Why Beauty is Truth (3 page)

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 rubble—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.