Why Beauty is Truth (2 page)

This epic discovery created the second theme of this book: that of a
group
—a mathematical “calculus of symmetry.” Galois took an ancient mathematical tradition, algebra, and reinvented it as a tool for the study of symmetry.

At this stage of the book, words like “group” are unexplained jargon. When the meaning of such words becomes important to the story, I will explain them. But sometimes we just need a convenient term to keep track of various items of baggage. If you run into something that looks like jargon but is not immediately discussed, then it will be playing the role of a useful label, and the actual meaning won't matter very much.
Sometimes the meaning will emerge anyway if you keep reading. “Group” is a case in point, but we won't find out what it means until the middle of the book.

Our story also touches upon the curious significance of particular numbers in mathematics. I am not referring to the fundamental constants of physics but to mathematical constants like π (the Greek letter pi). The speed of light, for instance, might in principle be anything, but it happens to be 186,000 miles per second in our universe. On the other hand, π is slightly larger than 3.14159, and nothing in the world can change that value.

The unsolvability of the quintic equation tells us that like π, the number 5 is also very unusual. It is the smallest number for which the associated symmetry group fails the Galois test. Another curious example concerns the sequence of numbers 1, 2, 4, 8. Mathematicians discovered a series of extensions of the ordinary “real” number concept to complex numbers and then to things called quaternions and octonions. These are constructed from two copies of the real numbers, four copies, and eight copies, respectively. What comes next? A natural guess is 16, but in fact there are
no
further sensible extensions of the number system. This fact is remarkable and deep. It tells us that there is something special about the number 8, not in any superficial sense, but in terms of the underlying structure of mathematics itself.

In addition to 5 and 8, this book features appearances by several other numbers, most notably 14, 52, 78, 133, and 248. These curious numbers are the dimensions of the five “exceptional Lie groups,” and their influence pervades the whole of mathematics and much mathematical physics. They are key characters in the mathematical drama, while other numbers, seemingly little different, are mere bit players.

Mathematicians discovered just how special these numbers are when modern abstract algebra came into being at the end of the nineteenth century. What counts is not the numbers themselves but the role they play in the foundations of algebra. Associated with each of these numbers is a mathematical object called a Lie group with unique and remarkable properties. These groups play a fundamental role in modern physics, and they appear to be related to the deep structure of space, time, and matter.

That leads to our final theme: fundamental physics. Physicists have long wondered why space has three dimensions and time one—why we live in a four-dimensional space-time. The theory of superstrings, the most recent attempt to unify the whole of physics into a single coherent set of laws, has led physicists to wonder whether space-time might have extra “hidden” dimensions. This may sound like a ridiculous idea, but it has good historical precedents. The presence of additional dimensions is probably the least objectionable feature of superstring theory.

A far more controversial feature is the belief that formulating a new theory of space and time depends mainly on the
mathematics
of relativity and quantum theory, the two pillars on which modern physics rests. Unifying these mutually contradictory theories is thought to be a mathematical exercise rather than a process requiring new and revolutionary experiments. Mathematical beauty is expected to be a prerequisite for physical truth. This could be a dangerous assumption. It is important not to lose sight of the physical world, and whatever theory finally emerges from today's deliberations cannot be exempt from comparison with experiments and observations, however strong its mathematical pedigree.

At the moment, though, there are good reasons for taking the mathematical approach. One is that until a really convincing combined theory is formulated, no one knows what experiments to perform. Another is that mathematical symmetry plays a fundamental role in both relativity and quantum theory, two subjects where common ground is in short supply, so we should value whatever bits of it we can find. The possible structures of space, time, and matter are determined by their symmetries, and some of the most important possibilities seem to be associated with exceptional structures in algebra. Space-time may have the properties it has because mathematics permits only a short list of special forms. If so, it makes sense to look at the mathematics.

Why does the universe seem to be so mathematical? Various answers have been proposed, but I find none of them very convincing. The symmetrical relation between mathematical ideas and the physical world, like the symmetry between our sense of beauty and the most profoundly important mathematical forms, is a deep and possibly unsolvable mystery. None of us can say
why
beauty is truth, and truth beauty. We can only contemplate the infinite complexity of the relationship.

