Why Beauty is Truth Read online

  Painstakingly, Nabu led his two friends through the procedure that their math teacher had shown them the week before. “Add half of 2 to 24, getting 25. Take the square root, which is 5—”

  Gamesh threw up his hands, baffled. “I’ve never really grasped that stuff about square roots, Nabu.”

  “Aha!” said Nabu. “Now we’re getting somewhere!” His two friends looked at him as if he’d gone mad. “Your problem isn’t solving equations, Gamesh. It’s square roots!”

  “It’s both,” muttered Gamesh.

  “But square roots come first. You have to master the subject one step at a time, like the Father of the Tablet House keeps telling us.”

  “He also keeps telling us to stop getting dirt on our clothes,” protested Humbaba, “but we don’t take any notice of—”

  “That’s different. It’s—”

  “It’s no good!” wailed Gamesh. “I’ll never become a scribe, and my father will wallop me until I can’t sit down, and mother will give me that pleading look of hers and tell me I’ve got to work harder and think of the family. But I can’t get math into my head! Law, I can remember. It’s fun! I mean how about ‘If a gentleman’s wife has her husband killed on account of another man, they shall impale her on a stake’? That’s what I call worth learning. Not dumb stuff like square roots.” He paused for breath and his hands shook with emotion. “Equations, numbers—why do we bother?”

  “Because they’re useful,” replied Humbaba. “Remember all that legal stuff about cutting off slave’s ears?”

  “Yeah!” said Gamesh. “Penalties for assault.”

  “Destroy a common man’s eye,” prompted Humbaba, “and you must pay him—”

  “One silver mina,” said Gamesh.

  “And if you break a slave’s bone?”

  “You pay his master half the slave’s price in compensation.”

  Humbaba sprung his trap. “So, if the slave costs sixty shekels, then you have to be able to work out half of sixty. If you want to practice law, you need math!”

  “The answer’s thirty,” said Gamesh immediately.

  “See!” yelled Nabu. “You can do math!”

  “I don’t need math for that, it’s obvious.” The would-be lawyer flailed the air, seeking a way to express the depth of his feelings. “If it’s about the real world, Nabu, yes, I can do the math. But not artificial problems about square roots.”

  “You need square roots for land measurement,” said Humbaba.

  “Yes, but I’m not studying to become a tax collector, my father wants me to be a scribe,” Gamesh pointed out. “Like him. So I don’t see why I have to learn all this math.”

  “Because it’s useful,” Humbaba repeated.

  “I don’t think that’s the real reason,” Nabu said quietly. “I think it’s all about truth and beauty, about getting an answer and knowing that it’s right.” But the looks on his friends’ faces told him that they weren’t convinced.

  “For me it’s about getting an answer and knowing that it’s wrong,” sighed Gamesh.

  “Math is important because it’s true and beautiful,” Nabu persisted. “Square roots are fundamental for solving equations. They may not be much use, but that doesn’t matter. They’re important for themselves.”

  Gamesh was about to say something highly improper when he noticed the teacher walking into the classroom, so he covered his embarrassment with a sudden attack of coughing.

  “Good morning, boys,” said the teacher brightly.

  “Good morning, master.”

  “Let me see your homework.”

  Gamesh sighed. Humbaba looked worried. Nabu kept his face expressionless. It was better that way.

  Perhaps the most astonishing thing about the conversation upon which we have just eavesdropped—leaving aside that it is complete fiction—is that it took place around 1100 BCE, in the fabled city of Babylon.

  Might have taken place, I mean. There is no evidence of three boys named Nabu, Gamesh, and Humbaba, let alone a record of their conversation. But human nature has been the same for millennia, and the factual background to my tale of three schoolboys is based on rock-hard evidence.

  We know a surprising amount about Babylonian culture because their records were written on wet clay in a curious wedge-shaped script called cuneiform. When the clay baked hard in the Babylonian sunshine, these inscriptions became virtually indestructible. And if the building where the clay tablets were stored happened to catch fire, as sometimes happened—well, the heat turned the clay into pottery, which would last even longer.

