Finding Zero

by Amir D. Aczel

  The Kharosthi (top, 3rd c. BCE), Brahmi (middle, King Ashoka’s inscriptions, 3rd c. BCE), and Nana Ghat and Nasik Caves (bottom) numerals.

  So the Nana Ghat inscriptions are extremely important because they provide strong evidence that our numerals are a genuine Indian invention made during the distant past, and then perhaps developed further while spreading around the world. These numbers were preceded by earlier Indian numerals, the Brahmi script seen in monuments of King Ashoka (slightly different from the Nana Ghat numbers); these, in turn, are related to the Kharosthi script, another alphabet employed in writing Sanskrit and regional languages starting in the third century BCE in north India and Pakistan.

  Noting that all these scripts bearing early numbers are Indian has made experts conclude that our numerals originate in India. But what about the zero?

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  Admittedly, C. K. Raju is a bit of an extremist, and he probably enjoys the international notoriety he gets by making provocative announcements at conferences around the world, such as claiming that Euclid never existed or that Indians invented the calculus long ago. But he is, in fact, fighting to right a wrong: to bring back to the East some of the credit it deserves for inventions made there, such as the invention of the numerals and the zero. “Shunya—the zero—is clearly an Indian invention,” he told me with utter confidence in his voice when I saw him again. And indeed, the zero is an Eastern invention. But we can’t say for sure that it comes from India.

  However, a Western bias has long influenced the history of mathematics and science, and correcting historical misconceptions is, and has been, a daunting task. Raju is one of the major players in the enterprise to bring back to the East the credit it deserves in this area.

  One big problem with Indian antiquities in general—making it so difficult to arrive at definitive conclusions—is the thorny issue of dating. Determining the dates of artifacts and events in Indian history has proved to be very hard: Some scholarly estimates of dates of items and documents have varied by as much as a thousand years. This problem has stood in the way of establishing an Indian chronology and especially has presented great difficulty in settling disputes about who invented what, and when.

  A particular problem affecting the Indian history of numbers and mathematics has been a frequent lack of details in mathematical issues. Indian mathematicians have notoriously left out important parts of what a Western mind would expect in a mathematical proof—or in a reasoned quantitative argument—as well as dates of discoveries.

  Even in the twentieth century, the mathematical prodigy Srinivasa Ramanujan, who produced many important results, left out key details of his work. Ramanujan was born in 1887 in the village of Erode, near Madras (now Chennai), in the state of Tamil Nadu in southern India. As a young man, he was able to derive hundreds of incredibly important and difficult mathematical results. When he sent them in letters to the prominent British mathematician G. H. Hardy, the latter realized that these amazing mathematical facts were written by a genius, but because Ramanujan provided no details or proofs, Hardy could not tell what was new and correct, what was not correct, and what was correct but had been known before. Hardy, nonetheless, was very impressed with the treasure trove of mathematical identities Ramanujan had sent him and said that “they defeated me completely; I had never seen anything in the least like them before.” These equations had to be true, he concluded, because “if they weren’t, nobody would have had the imagination to invent them!”1

  Hardy was so taken with the unexpected mathematics of an unknown young mathematician from a faraway land that he invited Ramanujan to join him in Cambridge so they could work together. Eventually Ramanujan came to England, but he was gravely ill and, tragically, would not live long. A good example of his knowledge without proof is what happened between him and Hardy when he was in the hospital in Putney, in the United Kingdom.

  While he lay there suffering from an unknown illness—now believed to have been a parasitic liver infection; he would die at 32, in the prime of his career—Hardy came to visit him. Not knowing what to say, he remarked, “I came here in a taxi with a rather dull number: 1729.” At that moment, Ramanujan, weak as he was, jumped up in bed and exclaimed, “No, Hardy, no, Hardy! It is a very interesting number! It is the smallest number expressible as the sum of two cubes in two different ways.” (Because 1729 = 13 + 123, and 1729 = 103 + 93.) Ramanujan just knew thousands of such facts about numbers and equations, and never bothered with derivations. He had no need to prove anything.

