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