  A final covering of desert sand would preserve the records indefinitely. Which is how Babylon became the place where written history begins. The story of humanity’s understanding of symmetry—and its embodiment in a systematic and quantitative theory, a “calculus” of symmetry every bit as powerful as the calculus of Isaac Newton and Gottfried Wilhelm Leibniz—begins here too. No doubt it might be traced back further, if we had a time machine or even just some older clay tablets. But as far as recorded history can tell us, it was Babylonian mathematics that set humanity on the path to symmetry, with profound implications for how we view the physical world.

  Mathematics rests on numbers but is not limited to them. The Babylonians possessed an effective notation that, unlike our “decimal” system (based on powers of ten), was “sexagesimal” (based on powers of sixty). They knew about right-angled triangles and had something akin to what we now call the Pythagorean theorem—though unlike their Greek successors, the mathematicians of Babylon seem not to have supported their empirical findings with logical proofs. They used mathematics for the higher purpose of astronomy, presumably for agricultural and religious reasons, and also for the prosaic tasks of commerce and taxation. This dual role of mathematical thought—revealing order in the natural world and assisting in human affairs—runs like a single golden thread throughout the history of mathematics.

  What is most important about the Babylonian mathematicians is that they began to understand how to solve equations.

  Equations are the mathematician’s way of working out the value of some unknown quantity from circumstantial evidence. “Here are some known facts about an unknown number: deduce the number.” An equation, then, is a kind of puzzle, centered upon a number. We are not told what this number is, but we are told something useful about it. Our task is to solve the puzzle by finding the unknown number. This game may seem somewhat divorced from the geometrical concept of symmetry, but in mathematics, ideas discovered in one context habitually turn out to illuminate very different contexts. It is this interconnectedness that gives mathematics such intellectual power. And it is why a number system invented for commercial reasons could also inform the ancients about the movements of the planets and even of the so-called fixed stars.

  The puzzle may be easy. “Two times a number is sixty: what is the number we seek?” You do not have to be a genius to deduce that the unknown number is thirty. Or it may be much harder: “I multiply a number by itself and add 25: the result is ten times the number. What is the number we seek?” Trial and error may lead you to the answer 5—but trial and error is an inefficient way to answer puzzles, to solve equations. What if we change 25 to 23, for example? Or 26? The Babylonian mathematicians disdained trial and error, for they knew a much deeper, more powerful secret. They knew a rule, a standard procedure, to solve such equations. As far as we know, they were the first people to realize that such techniques existed.

  The mystique of Babylon stems in part from numerous Biblical references. We all know the story of Daniel in the lion’s den, which is set in Babylon during the reign of King Nebuchadnezzar. But in later times, Babylon became almost mythical, a city long vanished, destroyed beyond redemption, that perhaps had never existed. Or so it seemed until roughly two hundred years ago.

  For thousands of years, strange mounds had dotted the plains of what we now call Iraq. Knights returning from the Crusades brought back souvenirs dragged from the ru
bble—decorated bricks, fragments of undecipherable inscriptions. The mounds were clearly the ruins of ancient cities, but beyond that, little was known.

  In 1811, Claudius Rich made the first scientific study of the rubble mounds of Iraq. Sixty miles south of Baghdad, beside the Euphrates, he surveyed the entire site of what he soon determined must be the remains of Babylon, and hired workmen to excavate the ruins. The finds included bricks, cuneiform tablets, beautiful cylinder seals that produced raised words and pictures when rolled over wet clay, and works of art so majestic that whoever carved them must be ranked alongside Leonardo da Vinci and Michelangelo.

  Even more interesting, however, were the smashed cuneiform tablets that littered the sites. We are fortunate that those early archaeologists recognized their potential value, and kept them safe. Once the writing had been deciphered, the tablets became a treasure-trove of information about the lives and concerns of the Babylonians.