  Indian proofs at times contained fewer details than might have been necessary for a complete understanding of a piece of work. Perhaps the convention on using a more terse form has something to do with a different kind of logic. In the case of the numerals, a similar phenomenon is evident. There are very ancient Indian documents that, if dated correctly, could probably demonstrate Indian primacy in the invention of numbers. But most of them—unless they are inscribed in stone—are undateable or undated. In the case of copper or bronze plates, of which many survive, the dates are usually placed off the main text, to the side, and could very well have been faked or added later. They are generally untrustworthy.

  I went in search of one such plate, the Khandela inscription, which if found could shed important light on the story of the further development of the numerals in India. It might have been dated correctly and in a verifiable way—but nobody knew if it really existed. Still, I felt that I had to make an effort to find it.

  Rajasthan. The name evokes images of desert horsemen in multicolored dress galloping on barren hills; fortresses by deep mountain lakes; mounted elephants marching in line to a fairy-tale palace; and of course snake charmers. From Khajuraho I flew to Jaipur, in eastern Rajasthan, in search of a copper plate I was not at all sure existed. While the Khajuraho airport is just a small airstrip and a hut, Jaipur’s airport is larger and even has a restaurant or two. The city is on India’s “golden triangle” tourist route, and the increased traffic has spurred recent growth.

  Arriving late at night—later than expected due to dense fog in Delhi, where I had to change planes from Khajuraho—I sat inside the arrivals hall and looked out through the glass wall at people waiting to greet passengers. Then I caught sight of the limo driver holding a sign with my name; he had been sent by the hotel to pick me up.

  In Rajasthan, for the price of an urban hotel room in any major city in the world, you can stay in a palace. And that’s what I did—for my lodgings in Jaipur, I had booked the former coach house of the palace of the maharaja of Jaipur, which the present ruler rents out along with the entire palace with all its suites and rooms. The coach house was a bit cheaper than sleeping in the palace proper. Nevertheless, it was a charming place to stay, with Kashmiri carpets covering the floors, beautiful ebony cabinets, and a royal four-poster bed. It was quiet and peaceful, and I slept much better here than in the rundown Best Western in Khajuraho. The next morning, I hired the driver from the previous night to take me on a 50-mile drive to an old ruin to the northeast, where the Khandela inscription, a copper plate with early numerals and perhaps also a zero, was rumored to have been seen.2 If it was still there, it was attached to the inside wall of a ruined temple.

  We drove on a winding desert road and passed a lake with an island in its midst, and on it an ancient castle. On the shore, close to the road, a procession was taking place, with a succession of decorated elephants and camels carrying people. A small crowd gathered around an elderly man playing a wind instrument with his nostrils. We continued our slow ascent toward the ruins. During a stop on the way, I saw a handful of tourists standing around a snake charmer, the cobra moving its head as if to the rhythm of the piper’s music.

  We finally reached our destination, a ruined, deserted temple on top of a hill. The wind was blowing at the summit, kicking up dust. The ruined temple consisted of just two walls, and many stones that had fallen from the other walls
covered the ground. But a walk around the remains of this temple revealed no copper plate anywhere. I spent two hours searching in likely locations, but there was no inscription anywhere. Many artifacts in Indian history have disappeared, and that gave me some comfort to soothe my disappointment. My driver took me back to the maharaja’s coach house. Next I would go to find the earliest known zero in India. This inscription certainly existed.

  Before leaving Jaipur, I went to visit the Jantar Mantar (observatory; literally “instrument formula”) in this city—in which some prominent Indian mathematicians had worked many centuries ago. The Jantar Mantar of Jaipur is now used as a museum to explain early astronomy to the people of India and to visitors. I studied the sophisticated instruments on display. These devices predated telescopes, so no lenses were used, but they were remarkably advanced and could estimate with good accuracy the various angles to heavenly bodies; some were tracking instruments that could follow the movements of planets, the moon, and the sun over the entire year. I inspected the numerals shown on these instruments. What was on display here were the later devices used at this observatory, from the sixteenth and seventeenth centuries. Therefore, all the numerals were our modern ones, and included a zero.

