Review by Yves Pomeau in Mathematical Reviews (MathSciNet);
A new kind of science. (English. English summary)
Wolfram Media, Inc., Champaign, IL, 2002. xiv+1197 pp. ISBN 1-57955-008-8
According to the author, this book took about ten years to write. This explains (perhaps) why it is so thick and why so many things are under consideration here. The author has clearly absorbed an immense body of literature, as is evident from the 349 pages of notes, in fine print. The main text is 846 pages long. The first sentence made me jump to the ceiling: it says, "Three centuries ago science was transformed by the dramatic new idea that rules based on mathematical equations could be used to describe the natural world." I take this as referring to Isaac Newton's Principia, which started modern science, but without a single equation, differential or not (and contrary to an often expressed belief). This rather awkward implicit reference sets the stage: Wolfram puts his magnum opus on the same footing as the grandest work in science. He claims to have made a major discovery that opens the way to progress in various areas of science where what he calls "traditional mathematics" failed to bring significant progress. It is not very easy, however, to see the nature of this discovery (or these discoveries) and the kind of progress they can bring.
Chapter 2 reports computer experiments done on cellular automata (CA) and compares a display of the results with various patterns (page 43), the message being that a simple CA can reproduce things looking like mosaics made by various civilizations, from ancient Greece to medieval Islam and Renaissance Italy. This leads to an interesting Chapter 3, examining various CA as well as related simple (in their formulation) systems such as tag systems and Turing machines. Once these simple programs are run on fast modern computers, one observes that simple CA rules can yield visually complex patterns. This is not at all unexpected, since it is now well known that simple dynamical systems like coupled nonlinear ordinary differential equations and polynomial iterations may have a very complex behavior. Discrete systems, like the law for the generation of digits of the square root of two, for instance, have as well completely random outcomes. However, there is as yet no known method for actually proving that any of these discrete iterations yields a random result, and no such proof is to be found in this book.
Chapter 4 is about comparing traditional simulations of linear and nonlinear PDE's like the heat equation, the wave equation and a nonlinear version of it. The claim (as I understand it) at the end is that continuous systems like PDE's are far more difficult to study than CA. Particularly when complex behavior is found, CA provide a much better description than PDE's. I disagree with this statement; in particular, important studies have been made (not mentioned by Wolfram) on the so-called Kuramoto-Sivashinsky PDE, which is known to show a random behavior that becomes complex (in the present sense) when damping is added and a critical value of this damping is reached \ref[H. Chaté and P. Manneville, Phys. Rev. Lett. 58 (1987), no. 2, 112--115]. The work of Chaté and Manneville shows something that is absent in the present book, namely that for PDE's with variable coefficients, the "complex behavior" is at the borderline between simple and chaotic behavior, which most likely explains why in the sequel Wolfram only rather rarely finds (by visual inspection) complex behavior.
Chapter 6 is a central chapter, as it introduces the classification of CA according to their behavior with random initial conditions (page 231). In class 1, one has quick convergence to a simple fixed point: black or white dots everywhere. Class 2 shows fixed points but with a random structure in space. Class 3 automata are fully chaotic, and class 4 automata are chaotic but with some large scale structures. What is disappointing is that this classification is based (as are too many things in this book) on visual observation, without an attempt to be more precise. If one admits (and this seems to be logical) that class 4 is akin to the onset of spatiotemporal intermittency in the sense of Chaté and Manneville, there should be typical long range correlation in class 4 CA as well as various critical exponents.
