by Lucy Hawking
Vincent checked his watch. “Five-o-six,” he replied.
“Five! We haven’t got long! Hold on—what time is it in Switzerland?”
“Six-o-six,” said Vincent.
“Right, we have to work fast,” said George, speaking as quickly as he could. “Annie, you told me that the meeting of the Order is at seven thirty tonight. Reeper said that TOERAG has a bomb—a quantum mechanical bomb—and I bet they’ve primed it to go off when the meeting starts so that the Collider—and everyone near it—will be blown to kingdom come, and science will be set back by centuries.”
“A quantum mechanical bomb?” said Annie, looking almost as sick as George had been a few minutes earlier. “What’s that?”
“Well, I know what it is,” confessed George, “but I’m not sure how to turn it off. We’d better take this with us.” He picked up Pooky’s string of numbers. “I’m not sure, but it might be the code to defuse the bomb. Or one of them, anyway.”
“What makes you think Reeper is telling the truth?” demanded Annie.
“We can’t know for sure, but I think he’s on our side this time. And Eric’s side. Reeper wants to stop the Collider and everyone around it from being blown up by those weirdos we saw in the cellar when we were looking for somewhere to put Freddy.”
“How can you trust this Reeper guy?” threw in Vincent. “Hasn’t he always double-crossed you in the past?”
Annie had pulled her cell phone out of her pocket. She tried calling her dad, but she couldn’t get through. She couldn’t even leave a message.
“I don’t know if we can,” said George. “We’re taking a chance on him. But if we don’t do something, it’s likely that the Collider will explode during the meeting of the Order of Science this evening.”
“How can we get there in time?” cried Annie. “We’d need to travel through a portal to do that, and we haven’t got Cosmos!”
“There is another portal,” said George, finally working it out in his head and discovering the missing link he had been searching for since his visit to the Math Department, “and I know where it is!”
“Where?” said Annie in confusion. “I thought Cosmos was the only supercomputer in the world—apart from Pooky, who isn’t safe.”
“You’re right,” agreed George. “We can’t use Pooky again—we don’t know how, and his portal is no good anyway. But we do know how to use new Cosmos, which means we might be able to operate old Cosmos.”
“Old Cosmos … ?”
“Do you remember your dad’s lecture?” George’s brain was now working at the speed of light. “That crummy professor, Zuzubin, was there. He’s the one who told Eric he had to go to Switzerland, and he’s the one who called the emergency meeting of the Order of Science to Benefit Humanity.”
“So what?” said Annie. “What are you saying?”
“When we left the Math Department, Zuzubin didn’t follow us,” George continued. “He went down the stairs, instead of coming out.”
“And … ?”
“Your dad once told us that when he was a student at Foxbridge, old Cosmos—the first supercomputer—lived in the basement of the Math Department. And after your dad’s lecture, I saw Zuzubin go down the stairs to the basement when we were going out of the front door. And I saw him wearing a pair of yellow glasses, just like the ones Eric found when he fell into the black hole. Which means that someone has been traveling around the Universe, dropping stuff.”
“And to do that, they must have a supercomputer,” said Annie, catching on. “So you think that old Cosmos is in the basement of the Math Department and Zuzubin has been using him … ?”
“But Annie’s dad was a student, like, zillions of years ago,” Vincent pointed out. “Surely that computer’s been shut down by now.”
“That’s what we’re supposed to think,” said George. “We’re supposed to think that old Cosmos doesn’t work. But if he does, and he can send Zuzubin to look at black holes, he could also send us to the Collider in time to defuse the quantum mechanical bomb.”
“But why would Zuzubin keep a secret like that?” asked Annie.
“I don’t know …” George’s voice was full of foreboding. “But I think we’re about to find out. We need to get to the Math Department. As fast as we can. Zuzubin will be at the Large Hadron Collider for the meeting, so we should be able to try old Cosmos.”
He and Annie thumped down the stairs two at a time, sped out of the door, and got their bikes, with Vincent following closely behind. “What I don’t get … ,” Annie’s friend said as he hopped onto his skateboard. “Why math? What has math got to do with anything? It’s just a bunch of numbers on a blackboard that all add up to another number. What’s that got to do with the Universe anyway? What use is math to anyone?”
THE LATEST SCIENTIFIC THEORIES
HOW MATHEMATICS IS
SURPRISINGLY USEFUL
IN UNDERSTANDING
THE UNIVERSE
It is obvious that some things in our everyday world are simple and others complex. We know our Sun will come up day after day exactly on time, but the weather changes in annoying and haphazard ways—unless like me you live in Arizona, where it is almost always warm and sunny. So you can set your alarm clock the night before and be sure you will wake up at the right time of day, but if you choose your clothes ahead of time you might get it badly wrong.
Those things that are simple, regular, and dependable can be described by numbers, like the number of hours in a day, or the number of days in a year. We can also use numbers to describe complicated things like the weather—such as the highest temperature each day—but in that case it’s often hard to spot any patterns in the numbers.
