Nature's Numbers : The Understanding of Mathematics

THE VIRTUAL UNREALITY MACHINE
I have a dream.
I am surrounded by-nothing. Not empty space, for there is no space to be empty. Not blackness, for there is nothing to be black. Simply an absence, waiting to become a presence. I think commands: let there be space. But what kind of space? I have a choice: three-dimensional space, multidimensional space, even curved space.


I choose.
Another command, and the space is filled with an all pervading fluid, which swirls in waves and vortices, here a placid swell, there a frothing, turbulent maelstrom. I paint space blue, draw white streamlines in the fluid to bring out the flow patterns.

I place a small red sphere in the fluid. It hovers, unsupported, ignorant of the chaos around it, until I give the word. Then it slides off along a streamline. I compress myself to one hundredth of my size and will myself onto the surface of the sphere, to get a bird's-eye view of unfolding events. Every few seconds,

I place a green marker in the flow to record the sphere's passing. If I touch a marker, it blossoms like a time lapse film of a desert cactus when the rains come-and on every petal there are pictures, numbers, symbols. The sphere can also be made to blossom, and when it does, those pictures, numbers, and symbols change as it moves.

Dissatisfied with the march of its symbols, I nudge the sphere onto a different streamline, fine-tuning its position until I see the unmistakable traces of the singularity I am seeking. I snap my fingers, and the sphere extrapolates itself into its own future and reports back what it finds. Promising ... Suddenly there is a whole cloud of red spheres, all being carried along by the fluid, like a shoal of fish that quickly spreads, swirling, putting out tendrils, flattening into sheets. Then more shoals of spheres join the game-gold, purple, brown, silver, pink .... I am in danger of running out of colors. Multicolored sheets intersect in a complex geometric form. I freeze it, smooth it, paint it in stripes. I banish the spheres with a gesture. I call up markers, inspect their unfolded petals, pull some off and attach them to a translucent grid that has materialized like a landscape from thinning mist.

Yes!
I issue a new command. "Save. Title: A new chaotic phenomenon in the three-body problem. 
Date: today."

Space collapses back to nonexistent void. Then, the morning's research completed, I disengage from my Virtual Unreality Machine and head off in search of lunch.

This particular dream is very nearly fact. We already have Virtual Reality systems that simulate events in "normal" space. I call my dream Virtual Unreality because it simulates anything that can be created by the mathematician's fertile imagination. Most of the bits and pieces of the Virtual Unreality Machine exist already. There is computer-graphics software that can "fly" you through any chosen geometrical object, dynamical-systems software that can track the evolving state of any chosen equation, symbolic-algebra software that can take the pain out of the most horrendous calculations- and get them right. It is only a matter of time before mathematicians will be able to get inside their own creations.

But, wonderful though such technology may be, we do not need it to bring my dream to life. The dream is a reality now, present inside every mathematician's head. This is what mathematical creation feels like when you're doing it. I've resorted to a little poetic license: the objects that are found in the mathematician's world are generally distinguished by symbolic labels or names rather than colors. But those labels are as vivid as colors to those who inhabit that world. In fact, despite its colorful images, my dream is a pale shadow of the world of imagination that every mathematican inhabits-a world in which curved space, or space with more than three dimensions, is not only commonplace but inevitable. You probably find the images alien and strange, far removed from the algebraic symbolism that the word "mathematics" conjures up. Mathematicians are forced to resort to written symbols and pictures to describe their world-even to each other. But the symbols are no more that world than musical notation is music.

Over the centuries, the collective minds of mathematicians have created their own universe. I don't know where it is situated- I don't think that there is a "where" in any normal sense of the word-but I assure you that this mathematical universe seems real enough when you're in it. And, not despite its peculiarities but because of them, the mental universe of mathematics has provided human beings with many of their deepest

Chapter 1 : The Natural Order
Chapter 2 : What Mathematics is For
Chapter 3 : What Mathematics is About
Chapter 4 : The Constants of Change
Chapter 5 : From Violins to Videos

Chapter 6 : Broken Symmetry
Chapter 7 : The Rhythm of Life
Chapter 8 : Do Dice Play God
Chapter 9 : Drops Dynamics and Daisies

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The Natural Order : Nature's Numbers Chapter 1

We live in a universe of patterns. Every night the stars move in circles across the sky. The seasons cycle at yearly intervals. No two snowflakes are ever exactly the same, but they all have sixfold symmetry. Tigers and zebras are covered in patterns of stripes, leopards and hyenas are covered in patterns of spots. Intricate trains of waves march across the oceans; very similar trains of sand dunes march across the desert. Colored arcs of light adorn the sky in the form of rainbows, and a bright circular halo sometimes surrounds the moon on winter nights. Spherical drops of water fall from clouds.
Human mind and culture have developed a formal system of thought for recognizing, classifying, and exploiting patterns. We call it mathematics. By using mathematics to organize and systematize our ideas about patterns, we have discovered a great secret: nature's patterns are not just there to be admired, they are vital clues to the rules that govern natural processes. Four hundred years ago, the German astronomer Johannes Kepler wrote a small book, The Six-Cornered Snowflake, as a New Year's gift to his sponsor. In it he argued that snowflakes must be made by packing tiny identical units together. This was long before the theory that matter is made of atoms had become generally accepted. Kepler performed no experiments; he just thought very hard about various bits and pieces of common knowledge. His main evidence was the sixfold symmetry of snowflakes, which is a natural consequence of regular packing. If you place a large number of identical coins on a table and try to pack them as closely as possible, then you get a honeycomb arrangement, in which every coin-except those at the edges-is surrounded by six others, arranged in a perfect hexagon.



The regular nightly motion of the stars is also a clue, this time to the fact that the Earth rotates. Waves and dunes are clues to the rules that govern the flow of water, sand, and air. The tiger's stripes and the hyena's spots attest to mathematical regularities in biological growth and form. Rainbows tell us about the scattering of light, and indirectly confirm that raindrops are spheres. Lunar haloes are clues to the shape of ice crystals.

There is much beauty in nature's clues, and we can all recognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things. Mathematics is to nature as Sherlock Holmes is to evidence. When presented with a cigar butt, the great fictional detective could deduce the age, profession, and financial state of its owner. His partner, Dr. Watson, who was not as sensitive to su~h matters, could only look on in baffled admiration, until the master revealed his chain of impeccable logic. When presented with the evidence of hexagonal snowflakes, mathethematicians can deduce the atomic geometry of ice crystals. If you are a Watson, it is just as baffling a trick, but I want to show you what it is like if you are a Sherlock Holmes. Patterns possess utility as well as beauty. Once we have learned to recognize a background pattern, exceptions suddenly stand out. The desert stands still, but the lion moves. Against the circling background of stars, a small number of stars that move quite differently beg to be singled out for special attention. The Greeks called them planetes, meaning "wanderer," a term retained in our word "planet." It took a lot longer to understand the patterns of planetary motion than it did to work out why stars seem to move in nightly circles. One difficulty is that we are inside the Solar System, moving along with it, and things that look simple from outside often look much more complicated from inside. The planets were clues to the rules behind gravity and motion.

We are still learning to recognize new kinds of pattern. Only within the last thirty years has humanity become explicitly aware of the two types of pattern now known as fractals and chaos. Fractals are geometric shapes that repeat their structure on ever-finer scales, and I will say a little about them toward the end of this chapter; chaos is a kind of apparent randomness whose origins are entirely deterministic, and I will say a lot about that in chapter 8. Nature "knew about" these patterns billions of years ago, for clouds are fractal and weather is chaotic. It took humanity a while to catch up.

The simplest mathematical objects are numbers, and the simplest of nature's patterns are numerical. The phases of the moon make a complete cycle from new moon to full moon and back again every twenty-eight days. The year is three hundred and sixty-five days long-roughly. People have two legs, cats have four, insects have six, and spiders have eight. Starfish have five arms (or ten, eleven, even seventeen, depending on the species). Clover normally has three leaves: the superstition that a four-leaf clover is lucky reflects a deepseated belief that exceptions to patterns are special. A very curious pattern indeed occurs in the petals of flowers. In nearly all flowers, the number of petals is one of the numbers that occur in the strange sequence 3, 5, 8, 13, 21, 34, 55, 89. For instance, lilies have three petals, buttercups have five, many delphiniums have eight, marigolds have thirteen, asters have twenty-one, and most daisies have thirty-four, fifty-five, or eighty-nine. You don't find any other numbers anything like as often. There is a definite pattern to those numbers, but one that takes a little digging out: each number is obtained by adding the previous two numbers together. For example, 3 + 5 = 8, 5 + 8 = 13, and so on. The same numbers can be found in the spiral patterns of seeds in the head of a sunflower. This particular pattern was noticed many centuries ago and has been widely studied ever since, but a really satisfactory explanation was not given until 1993. It is to be found in chapter 9.

Numerology is the easiest-and consequently the most dangerous-method for finding patterns. It is easy because anybody can do it, and dangerous for the same reason. The difficulty lies in distinguishing significant numerical patterns from accidental ones. Here's a case in point. Kepler was fascinated with mathematical patterns in nature, and he devoted much of his life to looking for them in the behavior of the planets. He devised a simple and tidy theory for the existence of precisely six planets (in his time only Mercury, Venus, Earth, Mars, Jupiter, and Saturn were known). He also discovered a very strange pattern relating the orbital period of a planet-the time it takes to go once around the Sun-to its distance from the Sun. Recall that the square of a number is what you get when you multiply it by itself: for example, the square of 4 is 4 x 4 = 16. Similarly, the cube is what you get when you multiply it by itself twice: for example, the cube of 4 is 4 x 4 x 4 = 64. Kepler found that if you take the cube of the distance of any planet from the Sun and divide it by the square of its orbital period, you always get the same number. It was not an especially elegant number, but it was the same for all six planets.

