One should hesitate to review a book that threatens pulling a muscle just by lifting it. Apart from the danger to one’s health, it likely will turn out to have been written by: 1) the unabomber or some other crank; 2) a fellow who unwisely declined the services of an editor; or 3) a genius who ranges over subjects way beyond anyone’s (that is, my) ability to comment intelligently. Stephen Wolfram’s A New Kind of Science fits firmly into categories 2 and 3. I’m still undecided about 1.
Wolfram fills almost twelve hundred pages (not counting the index) with discussions of subjects as disparate as free will, the fundamental laws of physics, evolution, the free market, extraterrestrials, and much, much more. In fact, he not only discusses them, but places them in relationship to each other. For example, the index includes entries for “Economics, and extraterrestrial trade” and “Economics, and free will.” The sheer breadth of the book elicited several waves of reaction in this reader. At first, a sense of awe that any person could know so much about so much. Next, excitement at the prospect that Wolfram might really be on to something fundamental. But, finally, déj vu as the subject eventually turned to topics I know something about.
Wolfram is a prodigy even among geniuses, being the youngest ever recipient of a MacArthur Foundation award. He published his first physics paper at age fifteen, earned a Ph.D. from Caltech in a single year, and joined the faculty there at the age of twenty. A few years later he headed to the Institute for Advanced Studies at Princeton. By the age of thirty he had written a computer program called “Mathematica” that helps engineers and scientists deal more easily with complicated mathematics. Marketing the software has made him independently wealthy, requiring neither an academic position nor the largesse of government research grants to pursue his interests.
With a start like that it’s understandable that Wolfram is enormously selfconfident, thinking he can entirely change the way science views the world”pretty much by himself, thank you very much. For the past decade he has closeted himself in his study, working on his book and neither publishing his interim results nor attending scientific meetings. Yet the word leaked out that he was writing a putative scientific blockbuster, and, given Wolfram’s reputation, in some circles the book has been very eagerly anticipated.
The Wolfram phenomenon has gotten quite a ride from the media, too. A LexisNexis search on the word “wolfram” shows 193 articles in the past six months, including stories in Newsweek , Business Week , and the Guardian , Telegraph , and Times of London. The New York Times has run three articles in the past few months, one from the science desk, another from the arts and ideas desk, plus an official book review. The leading science journal Nature ran an article plus a book review.
Certainly the bulk of the attention is due to Wolfram’s impressive biography and mysterious ways. But surely some of the anticipation can be traced to widespread awareness that something big is missing from our understanding of the world. Too many problems that should have been solved by now remain intractable”problems like forecasting the weather or the course of epidemics, understanding the origin of life or the behavior of economic markets, or even predicting things as trivial as how a rising column of smoke will curl. There must be some simple trick or basic insight, the thinking goes, that is being overlooked. Once we figure out the trick, everything will fall into place. In the past few decades there have been a number of candidates for this New Insight”catastrophe theory, chaos theory, complexity theory. All claimed to have wideranging implications, from physics to economics and social interactions. All have been trumpeted for a while. None have lived up to the hype.
Although Wolfram draws connections to virtually every corner of the universe, A New Kind of Science is basically a book about computing. In particular, Wolfram investigates a type of computer program called a cellular automaton (CA). To get the gist of what a CA does, imagine a very large piece of graph paper, with a grid of squares. In the middle of the top line color one square black. Now move down a line and choose a simple rule to decide if the square underneath should be blackened or not. For example, one rule might be to color a square black on the next line if the square directly above it was itself black, its left hand neighbor was also black, and its right hand neighbor was white. Now move down another line and apply the same rule again. Repeat forever. There are exactly 256 such rules (Wolfram provides pictures of over half of them), where the outcome depends only on nearestneighbors. Most of the rules yield pretty simple patterns”a solid black triangle, a triangle with alternating white and black squares, and so on.
