My father was up on the bridge, and while I was finishing my breakfast—my mother and sister were still asleep—he came down and into the cabin. “You’re up early,” he said. “Yes,” I answered excitedly, “Laci is taking me to Athens to look at ancient Greek numbers!” My father nodded. I wasn’t sure that he understood—or appreciated—how important to me my relationship with Laci was: that I would even give up time with my family to go look at some numbers with him.
Laci came, and we went down the gangway to the pier and into a cab that was waiting for us. We drove through the heavy morning traffic of greater Athens, climbing up the densely populated hills between Piraeus and the Greek capital. The pollution was heavy—the Athens area, like Los Angeles, suffers from a thermal inversion pattern that traps gasses and particles—and the bad air made me cough frequently. In about an hour we came to the center of Athens and the cab dropped us off at the Plaka, the city square below the acropolis, full of shops and boutiques and cafes. From there, we climbed up the steep path of the acropolis hill to the Parthenon, the fifth-century BCE temple to the Greek goddess Athena.
We walked slowly up the white stone path. The air was clear and crisp here, and crocuses were blooming; I could smell the fresh scent of the pine trees around us. At the very top, we paid the entrance fee and went into the ruins of the ancient acropolis. We then slowly climbed up the stone stairs to the famous Parthenon. I stopped from time to time to catch my breath and to admire the incredible view of the temple from below as I climbed toward it.
“Beautiful, isn’t it?” asked Laci.
“Yes,” I said. “It’s a very pretty building with these columns.”
“These are made of marble,” he said. “But you know why you find it so beautiful?” I said I didn’t know. “It’s the proportion,” he said. “The Parthenon follows the ancient Greek proportion called the golden ratio. The ratio of its length to its height is about 1.618, a number that seems to characterize many things we consider beautiful—and it also appears in nature.” I was fascinated. He explained that the golden ratio came from a mathematical series of numbers called the Fibonacci sequence, and he showed me how it is obtained, each number being the sum of the two previous ones: 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. If you divide a number by its predecessor, you get something approximating the golden ratio of 1.618. (For example, the sequence continues with 55 and 89. The ratio of 89 to 55 is 1.61818 . . .) I found this idea enthralling.
We went into the Parthenon and saw some statues, beautiful likenesses of the goddess Athena still with traces of ancient red and gold paint on her face. Laci bent down and pointed to the pedestal of one of the statues and showed me letters carved in the marble. “Look,” he said. “This is what I wanted you to see.” I saw a letter I did not recognize—it was Greek. “Greek letters were not only used for writing, but also for numbers,” Laci said. The letter he was pointing to, he explained, was the Greek delta. It stood for the number four. “This was the fourth statue out of the assemblage that once stood here,” he said.
After spending an hour admiring this monument of ancient Greek civilization, we left the temple and began to descend slowly. We stopped to sit down on a flat marble slab that was once part of this columned edifice. From here we could admire the Parthenon. It was hot, and we drank water from plastic bottles Laci had bought. After a few moments, he took out a notebook and showed me how the Greeks used letters as their numerals, and how their arithmetic, using only letters and no zero, worked. Here is what he wrote:
The Greek letters used as numbers (including the archaic letters digamma, koppa, and sampi).
The Greek alphabet of that time, the fifth century BCE—the height of classical Greece—included letters that had been extinct since antiquity: digamma () for 6; koppa () for 90; and sampi () for 900. So by the fifth century BCE the Greeks had revived letters in an alphabet they no longer used for writing, just so they would have enough symbols for numbers.
Laci pointed out that the custom of using letters for numbers was rooted in Phoenicia, from which the Hebrew alphabet hails as well. Some Orthodox Jews to this day, he explained, have watches with faces displaying Hebrew letters for the numbers from 1 to 12.
We then went to the museum of Athens, one of the greatest archaeological museums in the world. Among beautiful statues of gods and goddesses I saw several stone inscriptions with numbers represented as letters.