Chapter 8 is concerned with applications to everyday systems (meaning physical systems unrelated to grand physical questions, like the unification of gravitation and quantum mechanics and the intimate structure of the universe, something considered later in Chapter 9). The three examples are the growth of snow flakes, the breaking of materials and fluid flows. The representation of the breaking of materials by CA does not differ so much from the molecular dynamics simulations of the same process, and they run into the same difficulty, namely that any phenomenon they can display occurs on a length scale of the same order of magnitude as the mesh, that is molecular order of magnitude, smaller than the observed roughness. The case of fluid flows is probably more interesting. It has been shown that CA can model fluid flows on large scales. Wolfram does not provide any in-depth discussion of this model, called the FHP (Frisch-Hasslacher-Pomeau) model in the literature (a term he does not use), after its inventors. The history of the CA fluid in the final notes (pages 999 and 1000) is quite distorted, since the only paper submitted and published in 1985 on this topic was by myself with two coworkers \ref[D. d'Humières, P. Lallemand and Y. Pomeau, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 301, (1985), 1391 (submittted Nov. 18, 1985); per revr.] and we were completely unaware at the time of any work done by Wolfram; his publications on this come later. His claims to have done something on FHP in the summer of 1985 are not substantiated by any publication. This brings me incidentally to a major weakness of this book: no reference to any work is made either in the main text (which may have some rationale) or in the fat note section (which is very bad).
Chapter 9, "Fundamental physics", is perhaps the most ambitious in this book. Wolfram tries to formulate the known laws of physics by using CA. This becomes a rather curious exercise in developing complex CA. The CA that could model the physical world seem so complex that I cannot see any obvious interest in them. The major difficulty is twofold: the laws of physics have some invariance properties that are very hard to reconcile with CA. The FHP model of fluid flows shows it well: one has to choose the CA very carefully to get the desired Galilean invariance of an ordinary fluid. This is with a classical (nonrelativistic) system, where time is the same everywhere. This unique time allows the CA to be advanced all at once. But things get very nasty if one insists that no information travel faster than the speed of light, as in our real world: a global updating at each time step is in obvious disagreement with this requirement. Wolfram introduces a model with a kind of mobile automaton that travels at the speed of light and updates the sites it visits. Perhaps this satisfies the requirement of causality, but it is not clear that such a model could for instance represent the propagation of light waves, the problem for which special relativity was invented by H. Lorenz. The final section on quantum phenomena in this chapter is even more speculative, and cannot be taken as a serious basis for future investigation.
The following chapters are more concerned with fundamental issues of computability and various mathematical results connected with these. The main claim (and the only one with a proof in a mathematical sense in the whole book) is that one of the CA rules, rule number 110, gives a universal computer. This means that, given a computer program, it can be transformed into a set of initial conditions for CA number 110 that, once they are run, are equivalent to the running of the program on any computer. This rather arcane result in computer science is certainly interesting. The final chapters, 12 and 13, build upon this to claim that simple CA can explain everything, since they belong to the same universality class (i.e., are computationally as powerful) as the most sophisticated computers, including the human brain.
It is not easy to give an objective assessment of such a daring book. The most difficult thing for me is: is the purpose of this book to show us (as in the fluid dynamics section) that CA can be used as a cheap (or moderately cheap) alternative to more straightforward methods of solving the PDE's of "traditional mathematics", or is it that the fundamental structure of things in the outside world is described by CA? This question leads to another one (not looked at in the book): if the universe at small scale is a CA, can experiments show this? At the moment, I have the impression that the CA representation of Nature belongs in the league of many (if not most) attempts made nowadays to explain the structure of the world by arguments based on the esthetic appeal of theories, without worrying too much about any experimentally testable verification. The CA representation of Nature suffers the same drawbacks as all present-day "theories of everything": starting from a theory without any small or large parameter, it is impossible to explain the occurrence of extremely large or small dimensionless numbers in fundamental physics (the most impressive one being the value of Einstein's gravitational constant: it must almost exactly cancel the contribution of the vacuum fluctuations to one part in 10 to the power 130). If one sticks to the esthetic point of view, I believe one must be very careful: esthetics is quite often a way of condoning worthless academic art, not Monet or Cézanne. The book by Wolfram surely cannot be dubbed academic, and I doubt seriously that it is the beginning of a new era in science: no physical phenomenon is explained that was not known before, and the fact that CA rule 110 is universal can hardly be considered as the beginning of a new era in mathematics either.
Reviewed by Yves Pomeau
American Mathematical Society American Mathematical Society
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