An interesting question arises: Why use a base of 60? We have ten fingers, so base 10 makes sense, and if you insist on also counting toes, a base 20 may be useful. But 60? In 1927, the prominent Austrian American historian of science Otto Neugebauer suggested that the choice of the large base of 60 was made to address an important practical problem in using numbers in Babylonia. Often, fractions of a whole, such as 1⁄2, 1⁄3, 3⁄4, and 2⁄3, were required as measures: Perhaps someone wanted to buy half a loaf of bread, or a third of a wheel of cheese, or two-thirds of a shepherd’s pie. How could the numbers 1⁄2, 1⁄3, 3⁄4, and 2⁄3—the most commonly used fractions—be reconciled with a natural system using ten numbers abstracted from fingers? Neugebauer’s answer was that 60 is a good solution since this number is divisible by 2, 3, 4, and 10, and for this reason it was chosen as the base for the entire system. Another hypothesis is that the Babylonians knew five planets (Mercury, Venus, Mars, Saturn, and Jupiter) and that they chose their base, for cosmological reasons, to be the product of this number and the 12 (lunar) months of the year.3
I already knew something about the Greco-Roman number system from visiting Greece and Rome. This system, too, lacked a zero, and with it the ability of the numbers to cycle so that the same signs could be used over and over again to mean different things. The Greco-Roman system, like the Babylonian and Egyptian, had to fade away, remaining only as an elegant way of commemorating official dates or representing time on clock and watch faces.
Then, in the thirteenth century, a system of numbers consisting of nine numerals and a round zero appeared in Europe. This innovation became popular and within a few decades took hold in all segments of educated society. Merchants, bankers, engineers, and mathematicians found that it improved their lives because they could make quicker calculations with fewer errors.
It is believed that Leonardo of Pisa (ca. 1170–1250), better known as Fibonacci (of the famous Fibonacci sequence), was the one to bring the Hindu-Arabic numerals to Europe. He did it through his book, Liber Abaci (the book of the abacus), published in 1202 and circulated widely throughout the continent. This mathematical volume described the nine Indian figures—the digits 1 through 9—and a symbol, 0, for what Fibonacci called zephirum, meaning zero. The root of the Latin zephirum has been traced to the Arabic word for zero, sifr. So a linguistic connection is found here from the Arab zero to the new European one. And the author clearly refers to the nine digits as “Indian.” We thus have in this one source an indication both for an Indian and for an Arabic origin for our modern numbers.
This number system introduced into Europe in the late Middle Ages was far superior to the Roman one used until then. It allowed an immense economy of notation so that the same digit, for example 4, can be used to convey itself or forty (40) when followed by a zero, or four hundred and four when written as 404, or four thousand when written as a 4 followed by three zeros (4000). The power of the Arabic, or Hindu, or Hindu-Arabic number system is incomparable as it allows us to represent numbers efficiently and compactly, enabling us to perform complicated arithmetic calculations that could not have been easily done before.
But the real origin of our amazing number system based on nine digits plus a place-holding zero remained a mystery. The nine numerals had been conjectured to originate in India, as Fibonacci had implied, but there was no unambiguous scholarly proof of this belief. And the zero: Was it Arabic, or Indian, or did it come from some other place? I still didn’t know.
3
I eventually built a career as a mathematician and statistician. For several years, I was a professor of mathematics at the University of Alaska in Juneau, and there, in 1984, Debra and I got married just below the Mendenhall Glacier, surrounded by Douglas fir trees, with occasional brown bears coming in from the forest to feed on the salmon making their way up the Mendenhall River. We had met at the university a few months earlier, when I helped Debra with a statistical problem about how well the university attracted and retained students from a diverse population of Alaska natives and Americans who had moved there from the Lower 48, and we fell in love.
Some years after our wedding by the glacier, we moved to Boston and I started teaching at Bentley University. Debra took a position running a program at MIT, our daughter Miriam was born, and I wrote a number of popular books on the history of mathematics and science.
