Almost twenty years ago, I heard Stephen Wolfram speak at a Gordon Conference

Almost twenty years ago, I heard Stephen Wolfram speak at a Gordon Conference about cellular automata (CA) and how they can be used to model seashell patterns. He had recently made a splash with a number of important papers applying CA to a variety of physical and biological systems. His early work on CA focused on a classification of the behavior of simple one-dimensional systems and he published some interesting papers suggesting that CA fall into four different classes. This was original work but from a mathematical point of view, hardly rigorous. What he was essentially claiming was that even if one adds more layers of complexity to the simple rules, one does not gain anything beyond these four simples types of behavior (which I will describe below). After a two decade hiatus from science (during which he founded Wolfram Science, the makers of Mathematica), Wolfram has self-published his magnum opus A New Kind of Science which continues along those lines of thinking. The book itself is beautiful to look at with almost a thousand pictures, 850 pages of text and over 300 pages of detailed notes. Indeed, one might suggest that the main text is for the general public while the notes (which exceed the text in total words) are aimed at a more technical audience. The book has an associated web site complete with a glowing blurb from the publisher, quotes from the media and an interview with himself to answer questions about the book. The copyright notice in this book is somewhat unusual – essentially saying that if you use any of the ideas in the book, you had better cite him or his lawyers will see you. (It should be noted that there are no citations in this book.) Wolfram makes many points in his book, but I will focus on three of them: (i) complex behavior can be generated by simple rules; (ii) modern mathematical methods are ill-equipped to deal with complex systems such as seen in biology and social sciences; and (iii) all systems can be viewed as computations and furthermore, even the most complex ones are equivalent to certain simple rules. This latter notion is the core idea in the book. The book begins with an outline of what will come later. Here Wolfram describes his main thesis: The Principle of Computational Equivalence. He bases almost everything in the book on countless computer experiments using cellular automata (CA). One should recall that a CA consists of a lattice of sites each of which has an associated state (often just 0 or 1). Each site is updated according to a rule which generally depends on some small neighborhood of nearby sites. For years CA were viewed mainly as a parlor game for producing pretty pictures. One of the goals of the present book is to claim that rather than just a simple game, these systems are how we should all be doing science. As an example consider a one-dimensional lattice with two states which are updated according to their current values and the two neighboring sites. There will be 256 possible rules so it is not at all difficult to exhaustively explore each possible rule. Wolfram provides a clever numerical code for specifying these rules. He is partial to two of these rules, 30 and 110. The binary representation of 110 is 01101110. The set of three digits consisting of your state and your two neighbors encodes a binary number from 0 to 7. Thus if this number is 0,4, or 7, then your new state will be zero otherwise, it will be 1. That is, {000, 100,