Which of these numerological observations is the more significant? The verdict of posterity is that it is the second one, the complicated and rather arbitrary calculation with squares and cubes. This numerical pattern was one of the key steps toward Isaac Newton's theory of gravity, which has explained all sorts of puzzles about the motion of stars and planets. In contrast, Kepler's neat, tidy theory for the number of planets has been buried without trace. For a start, it must be wrong, because we now know of nine planets, not six. There could be even more, farther out from the Sun, and small enough and faint enough to be undetectable. But more important, we no longer expect to find a neat, tidy theory for the number of planets. We think that the Solar System condensed from a cloud of gas surrounding the Sun, and the number of planets presumably depended on the amount of matter in the gas cloud, how it was distributed, and how fast and in what directions it was moving. An equally plausible gas cloud could have given us eight planets, or eleven; the number is accidental, depending on the initial conditions of the gas cloud, rather than universal, reflecting a general law of nature.

The big problem with numerological pattern-seeking is that it generates millions of accidentals for each universal. Nor is it always obvious which is which. For example, there are three stars, roughly equally spaced and in a straight line, in the belt of the constellation Orion. Is that a clue to a significant law of nature? Here's a similar question. 10, Europa, and Ganymede are three of Jupiter's larger satellites. They orbit the planet in, respectively, 1.77, 3.55, and 7.16 days. Each of these numbers is almost exactly twice the previous one. Is that a significant pattern? Three stars in a row, in terms of position; three satellites "in a row" in terms of orbital period. Which pattern, if either, is an important clue? I'll leave you to think about that for the moment and return to it in the next chapter.

In addition to numerical patterns, there are geometric ones. In fact this book really ought to have been called Nature's Numbers and Shapes. I have two excuses. First, the title sounds better without the "and shapes." Second, mathematical shapes can always be reduced to numbers-which is how computers handle graphics. Each tiny dot in the picture is stored and manipulated as a pair of numbers: how far the dot is along the screen from right to left, and how far up it is from the bottom. These two numbers are called the coordinates of the dot. A general shape is a collection of dots, and can be represented as a list of pairs of numbers. However, it is often better to think of shapes as shapes, because that makes use of our powerful and intuitive visual capabilities, whereas complicated lists of numbers are best reserved for our weaker and more laborious symbolic abilities.

Until recently, the main shapes that appealed to mathematicians were very simple ones: triangles, squares, pentagons, hexagons, circles, ellipses, spirals, cubes, spheres, cones, and so on. All of these shapes can be found in nature, although some are far more common, or more evident, than others. The rainbow, for example, is a collection of circles, one for each color. We don't normally see the entire circle, just an arc; but rainbows seen from the air can be complete circles. You also see circles in the ripples on a pond, in the the human eye, and on butterflies' wings.

Talking of ripples, the flow of fluids provides an inexhaustible supply of nature's patterns. There are waves of many different kinds-surging toward a beach in parallel ranks, spreading in a V-shape behind a moving boat, radiating outward from an underwater earthquake. Most waves are gregarious creatures, but some-such as the tidal bore that sweeps up a river as the energy of the incoming tide becomes confined to a tight channel-are solitary. There are swirling spiral whirlpools and tiny vortices. And there is the apparently structureless, random frothing of turbulent flow, one of the great enigmas of mathematics and physics. There are similar patterns in the atmosphere, too, the most dramatic being the vast spiral of a hurricane as seen by an orbiting astronaut.

There are also wave patterns on land. The most strikingly mathematical landscapes on Earth are to be found in the great ergs, or sand oceans, of the Arabian and Sahara deserts. Even when the wind blows steadily in a fixed direction, sand dunes form. The simplest pattern is that of transverse dunes, which-just like ocean waves-line up in parallel straight rows at right angles to the prevailing wind direction. Sometimes the rows themselves become wavy, in which case they are called barchanoid ridges; sometimes they break up into innumerable shield-shaped barchan dunes. If the sand is slightly moist, and there is a little vegetation to bind it together, then you may find parabolic dunes-shaped like a U, with the rounded end pointing in the direction of the wind. These sometimes occur in clusters, and they resemble the teeth of a rake. If the wind direction is variable, other forms become possible. For example, clusters of star-shaped dunes can form, each having several irregular arms radiating from a central peak. They arrange themselves in a random pattern of spots.

Nature's love of stripes and spots extends into the animal kingdom, with tigers and leopards, zebras and giraffes. The shapes and patterns of animals and plants are a happy hunting ground for the mathematically minded. Why, for example, do so many shells form spirals? Why are starfish equipped with a symmetric set of arms? Why do many viruses assume regular geometric shapes, the most striking being that of an icosahedron-a regular solid formed from twenty equilateral triangles? Why are so many animals bilaterally symmetric? Why is that symmetry so often imperfect, disappearing when you look at the detail, such as the position of the human heart or the differences between the two hemispheres of the human brain? Why are most of us right-handed, but not all of us?

In addition to patterns of form, there are patterns of movement. In the human walk, the feet strike the ground in a regular rhythm: left-right-Ieft-right-Ieft-right. When a four-legged creature-a horse, say-walks, there is a more complex but equally rhythmic pattern. This prevalence of pattern in locomotion extends to the scuttling of insects, the flight of birds, the pulsations of jellyfish, and the wavelike movements of fish, worms, and snakes. The sidewinder, a desert snake, moves rather like a single coil of a helical spring, thrusting its body forward in a series of S-shaped curves, in an attempt to minimize its contact with the hot sand. And tiny bacteria propel themselves along using microscopic helical tails, which rotate rigidly, like a ship's screw.

Finally, there is another category of natural pattern-one that has captured human imagination only very recently, but dramatically. This comprises patterns that we have only just learned to recognize-patterns that exist where we thought everything was random and formless. For instance, think about the shape of a cloud. It is true that meteorologists classify clouds into several different morphological groups-cirrus, stratus, cumulus, and so on-but these are very general types of form, not recognizable geometric shapes of a conventional mathematical kind. You do not see spherical clouds, or cubical clouds, or icosahedral clouds. Clouds are wispy, formless, fuzzy clumps. Yet there is a very distinctive pattern to clouds, a kind of symmetry, which is closely related to the physics of cloud formation. Basically, it is this: you can't tell what size a cloud is by looking at it. If you look at an elephant, you can tell roughly how big it is: an elephant the size of a house would collapse under its own weight, and one the size of a mouse would have legs that are uselessly thick. Clouds are not like this at all. A large cloud seen from far away and a small cloud seen close up could equally plausibly have been the other way around. They will be different in shape, of course, but not in any manner that systematically depends on size.

This "scale independence" of the shapes of clouds has been verified experimentally for cloud patches whose sizes vary by a factor of a thousand. Cloud patches a kilometer across look just like cloud patches a thousand kilometers across. Again, this pattern is a clue. Clouds form when water undergoes a "phase transition" from vapor to liquid, and physicists have discovered that the same kind of scale invariance is associated with all phase transitions. Indeed, this statistical self-similarity, as it is called, extends to many other natural forms. A Swedish colleague who works on oil-field geology likes to show a slide of one of his friends standing up in a boat and leaning nonchalantly against a shelf of rock that comes up to about his armpit. The photo is entirely convincing, and it is clear that the boat must have been moored at the edge of a rocky gully about two meters deep. In fact, the rocky shelf is the side of a distant fjord, some thousand meters high. The main problem for the photographer was to get both the foreground figure and the distant landscape in convincing focus.

Nobody would try to play that kind of trick with an elephant. However, you can play it with many of nature's shapes, including mountains, river networks, trees, and very possibly the way that matter is distributed throughout the entire universe. In the term made famous by the mathematician Benoit Mandelbrot, they are all fractals. A new science of irregularity- fractal geometry-has sprung up within the last fifteen years. I'm not going to say much about fractals, but the dynamic process that causes them, known as chaos, will be prominently featured.

Thanks to the development of new mathematical theories, these more elusive of nature's patterns are beginning to reveal their secrets. Already we are seeing a practical impact as well as an intellectual one. Our newfound understanding of nature's secret regularities is being used to steer artificial satellites to new destinations with far less fuel than anybody had thought possible, to help avoid wear on the wheels of locomotives and other rolling stock, to improve the effectiveness of heart pacemakers, to manage forests and fisheries, even to make more efficient dishwashers. But most important of all, it is giving us a deeper vision of the universe in which we live, and of our own place in it

Chapter 1 : The Natural Order
Chapter 2 : What Mathematics is For
Chapter 3 : What Mathematics is About
Chapter 4 : The Constants of Change
Chapter 5 : From Violins to Videos

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What Mathematics is For : Nature's Numbers Chapter 2

We've now established the uncontroversial idea that nature is full of patterns. But what do we want to do with them? One thing we can do is sit back and admire them. Communing with nature does all of us good: it reminds us of what we are. Painting pictures, sculpting sculptures, and writing poems are valid and important ways to express our feelings about the world and about ourselves. The entrepreneur's instinct is to exploit the natural world. The engineer's instinct is to change it. The scientist's instinct is to try to understand it-to work out what's really going on. The mathematician's instinct is to structure that process of understanding by seeking generalities that cut across the obvious subdivisions. There is a little of all these instincts in all of us, and there is both good and bad in each instinct.

