Those of us interested in the position of mathematics in public culture could not help but be impressed a few weeks back, when the No. 1 spot on Amazon's best-seller list was briefly occupied not by Nora Roberts or The Nanny Diaries but by Stephen Wolfram's A New Kind of Science: a math book! The physicist-turned-reclusive entrepreneur has produced a 1,263-page account of his attempt to reformulate all of science on the foundation of mathematical formalisms known as cellular automata. And I do mean all of science. There is something to admire about a book one column of whose index includes "Textile making, rules in," "Theology, and form of extraterrestrials," and "Theorems, automated proving of." (Alas, one must pass to the next column to attain "Theory of universe—SEE Ultimate theory of physics.")
A New Kind of Science, as will not surprise anyone who has tried to lift it, is really two books in one. First of all, Wolfram has produced a thorough and often enjoyable account of the behavior of cellular automata. A cellular automaton, or CA, is a computer program—but not one like VisiCalc or Quake, designed with a specific purpose in mind. A CA does just one thing: You give it an input, and it produces an output. Give that output back to the program as input, and you've got a new output. And so on. Cellular automata are typically simple enough to be implemented in a line or two of code; yet from these simple instructions, repeated thousands of times, they can produce drastically complex behavior. The best-known CA is John Conway's Game of Life. Before you decide that "Life" is too grand a name for a two-line computer program, play with this Java applet.
As simple as CAs are, they come in a daunting variety of flavors. Some of the programs produce very simple, easy-to-predict output; some produce results that look, to the naked eye, completely random. And others, such as the Game of Life, are somewhere in between, producing results that display visible structure and yet are extremely difficult to predict. Wolfram is a kind of zoologist. He's spent two decades watching these creatures in their natural habitat (his computer), tracking their lives and deaths through millions of generations, and painstakingly classifying which species are best at certain kinds of tasks. It's an impressive piece of work.
The problem is Wolfram thinks he's the other kind of zoologist. He's convinced that cellular automata, and other simple computational machines, can explain the complexity of living beings—not to mention quantum mechanics and relativity, fluid turbulence, free will, and the behavior of financial markets. Newton's revolution in mathematics and physics, founded on his theory of calculus, stands here as a 300-year wrong turn. Our big mistake was inventing physics before computer science—because computation, Wolfram says, underlies everything. Wolfram's attempt to explain and support this grand conclusion makes up the larger, and less appealing, part of A New Kind of Science.
This space won't admit a full review of Wolfram's work; I'll say only that the book is surprisingly unsurprising. Wolfram sets himself up in opposition to what he sees as "traditional science," where, he says, "whenever a phenomenon is encountered that seems complex it is taken almost for granted that the phenomenon must be the result of some underlying mechanism that is itself complex." "Traditional scientists," in Wolfram's description, are sickly creatures indeed; and we are very lucky that real scientists resemble them in almost no particular. Physicists know well that simple differential equations (calculus, that is) can produce unpredictably complex behavior. Even in Wolfram's own field of cellular automata, it's decades-old news that a few simple rules (for instance, those of Conway's Game of Life) can produce behavior as complicated as the most sophisticated digital computer. Again and again, Wolfram commands us to marvel at assertions that most scientists would accept without blinking.
Wolfram, for instance, claims to have overturned the basic beliefs of mainstream biology. There, he says, "... it has come to be assumed that essentially every feature of every organism can be explained on the basis of it somehow maximizing the fitness of the organism." Compare that with Stephen Jay Gould: "... [S]election cannot suffice as a full explanation for many aspects of evolution; for other types and styles of causes become relevant, or even prevalent, in domains both far above and far below the traditional Darwinian locus of the organism. These additional principles are as directionless, nonteleological, and materialistic as natural selection itself. …" Wolfram makes a good case that natural selection, operating to maximize fitness, may not explain the complex features of typical organisms. That's science, but it's not new. Biologists know very well that natural selection can't predict the particular pattern of stripes on a particular zebra, just as physicists know that the solution to a differential equation cannot, in practice, predict next year's weather.
But neither can Wolfram.