  The earliest view that our numbers, with zero, came from India appears to have been proposed in a scholarly publication by the renowned German historian of science Moritz Cantor. In a publication of 1891, Cantor says, “This kind of conscious juggling with the notions of positional arithmetic with the zero is most easily explained in the home of these notions, which home for us is India and this we may affirm even if there is question of a second home. We mean if both notions were born in Babylon, of which there is great probability, and were carried over into India in a very undeveloped state.”3

  Louis C. Karpinski of the University of Michigan quoted Cantor’s groundbreaking passage on the putative origin of numbers in his article in Science on June 21, 1912. He had translated Cantor’s German into English.4 Then his own article went on to discredit any notion that the numbers may have originated in Babylonia, in any form, because the Babylonians used a sexagesimal system, building their numbers using a very large base: 60. He pointed out that the Babylonians did not use any place-holding zero. His conclusion was that the numbers had to have originated in India. But what was the evidence that the numbers—and especially the all-important zero—were invented in India?

  Neither Cantor nor Karpinski presented any such definitive proof. Karpinski said in his article that “[an] early document referring to the Hindu numerals has been published. This document is of prime importance because, being written in 662 A.D., it antedates by more than two centuries the earliest known appearance in the ninth century of the numerals in Europe.”5 Surprisingly—for an article in the prestigious journal Science—Karpinski doesn’t tell us what this document is. In fact, if such a document existed today—and if one could prove definitively that it was written in the seventh century—it would be one of the most important documents in the history of science. Equally surprising is his statement that the numerals arrived in Europe in the ninth century, again without any proof.

  So in the meantime, lacking convincing evidence that the numbers including zero were of Indian origin, many scholars in Europe remained as skeptical as ever about any Eastern origins whatsoever—some claiming, as we will see, that the numbers and the zero were either invented by the Europeans themselves or by the Arabs.

  Perhaps Karpinski was alluding to the famous Bakhshali manuscript. This mathematical document, written on birch bark, was discovered in the 1800s in the ground near the village of Bakhshali, not far from Peshawar in present-day Pakistan. It clearly is very ancient, and the bark on which it was written, 70 leaves of it, is so fragile that no one has been allowed to touch it for fear it will disintegrate. Today this ancient document is on display at the Bodleian Library at Oxford. Because the Bakhshali cannot be touched, no samples of it can be taken for radiocarbon analysis—which could reveal its actual age with excellent accuracy—and so we still cannot tell how old it is. Many scholars believe, based on linguistic and textual analysis, that it was created between the eighth and twelfth centuries CE, while others place it much earlier, anywhere from 200 BCE to 300 CE. But without radiocarbon analysis, it is not possible to date it definitively.

  The manuscript contains a wealth of mathematical writings, from early equations to ways of estimating square roots to the uses of negative numbers. Most importantly, the Bakhshali uses a symbol for zero. If it could be dated to the second or third century, or even the fourth, it would establish that zero—and with it our entire number system—was invented very early in India. Should the British authorities ever allow a simple, hardly invasive procedure to be undertaken, in which a tiny amount of the bark is analyzed in a radiocarbon lab, we would know the real age of this extremely important artifact. Until then, the date of the most important Indian artifact in the history of mathematics remains very doubtful.