I want to show you what the mathematical instinct has done for human understanding, but first I want to touch upon the role of mathematics in human culture. Before you buy something, you usually have a fairly clear idea of what you want to do with it. If it is a freezer, then of course you want it to preserve food, but your thoughts go well beyond that. How much food will you need to store? Where will the freezer have to fit? It is not always a matter of utility; you may be thinking of buying a painting, You still ask yourself where you are going to put it, and whether the aesthetic appeal is worth the asking price. It is the same with mathematics-and any other intellectual worldview, be it scientific, political, or religious. Before you buy something, it is wise to decide what you want it for.

So what do we want to get out of mathematics?

Each of nature's patterns is a puzzle, nearly always a deep one. Mathematics is brilliant at helping us to solve puzzles. It is a more or less systematic way of digging out the rules and structures that lie behind some observed pattern or regularity, and then using those rules and structures to explain what's going on. Indeed, mathematics has developed alongside our understanding of nature, each reinforcing the other. I've mentioned Kepler's analysis of snowflakes, but his most famous discovery is the shape of planetary orbits. By performing a mathematical analysis of astronomical observations made by the contemporary Danish astronomer Tycho Brahe, Kepler was eventually driven to the conclusion that planets move in ellipses. The ellipse is an oval curve that was much studied by the ancient Greek geometers, but the ancient astronomers had preferred to use circles, or systems of circles, to describe orbits, so Kepler's scheme was a radical one at that time. People interpret new discoveries in terms of what is important to them. The message astronomers received when they heard about Kepler's new idea was that neglected ideas from Greek geometry could help them solve the puzzle of predicting planetary motion. It took very little imagination for them to see that Kepler had made a huge step forward. All sorts of astronomical phenomena, such as eclipses, meteor showers, and comets, might yield to the same kind of mathematics. The message to mathematicians was quite different. It was that ellipses are really interesting curves. It took very little imagination for them to see that a general theory of curves would be even more interesting. Mathematicians could take the geometric rules that lead to ellipses and modify them to see what other kinds of curve resulted.

Similarly, when Isaac Newton made the epic discovery that the motion of an object is described by a mathematical relation between the forces that act on the body and the acceleration it experiences, mathematicians and physicists learned quite different lessons. However, before I can tell you what these lessons were I need to explain about acceleration. Acceleration is a subtle concept: it is not a fundamental quantity, such as length or mass; it is a rate of change. In fact, it is a "second order" rate of change-that is, a rate of change of a rate of change. The velocity of a body-the speed with which it moves in a given direction-is just a rate of change: it is the rate at which the body's distance from some chosen point changes. If a car moves at a steady speed of sixty miles per hour, its distance from its starting point changes by sixty miles every hour. Acceleration is the rate of change of velocity. If the car's velocity increases from sixty miles per hour to sixty-five miles per hour, it has accelerated by a definite amount. That amount depends not only on the initial and final speeds, but on how quickly the change takes place. If it takes an hour for the car to increase its speed by five miles per hour, the acceleration is very small; if it takes only ten seconds, the acceleration is much greater.

I don't want to go into the measurement of accelerations. My point here is more general: that acceleration is a rate of change of a rate of change. You can work out distances with a tape measure, but it is far harder to work out a rate of change of a rate of change of distance. This is why it took humanity a long time, and the genius of a Newton, to discover the law of motion. If the pattern had been an obvious feature of distances, we would have pinned motion down a lot earlier in our history.

In order to handle questions about rates of change, Newton- and independently the German mathematician Gottfried Leibniz-invented a new branch of mathematics, the calculus. It changed the face of the Earth-literally and metaphorically. But, again, the ideas sparked by this discovery were different for different people. The physicists went off looking for other laws of nature that could explain natural phenomena in terms of rates of change. They found them by the bucketfulheat, sound, light, fluid dynamics, elasticity, electricity, magnetism. The most esoteric modern theories of fundamental particles still use the same general kind of mathematics, though the interpretation-and to some extent the implicit worldview-is different. Be that as it may, the mathematicians found a totally different set of questions to ask. First of all, they spent a long time grappling with what "rate of change" really means. In order to work out the velocity of a moving object, you must measure where it is, find out where it moves to a very short interval of time later, and divide the distance moved by the time elapsed. However, if the body is accelerating, the result depends on the interval of time you use. Both the mathematicians and the physicists had the same intuition about how to deal with this problem: the interval of time you use should be as small as possible. Everything would be wonderful if you could just use an interval of zero, but unfortunately that won't work, because both the distance traveled and the time elapsed will be zero, and a rate of change of DID is meaningless. The main problem with nonzero intervals is that whichever one you choose, there is always a smaller one that you could use instead to get a more accurate answer. What you would really like is to use the smallest possible nonzero interval of time-but there is no such thing, because given any nonzero number, the number half that size is also nonzero. Everything would work out fine if the interval could be made infinitely small-"infinitesimal." Unfortunately, there are difficult logical paradoxes associated with the idea of an infinitesimal; in particular, if we restrict ourselves to numbers in the usual sense of the word, there is no such thing. So for about two hundred years, humanity was in a very curious position as regards the calculus. The physicists were using it, with great success, to understand nature and to predict the way nature behaves; the mathematicians were worrying about what it really meant and how best to set it up so that it worked as a sound mathematical theory; and the philosophers were arguing that it was all nonsense. Everything got resolved eventually, but you can still find strong differences in attitude.

The story of calculus brings out two of the main things that mathematics is for: providing tools that let scientists calculate what nature is doing, and providing new questions for mathematicians to sort out to their own satisfaction. These are the external and internal aspects of mathematics, often referred to as applied and pure mathematics (I dislike both adjectives, and I dislike the implied separation even more). It might appear in this case that the physicists set the agenda: if the methods of calculus seem to be working, what does it matter why they work? You will hear the same sentiments expressed today by people who pride themselves on being pragmatists. I have no difficulty with the proposition that in many respects they are right. Engineers designing a bridge are entitled to use standard mathematical methods even if they don't know the detailed and often esoteric reasoning that justifies these methods.

But I, for one, would feel uncomfortable driving across that bridge if I was aware that nobody knew what justified those methods. So, on a cultural level, it pays to have some people who worry about pragmatic methods and try to find out what really makes them tick. And that's one of the jobs that mathematicians do. They enjoy it, and the rest of humanity benefits from various kinds of spin-off, as we'll see. In the short term, it made very little difference whether mathematicians were satisfied about the logical soundness of the calculus. But in the long run the new ideas that mathematicians got by worrying about these internal difficulties turned out to be very useful indeed to the outside world. In Newton's time, it was impossible to predict just what those uses would be, but I think you could have predicted, even then, that uses would arise. One of the strangest features of the relationship between mathematics and the "real world," but also one of the strongest, is that good mathematics, whatever its source, eventually turns out to be useful. There are all sorts of theories why this should be so, ranging from the structure of the human mind to the idea that the universe is somehow built from little bits of mathematics. My feeling is that the answer is probably quite simple: mathematics is the science of patterns, and nature exploits just about every pattern that there is. I admit that I find it much harder to offer a convincing reason for nature to behave in this manner. Maybe the question is back to front: maybe the point is that creatures able to ask that kind of question can evolve only in a universe with that kind of structure .•

Whatever the reasons, mathematics definitely is a useful way to think about nature. What do we want it to tell us about the patterns we observe? There are many answers. We want to understand how they happen; to understand why they happen, which is different; to organize the underlying patterns and regularities in the most satisfying way; to predict how nature will behave; to control nature for our own ends; and to make practical use of what we have learned about our world. Mathematics helps us to do all these things, and often it is indispensable.

For example, consider the spiral form of a snail shell. How the snail makes its shell is largely a matter of chemistry and genetics. Without going into fine points, the snail's genes include recipes for making particular chemicals and instructions for where they should go. Here mathematics lets us do the molecular bookkeeping that makes sense of the different chemical reactions that go on; it describes the atomic structure of the molecules used in shells, it describes the strength and rigidity of shell material as compared to the weakness and pliability of the snail's body, and so on. Indeed, without mathematics we would never have convinced ourselves that matter really is made from atoms, or have worked out how the atoms are arranged. The discovery of genes-and later of the molecular structure of DNA, the genetic material-relied heavily on the existence of mathematical clues. The monk Gregor Mendel noticed tidy numerical relationships in how 'This explanation. and others. are discussed in The Collapse of Chaos. by Jack Cohen and Ian Stewart (New York: Viking. 1994).

the proportions of plants with different characters, such as seed color, changed when the plants were crossbred. This led to the basic idea of genetics-that within every organism is some cryptic combination of factors that determines many features of its body plan, and that these factors are somehow shuffled and recombined when passing from parents to offspring. Many different pieces of mathematics were involved in the discovery that DNA has the celebrated double-helical structure. They were as simple as Chargaff's rules: the observation by the Austrian-born biochemist Erwin Chargaff that the four bases of the DNA molecule occur in related proportions; and they are as subtle as the laws of diffraction, which were used to deduce molecular structure from X-ray pictures of DNA crystals.

