time. But I wanted to see you. And I have something that may interest you.” We sat down in the spacious living room that opened to the balcony with its views of the mountains and talked about the old days on the ship, about the mountains, and about mathematics and the birth of the numbers. “You told me something when we parted on the ship so long ago—in 1972,” I said. He looked surprised. “The name was George Cœdès,” I said, spelling it out. “He was the French archaeologist who found the first known zero in Asia.”
“Ah yes,” he said. “I vaguely remember something now. So he found it, right.”
“But it was lost, you know,” I added. “The Khmer Rouge—”
“Ah, yes, they destroyed everything, I’ve heard. So it is gone now?”
“Well—I did manage to find it,” I said.
“Find the first zero?” There was a glint in his old yet still keen eyes.
“Yes. Let me show it to you.” I turned on my PC and showed him the photographs of K-127. “This is the oldest zero in history,” I said. “I found it after so many years of searching. And it was indeed Cœdès who first published it in 1931, debunking those old claims about the zero being a Western or an Arab invention.”
Laci sat on the couch across from me, smiling. “So, my friend,” he said, “you found the earliest known zero. Congratulations! That’s really something. What will you do next?”
“We still don’t know where the numerals as a whole came from. Someone should look into the Indian numbers: Ashoka’s, the Nana Ghat’s, the Kharosthi. There may be place for good research there, and to see whether, indeed, Aramaic letters have led to the numerals. But as a mathematician, I suppose you aren’t interested in this kind of work.”
“No,” he said, “your idea about the origin of the concept of zero, coming from the Buddhist void, is more interesting to me. Maybe some philosophers will follow with this thread.” He paused, and after a moment continued. “But what you’ve achieved is significant—and I’m so glad that a casual conversation with me so long ago led you to this fruitful search.” He was clearly pleased and stood up. “You did it, you did it, I am so proud of you!” He held my hand. “This calls for a drink.” He excitedly called for the woman—girlfriend or wife, I never found out—and she brought us whiskey on ice.
Then she opened a jar of black Russian caviar and spread it on little toasts for the three of us. Real Caspian sturgeon—I knew it. It must have cost a pretty penny. “I remember eating caviar on the ship,” I said. “But that deep-pocketed shipping company, Zim Passenger Lines, which could afford to decorate the ship’s halls with original oils by Chagall and Miró—and then went bankrupt because it had spent so much money on such luxuries—paid for it all.”
“Ah, don’t worry.” Laci straightened up and looked at me. “We get a lot of it here.” The woman laughed knowingly, and he walked over from the living room to the adjacent kitchen and opened a large refrigerator, just so I could see what was inside: many more jars of Caspian caviar. And the bar was stocked with a lot of expensive liquor: scotch whisky, Calvados, Drambuie, Grand Marnier, sake. I looked at him a little puzzled.
“Well,” he said after a moment, “you saw the French customs checkpoint just up the road, right? You couldn’t have missed it.” I didn’t understand. “You know that Andorra is one of the last tax havens in the world, don’t you?” I nodded. A vague notion began to surface in my mind. There was a moment’s silence. He looked at me, and then he said, “You know, late at night, there is nobody there at the checkpoint. And this house is at exactly the right place—”
“Just like my mother’s suitcase,” I interrupted.
The thinnest of smiles spread across those old lips. “Just like your mother’s suitcase,” he said.
Notes
Chapter 1
1.An analysis of the forms of Latin numerals, and a new theory about their being derived from Etruscan signs, is well presented in Paul Keyser, “The Origin of the Latin Numerals from 1 to 1000,”
American Journal of Archaeology 92 (October 1988): 529–46.
Chapter 2
1.Georges Ifrah, The Universal History of Numbers (New York: Wiley, 2000) has a number of pictures of ancient bones with markings.
2.Thomas Heath, A History of Greek Mathematics, Vol. I (New York: Dover, 1981), 7.
3.For a modern description of this issue, including later contentions by other scholars, see Georges Ifrah, Universal History of Numbers, 91.
Chapter 3
1.A good description of the Mayan numerals, calendar, and the zero glyph can be found in Charles C. Mann, 1491: New Revelations of the Americas Before Columbus (New York: Knopf, 2005), 22–23, 242–47.
2.Georges Ifrah, The Universal History of Numbers (New York: Wiley, 2000), 360.
Chapter 4
1.Saint Augustine, The City of God (New York: Modern Library, 2000), 363.
2.Chapter 18, Verse 8 of the Mulamadhyamakakarika, written by the prominent Buddhist monk and scholar Nagarjuna in the second century CE.
Chapter 5
1.Louise Nicholson, India (Washington, DC: National Geographic, 2014), 110.
2.David Eugene Smith, History of Mathematics, Volume 2: Special Topics in Elementary Mathematics (Boston: Ginn and Company, 1925), 594.
3.Takao Hayashi has referred me to Alexander Cunningham, “Four Reports Made During the Years 1862–1865,” Archaeological Survey of India 2 (1871): 434.
Chapter 6
1.Mark Zegarelli, Logic for Dummies (New York: Wiley, 2007), 20–21.
2.Ibid., 22–23.
3.F. E. J. Linton, “Shedding Some Localic and Linguistic Light on the Tetralemma Conundrums,” manuscript, http://tlvp.net/~fej.math.wes/SIPR_AMS-IndiaDoc-MSIE.htm.
Finding Zero Page 17