In 2008, I got a call from my friend Dr. Andres Roemer, a Mexican intellectual who had studied policy at Harvard, got his PhD at Berkeley, and was the host of popular television shows produced in Mexico and aired throughout the Spanish-speaking world. Andres invited me to speak about probability theory at an international conference he was inaugurating. After the conference, Debra and I went to see the National Museum of Anthropology in Mexico City. Unexpectedly, this visit to a museum rekindled my childhood interest in the origins of numbers.
In the main hall of the museum, on the wall facing the entering visitor, we were greeted by the stunning Aztec Stone of the Sun. A circular stone 12 feet in diameter and weighing more than 24 tons, this artifact, attached to the museum’s wall, bears a central face, believed to represent the Aztec god of the sun, Tonatiuh. Around it are markings and designs that have never been deciphered. This might have served as an ancient calendar. This curious archaeological find reminded me of something I had seen decades earlier with Laci on our visit to Athens.
The enigmatic Aztec Stone of the Sun at Mexico’s National Museum of Anthropology.
Just below the Acropolis, at the periphery of the Plaka, there stands an ancient Greek tower dated to the second century BCE. This tower is octagonal in shape to represent the eight winds that navigators have recognized since antiquity—each wind blowing from one of the compass directions: north, northeast, east, southeast, south, southwest, west, and northwest. Could there be a connection? I wondered: Could the Aztec Stone of the Sun also represent the eight cardinal directions, just as the Tower of the Winds had?
Debra and I marveled at this ancient Aztec artifact with its perfect geometrical design and intricate markings. We wondered what it truly represented and what its purpose might have been. We talked quietly about it for almost half an hour, standing mesmerized by this mysterious stone carved by mathematically inclined people centuries ago. A nearby display explained that the Aztec Stone was likely made in the fifteenth century and had been discovered right in the center of Mexico City.
Some years earlier, I was surprised to learn from an anthropologist friend that much scientific research is actually done inside museums—not just in the field, as one might expect. Museums present collections of artifacts that have been cleaned and prepared for display and are usually shown within the context of similar items that are related to them by date or type or location of discovery, or all of these. This practice facilitates analysis by experts as much as admiration by the general public. Recently, many great museums around the world have begun enhancing their exhibits with video presentations about topics related to their displays, which serve an important public-education need.
And when Debra and I went upstairs from the Aztec Stone, we serendipitously found an ongoing video presentation on Mesoamerican mathematics. I was fascinated to learn that two millennia ago, the Mayans had devised a sophisticated calendar using glyphs for numbers—including a zero. The Maya numerals go back to 37 BCE, and they are simple to write. The numbers 1 through 4 are dots, 5 is a bar, and 10 is two bars one on top of the other; and for zero there is a crescent-moon glyph.
In fact, the Maya invented four kinds of calendars. One was the Long Count calendar, which represented days from a starting point that corresponds, in our modern calendar, to August 11, 3114 BCE—the date of the creation of the universe, according to Mayan mythology. The counting of days from creation used a mixed base-20 and base-18 number system. One of the digits in this calendar would reset to zero after reaching 18; otherwise the calendar used the vigesimal (base 20) num
bers. Surprisingly, the ancient Maya, who inhabited the Yucatan Peninsula in Mexico and parts of Central America, apparently understood the concept of zero as early as the first century BCE.
The Maya also had a Short Count, a cyclical calendar with 260 days (20 times 13 days), which was their sacred calendar. At the end of the cycle, monuments were erected to commemorate the fulfillment of the period. A third Mayan calendar had 360 days—close to the usual solar year of 365.24 days. It was for the construction of this calendar that the usual base-20 Mayan number system was adjusted to also employ a base of 18, because 360 equals 20 times 18. Had they used only multiples of 20, their year would have been forced to have 400 (20 × 20) days.1
Yet a fourth calendar used by the Maya was based on cycles of the planet Venus. The Mayans were astute observers of the sky and had long ago noticed that Venus would rise with the sun (we call this a heliacal rising) every 584 days—so the calendar based on Venus reset itself to zero after 584 days. The Mayan calendars, and the predominantly vigesimal Mayan number system with zero, are some of the most intriguing discoveries in the history of science. In 2012 there was worldwide panic in some circles of society fearing the end of the world because one of the Mayan calendars reset itself to zero. Of course, nothing happened; our planet continued to revolve around the sun, and this fear turned out to be as unfounded as the similar Y2K worry of a dozen years earlier.
