A Mathematician Looks at Wolfram’s New Kind of Science Lawrence Gray A New Kind of Science properties. His ac- Stephen Wolfram tivities in this direc- Wolfram Media, Inc., 2002 tion were inter- ISBN 1-579-955008-8, $44.95 rupted when he became occupied n May 2002 Stephen Wolfram finally unveiled with the develop- his self-proclaimed masterpiece, A New Kind ment and promotion of Science (hereinafter referred to as ANKS). of Mathematica. But Published by Wolfram’s own company, the he felt that the ideas I1,280-page volume contains his thoughts on in several of his CA everything from the physics of the universe to the papers had never re- mysteries of human behavior, all based on the ally been “absorbed” results of several years of analyzing the graphical by other scientists output of some very simple computer programs. (ANKS, p. 882), so in The scope of the book is impressive, covering a 1991 he began work bewildering variety of mathematical models and on the book that he hopes will start a scientific rev- illustrated by 973 high-resolution black and white olution. pictures. There are whole chapters devoted to Do we need this revolution? According to Wol- biology, physics, and human perception, with fram, “traditional” mathematics and science are shorter sections touching on such unexpected doomed: mathematics because of its emphasis on subjects as free will and extraterrestrial art. The rigorous proof, and science because of its prefer- extensive historical and technical notes at the end ence for models that can make accurate predictions. of the book (349 pages of small print) provide He says that the most interesting problems presented fascinating background material. by nature are likely to be formally undecidable or The primary mathematical focus of the book is computationally irreducible (ANKS, pp. 7, 794–5, a class of discrete-time dynamical systems called and 1138), rendering proofs and predictions cellular automata, or “CAs”. (See the next section impossible. Mathematicians and scientists have for definitions and examples.) Back in the 1980s, managed to keep busy only by carefully choosing Wolfram introduced several ideas that had a sig- to work on the relatively small set of problems nificant impact on CA research, and he also dis- that have simple solutions (ANKS, p. 3). covered a number of specific CAs with intriguing There is more: most mathematical models in science are based on the assumption that time and Lawrence Gray is professor of mathematics at the School space are continuous, whereas Wolfram says that of Mathematics, University of Minnesota, Minneapolis. His time and space are discrete. He would have us email address is [email protected]. 200 NOTICES OF THE AMS VOLUME 50, NUMBER 2 abandon models based on calculus and Euclidean suggest that such models are out there somewhere, geometry in favor of discrete systems like CAs and Wolfram provides many examples from all (ANKS, p. 8). Indeed, he sees the entire universe areas of science to try to show us that they can as a CA-like system that likely follows a simple actually be found. dynamical rule, and the better part of Chapter 9 Am I convinced? Not really. Wolfram’s brand of consists of some clever speculation on the exact computer experimentation is a potentially power- nature of such a rule. ful scientific tool. Indeed, I find that by far the To make his argument convincing, Wolfram most valuable aspect of the book is that it brings needed a simple CA that was capable of highly together so many interesting examples of CAs and complex behavior. Enter the CA known as “Rule 110”, related models that first found the light of day in whose dynamical rule is about as simple as possi- one of his computer searches. But can he really ble, as you will see later in this review. The model justify statements like this: “…the new kind of sci- was discovered by Wolfram in the 1980s, when ence that I develop in this book is for the first time he conjectured that its behavior was “universal”, able to make meaningful statements about even im- meaning that it could be used to simulate a mensely complex behavior” (ANKS, p. 3)? universal Turing machine. The conjecture was Wolfram loves to tell us why other scientific proved in the 1990s by Matthew Cook, a former theories cannot handle complexity. But in these employee of Wolfram Research. discussions, he badly mischaracterizes his com- Rule 110 is featured prominently throughout petition. Here is a typical example: “The field of ANKS, and it provides the primary motivation for nonlinear dynamics is concerned with analyzing Wolfram’s scientific philosophy, which is that the more complicated equations [than linear ones]. Its key to understanding complex behavior can be greatest success has been with so-called soliton found in very simple discrete systems. He has been equations for which careful