The question of why snails have spiral shells has a very different character. It can be asked in several contexts-in the short-term context of biological development, say, or the longterm context of evolution. The main mathematical feature of the developmental story is the general shape of the spiral. Basically, the developmental story is about the geometry of a creature that behaves in much the same way all the time, but keeps getting bigger. Imagine a tiny animal, with a tiny protoshell attached to it. Then the animal starts to grow. It can grow most easily in the direction along which the open rim of the shell points, because the shell gets in its way if it tries to grow in any other direction. But, having grown a bit, it needs to extend its shell as well, for self-protection. So, of course, the shell grows an extra ring of material around its rim. As this process continues, the animal is getting bigger, so the size of the rim grows. The simplest result is a conical shell, such as you find on a limpet. But if the whole system starts with a bit of a twist, as is quite likely, then the growing edge of the shell rotates slowly as well as expanding, and it rotates in an off-centered manner. The result is a cone that twists in an ever-expanding spiral. We can use mathematics to relate the resulting geometry to all the different variables-such as growth rate and eccentricity of growth-that are involved.

If, instead, we seek an evolutionary explanation, then we might focus more on the strength of the shell, which conveys an evolutionary advantage, and try to calculate whether a long thin cone is stronger or weaker than a tightly coiled spiral. Or we might be more ambitious and develop mathematical models of the evolutionary process itself, with its combination of random genetic change-that is, mutations-and natural selection. A remarkable example of this kind of thinking is a computer simulation of the evolution of the eye by Daniel Nilsson and Susanne Pelger, published in 1994. Recall that conventional evolutionary theory sees changes in animal form as being the result of random mutations followed by subsequent selection of those individuals most able to survive and reproduce their kind. When Charles Darwin announced this theory, one of the first objections raised was that complex structures (like an eye) have to evolve fully formed or else they won't work properly (half an eye is no use at all), but the chance that random mutation will produce a coherent set of complex changes is negligible. Evolutionary theorists quickly responded that while half an eye may not be much use, a halfdeveloped eye might well be. One with a retina but no lens, say, will still collect light and thereby detect movement; and any way to improve the detection of predators offers an evolutionary advantage to any creature that possesses it. What we have here is a verbal objection to the theory countered by a verbal argument. But the recent computer analysis goes much further.

It starts with a mathematical model of a flat region of cells, and permits various types of "mutation." Some cells may become more sensitive to light, for example, and the shape of the region of cells may bend. The mathematical model is set up as a computer program that makes tiny random changes of this kind, calculates how good the resulting structure is at detecting light and resolving the patterns that it "sees," and selects any changes that improve these abilities. During a simulation that corresponds to a period of about four hundred thousand years-the blink of an eye, in evolutionary termsthe region of cells folds itself up into a deep, spherical cavity with a tiny iris like opening and, most dramatically, a lens. 

Moreover, like the lenses in our own eyes, it is a lens whose refractive index-the amount by which it bends light-varies from place to place. In fact, the pattern of variation of refractive index that is produced in the computer simulation is very like our own. So here mathematics shows that eyes definitely can evolve gradually and naturally, offering increased survival value at every stage. 

More than that: Nilsson and Pelger's work demonstrates that given certain key biological faculties (such as cellular receptivity to light, and cellular mobility), structures remarkably similar to eyes will form-all in line with Darwin's principle of natural selection. The mathematical model provides a lot of extra detail that the verbal Darwinian argument can only guess at, and gives us far greater confidence that the line of argument is correct. I said that another function of mathematics is to organize the underlying patterns and regularities in the most satisfying way. To illustrate this aspect, let me return to the question raised in the first chapter. Which-if either-is significant:

FIGURE 1.
Computer model of the evolution of an eye. Each step in the computation corresponds to about two hundred years of biological evolution.

The three-in-a-row pattern of stars in Orion's belt, or the threein- a-row pattern to the periods of revolution of Jupiter's satellites? Orion first. Ancient human civilizations organized the stars in the sky in terms of pictures of animals and mythic heroes. In these terms, the alignment of the three stars in Orion appears significant, for otherwise the hero would have no belt from which to hang his sword. However, if we use three-dimensional geometry as an organizing principle and place the three stars in their correct positions in the heavens, then we find that they are at very different distances from the Earth. Their equispaced alignment is an accident, depending on the position from which they are being viewed. Indeed, the very word "constellation" is a misnomer for an arbitrary accident of viewpoint.

The numerical relation between the periods of revolution of 10, Europa, and Ganymede could also be an accident of viewpoint. How can we be sure that "period of revolution" has any significant meaning for nature? However, that numerical relation fits into a dynamical framework in a very significant manner indeed. It is an example of a resonance, which is a relationship between periodically moving bodies in which their cycles are locked together, so that they take up the same relative positions at regular intervals. This common cycle time is called the period of the system. The individual bodies may have different-but related-periods. We can work out what this relationship is. When a resonance occurs, all of the participating bodies must return to a standard reference position after a whole number of cycles-but that number can be different for each. So there is some common period for the system, and therefore each individual body has a period that is some whole-number divisor of the common period. In this case, the common period is that of Ganymede, 7.16 days. The period of Europa is very close to half that of Ganymede, and that of 10 is close to one-quarter. 10 revolves four times around Jupiter while Europa revolves twice and Ganymede once, after which they are all back in exactly the same relative positions as before. This is called a 4:2:1 resonance.

The dynamics of the Solar System is full of resonances. The Moon's rotational period is (subject to small wobbles caused by perturbations from other bodies) the same as its period of revolution around the Earth-a 1:1 resonance of its orbital and its rotational period. Therefore, we always see the same face of the Moon from the Earth, never its "far side." Mercury rotates once every 58.65 days and revolves around the Sun every 87.97 days. Now, 2 x 87.97 = 175.94, and 3 x 58.65 = 175.95, so Mercury's rotational and orbital periods are in a 2:3 resonance. (In fact, for a long time they were thought to be in 1:1 resonance, both being roughly 88 days, because of the difficulty of observing a planet as close to the Sun as Mercury is. This gave rise to the belief that one side of Mercury is incredibly hot and the other incredibly cold, which turns out not to be true. A resonance, however, there is-and a more interesting one than mere equality.)

In between Mars and Jupiter is the asteroid belt, a broad zone containing thousands of tiny bodies. They are not uniformly distributed. At certain distances from the Sun we find asteroid "beltlets"; at other distances we find hardly any. The explanation-in both cases-is resonance with Jupiter. The Hilda group of asteroids, one of the beltlets, is in 2:3 resonance with Jupiter. That is, it is at just the right distance so that all of the Hilda asteroids circle the Sun three times for every two revolutions of Jupiter. The most noticeable gaps are at 2:1, 3:1, 4:1, 5:2, and 7:2 resonances. You may be worried that resonances are being used to explain both clumps and gaps, The reason is that each resonance has its own idiosyncratic dynamics; some cause clustering, others do the opposite. It all depends on the precise numbers.

Another function of mathematics is prediction. By understanding the motion of heavenly bodies, astronomers could predict lunar and solar eclipses and the return of comets. They knew where to point their telescopes to find asteroids that had passed behind the Sun, out of observation?-l contact. Because the tides are controlled mainly by the position of the Sun and Moon relative to the Earth, they could predict tides many years ahead. (The chief complicating factor in making such predictions is not astronomy: it is the shape of the continents and the profile of the ocean depths, which can delay or advance a high tide. 

However, these stay pretty much the same from one century to the next, so that once their effects have been understood it is a routine task to compensate for them.) In contrast, it is much harder to predict the weather. We know just as much about the mathematics of weather as we do about the mathematics of tides, but weather has an inherent unpredictability. Despite this, meteorologists can make effective short-term predictions of weather patternssay, three or four days in advance. The unpredictability of the weather, however, has nothing at all to do with randomnessa topic we will take up in chapter 8, when we discuss the concept of chaos.

The role of mathematics goes beyond mere prediction. Once you understand how a system works, you don't have to remain a passive observer. You can attempt to control the system, to make it do what you want. It pays not to be too ambitious: weather control, for example, is in its infancy-we can't make rain with any great success, even when there are rainclouds about. Examples of control systems range from the WHAT MATHEMATiCS is FOR 27 thermostat on a boiler, which keeps it at a fixed temperature, to the medieval practice of coppicing woodland. Without a sophisticated mathematical control system, the space shuttle would fly like the brick it is, for no human pilot can respond quickly enough to correct its inherent instabilities. The use of electronic pacemakers to help people with heart disease is another example of control.

These examples bring us to the most down-to-earth aspect of mathematics: its practical applications-how mathematics earns its keep. Our world rests on mathematical foundations, and mathematics is unavoidably embedded in our global culture. The only reason we don't always realize just how strongly our lives are affected by mathematics is that, for sensible reasons, it is kept as far as possible behind the scenes. When you go to the travel agent and book a vacation, you don't need to understand the intricate mathematical and physical theories that make it possible to design computers and telephone lines, the optimization routines that schedule as many flights as possible around any particular airport, or the signal-processing methods used to provide accurate radar images for the pilots. When you watch a television program, you don't need to understand the three-dimensional geometry used to produce special effects on the screen, the coding methods used to transmit TV signals by satellite, the mathematical methods used to solve the equations for the orbital motion of the satellite, the thousands of different applications of mathematics during every step of the manufacture of every component of the spacecraft that launched the satellite into position. When a farmer plants a new strain of potatoes, he does not need to know the statistical theories of genetics that identified which genes made that particular type of plant resistant to disease.

