Chaos: Making a New Science

by James Gleick

The most passionate advocates of the new science go so far as to say that twentieth-century science will be remembered for just three things: relativity, quantum mechanics, and chaos.

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.”

Tiny differences in input could quickly become overwhelming differences in output—a phenomenon given the name “sensitive dependence on initial conditions.”

The Butterfly Effect acquired a technical name: sensitive dependence on initial conditions.

“I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives.”

THE LABORATORY MOUSE of the new science was the pendulum: emblem of classical mechanics, exemplar of constrained action, epitome of clockwork regularity.

Physicists assumed that any behavior they could actually observe regularly would have to be stable, since in real systems tiny disturbances and uncertainties are unavoidable. You never know the parameters exactly. If you want a model that will be both physically realistic and robust in the face of small perturbations, physicists reasoned that you must surely want a stable model.

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.

The result of a mathematical development should be continuously checked against one’s own intuition about what constitutes reasonable biological behavior. When such a check reveals disagreement, then the following possibilities must be considered: A mistake has been made in the formal mathematical development; The starting assumptions are incorrect and/or constitute a too drastic oversimplification; One’s own intuition about the biological field is inadequately developed; A penetrating new principle has been discovered.

Faculty members are familiar with a certain kind of person who looks to the mathematicians like a good physicist and looks to the physicists like a good mathematician. Very properly, they do not want that kind of person around.”

“The mathematical intuition so developed ill equips the student to confront the bizarre behaviour exhibited by the simplest of discrete nonlinear systems,”

The École Normale and École Polytechnique were elite schools with no parallel in American education. Together they prepared fewer than 300 students in each class for careers in the French universities and civil service.

the group began to write an enormous treatise, more and more fanatical in style, meant to set the discipline straight. Logical analysis was central. A mathematician had to begin with solid first principles and deduce all the rest from them. The group stressed the primacy of mathematics among sciences, and also insisted upon a detachment from other sciences. Mathematics was mathematics—it could not be valued in terms of its application to real physical phenomena. And above all, Bourbaki rejected the use of pictures. A mathematician could always be fooled by his visual apparatus. Geometry was untrustworthy. Mathematics should be pure, formal, and austere.

Clouds are not spheres, Mandelbrot is fond of saying. Mountains are not cones. Lightning does not travel in a straight line. The new geometry mirrors a universe that is rough, not rounded, scabrous, not smooth. It is a geometry of the pitted, pocked, and broken up, the twisted, tangled, and intertwined.

it becomes apparent that the Koch curve has some interesting features. For one thing, it is a continuous loop, never intersecting itself, because the new triangles on each side are always small enough to avoid bumping into each other. Each transformation adds a little area to the inside of the curve, but the total area remains finite, not much bigger than the original triangle, in fact. If you drew a circle around the original triangle, the Koch curve would never extend beyond it. Yet the curve itself is infinitely long, as long as a Euclidean straight line extending to the edges of an unbounded universe. Just as the first transformation replaces a one-foot segment with four four-inch segments, every transformation multiplies the total length by four-thirds. This paradoxical result, infinite length in a finite space, disturbed many of the turn-of–the-century mathematicians who thought about it. The Koch curve was monstrous, disrespectful to all reasonable intuition about shapes and—it almost went without saying—pathologically unlike anything to be found in nature.

Several chaos-minded cardiologists found that the frequency spectrum of heartbeat timing, like earthquakes and economic phenomena, followed fractal laws, and they argued that one key to understanding heartbeat timing was the fractal organization of the His-Purkinje network, a labyrinth of branching pathways organized to be self-similar on smaller and smaller scales.

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.

“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.”

It was not mathematics; he was not proving anything. He was studying numbers, yes, but numbers are to a mathematician what bags of coins are to an investment banker: nominally the stuff of his profession, but actually too gritty and particular to waste time on. Ideas are the real currency of mathematicians.

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.

The education of a physicist depends on a system of mentors and protégés. Established professors get research assistants to help with laboratory work or tedious calculations. In return the graduate students and postdoctoral fellows get shares of their professors’ grant money and bits of publication credit. A good mentor helps his student choose problems that will be both manageable and fruitful. If the relationship prospers, the professor’s influence helps his protégé find employment. Often their names will be forever linked.

The paragon of a complex dynamical system and to many scientists, therefore, the touchstone of any approach to complexity is the human body. No object of study available to physicists offers such a cacophony of counterrhythmic motion on scales from macroscopic to microscopic: motion of muscles, of fluids, of currents, of fibers, of cells. No physical system has lent itself to such an obsessive brand of reductionism: every organ has its own micro-structure and its own chemistry, and student physiologists spend years just on the naming of parts. Yet how ungraspable these parts can be! At its most tangible, a body part can be a seemingly well-defined organ like the liver. Or it can be a spatially challenging network of solid and liquid like the vascular system. Or it can be an invisible assembly, truly as abstract a thing as “traffic” or “democracy,” like the immune system, with its lymphocytes and T4 messengers, a miniaturized cryptography machine for encoding and decoding data about invading organisms. To study such systems without a detailed knowledge of their anatomy and chemistry would be futile, so heart specialists learn about ion transport through ventricular muscle tissue, brain specialists learn the electrical particulars of neuron firing, and eye specialists learn the name and place and purpose of each ocular muscle. In the 1980s chaos brought to life a new kind of physiology, built on the idea that mathematical tools could help scientists understand global complex...

A considerable obstacle, he felt, was the uncomfortable antipathy of many physiologists to mathematics. “In 1986 you won’t find the word fractals in a physiology book,” he said. “I think in 1996 you won’t be able to find a physiology book without it.”

Modeling any one piece of the heart’s behavior would strain a supercomputer; modeling the whole interwoven cycle would be impossible. Computer modeling of the kind that seems natural to a fluid dynamics expert designing airplane wings for Boeing or engine flows for the National Aeronautics and Space Administration is an alien practice to medical technologists.

By changing the patterns of fluid flow in the heart, artificial valves create areas of turbulence and areas of stagnation; when blood stagnates, it forms clots; when clots break off and travel to the brain, they cause strokes. Such clotting was the fatal barrier to making artificial hearts. Only in the mid–1980s, when mathematicians at the Courant Institute of New York University applied new computer modeling techniques to the problem, did the design of heart valves begin to take full advantage of available technology.