  The tablets and other remains tell us that the history of ancient Mesopotamia was lengthy and complex, involving many different cultures and states. It is customary to employ the word “Babylonian” to refer to them all, as well as to the specific culture that was centered upon the city of Babylon. However, the heart of Mesopotamian culture moved repeatedly, with Babylon both coming into, and falling out of, favor. Archaeologists divide Babylonian history into two main periods. The Old Babylonian period runs from about 2000 to 1600 BCE, and the Neo-Babylonian period runs from 625 to 539 BCE. In between are the Old Assyrian, Kassite, Middle Assyrian, and Neo-Assyrian periods, when Babylon was ruled by outsiders. Moreover, Babylonian mathematics continued in Syria, throughout the period known as Seleucid, for another five hundred years or more.

  The culture itself was much more stable than the societies in which it resided, and it remained mostly unchanged for some 1200 years, sometimes temporarily disrupted by periods of political upheaval. So any particular aspect of Babylonian culture, other than some specific historical event, probably came into existence well before the earliest known record. In particular, there is evidence that certain mathematical techniques, whose first surviving records date to around 600 BCE, actually existed far earlier. For this reason, the central character in this chapter—an imaginary scribe to whom I shall give the name Nabu-Shamash and whom we have already met during his early training in the brief vignette about three school friends—is deemed to have lived sometime around 1100 BCE, being born during the reign of King Nebuchadnezzar I.

  All the other characters that we will meet as our tale progresses were genuine historical figures, and their individual stories are well documented. But among the million or so clay tablets that have survived from ancient Babylon, there is little documented evidence about specific individuals other than royalty and military leaders. So Nabu-Shamash has to be a pastiche based on plausible inferences from what we have learned about everyday Babylonian life. No new inventions will be attributed to him, but he will encounter all those aspects of Babylonian knowledge that play a role in the story of symmetry. There is good evidence that all Babylonian scribes underwent a thorough education, with mathematics as a significant component.

  Our imaginary scribe’s name is a combination of two genuine Babylonian names, the scribal god Nabu and the Sun god Shamash. In Babylonian culture it was not unusual to name ordinary people after gods, though perhaps two god-names would have been considered a bit extreme. But for narrative reasons we are obliged to call him something more specific, and more atmospheric, than merely “the scribe.”

  When Nabu-Shamash was born, the king of Babylonia was Nebuchadnezzar I, the most important monarch of the Second Dynasty of Isin. This was not the famous Biblical king of the same name, who is usually referred to as Nebuchadnezzar II; the Biblical king was the son of Nabopolassar, and he reigned from 605 to 562 BCE.

  Nebuchadnezzar II’s reign represented the greatest flowering of Babylon, both materially and in regional power. The city also flourished under his earlier namesake, as Babylon’s power extended to encompass Akkad and the mountainous lands to the north. But Akkad effectively seceded from Babylon’s control during the reigns of Ahur-resh-ishi and his son Tiglath-Pileser I, and it strengthened its own security by taking action against the mountain and desert tribes that surrounded it on three sides. So Nabu-Shamash’s life began during a stable period of Babylonian history, but by the time he became a young man, Babylon’s star was beginning to wane, and life was becoming more turbulent.

  Nabu-Shamash was born into a typical “upper-class” household in the Old City of Babylon, not far from the Libil-hegalla canal and close to the justly famed Ishtar Gate, a ceremonial entrance decorated with colored ceramic bricks in fanciful forms—bulls, lions, even dragons. The road through the Ishtar Gate was impressive, reaching a width of 20 meters; it was paved with limestone flags on top of a bed of asphalt, with a brick foundation. Its name was “May the enemy not have victory”—rather typical of Babylon’s main street names—but it is generally known as the Processional Way, being used by the priests to parade the god Marduk through the city when ceremony so decreed.