But somebody had to understand all these things in the past, otherwise airliners, television, spacecraft, and diseaseresistant potatoes wouldn't have been invented. And somebody has to understand all these things now, too, otherwise they won't continue to function. And somebody has to be inventing new mathematics in the future, able to solve problems that either have not arisen before or have hitherto proved intractable, otherwise our society will fall apart when change requires solutions to new problems or new solutions to old problems. If mathematics, including everything that rests on it, were somehow suddenly to be withdrawn from our world, human society would collapse in an instant. And if mathematics were to be frozen, so that it never went a single step farther, our civilization would start to go backward. We should not expect new mathematics to give an immediate dollars-and-cents payoff. The transfer of a mathematical idea into something that can be made in a factory or used in a home generally takes time. Lots of time: a century is not unusual. In chapter 5, we will see how seventeenth-century interest in the vibrations of a violin string led, three hundred years later, to the discovery of radio waves and the invention of radio, radar, and television. It might have been done quicker, but not that much quicker. If you think-as many people in our increasingly managerial culture do-that the process of scientific discovery can be speeded up by focusing on the application as a goal and ignoring "curiosity-driven" research, then you are wrong. In fact that very phrase, "curiosity-driven research," was introduced fairly recently by unimaginative bureaucrats as a deliberate put-down. Their desire for tidy projects offering guaranteed short-term profit is much too simpleminded, because goal-oriented research can deliver only predictable results. You have to be able to see the goal in order to aim at it. But anything you can see, your competitors can see, too. The pursuance of safe research will impoverish us all. The really important breakthroughs are always unpredictable. It is their very unpredictability that makes them important: they change our world in ways we didn't see coming.

Moreover, goal-oriented research often runs up against a brick wall, and not only in mathematics. For example, it took approximately eighty years of intense engineering effort to develop the photocopying machine after the basic principle of xerography had been discovered by scientists. The first fax machine was invented over a century ago, but it didn't work fast enough or reliably enough. The principle of holography (three-dimensional pictures, see your credit card) was discovered over a century ago, but nobody then knew how to produce the necessary beam of coherent light-light with all its waves in step. This kind of delay is not at all unusual in industry, let alone in more intellectual areas of research, and the impasse is usually broken only when an unexpected new idea arrives on the scene.

There is nothing wrong with goal-oriented research as a way of achieving specific feasible goals. But the dreamers and the mavericks must be allowed some free rein, too. Our world is not static: new problems constantly arise, and old answers often stop working. Like Lewis Carroll's Red Queen, we must run very fast in order to stand still.

Chapter 1 : The Natural Order
Chapter 2 : What Mathematics is For
Chapter 3 : What Mathematics is About
Chapter 4 : The Constants of Change
Chapter 5 : From Violins to Videos

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What Mathematics Is About : Nature's Numbers Chapter 3

When we hear the word "mathematics," the first thing that springs to mind is numbers. Numbers are the heart of mathematics, an all-pervading influence, the raw materials out of which a great deal of mathematics is forged. But numbers on their own form only a tiny part of mathematics. I said earlier that we live in an intensely mathematical world, but that whenever possible the mathematics is sensibly tucked under the rug to make our world "user-friendly." However, some mathematical ideas are so basic to our world that they cannot stay hidden, and numbers are an especially prominent example. Without the ability to count eggs and subtract change, for instance, we could not even buy food. And so we teach arithmetic. To everybody. Like reading and writing, its absence is a major handicap. And that creates the overwhelming impression that mathematics is mostly a matter of numbers-which isn't really true. The numerical tricks we learn in arithmetic are only the tip of an iceberg. We can run our everyday lives without much more, but our culture cannot run our society by using such limited ingredients. Numbers are just one type of object that mathematicians think about. In this chapter, I will try to show you some of the others and explain why they, too, are important.

Inevitably my starting point has to be numbers. A large part of the early prehistory of mathematics can be summed up as the discovery, by various civilizations, of a wider and wider range of things that deserved to be called numbers. The simplest are the numbers we use for counting. In fact, counting began long before there were symbols like 1, 2, 3, because it is possible to count without using numbers at all-say, by counting on your fingers. You can work out that "I have two hands and a thumb of camels" by folding down fingers as your eye glances over the camels. You don't actually have to have the concept of the number "eleven" to keep track of whether anybody is stealing your camels. You just have to notice that next time you seem to have only two hands of camels-so a thumb of camels is missing.

You can also record the count as scratches on pieces of wood or bone. Or you can make tokens to use as countersclay disks with pictures of sheep on them for counting sheep, or disks with pictures of camels on them for counting camels. As the animals parade past you, you drop tokens into a bagone token for each animal. The use of symbols for numbers probably developed about five thousand years ago, when such counters were wrapped in a clay envelope. It was a nuisance to break open the clay covering every time the accountants wanted to check the contents, and to make another one when they had finished. So people put special marks on the outside of the envelope summarizing what was inside. Then they realized that they didn't actually need any counters inside at all: they could just make the same marks on clay tablets. It's amazing how long it can take to see the obvious. But of course it's only obvious now.

The next invention beyond counting numbers was fractions- the kind of number we now symbolize as 2/3 (two thirds) or 22/7 (twenty-two sevenths-or, equivalently, three and one-seventh). You can't count with fractions-although two-thirds of a camel might be edible, it's not countable-but you can do much more interesting things instead. In particular, if three brothers inherit two camels between them, you can think of each as owning two-thirds of a camel-a convenient legal fiction, one with which we are so comfortable that we forget how curious it is if taken literally.

Much later, between 400 and 1200 AD, the concept of zero was invented and accepted as denoting a number. If you think that the late acceptance of zero as a number is strange, bear in mind that for a long time "one" was not considered a number because it was thought that a number of things ought to be several of them. Many history books say that the key idea here was the invention of a symbol for "nothing." That may have been the key to making arithmetic practical; but for mathematics the important idea was the concept of a new kind of number, one that represented the concrete idea "nothing." Mathematics uses symbols, but it no more is those symbols than music is musical notation or language is strings of letters from an alphabet. Carl Friedrich Gauss, thought by many to be the greatest mathematician ever to have lived, once said (in Latin) that what matters in mathematics is "not notations, but notions." The pun "non notationes, sed notiones" worked in Latin, too.

The next extension of the number concept was the invention of negative numbers. Again, it makes little sense to think of minus two camels in a literal sense; but if you owe somebody two camels, the number you own is effectively diminished by two. So a negative number can be thought of as representing a debt. There are many different ways to interpret these more esoteric kinds of number; for instance, a negative temperature (in degrees Celsius) is one that is colder than freezing, and an object with negative velocity is one that is moving backward, So the same abstract mathematical object may represent more than one aspect of nature.

Fractions are all you need for most commercial transactions, but they're not enough for mathematics. For example, as the ancient Greeks discovered to their chagrin, the square root of two is not exactly representable as a fraction. That is, if you multiply any fraction by itself, you won't get two exactly. You can get very close-for example, the square of 17/12 is 289/144, and if only it were 288/144 you would get two. But it isn't, and you don't-and whatever fraction you try, you never will. The square root of two, usually denoted .,,)2, is therefore said to be "irrational." The simplest way to enlarge the number system to include the irrationals is to use the socalled real numbers-a breathtakingly inappropriate name, inasmuch as they are represented by decimals that go on forever, like 3.14159 ... , where the dots indicate an infinite number of digits. How can things be real if you can't even write them down fully? But the name stuck, probably because real numbers formalize many of our natural visual intuitions about lengths and distances.

The real numbers are one of the most audacious idealizations made by the human mind, but they were used happily for centuries before anybody worried about the logic behind them. Paradoxically, people worried a great deal about the next enlargement of the number system, even though it was entirely harmless. That was the introduction of square roots for negative numbers, and it led to the "imaginary" and "complex" numbers. A professional mathematican should never leave home without them, but fortunately nothing in this book will require a knowledge of complex numbers, so I'm going to tuck them under the mathematical carpet and hope you don't notice. However, I should point out that it is easy to interpret an infinite decimal as a sequence of ever-finer approximations to some measurement-say, of a length or a weight-whereas a comfortable interpretation of the square root of minus one is more elusive.

In current terminology, the whole numbers 0, 1, 2, 3, ... are known as the natural numbers. If negative whole numbers are included, we have the integers. Positive and negative fractions are called rational numbers. Real numbers are more general; complex numbers more general still. So here we have five number systems, each more inclusive than the previous: natural numbers, integers, rationals, real numbers, and complex numbers. In this book, the important number systems will be the integers and the reals. We'll need to talk about rational numbers every so often; and as I've just said, we can ignore the complex numbers altogether. But I hope you understand by now that the word "number" does not have any immutable god-given meaning. More than once the scope of that word was extended, a process that in principle might occur again at any time.

However, mathematics is not just about numbers. We've already had a passing encounter with a different kind of object of mathematical thought, an operation; examples are addition, subtraction, multiplication, and division. In general, an operation is something you apply to two (sometimes more) mathematical objects to get a third object. I also alluded to a third type of mathematical object when I mentioned square roots. If you start with a number and form its square root, you get another number. The term for such an "object" is function. You can think of a function as a mathematical rule that starts with a mathematical object-usually a number-and associates to it another object in a specific manner. Functions are often defined using algebraic formulas, which are just shorthand ways to explain what the rule is, but they can be defined by any convenient method. Another term with the same meaning as "function" is transformation: the rule transforms the first object into the second. This term tends to be used when the rules are geometric, and in chapter 6 we will use transformations to capture the mathematical essence of symmetry.