Our ancestors noticed many patterns in nature: not just day and night, but the seasons, the movements of the Moon, stars, and planets in the sky, and the rise and fall of the tides. Sometimes they used numbers to describe the patterns; sometimes they used songs or poetry instead. Many ancient peoples went to a lot of effort with numbers and patterns to describe the movement of heavenly bodies. They liked to predict eclipses—scary but exciting events where the Moon blots out the light of the Sun and you can see the stars in the daytime. Knowing when an eclipse would happen required lots of boring calculations, and they didn’t always get it right. But when they did, people were impressed.
Long ago, nobody knew why numbers and simple patterns occur so often in nature. But about four hundred years ago, some people began to study the patterns more carefully. Especially in Europe, there were beautiful and quite skillfully made instruments to help observe and measure things accurately. People had clocks and sundials and all sorts of metal gadgets for distances and angles and times. Eventually they had small telescopes, too. These curious people called themselves “natural philosophers”—and were what we would now call scientists.
One thing natural philosophers puzzled over was motion. At first, there seemed to be two sorts: stars and planets moving in the sky, and objects moving around on Earth. Everybody knows that when you throw a ball it travels in a curved path, and it doesn’t take too many tries to see that the curve is always the same if the ball is thrown at the same speed and angle.
Of course, our ancestors were well aware that moving objects followed simple predictable paths. They knew it because their lives depended on it. Hunters needed to be sure that when a stone left a sling or an arrow left a bow, it would behave the same way today as it did yesterday. In Australia, the ancient people known as Aboriginals were so ingenious they could make a flat stick called a boomerang, and when it was thrown it would follow a special path that would cause it to curve back toward the thrower.
By the sixteenth century, mathematics had gone some way beyond simple arithmetic, to include algebra and other fancy methods, and the natural philosophers were able to write down equations to describe many of the patterns found in nature. In particular, they could write equations to describe curves like the paths of arrows and balls. For example, a simple equation describes a ci
rcle, a slightly different one a squashed circle called an ellipse, and yet another describes the curve of a rope hanging between two poles. Using this more advanced mathematics, a huge variety of patterns and shapes could be described not just in words, but in symbols and equations, written on paper and printed for other scientists and mathematicians to study.
Useful though all this was, it was still just a description of patterns in nature, not an explanation. The big breakthrough began with the work of Galileo Galilei in Italy in the early seventeenth century. Everybody knows that when an object is dropped from a height, it rushes toward the ground faster and faster. Galileo wanted to make this precise: How much faster does it go after one second, two seconds, three seconds … ? Was there a pattern? He found the answer by experimenting—he tried dropping things and timing them. He rolled balls down slopes so everything happened more slowly and easily. Then he sat down with all the measurements and did some arithmetic and algebra, until he found a single formula that correctly describes the way that all falling bodies accelerate; that is, go faster and faster as they fall.
Galileo’s formula is pretty simple: If the object is dropped from rest its speed increases in proportion to the time it has been falling. This means that when the object has been falling for two seconds it’s going exactly twice as fast as it was at one second. And there’s more. If the object is thrown from a height at an angle instead of just dropped, it will still fall in the same way but it will also move horizontally, and Galileo’s formula says that the shape of the path the object follows is a parabola—one of the curves mathematicians already knew about from studying geometry.
The decisive step came when Isaac Newton in England worked out how objects like balls change their motion (that is, accelerate or decelerate) when they are pushed or pulled by forces. He wrote down a very simple equation to describe it.
In the case of Galileo’s falling objects, the force concerned is, of course, gravity. We feel the force of gravity all the time. Newton suggested that the Earth pulls everything downward, toward its center, with a force proportional to the amount of matter the object contains (known as its mass). Newton’s equation connecting force and acceleration then explained Galileo’s formula for falling bodies.
But this was just the start. Newton also suggested that not just the Earth, but every object in the Universe—including the Sun, Moon, planets, stars, and even people—pull on every other object with a force of gravity that gets weaker with distance in a precise way, called an inverse square. That’s a fancy way of saying that at twice the distance from the center of the Earth (or the Sun, or the Moon) the force is one-quarter as strong; at three times the distance it is one-ninth, and so on.
Using this formula plus his equation for how force and acceleration are related, Newton was able to do some cool mathematics (some of which he invented) to work out how planets and comets move around the Sun, pulled by the Sun’s gravity. He also calculated how the Moon goes around the Earth. And the numbers all came out right! More than that, the shapes of the orbits were also correctly described by his calculations. For example, astronomers had measured that the orbits of the planets are ellipses, and the great Newton showed they should be—from his calculations! No wonder everybody thought he was a hero and a genius. The government was so pleased they put him in charge of printing all England’s money.
The really important thing about Newton’s work on motion and gravitation is deeper, however. He proposed that his formula for gravity and his equation for force and acceleration were laws of nature. That is, they should be the same everywhere in the Universe and at all times, and can never change—rather like God, whom Newton believed in. Before Newton, some people thought the motions of objects on Earth, like balls and boats and birds, had nothing to do with the motions of bodies in the sky, like the Moon and planets. Now we knew they all obeyed the same laws. While other scientists had described motion, Newton explained it in terms of mathematical laws.
