Finding Zero (20 page)

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
.

4.
See Pierre Cartier, “A Mad Day's Work,”
Bulletin of the American Mathematical Society
38,
no. 4 (2001): 393.

5.
Ibid., 395; italics in the original.

6.
F. E. J. Linton, “Shedding Some Localic and Linguistic Light on the Tetralemma Conundrums.”

7.
More on this can be found in C. K. Raju, “Probability in India,” in
Philosophy of Statistics,
Dov Gabbay, Paul Thagard, and John Woods, eds. (San Diego: North Holland, 2011), 1175.

Chapter 7

1.
Kim Plofker,
Mathematics in India
(Princeton, NJ: Princeton University Press, 2009), 5.

2.
John Keay,
India: A History
(New York: Grove Press, 2000), 29.

3.
Ibid., 30.

4.
Ibid., 30.

5.
John McLeish,
The Story of Numbers
(New York: Fawcett Colombine, 1991), 115.

6.
Ibid., 116.

7.
David Eugene Smith,
History of Mathematics,
volume 2: Special Topics in Elementary Mathematics
(Boston: Ginn and Company, 1925), 65.

8.
M. E. Aubet,
The Phoenicians and the West
(Cambridge: Cambridge University Press, 2001).

Chapter 8

1.
Robert Kanigel,
The Man Who Knew Infinity
(New York: Washington Square, 1991), 168.

2.
There is a rare reference to this plate, which may have had an early zero in it, in
Epigraphia Indica
34 (1961–1962).

3.
Moritz Cantor,
Vorlesungen uber Geschichte der Mathematik
vol. 1 (Leipzig: Druck & Teubner, 1891), 608.

4.
Louis C. Karpinski, “The Hindu-Arabic Numerals,”
Science
35, no. 912 (June 21, 1912): 969–70.

5.
Ibid., 969.

6.
G. R. Kaye, “Indian Mathematics,”
Isis
2, No. 2 (September 1919): 326.

7.
Ibid., 328.

8.
I am grateful to Takao Hayashi for this information. He discusses the lost Khandela tablet in his book in Japanese,
Indo no sugaku
[Mathematics in India] (Tokyo: Chuo koron she, 1993), 28–29.

Chapter 10

1.
A good modern source is Pich Keo,
Khmer Art in Stone,
5th ed. (Phnom Penh: National Museum of Cambodia, 2004).

2.
George Cœdès, “A propos de l'origine des chiffres arabes,”
Bulletin of the School of Oriental Studies
(University of London) 6, no. 2 (1931).

3.
Ibid.

4.
Ibid., 328.

Chapter 11

1.
Chou Ta-kuan, “Recollections of the Customs of Cambodia,” translated into French by Paul Pelliot in
Bulletin de l'École Française d'Extrême-Orient,
123, no. 1 (1902): 137–77. Reprinted in English in
The Great Chinese Travelers,
Jeannette Mirsky ed. (Chicago: University of Chicago Press, 1974), 204–6.

2.
Ismail Kushkush, “A Trove of Relics in War-Torn Land,”
International Herald Tribune,
April 2, 2013, 2.

3.
C. K. Raju, “Probability in India,” in
Philosophy of Statistics,
Dov Gabbay, Paul Thagard, and John Woods, eds. (San Diego: North Holland, 2011), 1176.

4.
Nagarjuna,
The Fundamental Wisdom of the Middle Way
(Oxford, UK: Oxford University Press, 1995), 3.

5.
Ibid., 73.

6.
Thich Nhat Hanh,
The Heart of the Buddha's Teaching
(New York: Broadway, 1999), 146–48.

7.
George Cœdès, “A propos de l'origine des chiffres arabes,”
Bulletin of the School of Oriental Studies
(University of London) 6, no. 2 (1931) 323–28.

Chapter 16

1.
This comes from Graham Priest, “The Logic of the
Catuskoti,
”
Comparative Philosophy
1, no. 2 (2010): 24.

2.
T. Tillemans, “Is Buddhist Logic Non-Classical or Deviant,” 1999, 189, quoted in Graham Priest, “the Logic of the
Catuskoti,
” 24.

3.
S. Rhadakrishnan and C. Moore, eds.,
A Sourcebook on Indian Philosophy
(Princeton, NJ: Princeton University Press, 1957), quoted in Graham Priest, “The Logic of the
Catuskoti,
” 25. Priest explains that “saint” is a poor translation and that what it means is someone who has reached enlightenment, a Buddha (or Tathagata).

4.
Graham Priest, “The Logic of the
Catuskoti,
” 28.

Chapter 17

1. For more on the story of Georg Cantor and the various levels of infinity, see Amir D. Aczel,
The Mystery of the Aleph
(New York: Washington Square Books, 2001).

Chapter 22

1.
For accurate radiocarbon dating of the Thera explosion see Amir D. Aczel, “Improved Radiocarbon Age Estimation Using the Bootstrap,”
Radiocarbon
37, no. 3 (1995): 845–49.

Chapter 24

1.
I heard this story from another well-known mathematician and friend of Kakutani, Janos Aczel (no relation; it's a common Hungarian last name).

Chapter 26

1.
George Cœdès, “A propos de l'origine des chiffres arabes,”
Bulletin of the School of Oriental Studies
(University of London) 6, no. 2 (1931): 326.

2.
Ibid., 327.

Chapter 28

1. Some details about the house and its location have been changed to protect the occupants' privacy.

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The Universal History of Numbers.
New York: Wiley, 2000. This is a well-recognized, comprehensive work on the history of numbers and is much quoted. It is, however, neither very scholarly nor based on original research. The fact that it receives continuing attention only points to the need for a very serious and deep analysis of this crucial step in humanity's intellectual history.

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The Nothing that Is: A Natural History of Zero.
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India: A History.
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American Journal of Archaeology
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Mann, Charles C.
1491: New Revelations of the Americas Before Columbus.
New York: Knopf, 2005. Good description of the Mayan numerals and zero glyph.

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The Story of Numbers.
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Nagarjuna.
The Fundamental Wisdom of the Middle Way.
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The Exact Sciences in Antiquity.
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The Heart of the Buddha's Teaching.
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India.
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Pich Keo.
Khmer Art in Stone.
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Mathematics in India.
Princeton, NJ: Princeton University Press, 2009. An excellent comprehensive source on the general developments in mathematics in India since antiquity.

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Catuskoti.
”
Comparative Philosophy
1, no. 2 (2010): 24–54.

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Philosophy of Statistics,
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History of Mathematics, volume 2: Special Topics in Elementary Mathematics.
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The Hindu-Arabic Numerals.
Boston: Gin and Company, 1911.

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