Mitchell Feigenbaum and the End of Chaoplexity
Feigenbaum, 1944-2019, didn’t just look like a genius, he was one, according to his fellow chaoplexologists. Ingbert Gruttner took this photo in 1987.
January 21, 2024. My previous column, “The Chaoplexity Delusion,” critiques recent calls for a revival of chaos and complexity, which in my 1996 book The End of Science I lumped together into a single term: chaoplexity. Given the response to “The Chaoplexity Delusion,” I thought readers might enjoy the following excerpt from The End of Science, in which I describe my 1994 encounter with Mitchell Feigenbaum (1944-2019), arguably the greatest of the chaoplexologists. —John Horgan
It was my meeting with Mitchell Feigenbaum that finally convinced me that chaoplexity is a doomed enterprise. Feigenbaum was perhaps the most compelling character in James Gleick’s book Chaos—and in the field as a whole.
Trained as a particle physicist, Feigenbaum became entranced with questions beyond the scope of that or any other field, questions about turbulence, chaos, and the relation between order and disorder. In the mid-1970s, when he was a postdoc at Los Alamos National Laboratory, he discovered a hidden order, called period doubling, underlying the behavior of a wide variety of nonlinear mathematical systems. The period of a system is the time it takes to return to its original state.
Feigenbaum found that the period of some nonlinear systems keeps doubling as they evolve and therefore rapidly approaches infinity (or eternity). Experiments confirmed that some simple real-world systems (although not as many as initially hoped) demonstrate period doubling. For example, as one gradually opens up a faucet, the water demonstrates period doubling as it progresses from a steady drip, drip, drip toward a turbulent gush. The mathematician David Ruelle has called period doubling a work of “particular beauty and significance” that “stands out in the theory of chaos.”
Feigenbaum, when I met him in March 1994 at Rockefeller University, where he has a spacious office overlooking Manhattan’s East River, looked every bit the genius he was said to be. With his magnificent, oversized head and swept-back hair, he resembled Beethoven, though more handsome. Feigenbaum spoke clearly, precisely, with no accent, but with a strange kind of formality, as if English were a second language that he had mastered through sheer brilliance. (The speech of string theorist Edward Witten has this same quality.)
When amused, Feigenbaum did not smile so much as grimace: his already protuberant eyes bulged still farther from their sockets, and his lips peeled back to expose twin rows of brown, peg-like teeth stained by countless filterless cigarettes and espressos (both of which he consumed during our meeting). His vocal cords, cured by decades of exposure to these toxins, yielded a voice as rich and resonant as a basso profundo’s and a deep, villainous snicker.
Like many chaoplexologists, Feigenbaum could not resist ridiculing particle physicists for daring to think they could achieve a theory of everything. It is quite possible, he said, that particle physicists might one day develop a theory that adequately accounts for all of nature’s fundamental forces, including gravity. But calling such a theory final would be something else again. “A lot of my colleagues like the idea of final theories because they’re religious. And they use it as a replacement for God, which they don’t believe in. But they just created a substitute.”
A unified theory of physics would obviously not answer all questions, Feigenbaum said. “If you really believe that this is a path of understanding the world, I can ask immediately: how do I write down in this formalism what you look like, with all the hairs on your head?” He stared at me until my scalp prickled. “Now, one answer is, ‘That’s not an interesting problem.’’”
Against my will, I felt slightly offended.
“Another answer is, ‘Well, it’s okay, but we can’t do it.’ The right answer is obviously an alloy of those two complements. We have very few tools. We can’t solve problems like that.”
Moreover, particle physicists are overly concerned with finding theories that are merely true, in the sense that they account for available data; the goal of science should be to generate “thoughts in your head” that “stand a high chance of being new or exciting,” Feigenbaum explained. “That’s the desideratum.”
He added: “There isn’t any security by knowing that something is true, at least as far as I’m concerned. I’m thoroughly indifferent to that. I like to know that I have a way of thinking about things.” I began to suspect that Feigenbaum, like David Bohm, had the soul of an artist, a poet, even a mystic: he sought not truth, but revelation.
Feigenbaum noted that the methodology of particle physics—and physics generally—had been to look at the simplest possible aspects of reality, “where everything has been stripped away.” The most extreme reductionists had suggested that looking at more complex phenomena was merely “engineering.”
But as a result of advances in chaos and complexity, he said, “some of these things that one relegated to engineering are now regarded as reasonable questions to ask from a more theoretical viewpoint. Not just to get the right answer but to understand something about how they work. And that you can even make sense out of that last comment flies in the face of what it means for a theory to be finished.”
