— Making a New Science
Select Quotes
Prologue
WHERE CHAOS BEGINS, classical science stops. – Page 1
To some physicists chaos is a science of process rather than state, of becoming rather than being. – Page 1
Chaos breaks across the lines that separate scientific disciplines. Because it is a science of the global nature of systems, it has brought together thinkers from fields that had been widely separated. – Page 1
“Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurement process; and chaos eliminates the Laplacian fantasy of deterministic predictability.” – Page 6
Tiny differences in input could quickly become overwhelming differences in output—a phenomenon given the name “sensitive dependence on initial conditions.” – Page 6
The Butterfly Effect
you can always try to solve a problem by proving that no solution exists. – Page 13
Scientists marching under Newton’s banner actually waved another flag that said something like this: Given an approximate knowledge of a system’s initial conditions and an understanding of natural law, one can calculate the approximate behavior of the system. This assumption lay at the philosophical heart of science. As one theoretician liked to tell his students: – Page 15
The Butterfly Effect acquired a technical name: sensitive dependence on initial conditions. – Page 23
Nonlinearity means that the act of playing the game has a way of changing the rules. – Page 24
The shape signaled pure disorder, since no point or pattern of points ever recurred. Yet it also signaled a new kind of order. – Page 29
Revolution
To some the difficulty of communicating the new ideas and the ferocious resistance from traditional quarters showed how revolutionary the new science was. Shallow ideas can be assimilated; ideas that require people to reorganize their picture of the world provoke hostility. – Page 38
Centers and institutes were founded to specialize in “nonlinear dynamics” and “complex systems.” Chaos has become not just theory but also method, not just a canon of beliefs but also a way of doing science. – Page 38
Differential equations describe the way systems change continuously over time. – Page 46
Chaos and instability, concepts only beginning to acquire formal definitions, were not the same at all. A chaotic system could be stable if its particular brand of irregularity persisted in the face of small disturbances. – Page 48
The chaos Lorenz discovered, with all its unpredictability, was as stable as a marble in a bowl. You could add noise to this system, jiggle it, stir it up, interfere with its motion, and then when everything settled down, the transients dying away like echoes in a canyon, the system would return to the same peculiar pattern of irregularity as before. It was locally unpredictable, globally stable. – Page 48
For three centuries it had been a case of the more you know, the less you know. – Page 53
knowledge bred ignorance. – Page 53
The spot is a self-organizing system, created and regulated by the same nonlinear twists that create the unpredictable turmoil around it. It is stable chaos. – Page 55
Life’S Ups And Downs
“Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties.” – Page 80
A Geometry Of Nature
Economists, like biologists, dealt with a world of willful living beings. Economists studied the most elusive creatures of all. – Page 85
“Science would be ruined if (like sports) it were to put competition above everything else, and if it were to clarify the rules of competition by withdrawing entirely into narrowly defined specialties. – Page 89
The rare scholars who are nomads-by–choice are essential to the intellectual welfare of the settled disciplines.” – Page 89
The Noah and Joseph Effects push in different directions, but they add up to this: trends in nature are real, but they can vanish as quickly as they come. – Page 94
But Mandelbrot found that as the scale of measurement becomes smaller, the measured length of a coastline rises without limit, bays and peninsulas revealing ever-smaller subbays and subpeninsulas—at least down to atomic scales, where the process does finally come to an end. Perhaps. – Page 96
Fractional dimension becomes a way of measuring qualities that otherwise have no clear definition: the degree of roughness or brokenness or irregularity in an object. – Page 98
The claim was that the degree of irregularity remains constant over different scales. Surprisingly often, the claim turns out to be true. Over and over again, the world displays a regular irregularity. – Page 98
IN THE MIND’S EYE, a fractal is a way of seeing infinity. – Page 98
Using techniques originated by mathematicians early in the century and then all but forgotten, Mandelbrot could characterize the fractional dimension precisely. For the Koch curve, the infinitely extended multiplication by four-thirds gives a dimension of 1.2618. – Page 98
“When I came in this game, there was a total absence of intuition. One had to create an intuition from scratch. Intuition as it was trained by the usual tools—the hand, the pencil, and the ruler—found these shapes quite monstrous and pathological. The old intuition was misleading. – Page 98
“Intuition is not something that is given. I’ve trained my intuition to accept as obvious shapes which were initially rejected as absurd, and I find everyone else can do the same.” – Page 98
Self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern. – Page 103
Contacts between surfaces have properties quite independent of the materials involved. They are properties that turn out to depend on the fractal quality of the bumps upon bumps upon bumps. – Page 105
It happens that the equations of fluid flow are in many contexts dimensionless, meaning that they apply without regard to scale. – Page 108
Sometimes the textbook approach seems to dance around the truth: “In the gradual transition from one type of artery to another it is sometimes difficult to classify the intermediate region. Some arteries of intermediate caliber have walls that suggest larger arteries, while some large arteries have walls like those of medium-sized arteries. The transitional regions… are often designated arteries of mixed type.” – Page 109
DNA surely cannot specify the vast number of bronchi, bronchioles, and alveoli or the particular spatial structure of the resulting tree, but it can specify a repeating process of bifurcation and development. – Page 110