  The family home was built of mud brick, with walls six feet thick to keep out the sun. The external walls had few openings—mainly a doorway at street level—and rose to a height of three stories, with lighter materials, mainly wood, being used for the top floor. The family owned many slaves, who performed routine household tasks. Their quarters, along with the kitchen, were to the right of the entrance. The family rooms were to the left: a long living room, bedrooms, and a bathroom. There was no bathtub in Nabu-Shamash’s time, though some have survived from other eras. Instead, a slave would pour water over the bather’s head and body, approximating a modern shower. A central courtyard opened to the sky, and toward the back were storerooms.

  Nabu-Shamash’s father was an official in the court of a king, name unknown, whose reign preceded Nebuchadnezzar I. His duties were largely bureaucratic: he was responsible for administering an entire district, ensuring that law and order were maintained, that the fields were properly irrigated, and that all necessary taxes were collected and paid. Nabu-Shamash’s father had also been trained as a scribe, because literacy and numeracy were basic skills for anyone in the Babylonian equivalent of the civil service.

  According to a decree attributed to the god Enlil, every man should follow in his father’s footsteps, and Nabu-Shamash was expected to do just that. However, scribal abilities also opened up other career paths, notably that of priest, so his training paved the way to a choice of professions.

  We know what Nabu-Shamash’s education was like because extensive records, written in Sumerian by people who were trained as scribes, have survived from roughly the period concerned. These records make it plain that Nabu-Shamash was fortunate in his choice of parentage, for only the sons of the well-to-do could hope to enter the scribal schools. In fact, the quality of Babylonian education was so high that foreign nobles sent their sons to the city to be educated.

  The school was called the Tablet House, presumably referring to the clay tablets used for writing and arithmetic. It had a head teacher, referred to as the “Expert” and as the “Father of the Tablet House.” There was a class teacher, whose main task was to make the boys behave themselves; there were specialist teachers in Sumerian and mathematics. There were prefects, called “Big Brothers,” whose job included keeping order. Like all students, Nabu-Shamash lived at home and went to school during the day, for around 24 days each 30-day month. He had three days off for recreation, and a further three for religious festivals.

  Nabu-Shamash began his studies by mastering the Sumerian language, especially its written form. There were dictionaries and grammatical texts to be studied, and long lists to be copied—legal phrases, technical terms, names. Later, he progressed to mathematics, and it was then that his studies became central to our tale.

  What did Nabu-Shamash learn? For everyone but philosophers, logicians, and professional mathematicians who are being pedanti
c, a number is a string of digits. Thus the year in which I write this sentence is 2006, a string of four digits. But as the pedants will jump to remind us, this string of digits is not the number at all but only its notation, and a rather sophisticated form of notation at that. Our familiar decimal system employs just ten digits, the symbols 0 through 9, to represent any number, however large. An extension of that system also permits the representation of very small numbers; more to the point, it permits the representation of numerical measurements to very high levels of precision. Thus the speed of light, according to the best current observations, is approximately 186,282.397 miles per second.

  We are so familiar with this notation that we forget how clever it is—and how difficult to grasp when we first encounter it. The key feature on which all else rests is this: the numerical value of a symbol, such as 2, depends on where it is placed relative to the other symbols. The symbol 2 does not have a fixed meaning independent of its context. In the number representing the speed of light, the digit “2” immediately before the decimal point does indeed mean “two.” But the other occurrence of “2” in that number means “two hundred.” In the date 2006, that same digit means “two thousand.”

  We would be exceedingly unhappy to have a system of writing in which the meaning of a letter depended on where it occurred in a word. Imagine, for instance, what reading would be like if the two a’s in “alphabet” had totally different meanings. But positional notation for numbers is so convenient and powerful that we find it hard to imagine that anyone really used any other method.

  It was not always thus. Our present notation dates back no more than 1500 years, and was first introduced into Europe a little more than 800 years ago. Even today, different cultures use different symbols for the same decimal digits—look at any Egyptian banknote. But ancient cultures wrote numbers in all sorts of strange ways. The most familiar to us is probably the Roman system, in which 2006 becomes MMVI. In ancient Greek it would be ζ. In place of our 2, 20, 200, and 2000, the Romans wrote II, XX, CC, and MM, and the Greeks wrote β, κ, α, and .