Stephen Wolfram, A New Kind of Science

The Bactra Review: Occasional and eclectic book reviews by Cosma Shalizi 132 Stephen Wolfram, A New Kind of Science by Stephen Wolfram Wolfram Media, 2002 A Rare Blend of Monster Raving Egomania and Utter Batshit Insanity Attention conservation notice: Once, I was one of the authors of a paper on cellular automata. Lawyers for Wolfram Research Inc. threatened to sue me, my co-authors and our employer, because one of our citations referred to a certain mathematical proof, and they claimed the existence of this proof was a trade secret of Wolfram Research. I am sorry to say that our employer knuckled under, and so did we, and we replaced that version of the paper with another, without the offending citation. I think my judgments on Wolfram and his works are accurate, but they're not disinterested. With that out of the way: it is my considered, professional opinion that A New Kind of Science shows that Wolfram has become a crank in the classic mold, which is a shame, since he's a really bright man, and once upon a time did some good math, even if he has always been arrogant. As is well-known (if only from his own publicity), Wolfram was a child prodigy in mathematics, who got his Ph.D. in theoretical physics at a tender age, and then, in the early and mid-1980s, was part of a wave of renewed interest in the subject of cellular automata. The constant reader of these reviews will recall that these are mathematical systems which are supposed to be toy models of physics. Space consists of discrete cells arranged in a regular lattice (like a chess-board, or a honeycomb), time advances in discrete ticks. At each time, each cell is in one of a finite number of states, which it changes according to a preset rule, after examining the states of its neighbors and its own state. A physicist would call a CA a fully-discretized classical field theory; a computer scientist would say each cell is a finite-state transducer, and the whole system a parallel, distributed model of computation. They were introduced by the great mathematician John von Neumann in the 1950s to settle the question of whether a machine could reproduce itself (answer: yes), and have since found a productive niche in modeling fluid mechanics, pattern formation, and many kinds of self-organizing system. After the foundational work of von Neumann and co., there was a long fallow period in the study of CAs, when publications slowed to a trickle, and people were more likely to think of themselves as studying the statistical mechanics of spin systems, or the ergodic properties of interacting particle systems, than cellular automata as such. The major exception was a popular CA invented by John Conway, the Game of Life, or just Life, which spawned a dedicated following, trying to fathom how such a ridiculously simple set of rules could produce such monstrously complicated results. In the late 1970s, mathematicians and physicists began to become increasingly interested in CAs as such, largely owing to the advent of (comparatively) cheap and powerful desktop computers, which let people simulate and visualize CAs. There was a school of thought --- obscure, but surprisingly widely known --- which, following the physicist Ed Fredkin, thought that the universe as a whole might in some sense be a CA. Many people participated in this revival, in many places --- prominent names include, alphabetically, Crutchfield, Durrett, Farmer, Frisch, Goles, Grassberger, Liggett, Margolus, Packard, Toffoli, Vichniac, etc. Wolfram's first paper on CAs, published in 1983, was titled "The Statistical Mechanics of Cellular Automata". It focused its attention on particular simple --- he said "elementary" --- CAs: one spatial dimension, two possible states for each cell, and a neighborhood consisting of the sites to the immediate right and left of a given cell. There are 8 possible configurations for such neighborhoods, and so 256 possible elementary CA rules; in the paper, Wolfram introduced a useful scheme for referring to those rules, and others, by number, so that we speak of rule 18, rule 22, rule 90, rule 110 (of which much more below), etc. Beyond that, the paper largely consisted of calculating the entropy of configurations generated by different rules, and saying that, while the rules were simple, the patterns they could generate were complicated and intriguing. Well, and so they were; and so said many other people at the first major modern conference on CAs, organized by Farmer, Toffoli and Wolfram at Los Alamos in 1983. Wolfram went on to publish a bunch more papers on CAs over the next few years: probably the most noteworthy are "Computation Theory of Cellular Automata" (1984), where he used a familiar device of elementary computer science (regular languages and their equivalent finite automata) to characterize the set of configurations it is possible for a CA to produce, and "Universality and Complexity in Cellular Automata", where he proposed a four-fold