Quotes from James Gleick

Shallow ideas can be assimilated; ideas that require people to reorganize their picture of the world provoke hostility.

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.

Christopher Scholz, a Columbia University professor specializing in the form and structure of the solid earth, first started thinking about fractals.

Ruelle had heard talks by Steve Smale about the horseshoe map and the chaotic possibilities of dynamical systems. He had also thought about fluid turbulence and the classic Landau picture. He suspected that these ideas were related—and contradictory.

The equations of fluid flow are nonlinear partial differential equations, unsolvable except in special cases. Yet Ruelle worked out an abstract alternative to Landau's picture, couched in the language of Smale, with images of space as a pliable material to be squeezed, stretched, and folded into shapes like horseshoes.

The fractal dimension of a metal's surface, for example, often provides information that corresponds to the metal's strength.

Galileo saw the regularity because he already had a theory that predicted it. He understood what Aristotle could not: that a moving object tends to keep moving, that a change in speed or direction could only be explained by some external force, like friction.

As the physicist Murray Gell-Mann once remarked: "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.

designed an apparatus to measure how well carbon dioxide conducted heat around the critical point where it turned from vapor to liquid. Most people thought that the thermal conductivity would change slightly. Swinney found that it changed by a factor of 1,000.

Most seductive of all was an image that the authors called a strange attractor.

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. The history of the system time can be charted by the moving point, tracing its orbit through phase space with the passage of time.

The great quantum theorist Richard P. Feynman expressed this feeling. "It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do?

Physical laws provided a trivial explanation for their shrinking. On second thought the connection between shrinking and loss of meaning was not so obvious. Why should it be that as things become small they also become incomprehensible?

Mathematical Ideas in Biology

Clearly—or almost clearly—the brain does not own any direct copies of stuff in the world. There is no library of forms and ideas against which to compare the images of perception. Information is stored in a plastic way, allowing fantastic juxtapositions and leaps of imagination. Some chaos exists out there, and the brain seems to have more flexibility than classical physics in finding the order in it.

pattern, especially pattern that appeared on different scales at the same time. They had a taste for randomness and complexity, for jagged edges and sudden leaps. Believers in chaos—and they sometimes call themselves believers, or converts, or evangelists—speculate about determinism and free will, about evolution, about the nature of conscious intelligence. They feel that they are turning back a trend in science toward reductionism, the analysis of systems

In fact, he argued, any coastline is—in a sense—infinitely long. In another sense, the answer depends on the length of your ruler.

The simulated weather in Edward Lorenz's new electronic computer changed slowly but certainly

Fluid dynamicists just did not believe them. They were not accustomed to experiments in the precise style of phase-transition physics.

the laws of Newton, appropriate tools for a clockmaker deity who could create a world and set it running for eternity.

Those who made such models took for granted that, from present to future, the laws of motion provide a bridge of mathematical certainty. Understand the laws and you understand the universe. That was the philosophy behind modeling weather on a computer.

But the experiment had never stopped. "There was the transition, very well defined," Swinney said. "So that was great. Then we went on, to look for the next one." There the expected Landau sequence broke down. Experiment failed to confirm theory. At the next transition the flow jumped all the way to a confused state with no distinguishable cycles at all. No new frequencies, no gradual buildup of complexity. "What we found was, it became chaotic.

Where Newton was reductionist, Goethe was holistic. Newton broke light apart and found the most basic physical explanation for color. Goethe walked through flower gardens and studied paintings, looking for a grand, all-encompassing explanation. Newton made his theory of color fit a mathematical scheme for all of physics. Goethe, fortunately or unfortunately, abhorred mathematics.

Feigenbaum persuaded himself that Goethe had been right about color. Goethe's ideas resemble a facile notion, popular among psychologists, that makes a distinction between hard physical reality and the variable subjective perception of it. The colors we perceive vary from time to time and from person to person—that much is easy to say.