On the way back to the ship, Laci asked the driver to stop in an alley in Piraeus and had me wait in the cab while he went into what looked like a store selling discount electronics. When he came back, he had a small package in his hand. “It’s just a small transistor radio,” he said to me. These were the new popular products of the time, the early 1960s, and people were going crazy over them. You would see a person walking down the street listening to a transistor radio as often as today you might see someone talking on a cellular phone. “Just a present for someone,” he said. I thought nothing more about it at the time.
After leaving Piraeus that evening, the Theodor Herzl sailed to Naples, and the passengers spent the day visiting nearby Pompeii. Next, there was a stop at Civitavecchia, the modern-day port of Rome (in antiquity Rome’s port was Ostia). The passengers had a tour of the city, with an in-depth study of Rome and the history of its empire. For a boy interested in the history of numbers, this was a cruise to remember.
In Pompeii, Laci and I traced the Roman numerals—all Latin letters—that were used for house numbers in this ancient town. Its ruins were in a remarkable state of preservation as they had been covered by volcanic ash for almost two millennia after the catastrophic eruption of Vesuvius in 79 CE. At its museum we saw more numbers written using letters, and it was fun to read them and even try basic arithmetic.
Rome was a feast for a budding number-hunter. Roman numerals were everywhere. Most interesting were the milestones the Romans had placed along their straight-as-a-ruler roads, and I slowly became adept at reading the distances on these ancient markers, which I saw in museums and on the most famous of Roman roads, the Via Appia Antica.
Laci explained to me how the Romans devised their number system. He drew for me all the Roman numerals: I is one, V is five, X is ten, L is 50, C is 100, D is 500, and M is 1,000. He explained that this method made it necessary for the written numbers to grow, and grow, and grow . . . He showed me how a Roman might have had to multiply XVIII by LXXXII, and ultimately get the answer MCDLXXVI. For us today, this operation is simply 18 × 82 = 1,476, and we can do it quickly and efficiently. Laci challenged me to perform such a calculation in the Greco-Roman system and made me construct its multiplication table; it was so huge and complicated that it took me a week to do. Amazingly, he said, this inefficient numerical system remained in wide use in the West until the thirteenth century, when it was replaced by the numbers we know today.1
I learned more about numbers from this lover of mathematics than I ever did at school, and I was grateful to him. But the mystery of where the ten numerals we use today originally came from continued to haunt me. At the same time, as I matured through school and pursued adventures to discover numerals while traveling the Mediterranean aboard my father’s ship, I began to understand the abstract—and even more mysterious—concept of a number. I realized that 3, for example, stood for the idea of “threeness”—something that was shared by all things in the universe that were three. All of them were to be described, in their quality of threeness, by the unique symbol 3. Equally, 5 stood for the quality of “fiveness” shared by everything that was five in number. This fascinating discovery made me even more eager to find where the numerals came from since they actually stood for something even deeper and more alluring than I could have ever imagined as a young child. I wanted to dedicate my life to traveling the world in search of an answer to the origin of numbers. Who invented these wondrous ten numerals? I asked myself this question all the time, and also, Who ever came
up with the amazing idea that a concept of “threeness” or “seventeenness” or “three-hundred-and-five-ness” could be captured simply by a combination of ten signs arranged in certain ways?
2
In 1972, after high school and the obligatory service in the Israeli army, I was accepted as an undergraduate in mathematics at the University of California at Berkeley. I now made one more voyage on my father’s ship. By then, all the passenger ships had been sold because Zim Lines had lost too much money due to poor management of the company’s cruise division, called Zim Passenger Lines, and my father was now the captain of a small, slow, old cargo ship, the M.V. Yaffo, which sailed between the Mediterranean and the Americas. So I hitched a ride aboard my father’s ship, embarking in Haifa in late July for the long trip that would ultimately bring me to Berkeley.