Operations and functions are very similar concepts. Indeed, on a suitable level of generality there is not much to distinguish them. Both of them are processes rather than things. And now is a good moment to open up Pandora's box and explain one of the most powerful general weapons in the mathematician's armory, which we might call the "thingification of processes." (There is a dictionary term, reification, but it sounds pretentious.) Mathematical "things" have no existence in the real world: they are abstractions. But mathematical processes are also abstractions, so processes are no less "things" than the "things" to which they are applied. The thingification of processes is commonplace. In fact, I can make out a very good case that the number "two" is not actually a thing but a process-the process you carry out when you associate two camels or two sheep with the symbols "1, 2" chanted in turn. A number is a process that has long ago been thingified so thoroughly that everybody thinks of it as a thing. It is just as feasible-though less familiar to most of us-to think of an operation or a function as a thing. For example, we might talk of "square root" as if it were a thingand I mean here not the square root of any particular number, but the function itself. In this image, the square-root function is a kind of sausage machine: you stuff a number in at one end and its square root pops out at the other.

In chapter 6, we will treat motions of the plane or space as if they are things. I'm warning you now because you may find it disturbing when it happens. However, mathematicians aren't the only people who play the thingification game. The legal profession talks of "theft" as if it were a thing; it even knows what kind of thing it is-a crime. In phrases such as "two major evils in Western society are drugs and theft" we find one genuine thing and one thingified thing, both treated as if they were on exactly the same level. For theft is a process, one whereby my property is transferred without my agreement to somebody else, but drugs have a real physical existence.

Computer scientists have a useful term for things that can be built up from numbers by thingifying processes: they call them data structures. Common examples in computer science are lists (sets of numbers written in sequence) and arrays (tables of numbers with several rows and columns). I've already said that a picture on a computer screen can be represented as a list of pairs of numbers; that's a more complicated but entirely sensible data structure. You can imagine much more complicated possibilities-arrays that are tables of lists, not tables of numbers; lists of arrays; arrays of arrays; lists of lists of arrays of lists .... Mathematics builds its basic objects of thought in a similar manner. Back in the days when the logical foundations of mathematics were still being sorted out, Bertrand Russell and Alfred North Whitehead wrote an enormous three-volume work, Principia Mathematica, which began with the simplest possible logical ingredient-the idea of a set, a collection of things. They then showed how to build up the rest of mathematics. Their main objective was to analyze the logical structure of mathematics, but a major part of their effort went into devising appropriate data structures for the important objects of mathematical thought.

The image of mathematics raised by this description of its basic objects is something like a tree, rooted in numbers and branching into ever more esoteric data structures as you proceed from trunk to bough, bough to limb, limb to twig .... But this image lacks an essential ingredient. It fails to describe how mathematical concepts interact. Mathematics is not just a collection of isolated facts: it is more like a landscape; it has an inherent geography that its users and creators employ to navigate through what would otherwise be an impenetrable jungle. For instance, there is a metaphorical feeling of distance. Near any particular mathematical fact we find other, related facts. For example, the fact that the circumference of a circle is 1t (pi) times its diameter is very close to the fact that the circumference of a circle is 21t times its radius. The connection between these two facts is immediate: the diameter is twice the radius. In contrast, unrelated ideas are more distant from each other; for example, the fact that there are exactly six different ways to arrange three objects in order is a long way away from facts about circles. There is also a metaphorical feeling of prominence. Soaring peaks pierce the skyimportant ideas that can be used widely and seen from far away, such as Pythagoras's theorem about right triangles, or the basic techniques of calculus. At every turn, new vistas arise-an unexpected river that must be crossed using stepping stones, a vast, tranquil lake, an impassable crevasse. The user of mathematics walks only the well-trod parts of this mathematical territory. The creator of mathematics explores its unknown mysteries, maps them, and builds roads through them to make them more easily accessible to everybody else. The ingredient that knits this landscape together is proof Proof determines the route from one fact to another. To professional mathematicians, no statement is considered valid unless it is proved beyond any possibility of logical error. But there are limits to what can be proved, and how it can be proved. A great deal of work in philosophy and the foundations of mathematics has established that you can't prove everything, because you have to start somewhere; and even when you've decided where to start, some statements may be neither provable nor disprovable. I don't want to explore those issues here; instead, I want to take a pragmatic look at what proofs are and why they are needed.

Textbooks of mathematical logic say that a proof is a sequence of statements, each of which either follows from previous statements in the sequence or from agreed axiomsunproved but explicitly stated assumptions that in effect define the area of mathematics being studied. This is about as informative as describing a novel as a sequence of sentences, each of which either sets up an agreed context or follows credibly from previous sentences. Both definitions miss the essential point: that both a proof and a novel must tell an interesting story. They do capture a secondary point, that the story must be convincing, and they also describe the overall format to be used, but a good story line is the most important feature of all.

Very few textbooks say that. Most of us are irritated by a movie riddled with holes, however polished its technical production may be. I saw one recently in which an airport is taken over by guerrillas who shut down the electronic equipment used by the control tower and substitute their own. The airport authorities and the hero then spend half an hour or more of movie time-several hours of story time-agonizing about their inability to communicate with approaching aircraft, which are stacking up in the sky overhead and running out of fuel. It occurs to no one that there is a second, fully functioning airport no more than thirty miles away, nor do they think to telephone the nearest Air Force base. The story was brilliantly and expensively filmed-and silly.

That didn't stop a lot of people from enjoying it: their critical standards must have been lower than mine. But we all have limits to what we are prepared to accept as credible. If in an otherwise realistic film a child saved the day by picking up a house and carrying it away, most of us would lose interest. Similarly, a mathematical proof is a story about mathematics that works. It does not have to dot every j and cross every t; readers are expected to fill in routine steps for themselvesjust as movie characters may suddenly appear in new surroundings without it being necessary to show how they got there. But the story must not have gaps, and it certainly must not have an unbelievable plot line. The rules are stringent: in mathematics, a single flaw is fatal. Moreover, a subtle flaw can be just as fatal as an obvious one. Let's take a look at an example. I have chosen a simple one, to avoid technical background; in consequence, the proof tells a simple and not very significant story. I stole it from a colleague, who calls it the SHIP/DOCK Theorem. You probably know the type of puzzle in which you are given one word (SHIP) and asked to turn it into another word (DOCK) by changing one letter at a time and getting a valid word at every stage. You might like to try to solve this one before reading on: if you do, you will probably understand the theorem, and its proof, more easily. Here's one solution: 

There are plenty of alternatives, and some involve fewer words. But if you play around with this problem, you will eventually notice that all solutions have one thing in common: at least one of the intermediate words must contain two vowels.

O.K., so prove it.

I'm not willing to accept experimental evidence. I don't care if you have a hundred solutions and every single one of them includes a word with two vowels. You won't be happy with such evidence, either, because you will have a sneaky feeling that you may just have missed some really clever sequence that doesn't include such a word. On the other hand, you will probably also have a distinct feeling that somehow "it's obvious." I agree; but why is it obvious?

You have now entered a phase of existence in which most mathematicians spend most of their time: frustration. You know what you want to prove, you believe it, but you don't see a convincing story line for a proof. What this means is that you are lacking some key idea that will blow the whole problem wide open. In a moment I'll give you a hint. Think about it for a few minutes, and you will probably experience a much more satisfying phase of the mathematician's existence: illumination.

Here's the hint. Every valid word in English must contain a vowel. It's a very simple hint. First, convince yourself that it's true. (A dictionary search is acceptable, provided it's a big dictionary.) Then consider its implications .... O.K., either you got it or you've given up. Whichever of these you did, all professional mathematicians have done the same on a lot of their problems. Here's the trick. You have to concentrate on what happens to the vowels. Vowels are the peaks in the SHIP/DOCK landscape, the landmarks between which the paths of proof wind.

In the initial word SHIP there is only one vowel, in the third position. In the final word DOCK there is also only one vowel, but in the second position. How does the vowel change position? There are three possibilities. It may hop from one location to the other; it may disappear altogether and reappear later on; or an extra vowel or vowels may be created and subsequently eliminated.

The third possibility leads pretty directly to the theorem. Since only one letter at a time changes, at some stage the word must change from having one vowel to having two. It can't leap from having one vowel to having three, for exampIe. But what about the other possibilities? The hint that I mentioned earlier tells us that the single vowel in SHIP cannot disappear altogether. That leaves only the first possibility: that there is always one vowel, but it hops from position 3 to position 2. However, that can't be done by changing only one letter! You have to move, in one step, from a vowel at position 3 and a consonant at position 2 to a consonant at position 3 and a vowel at position 2. That implies that two letters must change, which is illegal. Q.E.D., as Euclid used to say. A mathematician would write the proof out in a much more formal style, something like the textbook model, but the important thing is to tell a convincing story. Like any good story, it has a beginning and an end, and a story line that gets you from one to the other without any logical holes appearing. Even though this is a very simple example, and it isn't standard mathematics at all, it illustrates the essentials: in particular, the dramatic difference between an argument that is genuinely convincing and a hand-waving argument that sounds plausible but doesn't really gel. I hope it also put you through some of the emotional experiences of the creative mathematician: frustration at the intractability of what ought to be an easy question, elation when light dawned, suspicion as you checked whether there were any holes in the argument, aesthetic satisfaction when you decided the idea really was O.K. and realized how neatly it cut through all the apparent complications. Creative mathematics is just like this-but with more serious subject matter.