His name appeared on a little list compiled by the Harvard historian of science I. Bernard Cohen. Cohen had scoured the annals of discovery for years, looking for scientists who had declared their own work to be “revolutions.” All told, he found just sixteen. Robert Symmer, a Scots contemporary of Benjamin Franklin whose ideas about electricity were indeed radical, but wrong. Jean-Paul Marat, known today only for his bloody contribution to the French Revolution. Von Liebig. Hamilton. Charles Darwin, of course. Virchow. Cantor. Einstein. Minkowski. Von Laue. Alfred Wegener—continental drift. Compton. Just. James Watson—the structure of DNA. And Benoit Mandelbrot. – Page 111
It was a legitimate question. If one scientist announces that a thing is probably true, and another demonstrates it with rigor, which one has done more to advance science? Is the making of a conjecture an act of discovery? Or is it just a cold-blooded staking of a claim? – Page 112
Strange Attractors
There was a story about the quantum theorist Werner Heisenberg, on his deathbed, declaring that he will have two questions for God: why relativity, and why turbulence. Heisenberg says, “I really think He may have an answer to the first question.” – Page 121
THEORISTS CONDUCT EXPERIMENTS with their brains. Experimenters have to use their hands, too. Theorists are thinkers, experimenters are craftsmen. The theorist needs no accomplice. The experimenter has to muster graduate students, cajole machinists, flatter lab assistants. The theorist operates in a pristine place free of noise, of vibration, of dirt. The experimenter develops an intimacy with matter as a sculptor does with clay, battling it, shaping it, and engaging it. The theorist invents his companions, as a naïve Romeo imagined his ideal Juliet. The experimenter’s lovers sweat, complain, and fart. – Page 125
“That’s true if you have an infinite amount of noise-free data.” And wheel dismissively back toward the blackboard, adding, “In reality, of course, you have a limited amount of noisy data.” – Page 126
“Always nonspecialists find the new things,” – Page 132
In phase space the complete state of knowledge about a dynamical system at a single instant in time collapses to a point. That point is the dynamical system—at that instant. At the next instant, though, the system will have changed, ever so slightly, and so the point moves. – Page 134
Universality
His friends speculated that he must be getting his vitamins from cigarettes.) – Page 179
UNIVERSALITY MADE THE DIFFERENCE between beautiful and useful. – Page 179
Knowledge was imperfect. Scientists were biased by the customs of their disciplines or by the accidental paths of their own educations. The scientific world can be surprisingly finite. – Page 182
The Experimenter
“The flecked river Which kept flowing and never the same way twice, flowing Through many places, as if it stood still in one.” – Page 196
Final cause is cause based on purpose or design: a wheel is round because that shape makes transportation possible. Physical cause is mechanical: the earth is round because gravity pulls a spinning fluid into a spheroid. The distinction is not always so obvious. – Page 200
In science, on the whole, physical cause dominates. Indeed, as astronomy and physics emerged from the shadow of religion, no small part of the pain came from discarding arguments by design, forward-looking teleology—the earth is what it is so that humanity can do what it does. – Page 200
In a computer experiment, when you generated your thousands or millions of data points, patterns made themselves more or less apparent. In a laboratory, as in the real world, useful information had to be distinguished from noise. In a computer experiment data flowed like wine from a magic chalice. In a laboratory experiment you had to fight for every drop. – Page 209
computer simulations help to build intuition or to refine calculations, but they do not give birth to genuine discovery. This, at any rate, is the experimenter’s creed. – Page 210
The Dynamical Systems Collective
a stream of data in ordinary language is less than random; each new bit is partly constrained by the bits before; thus each new bit carries somewhat less than a bit’s worth of real information. There was a hint of paradox floating in this formulation. The more random a data stream, the more information would be conveyed by each new bit. – Page 257
The mixing never reverses itself, even if you wait till the end of the universe, which is why the Second Law is so often said to be the part of physics that makes time a one-way street. – Page 257
Where is this information coming from? The heat bath of the microscales, billions of molecules in their random thermodynamic dance. Just as turbulence transmits energy from large scales downward through chains of vortices to the dissipating small scales of viscosity, so information is transmitted back from the small scales to the large—at any rate, this was how Shaw and his colleagues began describing it. – Page 260
“When you think about a variable, the evolution of it must be influenced by whatever other variables it’s interacting with. Their values must somehow be contained in the history of that thing. Somehow their mark must be there.” – Page 266
Inner Rhythms
Only the most naïve scientist believes that the perfect model is the one that perfectly represents reality. Such a model would have the same drawbacks as a map as large and detailed as the city it represents, – Page 278
a discipline that barely existed in the United States, theoretical biology. – Page 292
Simply put, a linear process, given a slight nudge, tends to remain slightly off track. A nonlinear process, given the same nudge, tends to return to its starting point. – Page 292
“When you reach an equilibrium in biology you’re dead,” he said. “If I ask you whether your brain is an equilibrium system, all I have to do is ask you not to think of elephants for a few minutes, and you know it isn’t an equilibrium system.” – Page 298
Chaos And Beyond
Simple systems give rise to complex behavior. Complex systems give rise to simple behavior. And most important, the laws of complexity hold universally, caring not at all for the details of a system’s constituent atoms. – Page 304
could not respond in the traditional way to the presence of unexpected fluctuations or oscillations—that is, by ignoring them. – Page 304
For them, chaos was the end of the reductionist program in science. Uncomprehension; resistance; anger; acceptance. – Page 304
considered chaos a poor name for his work, because it implied randomness. To him, the overriding message was that simple processes in nature could produce magnificent edifices of complexity without randomness. – Page 306
Afterword
It’s the best possible time to be alive, when almost everything you thought you knew was wrong. – Page 320