October 18, 2010 | 17
The passing of the mathematician Benoit Mandelbrot has triggered in me a wave of nostalgia for the 1980s, when Mandelbrot and other researchers seemed to be creating a scientific revolution. They hoped that sophisticated new mathematical techniques, plus increasingly powerful computers, could help them fathom a wide range of complex, nonlinear phenomena—from brains and immune systems to economies and climate—that had resisted analysis by the reductionist methods of the past.
The journalist James Gleick brilliantly described this research in his 1987 bestseller Chaos: Making a New Science. Mandelbrot, an applied mathematician who dabbled in a wide variety of fields, was a hero of Gleick’s book. Beginning in the 1960s Mandelbrot realized that many real-world phenomena—clouds, snowflakes, coastlines, stock market fluctuations, brain tissue—have similar properties. They display "self-similarity," patterns that recur at smaller and smaller scales; and they have fuzzy boundaries.
Mandelbrot found that he could model these phenomena with mathematical objects that he called fractals. The name refers to a property called fractional dimensionality: fractals are fuzzier than a line but never quite fill a plane. The most famous fractal is the Mandelbrot set, which is generated by repeatedly solving a simple mathematical function and plugging the answer back into it.
When plotted by a computer, the Mandelbrot set produces a very odd-looking object, resembling a warty snowman toppled on its side. As you look at the object with higher and higher resolution, you see that the snowman’s borders are as vague as the borders of a flame; that is what fractional dimensionality looks like. Certain patterns, such as the warty snowman, keep recurring at smaller scales with subtle variations.
Other mathematicians had explored similar phenomena since at least the early 20th century. Moreover, in 1990 several mathematicians claimed to have explored the Mandelbrot set before Mandelbrot did, a claim that I reported on in an article for Scientific American, "Who discovered the Mandelbrot set?" After I called him for a comment, Mandelbrot contacted my editor to protest the story’s publication, but I don’t blame him for that. He clearly deserved credit for drawing the attention of other researchers to fractals—including the Mandelbrot set—and pointing out their peculiar similarities to a host of natural phenomena.
What resulted, finally, from attempts to model the physical world with fractals? Or all the other trendy mathematical and computational methods described in Chaos and a host of sequels by other science writers? Let me remind you of some of the other buzzwords from that heady era: cellular automata, artificial life, genetic algorithms, self-organized criticality. All purported to be methods for modeling, and hence understanding, complex phenomena. All these approaches found applications, but none turned out to be as powerful as adherents had hoped.
Indeed, chaos theory and its successor, complexity (which was really just chaos in a glossy new wrapping), followed the same boom–bust cycle as two previous scientific movements: cybernetics and catastrophe theory. Cybernetics (a neologism coined from the Greek term kubernetes, or steersman) was conceived by the mathematician Norbert Wiener. In his 1948 book Cybernetics: Control and Communication in the Animal and the Machine he proclaimed that cybernetics could in principle model the operation of not only machines but also all biological phenomena, from single-celled organisms up through the economies of nation–states.
Cybernetics became extremely popular, especially in Russia, but by the 1960s it was already losing its luster. In 1961 the electrical engineer John Pierce sneered that the term cybernetics "has been used most extensively in the press and in popular and semiliterary, if not semiliterate, magazines." (Cybernetics lives on, of course, in the pop-culture term "cyberspace," coined by sci-fi writer William Gibson.)
The next big idea was catastrophe theory, a set of equations that the French mathematician Rene Thom claimed could model phenomena exhibiting abrupt, "catastrophic" discontinuities. Thom and his followers suggested that catastrophe theory could help to explain not only events such as earthquakes but also biological and social phenomena, such as the emergence of life, the metamorphosis of a caterpillar into a butterfly and the collapse of civilizations.
One reviewer of Thom’s 1972 book Structural Stability and Morphogenesis compared it with Newton’s Principia, arguably the most important scientific treatise of all time. By the late 1970s, critics were complaining that Thom’s work "provides no new information about anything" and is "exaggerated, not wholly honest." An essay in Nature called catastrophe theory "one of many attempts to deduce the world by thought alone," which is "a dream that cannot come true." (Scientific American is part of Nature Publishing Group.)
Chaos theory followed the same pattern. In 1991, just four years after Chaos was published, David Ruelle, who like Mandelbrot was a pioneer in mathematical modeling of chaotic systems, complained that chaos, "in spite of frequent triumphant announcements of ‘novel’ breakthroughs, has had a declining output of interesting discoveries."
Every now and then, some ambitious soul proclaims once again that he has created an all-powerful mathematical theory. One high-profile example was the physicist and mathematical-software mogul Stephen Wolfram, who declared that his 2002 self-published book A New Kind of Science would engender, well, a new kind of science. But Wolfram’s "new" approach to solving all scientific puzzles was just cellular automata, the computational modeling system invented by John Von Neumann in the 1950s.
Perhaps it is time to acknowledge that no single mathematical or computational system can model all of reality. Mandelbrot, who was not known for modesty, suggested as much. "Most emphatically, I do not consider the fractal point of view as a panacea," he wrote in his strange and wonderful 1977 book The Fractal Geometry of Nature. The similarity of a fractal to a natural phenomenon, he noted, did not necessarily yield deep insights into its underlying physical mechanisms.
But fractals have proved useful for financial modeling, image-compression and other applications. And Mandelbrot was absolutely right that computer-generated fractal images revealed "a world of pure plastic beauty unsuspected until now." I can still remember the thrill I got when I first saw the colored images of the Mandelbrot set, which shimmered with intimations of infinity. I will always be grateful to Mandelbrot for that.
Image credits: Wikicommons