A cargo ship is as different from a cruise ship as a truck is from a limousine. Like a working truck, a cargo ship can be dirty and dusty, but its cabins are roomy; with no passengers, there’s no need to cram many people into limited cabin space, and the crew is therefore more comfortable. But the flipside is that there is nothing to do: no cocktail parties or bars or ballrooms, and no exciting social gatherings. A voyage on a cargo ship can be lonely. But Laci was still my father’s steward, and I was very fond of him. We often talked about mathematics during this trip.
As an adult, I now understood that the mystery that had held my attention since early childhood was really two mysteries. One was where the first numerals originated: In which part of the world, and when, did people first invent the nine numerals, and zero, that with time evolved into the numbers that rule our world? And the second was a deeper conundrum, which I was now sophisticated enough to discern: How did humans abstract the concept of number? How did the idea of a number originate, and how did it develop and mature through history, to bring us to the digitally dominated society in which we live today?
Laci and I spent many hours together discussing this latter idea as the ship slowly made its way across the ocean. It was a far deeper discussion than I had ever had with him as a child, certainly, and he brought to it his full intellectual abilities as a one-time doctoral student of pure mathematics at the top Russian university, where mathematics has always been one of the most important fields of study. It was a pleasure to sit with Laci on deck chairs discussing mathematical concepts I was only beginning to understand and be fascinated by—building on the earlier ideas he had taught me as a child: numerals, numbers, prime numbers, and the mysterious Fibonacci sequence.
Having finally crossed the Atlantic, we docked for a few days in Halifax, Nova Scotia, and then continued to New York City; Charleston, South Carolina; and finally Miami, where I disembarked before the ship continued on to the Caribbean and South America. Before I left ship to fly to San Francisco and start my studies, Laci said one last thing to me in parting: “Remember how when you were little we used to talk about where the numerals came from? Maybe you’ll find out. I once read in a science magazine that a French archaeologist may have found something about the numbers in Asia decades ago—something important relating to the zero. But I don’t remember any of the details.”
Laci’s parting words intrigued me, but I had no opportunity to pursue this research further. At Berkeley, I had a full plate of math courses, challenging but often enjoyable, and I had to worry about grades and exams and learning to become a mathematician.
However, through my courses—mostly mathematics, but also anthropology, sociology, and philosophy—I learned a fair amount also about numbers and their development.
Numbers, as a concept, are much older than we might think. In 1960, a Belgian explorer named Jean de Heinzelin de Braucourt was surveying the region of Ishango, at the border of present-day Uganda and Congo (then the Belgian Congo), when he discovered a strange-looking bone: a baboon’s fibula bearing what looked like numerous tally marks. Analysis later concluded that these markings might evidence very early counting. The bone has three sets of identical notches, adding, respectively, to the totals 60, 48, and 60. The markings are grouped in several sets containing 5, 7, 9, 11, or 13 tally marks each. This bone was scientifically dated to about 20,000 years ago—the Paleolithic era—when humans lived in hunter-gatherer groups. The Ishango bone provides some of the earliest known evidence of a form of counting by humans who lived in Africa so long ago, and it is now displayed at the Royal Belgian Institute of Natural Sciences in Brussels.
The Ishango bone: a baboon fibula, about 20,000 years old, bearing notches believed to represent early evidence of counting by our species.
What does the Ishango bone represent? It seems that in prehistoric times, early humans roaming the bush in Africa used bones of dead animals as a way of drawing the first one-to-one correspondence between the number of animals they were able to hunt and notches they made on a piece of bone. This was not quite counting, but it was close. Anyone could see that a bone that had more such tally marks implied that its owner had hunted more animals—perhaps without really knowing what the actual number was. Of course all this is merely a hypothesis, but it is a likely one.