In practical terms, this was a huge leap forward, because now anyone could sit in a chair and work out how such-and-such an object would move, without ever seeing it, or even leaving the room. For example, you can calculate where a cannonball will land if it is fired at a certain speed and angle. You can work out how fast it would need to go to fly off the Earth and never come back. Using Newton’s simple equations, engineers can figure out exactly how to point a rocket to send a spacecraft to the Moon or Mars—before they even have the money to build the rocket.
All this made physics—the study of the basic laws of the Universe—a predictive science. Physicists could play with their equations and predict things that nobody knew before, like the existence of unknown planets. Uranus and Neptune were found after astronomers used Newton’s laws to work out where in the sky they should be, and we now use those laws to predict the existence of planets going around other stars.
Very soon physicists began applying the same ideas to other forces, like electricity and magnetism, and sure enough, they were found to obey simple mathematical laws, too. Then atoms and their nuclei were studied, and they also can be explained in detail with mathematical formulas. So there are now quite a number of equations in physics textbooks.
Some physicists wonder whether it will go on like this forever, or whether all the laws and equations can be merged in some way, into some super-duper law that contains all the others. Quite a lot of smart people have peered at the equations to look for links, and a few have been found that turned out to be right.
A famous example was when James Clerk Maxwell, a Scottish physicist in the nineteenth century, found that the laws of electricity and magnetism could be joined, and when he had done that he solved the equations and discovered that the combined electromagnetic force could generate waves of electromagnetism. When he worked out the speed of the waves from his equations, he found it was the same as the speed of light. Bingo! Light must be an electromagnetic wave, he said. The quest for a super-duper law combining all the forces goes on. It needs a really bright youngster to pull everything together.
When I was a schoolboy, I liked a pretty girl called Lindsay. One day I was doing a homework problem in physics. I had to calculate (that is, predict) what angle to throw a ball for it to go the maximum distance up a hill of a certain slope. Lindsay (who was studying liberal arts) sat opposite me in the school library, which was nice, although it made me a little nervous. She asked what I was doing, and when I described the problem she remarked in wonderment, “But how can you find out what a ball will do by writing things on a piece of paper?” At the time I thought this was a silly question. After all, this was my homework! But Lindsay had actually touched on a very deep issue. Why can we use simple mathematical laws to describe, and even predict, things that go on in the world around us? Where do the laws come from? That is, why does nature have laws at all? And even if for some reason there have to be laws of nature, why are they so simple (like the inverse square law of gravity)? We can imagine a universe with mathematical laws that are so subtle and complicated that even the brainiest human mathematician would be baffled.
Nobody knows why the Universe can be explained with simple mathematics, or why human brains are good enough to work it all out. Maybe we just got lucky? Some people think there is a Mathematician God who made the Universe that way. Scientists are not very interested in gods, though. Could it be that life will arise only if the Universe has simple mathematical laws, so nature has to be mathematical or we wouldn’t be here arguing about it? Perhaps there are many universes, each with laws different from our own, and maybe some with no laws to speak of at all. These other universes may be devoid of scientists and mathematicians. Or maybe not.
To be honest, it’s all a mystery, and most scientists think it’s not part of their job to worry about it. They just take the mathematical laws of nature as a fact, and get on with their calculations.
I’m not one of them. I lie awake at night turning it all over in my mind. I’d like an answer. But whethe
r or not there is a reason for the mathematical simplicity of the Universe, it’s clear that physics and mathematics are deeply interwoven, and that we will always need people who can do experiments and people who can do mathematics. And they had better keep talking to each other!
Paul
Chapter Fifteen
George and Annie pedaled furiously past Foxbridge’s curiously shaped citadels of learning, Vincent curving gracefully beside them on his skateboard. The town was full of old and beautiful buildings, where for centuries scholars had dreamed up great theories, explaining the Universe and all its wonders to a world that only sometimes wanted to know.
Some of the colleges looked like fortresses—for good reason. Throughout the ages they had at times been forced to lock their gates to keep out angry mobs, furious at some of the new ideas their scholars had propounded. Gravity, for example. The orbit of the Earth around the Sun, rather than the other way around. Evolution. The Big Bang. The double helix of DNA, and the possibilities of life in other universes. The walls of these colleges were thick, with tiny windows to protect those within from a real and often unfriendly world outside.
The three children scooted into the courtyard of the Math Department, throwing the bikes against the black railings and running up the steps to the front entrance. Today the glass doors just swung in the breeze, and no one stopped them as they dashed into the hallway. They were greeted only by the familiar smell of chalk dust and old socks, and by the distant clank of the tea tray being unloaded.
“Don’t take the elevator!” hissed Annie as Vincent started to press the button. “It’s too noisy! Let’s go down the stairs.”
Vincent parked his precious skateboard under the notice board in the hallway—he noticed ads for enticing events such as DOUBLY PERIODIC MONOPOLES: A 3D INTEGRABLE SYSTEM or THE EARLY UNIVERSE: TRANSITIONAL PHASES!—and the three of them tiptoed down the steps to the basement, George first, Annie next, and Vincent following behind.