Sometime in the early 1980s, I received an intriguing phone call from the Cambodian Embassy in Washington, D.C. It seems that in an article on the history of mathematics that I had published, I had casually mentioned that the zero of the Hindu Arabic numeral system could have possessed a Cambodian origin. I had come across this fact many years before in an obscure footnote. The officials at the Embassy wanted to know just “Where this zero could be found?” Of course, I had no idea and told them so. At this time, Cambodia was resurrecting itself as a nation in the wake of the Khmer Rouge rampage (1975–1979) and the existence of such a historically significant artifact could serve as a source of unifying national pride. I also was left wondering: “Where is this zero?”
In Finding Zero: A Mathematician’s Odyssey to Uncover the Origins of Numbers, Amir Aczel has solved this mystery. Aczel is a mathematician and popular science writer, perhaps best known for his award winning Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem (New York: Dell Publications, 1997). He too was plagued by the question of the origins of this “first zero” and set out to resolve it. This book documents his intellectual and physical quest for an answer. It takes the reader from the ancient temples of India across the cities and jungles of Southeast Asia and finally to the Angkor Wat complex in Cambodia.
The object of the search is identified as a stone stele uncovered in 1891, bearing a Khmer epigraphic inscription including the date for the Khmer year 605 (683ce) where the zero placeholder is represented by a dot. Its significance unrecognized at the time, the artifact was designated K-127 and stored away. Rediscovered in 1931 by the French archeologist Georges Cœdès (1886–1969), K-127’s inscription was deciphered to reveal the first known use of a symbol for zero that is directly related to our numbers, the “Hindu-Arabic system”.
In the disruptions of the Second World War, the following period of colonial transition and the resulting era of political unrest, K-127 and its significance became forgotten. Amir Aczel’s dogged formal search, which lasted four years, finally led him to an undistinguished storage shed within the Angkor Conservations, a complex outside of Siem Reap, Cambodia, the gateway town to Angkor Wat. There, he found stele K-127 and arranged for its transport and more prominent ensured survival in Cambodia’s National Museum in Phnom Penh.
Finding Zero is written for a popular audience and is therefore part travel adventure and part mathematical testament. It adequately conveys the dust, heat and confusion of a prolonged journey through South and Southeast Asia. But as a mathematical examination and explanation of zero, it might not satisfy all readers.
Aczel first defines zero as the point of reflection on the number line between the positive and negative numbers. I would think that, conceptually, zero arose as an operational result when one quantity was subtracted from itself, i.e. \(2-2=0\). Certainly this is how it was realized on the abacus and counting boards and an empty space left to represent it. The symbol came later as the cardinality of the null set and its systematic use as a placeholder even later in history.
Within the discussion, this evolution appears a bit clouded. For example, mention is made of the later Mayan use of a zero, but no consideration given for the Egyptian and Babylonian employ of placeholder zero within their much older mathematics. There is much mathematical substance touched upon in the text to intrigue the perceptive reader, however: the accomplishments and contributions of such mathematical personages as Leonardo of Pisa (Fibonacci), Pierre de Fermat, Georg Cantor, Srinivasa Ramanujan, Alexander Grothendieck, and Bertrand Russell are considered.
I found particularly interesting the author’s emerging conviction that the philosophical origins of zero and infinity resulted from oriental ritual religious practices and his discovery that there are other logics than the one we follow in the West. Buddhist logic would seem to ignore the “Law of Excluded Middle” and allow for degrees of truth and falseness. I have also found this to be prevalent in other traditional cultures. When the author queried an elder Buddhist monk as to the meaning of the Shunyata, the Buddhist void, he was informed: “Everything is not everything” (Seems like the null set to me.) Although, Finding Zero describes the search for a particular zero, it is an interesting and thought-provoking excursion into the history of mathematics.
Frank Swetz is Professor of Mathematics and Education, Emeritus, The Pennsylvania State University. He is the author of several books on the history of mathematics; his most recent book is Mathematical Expeditions: Exploring Word Problems across the Ages, Johns Hopkins University Press, 2012.