Predictions, in the end, are what A New Kind of Science most notably lacks. For all its bulk, the book presents very little evidence to back up its claims. In fact, it's often hard to tell precisely what the claims are. Wolfram has consciously avoided publishing the details of his work in scientific journals: "I gradually came to realize," he explains, "that technical papers scattered across the journals of all sorts of fields could never successfully communicate the kind of major new intellectual structure that I seemed to be beginning to build." But peer-reviewed publication offers a great advantage: It forces scientists to distinguish between what they have proved, what they have demonstrated by experiment, and what they can only suggest. Having failed to make this distinction, Wolfram has produced an immensely frustrating book. A New Kind of Science often won my interest, even my admiration, but failed—and in science, this failing cancels everything else—to win my trust.
Which brings us back to Amazon and the most interesting question about A New Kind of Science: Why did so many people buy it? Why was Wolfram profiled in Forbes, Le Monde, Wired, and no less than four sections of the New York Times? What sets him apart from every other striver with a big idea and no proof?
For one thing, Wolfram has credentials—he started as a child prodigy in particle physics and won a MacArthur "genius" grant at 21. He held tenured positions at Cal Tech and Illinois, then left academia to start a company that produces a widely used and appreciated scentific software package, Mathematica.[Correction, 7/4: Wolfram did not have tenure at Cal Tech. He was a senior research associate.]
But there's more to it than that. Wolfram is selling not just a theory, but a story—the story of a genius who retreats from the misunderstanding and jealousy of smaller minds. Some call the genius crazy, but he doesn't mind. Some raise technical objections to his theory. But the genius knows that true ideas are always simple and that technicalities are just the so-called experts' way of hiding the inadequacy of their ideas. He should read and give credit to the work of his peers? What they know, he doesn't need. Anyway, reading would waste time: time he needs to erect the intellectual monument history will thank him for.
Sound familiar? You've heard it a lot lately. It's the story of A Beautiful Mind: Ron Howard's cartoon version of John Nash is shown scribbling in the library a lot, but does his real work while helping his less inspired classmates pick up girls in bars. It's the story of the mathematician in Proof, who knocks off a spectacular theorem by spending a decade without colleagues, books, or conversation—and doesn't tell anyone. It's the story for which people might mistake the true stories of Andrew Wiles and Srinivasa Ramanujan, who carried out great work alone.
Wolfram tells the story when he recounts his version of the early history of cellular automata:
I started a research center and a journal, published a list of problems to attack, and worked hard to communicate the importance of the direction I was defining. ... [T]here seemed to be very little success in breaking away from traditional methods and intuition. And after a while I realized that if there was going to be any dramatic progress made, I was the one who was going to have to make it.
And he tells it in another chapter when he describes to Stephen Levy of Wired his method of approaching an unfamiliar scientific area: "One can devour lots of papers in very short amounts of time in the middle of the night. ... [U]sually in a few days it all starts to kind of crystallize and you realize that there really are only three ideas in this field, and two of them you don't believe."
And, with a few exceptions, journalists like the story nearly as much as Wolfram does. Levy fills in a piece Wolfram missed: "Not heard from yet are the voices of the establishment, which undoubtedly will have problems with the unconventional work and its author." In his New York Times book review, George Johnson writes, "[T]he artist returned to his garret to tinker, ignoring the bad vibes and hexes cast by jealous colleagues hoping to see him fall flat on his face." He goes on to commend Wolfram for eschewing scientific publication—or, as Johnson calls it, "unreadable papers published in fashionable zines." (Unreadable to whom? Fashionable to whom? And—"zines"??) "It's too bad," he concludes, "that more science isn't delivered this way."
I can't entirely blame the journalists; it's a good story. And we mathematicians have raised our profile greatly by telling it. But that publicity comes with a price. Wolfram's beliefs about cellular automata may or may not catch on. Frankly, I hope people do make new science out of his ideas. But I'm afraid it's more likely that people will learn different lessons from Wolfram's book and its treatment in the press. They'll learn that the scientific process is about protecting turf, not about producing knowledge, and that if something such as calculus seems difficult, it must be beside the point or wrong. I don't think Wolfram means to teach those lessons. But plenty of people already believe them. And if their view of science becomes the dominant one, "traditional scientists" will have an opponent more dangerous than Stephen Wolfram to contend with.