Proofs must be convincing to be accepted by mathematicians. There have been many cases where extensive numerical evidence suggested a completely wrong answer. One notorious example concerns prime numbers-numbers that have no divisors except themselves and 1. The sequence of primes begins 2, 3, 5, 7, 11, 13, 17, 19 and goes on forever. Apart from 2, all primes are odd; and the odd primes fall into two classes: those that are one less than a multiple of four (such as 3, 7, 11, 19) and those that are one more than a multiple of four (such as 5, 13, 17). If you run along the sequence of primes and count how many of them fall into each class, you will observe that there always seem to be more primes in the "one less" class than in the "one more" class. For example, in the list of the seven pertinent primes above, there are four primes in the first class but only three in the second. This pattern persists for numbers up to at least a trillion, and it seems entirely reasonable to conjecture that it is always true. However, it isn't.

By indirect methods, number theorists have shown that when the primes get sufficiently big, the pattern changes and the "one more than a multiple of four" class goes into the lead. The first proof of this fact worked only when the numbers got bigger than 10'10'10'10'46, where to avoid giving the printer kittens I've used the ' sign to indicate forming a power. This number is utterly gigantic. Written out in full, it would go 10000 ... 000, with a very large number of Os. If all the matter in the universe were turned into paper, and a zero could be inscribed on every electron, there wouldn't be enough of them to hold even a tiny fraction of the necessary zeros.

No amount of experimental evidence can account for the possibility of exceptions so rare that you need numbers that big to locate them. Unfortunately, even rare exceptions matter in mathematics. In ordinary life, we seldom worry about things that might occur on one occasion out of a trillion. Do you worry about being hit by a meteorite? The odds are about one in a trillion. But mathematics piles logical deductions on top of each other, and if any step is wrong the whole edifice may tumble. If you have stated as a fact that all numbers behave in some manner, and there is just one that does not, then you are wrong, and everything you have built on the basis of that incorrect fact is thrown into doubt. Even the very best mathematicians have on occasion claimed to have proved something that later turned out not to be so-their proof had a subtle gap, or there was a simple error in a calculation, or they inadvertently assumed something that was not as rock-solid as they had imagined. So, over the centuries, mathematicians have learned to be extremely critical of proofs. Proofs knit the fabric of mathematics together, and if a single thread is weak, the entire fabric may unravel.

Chapter 1 : The Natural Order
Chapter 2 : What Mathematics is For
Chapter 3 : What Mathematics is About
Chapter 4 : The Constants of Change
Chapter 5 : From Violins to Videos

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The Constants of Change : Nature's Numbers Chapter 4

For a good many centuries, human thought about nature has swung between two opposing points of view. According to one view, the universe obeys fixed, immutable laws, and everything exists in a well-defined objective reality. The opposing view is that there is no such thing as objective reality; that all is flux, all is change. As the Greek philosopher Heraclitus put it, "You can't step into the same river twice." The rise of science has largely been governed by the first viewpoint. But there are increasing signs that the prevailing cultural background is starting to switch to the second-ways of thinking as diverse as postmodernism, cyberpunk, and chaos theory all blur the alleged objectiveness of reality and reopen the ageless debate about rigid laws and flexible change.

What we really need to do is get out of this futile game altogether. We need to find a way to step back from these opposing worldviews-not so much to seek a synthesis as to see them both as two shadows of some higher order of reality- shadows that are different only because the higher order is being seen from two different directions. But does such a higher order exist, and if so, is it accessible? To many-especially scientists-Isaac Newton represents the triumph of rationality over mysticism. The famous economist John Maynard Keynes, in his essay Newton, the Man, saw things differently: In the eighteenth century and since, Newton came to be thought of as the first and greatest of the modern age of scientists, a rationalist, one who taught us to think on the lines of cold and untinctured reason. I do not see him in this light. I do not think that anyone who has pored over the contents of that box which he packed up when he finally left Cambridge in 1696 and which, though partly dispersed, have come down to us, can see him like that. Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago. Isaac Newton, a posthumous child born with no father on Christmas Day, 1642, was the last wonder-child to whom the Magi could do sincere and appropriate homage.

Keynes was thinking of Newton's personality, and of his interests in alchemy and religion as well as in mathematics and physics. But in Newton's mathematics we also find the first significant step toward a worldview that transcends and unites both rigid law and flexible flux. The universe may appear to be a storm-tossed ocean of change, but Newtonand before him Galileo and Kepler, the giants upon whose shoulders he stood-realized that change obeys rules. Not only can law and flux coexist, but law generates flux. Today's emerging sciences of chaos and complexity supply the missing converse: flux generates law. But that is another story, reserved for the final chapter.

Prior to Newton, mathematics had offered an essentially static model of nature. There are a few exceptions, the most obvious being Ptolemy'S theory of planetary motion, which reproduced the observed changes very accurately using a system of circles revolving about centers that themselves were attached to revolving circles-wheels within wheels within wheels. But at that time the perceived task of mathematics was to discover the catalogue of "ideal forms" employed by nature. The circle was held to be the most perfect shape possible, on the basis of the democratic observation that every point on the circumference of a circle lies at the same distance from its center. Nature, the creation of higher beings, is by definition perfect, and ideal forms are mathematical perfection, so of course the two go together. And perfection was thought to be unblemished by change.

Kepler challenged that view by finding ellipses in place of complex systems of circles. Newton threw it out altogether, replacing forms by the laws that produce them. Although its ramifications are immense, Newton's approach to motion is a simple one. It can be illustrated using the motion of a projectile, such as a cannonball fired from a gun at an angle. Galileo discovered experimentally that the path of such a projectile is a parabola, a curve known to the ancient Greeks and related to the ellipse. In this case, it forms an inverted V-shape. The parabolic path can be most easily understood by decomposing the projectile's motion into two independent components: motion in a horizontal direction and motion in a vertical direction. By thinking about these two types of motion separately, and putting them back together only when each has been understood in its own right, we can see why the path should be a parabola.

The cannonball's motion in the horizontal direction, parallel to the ground, is very simple: it takes place at a constant speed. Its motion in the vertical direction is more interesting. It starts moving upward quite rapidly, then it slows down, until for a split second it appears to hang stationary in the air; then it begins to drop, slowly at first but with rapidly increasing velocity. Newton's insight was that although the position of the cannonball changes in quite a complex way, its velocity changes in a much simpler way, and its acceleration varies in a very simple manner indeed. Figure 2 summarizes the relationship between these three functions, in the following example.

Suppose for the sake of illustration that the initial upward velocity is fifty meters per second (50 m/sec). Then the height of the cannonball above ground, at one-second intervals, is: 

        0, 45, 80, 105, 120, 125, 120, 105, 80, 45, 0. 

You can see from these numbers that the ball goes up, levels off near the top, and then goes down again. But the general pattern is not entirely obvious. The difficulty was compounded in Galileo's time-and, indeed, in Newton's because it was hard to measure these numbers directly. In actual fact, Galileo rolled a ball up a gentle slope to slow the whole process down. The biggest problem was to measure time accurately: the historian Stillman Drake has suggested that perhaps Galileo hummed tunes to himself and subdivided the basic beat in his head, as a musician does. The pattern of distances is a puzzle, but the pattern of velocities is much clearer. The ball starts with an upward velocity of 50 m/sec. One second later, the velocity has

Calculus in a nutshell. 

Three mathematical patterns determined by a cannonball: height, velocity, and acceleration. The pattern of heights, which is what we naturally observe, is complicated. Newton realized that the pattern of velocities is simpler, while the pattern of accelerations is simpler still. The two basic operations of calculus, differentiation and integration, let us pass from any of these patterns to any other. So we can work with the simplest, acceleration, and deduce the one we really want-height. decreased to (roughly) 40 m/sec; a second after that, it is 30 m/sec; then 20 m/sec, 10 m/sec, then a m/sec (stationary). A second after that, the velocity is 10 m/sec downward. Using negative numbers, we can think of this as an upward velocity of -10 m/sec. In successive seconds, the pattern continues: -20 m/sec, -30 m/sec, -40 m/sec, -50 m/sec. At this point, the cannonball hits the ground. So the sequence of velocities, measured at one-second intervals, is:

       50, 40, 30, 20, 10, 0, -10, -20, -30, -40, -50.

Now there is a pattern that can hardly be missed; but let's go one step further by looking at accelerations. The corresponding sequence for the acceleration of the cannonball, again using negative numbers to indicate downward motion, is

      -10, -10, -10, -10, -10, -10, -10, -10, -10, -10, -10.

I think you will agree that the pattern here is extremely simple. The ball undergoes a constant downward acceleration of 10 m/sec2 • (The true figure is about 9.81 m/sec2 , depending on whereabouts on the Earth you perform the experiment. But 10 is easier to think about.)

How can we explain this constant that is hiding among the dynamic variables? When all else is flux, why is the acceleration fixed? One attractive explanation has two elements. The first is that the Earth must be pulling the ball downward; that is, there is a gravitational force that acts on the ball. It is reasonable to expect this force to remain the same at different heights above the ground. Indeed, we feel weight because gravity pulls our bodies downward, and we still weigh the same if we stand at the top of a tall building. Of course, this appeal to everyday observation does not tell us what happens if the distance becomes sufficiently large-say the distance that separates the Moon from the Earth. That's a different story, to which we shall return shortly.