But some rules produce remarkably complex patterns. Some show nested arrangements of triangles within triangles, while others yield triangles in which the left side consists of straight lines, but the right side swirls with larger and smaller triangles in no discernible pattern. With other rules a regular pattern is suddenly cut off by a random jagged line of dots apparently coming out of left field. Wolfram stresses that, unlike previous ideas such as chaos theory, where randomness and uncertainty derived from the physical environment, the randomness of CAs is generated by a simple, regular, discrete, numerical process. We start with the simplicity of pure integers, yet somehow end up with capriciousness and unpredictability.
From this basic result, Wolfram draws a number of large conclusions. He claims that since the colors of squares in some CAs appear random, there is no way to predict ahead of time what color a given square will be when the computer program is run. This illustrates a principle he calls computational irreducibility. In short, the outcome of many types of processes cannot be calculated from an equation or set of equations. There is no mathematical shortcut. One simply has to watch the actual process unfold to see what will happen. This pretty much spells doom to longrange weather forecasting. Another result is that some CAs can act as “universal computers,” performing any calculation that a desktop computer can perform. For example, he shows that the output of a particular CA contains lines spaced according to the prime numbers.
Another big conclusion is called the Principle of Computational Equivalence (Wolfram’s capitalization). This is a little harder to grasp. Wolfram summarizes it this way: “Whenever one sees behavior that is not obviously simple”in essentiallyany system”it can be thought of as corresponding to a computation of equivalent sophistication.” One consequence, he thinks, is that our minds cannot grasp most complex processes because those processes have the same computational sophistication as our brains. Our thinking fails to keep up with the processes.
Much of this is interesting and, for all I know, may have important implications for mathematics and computational theory. But Wolfram wants his work to go beyond computation to explain all of nature. Here he is unconvincing. The problem is that there are no obvious physical rules in nature that correspond even to Wolfram’s simplest computer CAs, let alone to the more complex ones. The fact that a program running on a computer lights up pixels in a pattern reminiscent of, say, curling cigarette smoke does not at all mean that the same causes underlie both the computer image and the real smoke.
When Wolfram turns to biological evolution he, like proponents of complexity theory before him, rudely dismisses Darwinian theory as overblown. But, again like those before him, Wolfram defends his ideas by pointing to a few simple features of life that may possibly fit with his math while passing over in silence the more complex underlying features that don’t. He writes, for example, that the geometric patterns of coloration in butterflies and sea shells may arise from simple physical constraints rather than specific coding in DNA. Well, maybe so. But the pigments that cause the coloration are the products of enzymes that are made by ribosomes that contain scores of complex macromolecular components. Do CAs have anything to do with that? Wolfram give us no reason to think so.
These examples of overreaching, though disappointing, are at least understandable. A more serious problem arises at a more fundamental level. Wolfram emphasizes that underwriting much of his thinking is a peculiar presupposition: “All processes, whether they are produced by human effort or occur spontaneously in nature, can be viewed as computations.” And he does mean all processes. In his view, rocks rolling down a hill are computers, taking input at each step and updating the system according to a set of rules, just as a PC does. Indeed, “fluid turbulence in the gas around a star” has made “more computation than has by most measures ever been done throughout the whole course of human intellectual history.” Moreover, in the same physical, unknowing sense, the human brain is only a computer. The main reason Wolfram thinks his ideas can be applied to all of nature is because by this definition the universe itself is a computer.
Stripped of the computer talk, this is just good oldfashioned materialism: in either a desktop computer, the universe at large, or a human brain, there’s nothing but particles bouncing around. That’s fine with Wolfram, who repeatedly states that one of his goals is to remove all notions of purpose from science. But when a definition of computation explicitly repudiates the mind that comprehends the calculations, we quickly descend into absurdity. When prodded in a New York Times interview, Wolfram agreed that a bucket of nails rusting quietly in a corner is a universal computer, comparable in pertinent features to the human mind. So, according to his assumptions, in the end Wolfram’s own genius can be reduced to the equivalent of a bucket of rusty nails. One notes the irony that Wolfram wrote his book to overcome faulty premises in science.
Michael J. Behe is Professor of Biological Sciences at Lehigh University in Bethlehem, Pennsylvania.

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