But the Mayan system was isolated from the rest of the world, and it used glyphs—written or carved symbolic signs—that were not suitable for economy of notation. Their signs grew in number as the numbers they represented got larger, in a manner similar to that of the Roman system. The zero was not a perfect positional element as in our numbers, and the base changed, depending on need, from 20 to 18. Georges Ifrah calls the Mayan numbers “a failed system”—it was not one that survived the test of time.2 Encountering it, however, rekindled my passion to search for the origins of our versatile base-10 numbers and the first zero in the East—the ten numerals that became the basis for the all-powerful system that controls our modern world.
4
Over the next year—energized by learning about the Mayan numbers—I worked hard to solve the great mystery that had intrigued me my entire life: Where do numbers—our numbers, the nine familiar digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, plus the all-important 0—come from?
I knew from Laci and from my university courses and extensive reading and research I had done that the nine digits we use today were believed to have originated in India. I also knew that at some point in the past, the Indians learned to use a place-holding zero. But there were no absolutely known facts about the origins of the numerals and the birthplace of the zero and no details. Was all this true? Books and articles all pointed me in one direction: East. My experiences with Laci on the ship gave me a deep, lifelong desire to find things out for myself, to see the actual evidence, to witness history.
So I began to plan a trip to India, hoping to find some answers there. I spent much time learning about Hinduism, Buddhism, and Jainism. I read books about the East and the cultures of Asia, about customs, philosophy, and mathematics. I sensed that Eastern religions were the key to knowledge about Asian societies, and I felt that perhaps the origin of the numerals was embedded in these religious traditions.
And here is what I learned about these fascinating religions, which until then were almost completely foreign to me. Brahmanism, more commonly known as the precursor of modern Hinduism, began in India and has three main gods: Brahma, Vishnu, and Shiva. Each god has a feminine aspect called his Shakti, or consort. Parvati is Shiva’s Shakti; she is also called Uma or Durga. Lakshmi is Vishnu’s Shakti. She emerges radiant from the Sea of Milk, seated on a floating lotus and holding a budding lotus flower in each hand. She is the bearer of good fortune. Brahma is the creator of the worlds, but he is born from a lotus flower on Vishnu’s stomach once Lakshmi wakes him up by massaging his legs as he lies in eternal slumber on top of the sea serpent, Ananta—which means infinity. Here we already see that at the very point these gods were conceived, the key mathematical idea of infinity makes its debut in the form of an infinite quantity or extent, as embodied by Ananta, and in the form of an infinite past: eternity until Vishnu is awakened.
Now, Vishnu is the maintainer of the worlds, and Shiva is their destroyer. In keeping with this powerful image, Shiva is often portrayed as holding a trident; at times he is represented by a stylized phallus. For a dangerous god he seems rather benevolent: His chief interest, as evident from much of the art in which he appears, seems to be sex.
Vishnu has four (or sometimes eight) arms. His four arms carry symbols representing the four elements of the cosmos: earth, wind, fire, and water. These are surprisingly similar to the Greek elements of the universe, which are also earth, wind, fire, and water, but which include a fifth essence called just that: quintessence, from which we get the word quintessential.
Shiva is almost always portrayed as having a third, vertical eye on his forehead. And Brahma has four faces, one for each of the four cardinal directions: north, south, east, and west. These gods seem to capture much symbolism about life and nature. Together, the three gods form the Trimurti—reminiscent of the trinity. And in fact Vishnu and Shiva are sometimes depicted as a single amalgamation: a statue with four arms and a third eye, representing a god named Harihara (Vishnu is Hari, and Shiva is Hara). Together, the two gods, or three when Brahma is included, are seen as representations of a single supreme being.
