Review by
Yves Pomeau in Mathematical Reviews
(MathSciNet);

MR1920418
(2003i:37002)

Wolfram,
Stephen

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
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