More (#3) from Chaos:

Hubbard began using a computer to do what the orthodox techniques had not done. The computer would prove nothing. But at least it might unveil the truth so that a mathematician could know what it was he should try to prove. So Hubbard began to experiment. He treated Newton’s method not as a way of solving problems but as a problem in itself. Hubbard considered the simplest example of a degree-three polynomial, the equation x3– 1 =0. That is, find the cube root of 1. In real numbers, of course, there is just the trivial solution: 1. But the polynomial also has two complex solutions: –½ + i√3/2, and –½ – i√3/2. Plotted in the complex plane, these three roots mark an equilateral triangle, with one point at three o’clock, one at seven o’clock, and one at eleven o’clock. Given any complex number as a starting point, the question was to see which of the three solutions Newton’s method would lead to. It was as if Newton’s method were a dynamical system and the three solutions were three attractors. Or it was as if the complex plane were a smooth surface sloping down toward three deep valleys. A marble starting from anywhere on the plane should roll into one of the valleys—but which?

Hubbard set about sampling the infinitude of points that make up the plane. He had his computer sweep from point to point, calculating the flow of Newton’s method for each one, and color-coding the results. Starting points that led to one solution were all colored blue. Points that led to the second solution were red, and points that led to the third were green. In the crudest approximation, he found, the dynamics of Newton’s method did indeed divide the plane into three pie wedges. Generally the points near a particular solution led quickly into that solution. But systematic computer exploration showed complicated underlying organization that could never have been seen by earlier mathematicians, able only to calculate a point here and a point there. While some starting guesses converged quickly to a root, others bounced around seemingly at random before finally converging to a solution. Sometimes it seemed that a point could fall into a cycle that would repeat itself forever—a periodic cycle—without ever reaching one of the three solutions.

As Hubbard pushed his computer to explore the space in finer and finer detail, he and his students were bewildered by the picture that began to emerge. Instead of a neat ridge between the blue and red valleys, for example, he saw blotches of green, strung together like jewels. It was as if a marble, caught between the conflicting tugs of two nearby valleys, would end up in the third and most distant valley instead. A boundary between two colors never quite forms. On even closer inspection, the line between a green blotch and the blue valley proved to have patches of red. And so on—the boundary finally revealed to Hubbard a peculiar property that would seem bewildering even to someone familiar with Mandelbrot’s monstrous fractals: no point serves as a boundary between just two colors. Wherever two colors try to come together, the third always inserts itself, with a series of new, self-similar intrusions. Impossibly, every boundary point borders a region of each of the three colors.

And:

For... Peitgen the study of complexity provided a chance to create new traditions in science instead of just solving problems. “In a brand new area like this one, you can start thinking today and if you are a good scientist you might be able to come up with interesting solutions in a few days or a week or a month,” Peitgen said. The subject is unstructured.

“In a structured subject, it is known what is known, what is unknown, what people have already tried and doesn’t lead anywhere. There you have to work on a problem which is known to be a problem, otherwise you get lost. But a problem which is known to be a problem must be hard, otherwise it would already have been solved.”

Peitgen shared little of the mathematicians’ unease with the use of computers to conduct experiments. Granted, every result must eventually be made rigorous by the standard methods of proof, or it would not be mathematics. To see an image on a graphics screen does not guarantee its existence in the language of theorem and proof. But the very availability of that image was enough to change the evolution of mathematics. Computer exploration was giving mathematicians the freedom to take a more natural path, Peitgen believed. Temporarily, for the moment, a mathematician could suspend the requirement of rigorous proof. He could go wherever experiments might lead him, just as a physicist could. The numerical power of computation and the visual cues to intuition would suggest promising avenues and spare the mathematician blind alleys. Then, new paths having been found and new objects isolated, a mathematician could return to standard proofs. “Rigor is the strength of mathematics,” Peitgen said. “That we can continue a line of thought which is absolutely guaranteed — mathematicians never want to give that up. But you can look at situations that can be understood partially now and with rigor perhaps in future generations. Rigor, yes, but not to the extent that I drop something just because I can’t do it now.”

More (#2) from Chaos:

As a mathematics paper, Lorenz’s climate work would have been a failure—he proved nothing in the axiomatic sense. As a physics paper, too, it was seriously flawed, because he could not justify using such a simple equation to draw conclusions about the earth’s climate. Lorenz knew what he was saying, though. “The writer feels that this resemblance is no mere accident, but that the difference equation captures much of the mathematics, even if not the physics, of the transitions from one regime of flow to another, and, indeed, of the whole phenomenon of instability.” Even twenty years later, no one could understand what intuition justified such a bold claim, published in Tellus, a Swedish meteorology journal. (“Tellus! Nobody reads Tellus,” a physicist exclaimed bitterly.) Lorenz was coming to understand ever more deeply the peculiar possibilities of chaotic systems—more deeply than he could express in the language of meteorology.