classification of CAs based on their long-run behavior. Class I decay to a fixed, static configuration; class II to periodic oscillation; class III to seething, pseudo-random, chaotic gurp; class IV were supposed to have complicated ordered structures interacting in odd ways, and never really settle down. This scheme was popular for a while, but no one (including Wolfram) was ever able to make it any more precise, and it's proved basically worthless for understanding what CAs do; it was a Nice Try. (For more on the problems with this scheme, see Lawrence Gray's review of this book [PDF].) In the mid-1980s, Wolfram had a position at the University of Illinois-Urbana's Beckman Institute for complex systems. While there, he and collaborators developed the program Mathematica, a system for doing mathematics, particularly algebraic transformations and finding exact-form solutions, similar to a number of other products (Maple, Matlab, Macsyma, etc.), which began to appear around the same time. Mathematica was good at finding exact solutions, and also pretty good at graphics. Wolfram quit Illinois, took the program private, and entered into complicated lawsuits with both his former employee and his co-authors (all since settled). Wolfram has since retreated from normal scientific life, in to, on the one hand, tending the Mathematica empire, and, on the other, his peculiar scientific vision and method. The vision is of the universe as, if not exactly a CA, then a simple discrete program of some sort. The method has involved an enormous number of man-hours on the part of subordinates who are, as it were, enserfed to him, scanning the behavior of likely-looking CAs and signing over the rights to their discoveries to Wolfram; their efforts are supplemented by frequent lawsuits and threats of lawsuits against those whom Wolfram feels have infringed on his turf. (In 1986, for instance, Wolfram filed a patent on the idea of using CAs as discrete approximations to partial differential equations, long after the idea was commonplace in the field; it was, for instance, expounded at length in two papers in the 1983 conference proceedings he helped edit. His 1986 paper on the subject is, however, a very important contribution to the field, now known as lattice-gas hydrodynamics.) What, then, is the revelation Wolfram has been vouchsafed? What is this new kind of science? Briefly stated, it is the idea that we should give up trying on complicated, continuous models, using normal calculus or probability theory or the like, which try to represent the mechanisms by which interesting phenomena are produced, or at least to accurately reproduce the details of such phenomena. Instead we should look for simple, discrete models, like CAs ("simple programs", as he calls them) which qualitatively reproduce certain striking features of those phenomena. In addition to this methodological advice, there is the belief that the universe must in some sense be such a simple program --- as he has notoriously said, "four lines of Mathematica". Most of the bulk of this monstrously bloated book is dedicated to examples of this approach, i.e., to CA rules which produce patterns looking like the growths of corals or trees, or explanations of how simple CAs can be used to produce reasonably high-quality pseudo-random numbers, or the like. As the saying goes, there is much here that is new and true, but what is true is not new, and what is new is not true; and some of it is even old and false, or at least utterly unsupported. Let's start with the true things that aren't new. Wolfram refers incessantly to his "discovery" that simple rules can produce complex results. Now, the word "discovery" here is legitimate, but only in a special sense. When I took pre-calculus in high school, I came up with a method for solving systems of linear equations, independent of my textbook and my teacher: I discovered it. My teacher, more patient than I would be with adolescent arrogance, gently informed me that it was a standard technique, in any book on linear algebra, called "reduction to Jordan normal form", after the man who discovered it in the 1800s. Wolfram discovered simple rules producing complexity in just the same way that I discovered Jordan normal form. I am not going to dwell on the way that finding simple laws to account for multitudes of complex phenomena has been the highest aim of the exact sciences since at least Galileo and Newton. But this idea has been a driving force in mathematical logic and computer science since Alan Turing, A. N. Kolmogorov and Emil Post (he of the "tag" system, of which more later). Herbert Simon eloquently explained the importance of the idea for the study of adaptation, psychology and society in his famous 1969 book, The Sciences of the Artificial. In 1973, the physicist-turned-ecologist Robert May published a well-known paper in Nature on, as the title had it, the complicated dynamics of a simple equation. It was an idea that was very much in