My research in preparation for the voyage to India gradually convinced me that there must have been something about the philosophy of the East—perhaps Buddhism, Jainism, or Hinduism, or maybe some ideas from all three religions together—that made the Eastern mind more amenable to completing the number system on both of its extremes: adding a zero at one end and infinity on the other.
We know that in Europe, before the arrival of our numbers, there was no zero. Numbers could be added, subtracted, multiplied, and divided, but no one thought of a zero. If you subtracted 5 from 5, for example, then there was nothing—but not the something we now call a zero. The computation simply ended at that point. Equally, Europeans never contemplated extremely large numbers and infinity—as I knew the Jains of India had definitely done—with the exception of religious philosophizing about God’s infinite qualities. For example, in The City of God, St. Augustine writes: “Of God’s eternal and unchangeable knowledge and will, whereby all He has made pleased Him in the eternal design as well as in the actual result.”1 This is an oblique reference to infinity, to eternal time, but the concept is not developed any further. Similar references to God’s infinite qualities and working in infinite settings of time and space are found in Jewish theology, especially Kabbalah, but there, too, these notions are vague and not yet well formed.
I became convinced that it must have taken an Eastern mind, perhaps using a unique and very different kind of logic, to invent both the idea of a zero and the concept of infinity. While I could not know it at the time, in the East my hunch would turn out to be truer than I initially expected. I was eager to travel to Asia and hoped to find there and see with my own eyes—in an ancient manuscript or on a carved stone—some of the first numerals recorded in our distant past. The thought of actually finding archaeological proof for our mathematical origins filled me with indescribable excitement. India would be the first place to look.
I landed in foggy Delhi at 2 a.m. on January 10, 2011, in the dead of winter. As I learned the hard way, while the south of India is tropical, Delhi, in the north, can be awfully cold. It was damp and freezing as I entered the terminal. I was shivering, deathly tired, and somewhat disoriented from a very long flight from America. I had with me a small suitcase, a couple of books about Indian mathematics, and a notebook with the name of a person I hoped would help me address the mystery.
Two years earlier at an international conference on the history of
science in Sydney, Australia, I had met Professor C. K. Raju, perhaps the most unusual academic I have ever encountered. At that meeting, Raju had given a talk that elicited shock, skepticism, and even jeers. He claimed that mathematics was born in India and that much of what we in the West attribute to the mathematicians of ancient Greece had, in fact, been achieved earlier in India. He didn’t offer much by way of definitive historical proof, but he made up for it with his passion, his own conviction that he was right, and his considerable personal charm.
There was something about his talk—and personality—that made me believe that maybe, just maybe, this man was not talking nonsense, as it sounded to most of us at the time. I knew there were many documents on Indian history that were unknown in the West (ancient Indian documents number in the millions), and I felt there was a chance that at least some of the mathematical derivations and results Raju was claiming for India might indeed have first been discovered in the subcontinent and later transmitted to Greece or other places.
For one thing, Pythagoras himself might have traveled to India in the fifth century BCE, as we know he had visited Egypt and Phoenicia. I had spent some time chatting with Raju and we remained in touch. Now that I had come to the East, I had arranged to meet him in the lobby of the opulent Oberoi Hotel in the heart of New Delhi.
Everything about the East seemed ruled by a bizarre kind of logic. We were supposed to meet at the hotel lobby at 2 p.m.; I sat there for two hours drinking a succession of cups of Assam tea and was about to give up at 10 minutes to 4, and then Raju appeared. He didn’t apologize or explain his lateness—it seemed natural to him, I suppose, that 2 o’clock or 4 o’clock were pretty much the same thing. After some small talk about our mutual acquaintances from the Sydney conference, he started on a long monologue about science and mathematics, claiming for India many of the world’s discoveries of the past three millennia, from theorems we usually attribute to the Greek mathematician Euclid to facts about relativity for which we normally credit Einstein. Then, referring to my project, he said, “Correcting Western bias in the history of science would be something you should definitely do.” He then opened a book and showed me a verse in it, surprising me with the bizarre logic of the East:
Finding Zero Page 3