manipulation leads to pushing this idea for twenty years. But in ANKS we a property similar to linearity. But the kinds of find a much more provocative version, the “Prin- systems that I discuss in this book typically show ciple of Computational Equivalence”, which we are much more complex behavior, and have no such told is a “new law of nature” that “has vastly richer simplifying properties” (ANKS, pp. 15–6). He is par- implications than…any [other] single collection ticularly hard on chaos theory, which he more or of laws in science” (ANKS, pp. 720 and 726). The less reduces to a trivial observation about sensitive entire final chapter of the book is devoted to this dependence on initial conditions: “Indeed, all that principle, but, surprisingly, Wolfram does not it shows is that if there is complexity in the details provide us with a definitive statement of it. Here of the initial conditions, then this complexity will is my attempt, pieced together from various phrases eventually appear in the large-scale behavior of in Chapter 12 of ANKS: the system” (ANKS, p. 13). He claims to have ex- amples of dynamical systems that exhibit chaotic Except for those trajectories that are behavior without sensitive dependence on initial obviously simple, almost all of the conditions. But his examples are highly question- trajectories of a (natural or theoretical) able, as I will later explain when I discuss his no- dynamical system can be viewed as tion of “intrinsic randomness generation”. computations of equivalent sophistica- Wolfram provides very little hard evidence for tion, which is to say that they are the Principle of Computational Equivalence. The universal (see ANKS, pp. 716–9). key phrase “obviously simple” is pretty much left Thus, if you observe a trajectory of some system, undefined, except to say that it covers systems such as a CA or a differential equation or the that are attracted to periodic orbits or follow some weather or even a bucket of rusting nails, then other easily detectable pattern. Even when the prin- either you will see something that is obviously ciple is taken at face value, serious doubts about simple, or else arbitrarily complex computations both its validity and practical significance have will pass before your eyes. been raised (see the list of reviews given below). I Wolfram’s attitude toward traditional mathe- will raise a few more later on, when I discuss fault- matics and science is consistent with his principle. tolerant computation and universal CAs. After all, if everything nontrivial behaves like a Despite the provocative attitude and high-minded universal Turing machine, then it is a waste of time speculation, there is plenty to enjoy in the book, to try to find ways to predict anything that is not especially the very accessible and very extensive already obvious. He advises scientists to start doing coverage of so many different kinds of discrete what he has been doing for the past two decades, models. In addition to CAs, we find mobile cellular which is to systematically explore simple CAs and automata, Turing machines, substitution systems, related discrete systems, searching for models to sequential substitution systems, tag systems, cyclic match various interesting natural phenomena. The tag systems, register machines, and causal Principle of Computational Equivalence seems to networks. For each type of system, Wolfram presents FEBRUARY 2003N OTICES OF THE AMS 201 and carefully explains numerous examples, ex- criticisms and questions Wolfram’s rejection pertly illustrated by instructive and thought- of natural selection as a significant factor in provoking graphics. I am familiar with Turing evolution.) machines, for example, but had not seen their work- 6. “The World According to Wolfram” by Brian ings graphically depicted as in ANKS. Interesting Hayes, in The American Scientist, July–August statistics are given about the range of behaviors in 2002. (Counters some of Wolfram’s claims of these systems, based on Wolfram’s own computer discovery.) experiments. 7. “Blinded by Science” by Jordan Ellenberg, in It is also a lot of fun watching Wolfram find Slate, posted on July 2, 2002. (This is a most connections with the real world. Some of them are entertaining and intelligent review.) original to Wolfram, many are not, and it is un- 8. “Is the Universe a Universal Computer?” by fortunately not always so easy to determine which Melanie Mitchell, in Science, October 4, 2002. is which. But it is great having them all together in (Takes Wolfram to task for several of his one book. Particularly impressive is the chapter grandiose assertions.) where he attempts to capture modern physics, in- Each of these articles finds something significant cluding relativity theory and quantum mechanics, to praise in ANKS (clarity, enthusiasm, expert in a CA-like system.