The Ishango bone is certainly the best early example of pre-counting. But there is some evidence that European humans may also have used some kind of pre-counting: In addition to the Ishango bone, several animal bones with markings that are likely tally counts have been discovered in Europe and also dated to the Paleolithic.1
A later piece of evidence for something resembling counting comes from the mysterious Neolithic stone arrangements, believed to be about 6,000 years old, at Carnac on the coast of Brittany in France. Interestingly, the megaliths are found in groups, and the groups are often comprised of a prime number of megaliths: 7, 11, 13, and 17. Was this by chance, or does it represent a form of counting? Does it, perhaps, represent an even deeper understanding of numbers? We don’t know. Carnac is a mystery that archaeology has never been able to explain, despite attempts over many decades. Nobody knows why so many very heavy stones were placed in rows. Perhaps a connection exists with Stonehenge, where similar stones were placed in circles around the same time.
But Ishango and Carnac do not present what we consider numbers. In everyday life, numbers began with simple representations of a quantity by people using their fingers. Aristotle wrote two-and-a-half millennia ago, “Or is it because men were born with ten fingers and so, because they possess the equivalent of pebbles to the number of their fingers, come to use this number for counting everything else as well?”2 And since we also have ten toes, early societies used these as well to count beyond ten. Remnants of such a base-20 number system that existed long ago can still be seen in French, through words such as quatre-vingt (four twenties) for 80.
Clearly, the form of counting we use evolved from an accident of nature: our having five fingers on each hand and five toes on each foot. Indeed, in some languages, such as Old Khmer, five was used as a point to anchor the other numbers: After four comes five, then five-and-one, five-and-two, and so on to ten, which becomes the next anchor. When we look at the early European numbers, the Roman numerals, we see the same trend: The Roman numbers IV, V, VI, VII, VIII, are all anchored at five (V), and only after eight do we come to measure them in relation to ten: IX, X, XI, XII, XIII. So the use of five and ten as key numbers evolved in different parts of the world.
In ancient India, the number 10 was the anchor. Decimal numbers were evident very early on there, as early as the sixth century BCE as observed in some inscriptions. And once they learned the powers of ten—simply by using ten fingers, and then using ten fingers ten times, as in ten people, one for each finger, each holding up his or her ten fingers—the Indians of antiquity understood that this process can go on forever. Ten times ten people holding their fingers up was ten to the third power, and ten times ten times ten people with ten fingers each was ten to the fourth power, and so on without a limit.
During the Han Dyn
asty in China (206 BCE to 220 CE), a mathematical work titled Nine Chapters on the Mathematical Art appeared, employing both positive numbers, colored red, and negative ones, colored black. And in Egypt of the third century, a leading Greek mathematician named Diophantus obtained negative answers to some of his equations but immediately dismissed them as unrealistic. So the idea for negative numbers is quite old, but people did not understand such numbers. The double-entry bookkeeping system used in accounting today was developed in Europe in the thirteenth century in part to avoid using negative numbers. To define negative numbers requires the concept of zero.
Negative numbers are, in a sense, a reflection across zero of the positive numbers. You can see this if you draw the number line, starting at zero and going to the right to 1, 2, 3, and so on; and then extending the numbers to the left of zero to –1, –2, –3, and onward. The number –1 is a reflection of the number +1 through a mirror placed at zero. But the zero plays many other important roles in mathematics and its applications. And some crucial equations in physics, biology, engineering, economics, and other fields use zero as the key element (Maxwell’s equations in physics are a good example).
The ancient Babylonians developed a cumbersome number system based on 60 rather than 10, and without a real zero. In this system, ambiguities arose because of the lack of a zero, and they could only be resolved from the context. As a modern example, if someone told you that something costs six-ninety-five, you would understand it to be $6.95 if you were buying a magazine and $695 if you were purchasing an airplane ticket. With their cumbersome number system, the Babylonians were forced to make such assumptions all the time. But interestingly, vestiges of their ancient system still exist today: We have 60 seconds in a minute, 60 minutes in an hour, and the circle has 360 (6 × 60) degrees. All in all, however, the Babylonian system would be as inadequate today as doing calculations with one’s fingers and toes.
Finding Zero Page 2
