»  National Review

September 16th, 2002

  Not Quite Copernicus


A New Kind of Science
by Stephen Wolfram
Wolfram Media, Inc.; 1,197 pp. $44.95

I consider Stephen Wolfram to be a great benefactor of humanity. The foundation for this opinion is that I am a daily user of the Mathematica software package, Wolfram's brainchild and the source of his considerable fortune. This wonderful tool allows me to do experimental mathematics, of a type that would have been impossible as little as 15 years ago without much wearisome programming. I can try out formulas, conjectures, and theorems with actual numbers to see what they turn up; I can plot graphs in two and three dimensions; I can factorize, differentiate, integrate, solve equations, invert matrices, and extract eigenvalues to my happy heart's content.

As well as having created this splendid product and made himself rich by astutely marketing it, Wolfram is impressively credentialed as a working scientist. He was a child prodigy, publishing his first paper on particle physics at age 15, going on to acquire a Ph.D. from Caltech when he was just 20, and proceeding quickly to important results in quantum field theory and cosmology. Stephen Wolfram is, without much doubt, one of the smartest people on the planet.

All of which leaves us with a very interesting question: Why has this undoubtedly brilliant, worthily successful man written such a silly book?

To get any further with this review, I shall have to try to tell you what A New Kind of Science is about. It is about cellular automata. Imagine — or, better yet, place in front of you — a sheet of graph paper marked off in tiny squares. Using a pencil, black in as many as you please of the top row of squares. You can just black in one square if you like. That top row is now complete, and will undergo no further change. Now black in squares on the second row according to some rule. Here is a possible rule: Take each second-row square in turn, and look at the three first-row squares above it — the one directly above, and the ones to left and right above. If all three are black, or none are, or only the left-above one is, leave your second-row square empty. Otherwise, black it in. When you've finished marking up squares in the second row, do the third according to the same rule, now using the second row as your reference row. Repeat until you have worked your way down to the bottom of the sheet, and it is covered with a pattern of black and white squares.

You are looking at a cellular automaton. If you can't be bothered to carry out the exercise, pick up a copy of A New Kind of Science in your local bookstore and browse pages 32-38, where 3,200 rows of this particular automaton are shown, starting with a single black square in the first row. Plainly the pattern you see is determined by two things: the way you marked up the first row, and the rule you applied for generating subsequent rows.

Why is this interesting? Because the patterns generated by these cellular automata are more complex than they have any right to be, given the simplicity of the generating rules. The particular rule I used above — Wolfram calls it "Rule 110" — was expressed in 50 words; yet it generates strange, never-repeating patterns, in which long stretches of apparent regularity are suddenly, unpredictably fractured into randomness, then mysteriously restored. That "unpredictably" is a precise mathematical term. There is no short-cut, no formula, to tell you what the 3,000th row will look like. You just have to run the darn thing through 3,000 iterations of the rule.

That's good for half an hour's entertainment on a rainy day, and cellular automata have been doodled with in recreational-math circles since the 1950s; but "a new kind of science"? What's that all about? Well, consider the universe at large. At some point in time, it — all its untold bazillions of particles — is in a certain configuration. At some later instant, it is in a different configuration. How did it get from the one state to the other? The traditional answer is: by obeying certain very complicated physical laws, which you need a ton of calculus and a truckload of algebra to understand. No, no, says Wolfram: It's a cellular automaton!

Or consider biological evolution. At a certain point in its history, the typical member of a certain species has such-and-such a genetic code. A million years later, it has a different one. How did it evolve from the first to the second? By natural selection working on random mutations, says traditional biology. Wolfram is shaking his head again: cellular automaton!

It's not just physics and biology: Wolfram sees cellular automata everywhere. Computer science, mathematics, philosophy, psychology, cryptography … Cellular automata! That's the whole thing; that's the 1,197 pages. A good proportion of them are filled with pictures of cellular automata doing their peculiar thing; the rest are occupied by prose of excruciating badness, which Wolfram justifies in an appendix. (There are 340 pages of those, by the way.) He has, he tells us airily, not only invented a new kind of science, but a new kind of science writing, too. The man's vanity is staggering. "I fully expect that some of the terms and concepts I use … will end up seeming dated in a few decades." People will still be poring over this book, you see, in the year 2082, still turning up hitherto-unnoticed nuggets of truth in it, gasping with astonishment as they do so.

I think it more likely that the book will be forgotten in a few months. What science mainly requires of a theory is predictive power. Since the outcomes of the more interesting cellular automata are, as I pointed out above, essentially and mathematically un-predictable, this requirement opens a nasty epistemological hole in the entire schema. What would be the equivalent of the observations made during the 1919 solar eclipse, dramatically confirming Einstein's general theory of relativity? The author offers no ideas.

Even where this book is at its strongest, in the section on fundamental physics, while Wolfram's ideas might just possibly be true at some level, they do not explain anything that is not already quite adequately explained. In the life sciences, not even this much can be said. The complexity that Wolfram's cellular automata exhibit is actually nothing like the complexity of living organisms, in which entire systems — circulatory, lymphatic, digestive, nervous — nourish and support each other.

This whole ludicrous farrago was cooked up during ten years when the author shut himself in his study with his gadgets for long hours, running his business by videoconference, publishing nothing in peer-reviewed journals — he scoffs at peer review, in his preface — and accepting no help from proofreaders or editors. He has published and promoted the book himself, using his own company.

It shows. Among the dark companions of overweening Vanity are Carelessness and his infant offspring, Error. Wolfram's sphere of knowledge is much wider than mine, but there is a modest overlap; and when he is writing about something I know well, he is often wrong. "In the late 1960s and early 1970s, there developed the notion of operating systems …" This will be interesting news to surviving programmers of the IBM 1400 series (1960), the Leo III (1961 — the operating system was named "GEORGE") and the Burroughs D825 (1962). Incredibly, Wolfram even makes mistakes when writing about Mathematica. Speaking of the distribution of prime numbers, he says: "A somewhat better approximation is  LogIntegral[n],  equal to  Integrate[1/Log[t],{t,2,n}]."  The second of those expressions is indeed one way to define the logarithmic integral — the "continental" way, favored by Landau and other German number theorists. Unfortunately, Mathematica uses the "American" definition, which integrates from 0, not from 2, rendering Wolfram's "equal to" false. (I checked, in a conversation with Mathematica on this point. Taking n = 12, I asked Mathematica to evaluate  LogIntegral[n].  It returned the answer: 7.00055. I then asked Mathematica to evaluate Integrate[1/Log[t],{t,2,n}].  Back came the answer: 5.95538.)

So: Is Wolfram, as he plainly believes, the new Copernicus? Or is he merely a new Darwin or Einstein? Well, if it's comparisons you are seeking, the one that occurred to me was the astronomer in Dr. Johnson's Rasselas, who, after years of intense, solitary intellection, went quietly nuts.

In time some particular train of ideas fixes the attention, all other intellectual gratifications are rejected, the mind, in weariness or leisure, recurs constantly to the favorite conception, and feasts on the luscious falsehood whenever she is offended with the bitterness of truth.

The universe a vast cellular automaton? As Diderot remarked in a similar case: "Nothing is that simple; certainly not everything!"