And:

Modern economics relies heavily on the efficient market theory. Knowledge is assumed to flow freely from place to place. The people making important decisions are supposed to have access to more or less the same body of information. Of course, pockets of ignorance or inside information remain here and there, but on the whole, once knowledge is public, economists assume that it is known everywhere. Historians of science often take for granted an efficient market theory of their own. When a discovery is made, when an idea is expressed, it is assumed to become the common property of the scientific world. Each discovery and each new insight builds on the last. Science rises like a building, brick by brick. Intellectual chronicles can be, for all practical purposes, linear.

That view of science works best when a well-defined discipline awaits the resolution of a well-defined problem. No one misunderstood the discovery of the molecular structure of DNA, for example. But the history of ideas is not always so neat. As nonlinear science arose in odd corners of different disciplines, the flow of ideas failed to follow the standard logic of historians. The emergence of chaos as an entity unto itself was a story not only of new theories and new discoveries, but also of the belated understanding of old ideas. Many pieces of the puzzle had been seen long before — by Poincaré, by Maxwell, even by Einstein — and then forgotten. Many new pieces were understood at first only by a few insiders. A mathematical discovery was understood by mathematicians, a physics discovery by physicists, a meteorological discovery by no one. The way ideas spread became as important as the way they originated.

Each scientist had a private constellation of intellectual parents. Each had his own picture of the landscape of ideas, and each picture was limited in its own way. 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. No committee of scientists pushed history into a new channel — a handful of individuals did it, with individual perceptions and individual goals.

And:

In the age of computer simulation, when flows in everything from jet turbines to heart valves are modeled on supercomputers, it is hard to remember how easily nature can confound an experimenter. In fact, no computer today can completely simulate even so simple a system as Libchaber’s liquid helium cell. Whenever a good physicist examines a simulation, he must wonder what bit of reality was left out, what potential surprise was sidestepped. Libchaber liked to say that he would not want to fly in a simulated airplane—he would wonder what had been missed. Furthermore, he would say that 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. His experiment was so immaculate, his scientific goals so abstract, that there were still physicists who considered Libchaber’s work more philosophy or mathematics than physics. He believed, in turn, that the ruling standards of his field were reductionist, giving primacy to the properties of atoms. “A physicist would ask me, How does this atom come here and stick there? And what is the sensitivity to the surface? And can you write the Hamiltonian of the system?

“And if I tell him, I don’t care, what interests me is this shape, the mathematics of the shape and the evolution, the bifurcation from this shape to that shape to this shape, he will tell me, that’s not physics, you are doing mathematics. Even today he will tell me that. Then what can I say? Yes, of course, I am doing mathematics. But it is relevant to what is around us. That is nature, too.”

More (#1) from Chaos:

...for decades, Mandelbrot believes, he had to play games with his work. He had to couch original ideas in terms that would not give offense. He had to delete his visionary-sounding prefaces to get his articles published. When he wrote the first version of his book, published in French in 1975, he felt he was forced to pretend it contained nothing too startling. That was why he wrote the latest version explicitly as “a manifesto and a casebook.” He was coping with the politics of science.

“The politics affected the style in a sense which I later came to regret. I was saying, ‘It’s natural to…, It’s an interesting observation that….’ Now, in fact, it was anything but natural, and the interesting observation was in fact the result of very long investigations and search for proof and self-criticism. It had a philosophical and removed attitude which I felt was necessary to get it accepted. The politics was that, if I said I was proposing a radical departure, that would have been the end of the readers’ interest.

“Later on, I got back some such statements, people saying, ‘It is natural to observe…’ That was not what I had bargained for.”

Looking back, Mandelbrot saw that scientists in various disciplines responded to his approach in sadly predictable stages. The first stage was always the same: Who are you and why are you interested in our field? Second: How does it relate to what we have been doing, and why don’t you explain it on the basis of what we know? Third: Are you sure it’s standard mathematics? (Yes, I’m sure.) Then why don’t we know it? (Because it’s standard but very obscure.)

Mathematics differs from physics and other applied sciences in this respect. A branch of physics, once it becomes obsolete or unproductive, tends to be forever part of the past. It may be a historical curiosity, perhaps the source of some inspiration to a modern scientist, but dead physics is usually dead for good reason. Mathematics, by contrast, is full of channels and byways that seem to lead nowhere in one era and become major areas of study in another. The potential application of a piece of pure thought can never be predicted. That is why mathematicians value work in an aesthetic way, seeking elegance and beauty as artists do. It is also why Mandelbrot, in his antiquarian mode, came across so much good mathematics that was ready to be dusted off.

So the fourth stage was this: What do people in these branches of mathematics think about your work? (They don’t care, because it doesn’t add to the mathematics. In fact, they are surprised that their ideas represent nature.)