On the other hand, chaos, too, had generated too much hype. “It’s a fraud to have named the subject ‘chaos,’” he said. “Imagine one of my [particle physicist] colleagues has gone to a party and meets someone and the person is all bubbling over about chaos and tells him that this reductionist stuff is all bullshit. Well, it’s infuriating, because it’s completely stupid what the person has been told,” Feigenbaum said. “I think it’s regrettable that people are sloppy, and they end up serving as representatives.”
Some of his colleagues at the Santa Fe Institute, Feigenbaum added, also had too naive a faith in the power of computers. “The proof is in the pudding,” he said, and paused, as if considering how to proceed diplomatically.
“It’s very hard to see things in numerical experiments. That is, people want to have fancier and fancier computers to simulate fluids. There is something to be learned in simulating fluids, but unless you know what you’re looking for, you’re not going to see anything. Because after all, if I just look out the window, there’s an overwhelmingly better simulation than I could ever do on a computer.”
He nodded toward his window, beyond which the leaden East River oozed. “I can’t interrogate it quite as sharply, but there’s so much stuff in that numerical simulation that if I don’t know what to interrogate it about, I will have learned nothing.” For these reasons much of the recent work on nonlinear phenomena “has not led to answers. The reason for that is, these are truly hard problems, and one doesn’t have any tools. And the job should really be to do those insightful calculations which require some piece of faith and good luck as well. People don’t know how to begin doing these problems.”
I admitted I was often confused by the rhetoric of people in chaos and complexity. Sometimes they seemed to be delineating the limits of science—such as the butterfly effect—and sometimes they implied that they could transcend those limits.
“We are building tools!” Feigenbaum cried. “We don’t know how to do these problems. They are truly hard. Every now and then we get a little pocket where we know how to do it, and then we try to puff it out as far as it can go. And when it reaches the border of where it is going, then people wallow for a while, and then they stop doing it. And then one waits for some new piece of insight. But it is literally the business of enlarging the borders of what falls under the suzerainty of science. It is not being done from an engineering viewpoint. It isn’t just to give you the answer to some approximation.”
Staring at me, he continued, “I want to know why. Why does the thing do this?”
I asked if this enterprise could, well, fail.
“Of course!” Feigenbaum bellowed, and he laughed maniacally. He confessed he had been stymied of late. Up through the late 1980s, he had sought to refine a method for describing how a fractal object, such as a cloud, might evolve over time when perturbed by various forces. He wrote two long papers on the topic, which were published in 1988 and 1989 in an obscure physics journal.
“I have no idea how well they’ve been read,” Feigenbaum said defiantly. “In fact, I’ve never been able to give a talk on them.” The problem, he suggested, might be that no one could understand what he was getting at. (Feigenbaum was renowned for obscurity as well as for brilliance.) Since then, he added, “I haven’t had a further better idea to know how to proceed in this.”
In the meantime, Feigenbaum had turned to applied science. Engineering. He had helped a map-making company develop software for automatically constructing maps with minimal spatial distortion and maximum aesthetic appeal. He belonged to a committee that was redesigning U.S. currency to make it less susceptible to counterfeiting. (Feigenbaum came up with the idea of using fractal patterns that blur when photocopied.)
I noted that these sounded like what would be, for most scientists, worthy projects. But people familiar with Feigenbaum’s former incarnation as a leader of chaos theory, if they heard he now worked on maps and currency, might think…
“He’s not doing serious things anymore,” Feigenbaum said quietly, as if to himself.
Not only that, I added. What people might think was that if someone who was arguably the most gifted explorer of chaos could not proceed any further, then perhaps the field had run its course.
“There’s some truth to that,” he replied. He acknowledged that he hadn’t had any good ideas about how to extend chaos theory since 1989. “One is on the lookout for things that are substantial, and at the moment…” He paused. “I don’t have a thought. I don’t know.” He turned his large, luminous eyes once again toward the river beyond his window, as if seeking a sign.
Feeling guilty, I told Feigenbaum that I would love to see his last papers on chaos. Did he have any reprints? In response, Feigenbaum thrust himself from his chair and careened wildly toward a row of filing cabinets on the far side of his office. En route, he cracked his shin against a low-lying coffee table. Wincing, teeth clenched, Feigenbaum limped onward, wounded by his collision with the world. The scene was a grotesque inversion of Samuel Johnson’s famous stone-kicking episode. The malevolent-looking coffee table seemed to be gloating: “I refute Feigenbaum thus.”
Further Reading:
If you like this type of portraiture, check out The End of Science, 2015 edition, Rational Mysticism and Mind-Body Problems (the latter free online).