Why Beauty Is Truth

PREFACE

The date is 13 May 1832. In the dawn mist, two young Frenchmen face each other, pistols drawn, in a duel over a young woman. A shot is fired; one of the men falls to the ground, fatally wounded. He dies two weeks later, from peritonitis, aged 21, and is buried in the common ditch—an unmarked grave. One of the most important ideas in the history of mathematics and science very nearly dies with him.

The surviving duelist remains unknown; the one who died was Évariste Galois, a political revolutionary and a mathematical obsessive whose collected works fill a mere sixty pages. Yet Galois left a legacy that revolutionized mathematics. He invented a language to describe symmetry in mathematical structures, and to deduce its consequences.

Today that language, known as “group theory,” is in use throughout pure and applied mathematics, where it governs the formation of patterns in the natural world. Symmetry also plays a central role at the frontiers of physics, in the quantum world of the very small and the relativistic world of the very large. It may even provide a route to the long-sought “Theory of Everything,” a mathematical unification of those two key branches of modern physics. And it all began with a simple question in algebra, about the solutions of mathematical equations—finding an “unknown” number from a few mathematical clues.

Symmetry is not a number or a shape, but a special kind of transformation—a way to move an object. If the object looks the same after being transformed, then the transformation concerned is a symmetry. For instance, a square looks the same if it is rotated through a right angle.

This idea, much extended and embellished, is fundamental to today’s scientific understanding of the universe and its origins. At the heart of Albert Einstein’s theory of relativity lies the principle that the laws of physics should be the same in all places and at all times. That is, the laws should be symmetric with respect to motion in space and the passage of time. Quantum physics tells us that everything in the universe is built from a collection of very tiny “fundamental” particles. The behavior of these particles is governed by mathematical equations—“laws of nature”—and those laws again possess symmetry. Particles can be transformed mathematically into quite different particles, and these transformations also leave the laws of physics unchanged.

These concepts, and more recent ones at the frontiers of today’s physics, could not have been discovered without a deep mathematical understanding of symmetry. This understanding came from pure mathematics; its role in physics emerged later. Extraordinarily useful ideas can arise from purely abstract considerations—something that the physicist Eugene Wigner referred to as “the unreasonable effectiveness of mathematics in the natural sciences.” With mathematics, we sometimes seem to get more out than we put in.

Starting with the scribes of ancient Babylon and ending with the physicists of the twenty-first century, Why Beauty Is Truth tells how mathematicians stumbled upon the concept of symmetry, and how an apparently useless search for what turned out to be an impossible formula opened a new window on the universe and revolutionized science and mathematics. More broadly, the story of symmetry illustrates how the cultural influence and historical continuity of big ideas can be brought into sharp relief by occasional upheavals, both political and scientific.

The first half of the book may seem at first sight to have nothing to do with symmetry and precious little to do with the natural world. The reason is that symmetry did not become a dominant idea by the route one might expect, through geometry. Instead, the profoundly beautiful and indispensable concept of symmetry that mathematicians and physicists use today arrived via algebra. Much of this book, therefore, describes the search for solutions of algebraic equations. This may sound technical, but the quest is a gripping one, and many of the key players led unusual and dramatic lives. Mathematicians are human, even though they are often lost in abstract thought. Some of them may let logic rule their lives too much, but we shall see time and again that our heroes could in fact be all too human. We will see how they lived and died, read of their love affairs and duels, vicious priority disputes, sex scandals, drunkenness, and disease, and along the way we will see how their mathematical ideas unfolded and changed our world.

Beginning in the tenth century BCE and reaching its climax with Galois in the early nineteenth century, the narrative retraces the step-by-step conquest of equations—a process that eventually ground to a halt when mathematicians tried to conquer the so-called “quintic” equation, involving the fifth power of the unknown. Did the methods break down because there was something fundamentally different about the quintic equation? Or might there be similar, yet more powerful methods that would lead to formulas for its solution? Were mathematicians stuck because of a genuine obstacle, or were they just being obtuse?

It is important to understand that solutions to quintic equations were known to exist. The question was, can they always be represented by an algebraic formula? In 1821 the young Norwegian Niels Henrik Abel proved that the quintic equation cannot be solved by algebraic means. His proof, however, was rather mysterious and indirect. It proved that no general solution is possible, but it did not really explain why.

It was Galois who discovered that the impossibility of solving the quintic stems from the symmetries of the equation. If those symmetries pass the Galois test, so to speak—meaning that they fit together in a very specific way, which I will not explain just yet—then the equation can be solved by an algebraic formula. If the symmetries do not pass the Galois test, then no such formula exists.

The general quintic equation cannot be solved by a formula because it has the wrong kind of symmetries.

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.

1

THE SCRIBES OF BABYLON

Across 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.”

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.