A
cross the region that today we call Iraq run two of the most famous rivers in the world, and the remarkable civilizations that arose there owed their existence to those rivers. Rising in the mountains of eastern Turkey, the rivers traverse hundreds of miles of fertile plains, and merge into a single waterway whose mouth opens into the Persian Gulf. To the southwest they are bounded by the dry desert lands of the Arabian plateau; to the northeast by the inhospitable ranges of the Anti-Taurus and Zagros Mountains. The rivers are the Tigris and the Euphrates, and four thousand years ago they followed much the same routes as they do today, through what were then the ancient lands of Assyria, Akkad, and Sumer.

To archaeologists, the region between the Tigris and Euphrates is known as Mesopotamia, Greek for “between the rivers.” This region is often referred to, with justice, as the cradle of civilization. The rivers brought water to the plains, and the water made the plains fertile. Abundant plant life attracted herds of sheep and deer, which in turn attracted predators, among them human hunters. The plains of Mesopotamia were a Garden of Eden for hunter-gatherers, a magnet for nomadic tribes.

In fact, they were so fertile that the hunter-gatherer lifestyle eventually became obsolete, giving way to a far more effective strategy for obtaining food. Around 9000 BCE, the neighboring hills of the Fertile Crescent, a little to the north, bore witness to the birth of a revolutionary technology: agriculture. Two fundamental changes in human society followed hard on its heels: the need to remain in one location in order to tend the crops, and the possibility of supporting large populations. This combination led to the invention of the city, and in Mesopotamia we can still find archaeological
remains of some of the earliest of the world's great city-states: Nineveh, Nimrud, Nippur, Uruk, Lagash, Eridu, Ur, and above all, Babylon, land of the Hanging Gardens and the Tower of Babel. Here, four millennia ago, the agricultural revolution led inevitably to an organized society, with all the associated trappings of government, bureaucracy, and military power. Between 2000 and 500 BCE the civilization that is loosely termed “Babylonian” flourished on the banks of the Euphrates. It is named for its capital city, but in the broad sense the term “Babylonian” includes Sumerian and Akkadian cultures. In fact, the first known mention of Babylon occurs on a clay tablet of Sargon of Akkad, dating from around 2250 BCE, although the origin of the Babylonian people probably goes back another two or three thousand years.

We know very little about the origins of “civilization”—a word that literally refers to the organization of people into settled societies. Nevertheless, it seems that we owe many aspects of our present world to the ancient Babylonians. In particular, they were expert astronomers, and the twelve constellations of the zodiac and the 360 degrees in a circle can be traced back to them, along with our sixty-second minute and our sixty-minute hour. The Babylonians needed such units of measurement to practice astronomy, and accordingly had to become experts in the time-honored handmaiden of astronomy: mathematics.

Like us, they learned their mathematics at school.

“What's the lesson today?” Nabu asked, setting his packed lunch down beside his seat. His mother always made sure he had plenty of bread and meat—usually goat. Sometimes she put a piece of cheese in for variety.

“Math,” his friend Gamesh replied gloomily. “Why couldn't it be law? I can
do
law.”

Nabu, who was good at mathematics, could never quite grasp why his fellow students all found it so difficult. “Don't you find it boring, Gamesh, copying all those stock legal phrases and learning them by heart?”

Gamesh, whose strengths were stubborn persistence and a good memory, laughed. “No, it's easy. You don't have to
think.
”

“That's precisely why I find it boring,” his friend said, “whereas math—”

“—is impossible,” Humbaba joined in, having just arrived at the Tablet House, late as usual. “I mean, Nabu, what am I supposed to do with
this?
” He gestured at a homework problem on his tablet. “I multiply
a number by itself and add twice the number. The result is 24. What is the number?”

“Four,” said Nabu.

“Really?” asked Gamesh. Humbaba said, “Yes, I know, but how do you
get
that?”

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.”