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.
Bibliography
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Aubet, M. E. The Phoenicians and the West. Cambridge: Cambridge University Press, 2001.
Saint Augustine. The City of God. New York: The Modern Library, 2000.
Boyer, Carl B., and Uta Merrzbach. A History of Mathematics. 2nd ed. New York: Wiley, 1993. This is a standard scholarly source on Babylonian, Egyptian, Greek, and other early mathematics, including a description of the early Hindu numerals; it does not include the discoveries of the earliest zeros in Southeast Asia.
Briggs, Lawrence Palmer. “The Ancient Khmer Empire.” Transactions of the American Philosophical Society (1951): 1–295. Information on some now-lost inscriptions with early numerals from Cambodia.
Cajori, Florian. A History of Mathematical Notations. Vols. 1 and 2. New York: Dover, 1993. A reissue of a superb source of information on mathematical notations; it does not include the discoveries of the earliest numerals in Southeast Asia.
Cantor, Moritz. Vorlesungen uber Geschichte der Mathematik. Vol. 1. Berlin, 1907.
Cœdès, George. “A propos de l’origine des chiffres arabes.” Bulletin of the School of Oriental Studies (University of London) 6, no. 2 (1931): 323–28. This is the seminal paper by Cœdès, which changed the entire chronology of the evolution of our number system by reporting and analyzing the discovery, by Cœdès himself, of a Cambodian zero two centuries older than the accepted knowledge at that time.
Cœdès, George. The Indianized States of Southeast Asia. Hilo: University of Hawaii Press, 1996. A comprehensive, authoritative source on the history of Southeast Asia with references to the author’s work on discovering the earliest numerals.
Cunningham, Alexander. “Four Reports Made During the Years 1862–1865.” Archaeological Survey of India 2 (1871): 434.
Dehejia, Vidaya. Early Buddhist Rock Temples. Ithaca: Cornell University Press, 1972. An excellent description of Buddhist rock and cave inscriptions, including very early numerals.
Diller, Anthony. “New Zeros and Old Khmer.” Mon-Khmer Studies Journal 25 (1996): 125–32. A recent source on early zeros in Cambodia dated to the seventh century.
Durham, John W. “The Introduction of ‘Arabic’ Numerals in European Accounting.” Accounting Historians Journal 19 (December 1992): 25–55.
Emch, Gerard, et al., eds. Contributions to the History of Indian Mathematics. New Delhi: Hindustan Books, 2005.
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Hayashi, Takao. Indo no sug
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Kaye, G. R. “Notes on Indian Mathematics: Arithmetical Notation.” JASB, 1907.
Kaye, G. R. “Indian Mathematics.” Isis 2, no. 2 (September 1919): 326–56. Kaye’s now-notorious manuscript discrediting Indian priority over the invention of numerals.
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Index
The index that appeared in the print version of this title does not match the pages in your e-book. Please use the search function on your e-reading device to search for terms of interest. For your reference, the terms that appear in the print index are listed below.
