The theme of Wolfram’s best-selling tome can be summarized in very few words: “Simple processes can generate complex behavior.” This massive and intricate book, generated from such a simple seed, is a good illustration of the principle.
One task of science is to make sense of the complexity that fills the world. Wolfram’s theme leads to “a new kind of science” because classical science assumes that “complex descriptions are needed for complex behavior.” These two apothegms (due to me, not to Wolfram) embody two important contrasts.
First, classical science focuses on describing the constraints acting on a system, couched in the language of mathematics (notably, partial differential equations). Wolfram shifts our attention from closed-form descriptions of a system to sets of instructions that will generate the system’s behavior.
Second, classical science assumes “conservation of complexity,” which means that if a system’s behavior is complex, the laws and processes that generate it must be equally complex. Wolfram shows that extremely simple processes can generate the most complex behavior possible.
This view of the world is different from the mainstream, and Wolfram believes that he is the first to appreciate it. The title claims that it is new. The author repeatedly makes assertions in the text about his discoveries. The book does not cite a single publication on its many themes by any other scientist. To be sure, 350 pages of detailed background notes occasionally make statements such as, “In the mid-1960s Edgar Codd showed...” Citations that would enable the reader to judge the extent of earlier work, however, are lacking. The notes omit many important cases, namely, Gosper’s mid-1970s proof of the universality of life, and earlier work that the author does mention in passing in fact shows that his intuitions were anticipated. These mentioned works include Fredkin’s explorations of basic physics as a cellular automaton, beginning a year before Wolfram was born, and Lindenmayer’s exploration of symbol substitution systems as explanations of biological growth.
Chapter 1 reviews anticipations of Wolfram’s theme in different branches of science including physics, biology, mathematics, social science, computer science, artificial intelligence, artificial life, catastrophe theory, chaos theory, and so on, arguing in each case that the researchers were close, but missed the central insight. Chapter 2 describes the author’s main research tool in studying complexity: the one-dimensional cellular automaton. Consider a row of cells, each of which can be black or white. At each time step, each cell may change color, based on a rule whose inputs are the current colors of the cell and the current colors of some limited number of neighbors. For n neighbors and k colors, such a rule offers at most k(n+1) different cases, for example, eight different cases for two colors and two neighbors. If one prints the successive states of row one beneath the next, various patterns emerge. Such patterns fall into four groups. Some become static. Some generate periodic structures. Some produce apparently random displays. A few, including one called rule 110, generate persistent structures that move across the row in successive generations, and interact with one another when they collide. Rule 110 returns as the star of the show in chapter 11.
In subsequent chapters, Wolfram introduces his reader to a range of other simple processes, including substitution systems, tag systems, and register machines (chapter 3), numerical systems such as irrational numbers and the primes (chapter 4), and multi-dimensional systems (chapter 5). All these chapters start with relatively simple initial conditions, and show how arbitrarily complex patterns can arise. Chapter 6 works in the other direction. It begins with random initial conditions, and shows how some processes can reduce these to simpler conditions.
The next three chapters apply these insights from formal systems to natural systems, ranging from snowflakes, seashells, and plant forms to the fundamental physics of space and time. These chapters showcase the author’s assertion that constraint-based descriptions can capture only relatively simple forms, and that a process-based model is much better suited to explaining the richness of the real world. Even the structure of space and time can be explained as the execution of a cellular automaton (chapter 9).
The last three chapters draw more general morals from this wealth of detail. Chapter 10 argues that our sense of complexity depends on our own computational processes. Those processes are bounded by universal computation. Modern computer science recognizes a class of processes whose behavior is rich enough to emulate any other process. In general, there is no more efficient way to understand what such a process will do than to execute it. The lack of any compact summary of such systems leads us to perceive them as complex. If Wolfram was aware of Wegner’s work on interaction machines, he would modify his argument at this point, but his conclusions would not require much modification.
Chapter 11 reintroduces rule 110, and shows that it is computationally universal. The ground is now laid for chapter 12, the final and longest chapter in the book. This highly speculative chapter expounds “the principle of computational equivalence,” which says that almost all processes--natural or manmade--that are not obviously simple can be viewed as computations of equivalent sophistication. Wolfram applies this unifying concept to themes as diverse as free will, intractability, Goedel’s theorem, and extra-terrestrial intelligence.
This book deserves its bestseller status. It is mind stretching, overflowing with graphical displays, and replete with nooks and crannies for energetic explorers. A Web site provides Mathematica code for the many sim ulations presented, so that readers can explore these behaviors for themselves. Scientifically, Wolfram’s contribution is significant. His disciplined approach of exhaustive searches of simple computational mechanisms has yielded important new insights, and he has integrated a wide range of previous work. In spite of his frequent claims, this kind of science is not new. It has, however, been under-appreciated, a defect that this book will help overcome.