The second element of the explanation is the real breakthrough. We have a body moving under a constant downward force, and we observe that it undergoes a constant downward acceleration. Suppose, for the sake of argument, that the pull of gravity was a lot stronger: then we would expect the downward acceleration to be a lot stronger, too. Without going to a heavy planet, such as Jupiter, we can't test this idea, but it looks reasonable; and it's equally reasonable to suppose that on Jupiter the downward acceleration would again be constant- but a different constant from what it is here. The simplest theory consistent with this mixture of real experiments and thought experiments is that when a force acts on a body, the body experiences an acceleration that is proportional to that force. And this is the essence of Newton's law of motion. The only missing ingredients are the assumption that this is always true, for all bodies and for all forces, whether or not the forces remain constant; and the identification of the constant of proportionality as being related to the mass of the body. To be precise, Newton's law of motion states that mass x acceleration = force.

That's it. Its great virtue is that it is valid for any system of masses and forces, including masses and forces that change over time. We could not have anticipated this universal applicability from the argument that led us to the law; but it turns out to be so.

Newton stated three laws of motion, but the modern approach views them as three aspects of a single mathematical equation.
So I will use the phrase "Newton's law of motion" to refer to the whole package.

The mountaineer's natural urge when confronted with a mountain is to climb it; the mathematician's natural urge when confronted with an equation is to solve it. But how? Given a body's mass and the forces acting on it, we can easily solve this equation to get the acceleration. But this is the answer to the wrong question. Knowing that the acceleration of a cannonball is always -10 m/sec2 doesn't tell us anything obvious about the shape of its trajectory. This is where the branch of mathematics known as calculus comes in; indeed it is why Newton (and Leibniz) invented it. Calculus provides a technique, which nowadays is called integration, that allows us to move from knowledge of acceleration at any instant to knowledge of velocity at any instant. By repeating the same trick, we can then obtain knowledge of position at any instant. And that is the answer to the right question. As I said earlier, velocity is rate of change of position, and acceleration is rate of change of velocity. 

Calculus is a mathematical scheme invented to handle questions about rates of change. In particular, it provides a technique for finding rates of change-a technique known as differentiation. Integration "undoes" the effect of differentiation; and integrating twice undoes the effect of differentiating twice. Like the twin faces of the Roman god Janus, these twin techniques of calculus point in opposite directions. Between them, they tell you that if you know anyone of the functions-position, velocity, or acceleration-at every instant, then you can work out the other two.

Newton's law of motion teaches an important lesson: namely, that the route from nature's laws to nature's behavior need not be direct and obvious. Between the behavior we observe and the laws that produce it is a crevasse, which the human mind can bridge only by mathematical calculations. This is not to suggest that nature is mathematics-that (as the physicist Paul Dirac put it) "God is a mathematician." Maybe nature's patterns and regularities have other origins; but, at the very least, mathematics is an extremely effective way for human beings to come to grips with those patterns. All of the laws of physics that were discovered by pursuing Isaac Newton's basic insight-that change in nature can be described by mathematical processes, just as form in nature can be described by mathematical things-have a similar character. The laws are formulated as equations that relate not the physical quantities of primary interest but the rates at which those quantities change with time, or the rates at which those rates change with time. For example the "heat equation," which determines how heat flows through a conducting body, is all about the rate of change of the body's temperature; and the "wave equation," which governs the motion of waves in water, air, or other materials, is about the rate of change of the rate of change of the height of the wave. The physical laws for light, sound, electricity, magnetism, the elastic bending of materials, the flow of fluids, and the course of a chemical reaction, are all equations for various rates of change. Because a rate of change is about the difference between some quantity now and its value an instant into the future, equations of this kind are called differential equations. The term "differentiation" has the same origin. Ever since Newton, the strategy of mathematical physics has been to describe the universe in terms of differential equations, and then solve them.

However, as we have pursued this strategy into more sophisticated realms, the meaning of the word "solve" has undergone a series of major changes. Originally it implied finding a precise mathematical formula that would describe what a system does at any instant of time. Newton's discovery of another important natural pattern, the law of gravitation, rested upon a solution of this kind. He began with Kepler's discovery that planets move in ellipses, together with two other mathematical regularities that were also noted by Kepler. Newton asked what kind of force, acting on a planet, would be needed to produce the pattern that Kepler had found. In effect, Newton was trying to work backward from behavior to laws, using a process of induction rather than deduction. And he discovered a very beautiful result. The necessary force should always point in the direction of the Sun; and it should decrease with the distance from the planet to the Sun. Moreover, this decrease should obey a simple mathematical law, the inverse-square law. This means that the force acting on a planet at, say, twice the distance is reduced to one-quarter, the force acting on a planet at three times the distance is reduced to one-ninth, and so on. From this discovery-which was so beautiful that it surely concealed a deep truth about the world-it was a short step to the realization that it must be the Sun that causes the force in the first place. The Sun attracts the planet, but the attraction becomes weaker if the planet is farther away. It was a very appealing idea, and Newton took a giant intellectual leap: he assumed that the same kind of attractive force must exist between any two bodies whatsoever, anywhere in the universe. And now, having "induced" the law for the force, Newton could bring the argument full circle by deducing the geometry of planetary motion. He solved the equations given by his laws of motion and gravity for a system of two mutually attracting bodies that obeyed his inverse-square law; in those days, "solved" meant finding a mathematical formula for their motion. The formula implied that they must move in ellipses about their common center of mass. As Mars moves around the Sun in a giant ellipse, the Sun moves in an ellipse so tiny that its motion goes undetected. Indeed, the Sun is so massive compared to Mars that the mutual center of mass lies beneath the Sun's surface, which explains why Kepler thought that Mars moved in an ellipse around the stationary Sun. However, when Newton and his successors tried to build on this success by solving the equations for a system of three or more bodies-such as Moon/Earth/Sun, or the entire Solar System-they ran into technical trouble; and they could get out of trouble only by changing the meaning of the word "solve." They failed to find any formulas that would solve the equations exactly, so they gave up looking for them. Instead, they tried to find ways to calculate approximate numbers. For example, around 1860 the French astronomer Charles-Eugene Delaunay filled an entire book with a single approximation to the motion of the Moon, as influenced by the gravitational attractions of the Earth and the Sun. It was an extremely accurate approximation-which is why it filled a book-and it took him twenty years to work it out. When it was subsequently checked, in 1970, using a symbolic-algebra computer program, the calculation took a mere twenty hours: only three mistakes were found in Delaunay's work, none serious.

The motion of the Moon/Earth/Sun system is said to be a three-body problem-for evident reasons. It is so unlike the nice, tidy two-body problem Newton solved that it might as well have been invented on another planet in another galaxy, or in another universe. The three-body problem asks for a solution for the equations that describe the motion of three masses under inverse-square-Iaw gravity. Mathematicians tried to find such a solution for centuries but met with astonishingly little success beyond approximations, such as Delaunay's, which worked only for particular cases, like Moon/Earth/Sun. Even the so-called restricted three-body problem, in which one body has a mass so small that it can be considered to exert no force at all upon the other two, proved utterly intractable. It was the first serious hint that knowing the laws might not be enough to understand how a system behaves; that the crevasse between laws and behavior might not always be bridgeable.

Despite intensive effort, more than three centuries after Newton we still do not have a complete answer to the threebody problem. However, we finally know why the problem has been so hard to crack. The two-body problem is "integrable"- the laws of conservation of energy and momentum restrict solutions so much that they are forced to take a simple mathematical form. In 1994, Zhihong Xia, ofthe Georgia Institute of Technology, proved what mathematicians had long suspected: that a system of three bodies is not integrable. Indeed, he did far more, by showing that such a system can exhibit a strange phenomenon known as Arnold diffusion, first discovered by Vladimir Arnold, of Moscow State University, in 1964. Arnold diffusion produces an extremely slow, "random" drift in the relative orbital positions. This drift is not truly random: it is an example of the type of behavior now known as chaos-which can be described as apparently random behavior with purely deterministic causes. Notice that this approach again changes the meaning of "solve." First that word meant "find a formula." Then its meaning changed to "find approximate numbers." Finally, it has in effect become "tell me what the solutions look like." In place of quantitative answers, we seek qualitative ones. In a sense, what is happening looks like a retreat: if it is too hard to find a formula, then try an approximation; if approximations aren't available, try a qualitative description. But it is wrong to see this development as a retreat, for what this change of meaning has taught us is that for questions like the three-body problem, no formulas can exist. We can prove that there are qualitative aspects to the solution that a formula cannot capture. The search for a formula in such questions was a hunt for a mare's nest.

Why did people want a formula in the first place? Because in the early days of dynamics, that was the only way to work out what kind of motion would occur. Later, the same information could be deduced from approximations. Nowadays, it can be obtained from theories that deal directly and precisely with the main qualitative aspects of the motion. As we will see in the next few chapters, this move toward an explicitly qualitative theory is not a retreat but a major advance. For the first time, we are starting to understand nature's patterns in their own terms.

Chapter 1 : The Natural Order
Chapter 2 : What Mathematics is For
Chapter 3 : What Mathematics is About
Chapter 4 : The Constants of Change
Chapter 5 : From Violins to Videos

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