Aczel, Debra, 28–31, 81, 83, 113, 131–5, 138, 143, 149–51, 187, 219–20
Aczel, Ilana, 2, 7–9, 11, 52–4, 109–10, 112
aleph (Hebrew letter), 66
Alfred P. Sloan Foundation, 99–100, 176
algebraic geometry, 58–9, 160
aluf (Phoenician letter), 66
Ananta (sea serpent), 35, 100
Angkor (Cambodian empire), 90–5, 100–3, 115–16, 125–6
as capital, 207
culture, 205–6
fall of, 184
Mouhot and, 207
post-Angkor, 91, 184, 207
pre-Angkor, 91, 94, 102, 116, 204, 208
See also Siem Reap, Cambodia
Angkor Conservation, 163, 165–8, 172, 181, 210, 217
Angkor Miracle Resort Hotel, 167–8, 170
Angkor Wat (temple), 91–3, 123, 129, 139, 165, 169, 174–5, 205
anti-American sentiment, 119
Apsara (nymphs), 92, 129
Aramaic alphabet, 65–6, 223
Aristotle, 22, 54, 57
Ashoka, King (Indian emperor), 64, 66–7, 223
Augustine of Hippo: City of God, 37
Aztec Stone of the Sun, 29–31
Babylonian numbers, 24–5, 63, 65, 75, 86, 208, 213
Bakhshali manuscript, 76–7, 81, 88–9
Baray (artificial lakes), 100–1, 205
Bee, Mr. (tuk-tuk driver), 170–3
Bengal Engineers, 43
Bentley University, 28
Bhagavad Gita, 139
Bonaparte, Louis-Napoléon, 128
Bouillevaux, Charles Emile (French missionary), 92
Brahma (Hindu god), 35–6, 101–2, 129
Brouwer, Andy, 120, 122, 124–7, 131, 163
Buddha, 39, 55–7, 102–4, 141–2, 144, 154, 164, 204
statues, 101–2, 115, 123, 129, 131, 135, 139, 184, 186
Buddhism, 34, 36, 90, 102, 150, 206–7
dependent co-origination, 154
four logical possibilities, 57, 60–1, 105–6, 137, 139–42, 148, 152
Mahayana, 103
meditation, 137
Naga (seven-headed cobra), 103, 129
Nagarjuna (philosopher), 39–40, 55, 57, 60–1, 105–6, 136–7, 139–40, 152–5
Nana Ghat inscriptions, 64–5
Shunyata (void), 40–1, 78, 104, 105–6, 127, 135, 137–8, 142, 148, 152–3, 154–5, 161–2, 223
Theravada, 103
Thich Nhat Hanh, 106
Bulletin de l’École Française d’Extrême-Orient, 87–8
Burma (Myanmar), 91. See also Myanmar
Burt, Captain T. S., 43–4
çaka (dynasty), 95–7, 209–10, 214
Cambodia
Cambodian National Museum, 102, 128–9, 165, 203–4, 218–19
See also Angkor (Cambodian empire); Phnom Penh (Cambodian capital); Sambor on Mekong, Cambodia; Siem Reap, Cambodia
Cantor, Georg (mathematician), 145–8
Cantor, M
oritz, 74–5, 77
cargo ships, 18–19
Carnac stones, 22
Cartier, Pierre, 59
Casselman, Bill, 83, 186–8, 191, 197
catuskoti (four corners/possibilities, also tetralemma), 57, 60–1, 105–6, 137, 139–42, 148, 152
Chamroeun Chhan, 116–17, 166, 171–2, 189
Chandra (Hindu god), 103
Chao Phraya River, 113–15, 164
Chatur-bhuja temple, 78, 210
Chenla (ancient region, later Burma and Myanmar), 91–3, 205
China
Chou Ta-kuan, 101–2
Cultural Revolution, 98
Fu Nan kingdom, 91–3, 205
Han Dynasty, 23
Nine Chapters on the Mathematical Art, 23
three-by-three squares, 48
Chou Ta-kuan (also Zhou Daguan), 101–2
Cœdès, Georges (archaeologist), 83–4, 100, 102, 106–8, 124, 150, 187, 204
Angkor, 92
birth and education, 85–9
career and life, 114–15, 117, 119–20, 128, 150, 156, 176
death and final years, 108, 216–17
on Khmer number system, 214–15
translation of K-127, 93–7, 107, 166, 176, 207–8, 210
Communism, 98, 118, 120, 131, 216
Khmer Rouge, 98–9, 107–8, 125, 129, 150, 163, 166, 173, 176, 200, 202–3, 210, 222
Copernicus, 194
counting, 20–2, 137, 216
cruise ships, 1–2, 7, 10, 15, 18, 52
Cultural Revolution, 98
Cunningham, Alexander (archaeologist), 45–6
Dalida (French-Italian singer), 3–4
decimals
nonrepeating, 146
number systems, 23, 63, 212, 215
Dedekind, Richard (mathematician), 146
Descartes, René, 194–5
Devi (Hindu god), 102
Dieu, Eric, 115–16, 189–93
digamma (archaic Greek letter), 14
Diophantus (mathematician), 23
double-entry bookkeeping, 23
Dreyfus trial, 86
Dürer, Albrecht (artist), 49–50
Durga (Hindu goddess, also Parvati), 35, 45