  The British scholar G. R. Kaye was the first person to study the Bakhshali, at the start of the twentieth century, and he concluded that it was no older than the twelfth century. Therefore, he could argue for a European or Arab origin of our number system. He wrote, “The orientalists who exploited Indian history and literature about a century ago were not always perfect in their methods of investigation and consequently promulgated many errors . . . According to orthodox Hindu tradition, the ‘Surya Siddhanta,’ the most important Indian astronomical work, was composed over two million years ago!”6

  Kaye clearly dismissed the research of Moritz Cantor, Louis Karpinski, and others who like them believed that the numbers and the zero originated in India. He continued in his article with a scathing attack on all who argued that our numbers come from India and mocked Indian date estimates, including those for the Bakhshali that placed it at an early era. His article continued, “In the sixteenth century CE, Hindu tradition ascribed the invention of the ‘nine figures with the device for places to make them suffice for all numbers’ to ‘the beneficent creator of the universe’; and this was accepted as evidence of the very great antiquity of the system!”7

  While I did not know it at the time, Kaye would play a major role in my story. In the meantime, working under the assumption that the zero—the key to our entire number system—was an Eastern invention, I asked myself why this was so, and inexorably I had to link it with the unique logic that I perceived in Asia. My thesis was that the number system we use today developed in the East because of religious, spiritual, philosophical, and mystical reasons—not for the practical concerns of trade and industry as in the West. In particular, nothingness—the Buddhist concept of Shunyata—and the Jain concept of extremely large numbers and infinity played paramount roles.

  The earliest zero in India is found in the city of Gwalior southeast of Agra, famed home of the Taj Mahal. Gwalior’s history is steeped in legend. In 8 CE, Suraj Sen, the ruler of Madhya Pradesh, contracted a serious illness and was about to die. He was cured by a hermit named Gwalipa, and in gratitude, Sen founded a city and named it after the man who had saved his life. Gwalior has many temples built over the centuries, and it has a famous fort whose defense played a role in many conflicts throughout Indian history. The fort was almost impenetrable; it stands on a high plateau in the middle of the modern city, rising sharply to 300 feet above its surroundings. This made it very hard for enemies to reach it and breach its walls. In a Hindu place of worship called the Chatur-bhuja temple—“the temple to the four-armed god” (the four-armed god in Hindu tradition is Vishnu, the powerful maintainer of our world)—there is an inscription in Sanskrit on the wall that records that it was built in the year 933 of a calendar whose starting point was 57 BCE. This makes the year it was built 876 CE. The numerals 933 used here are surprisingly similar to our modern numbers. The inscription also records that the land grant for the temple had a length of 270 hastas (a measure of length). The 0 in 270 is th
e oldest zero that can be seen in India today.

  So by 876 CE, the Indians had the crucially important use of a place-holding zero at their disposal in a number system that from our modern vantage point was perfect. Their system would have enabled them to compute in a powerful, efficient, and unambiguous way. But would it be possible now to go still further back in time and find when the first zero had made its debut, the first exemplar of one of humanity’s greatest intellectual inventions? I wanted to see it with my own eyes, to touch it, to feel it.

  I left India with this remaining mystery unsolved. I learned much there, but nothing about where the key to the numbers—the primeval zero—came from, and when. If the oldest zero in India was from 876, then it was possible that it had come from Arabia—and had gone to Arabia from Europe—because the ninth century is well within the timeframe of extensive Arab sea trade. This was a time when Arab trade flourished, allowing for the possible transfer of goods—and ideas and information—across the realms the traders roamed, meaning between Europe and the East. Such transfer could well have taken place from east to west, or from west to east. And this was exactly the argument that the Western-biased Kaye had put forward in his lectures and articles. Lacking an earlier Eastern zero than that of Gwalior, Kaye’s argument that our numbers with the zero numeral were either European or Arabic in origin could not be countered or disproved.

  But if a zero could be found in the East that predated the emergence of Arab trade, this could provide strong support for the hypothesis that the zero was indeed an Eastern invention. This is why the Gwalior zero—important as it is—could not definitively determine who invented the most important component of our number system.

  The oldest zero was of course the Mayan zero—but it was confined to Mesoamerica and went nowhere from there. And the Gwalior zero was from the mid-ninth century—so it was no good as a historical landmark. If the Khandela inscription was ever to be rediscovered, it could bring the invention of the Indian zero down to 809 CE, as had been reported by people who claimed to have inspected it decades ago.8 But because its date was still late, this wouldn’t help much in finding a decisive, earlier zero that could settle the question of who invented the concept and the numeral.