A New Kind of Science?

A New Kind of Science
Stephen Wolfram
Wolfram Media, Inc.
Champaign IL 2002
$44.95 (1192 pages)
IBSN: I-57955-088-8

Early in the 1980s Stephen Wolfram began to work in earnest upon 
cellular automata.  These are a class of computer model which may be 
visualized as a set of memory locations, each containing one bit. 
These bits are updated in a succession of time steps.   In each step, 
the new value of each bit depends upon the values of neighboring 
bits.   Wolfram particularly studied the class of automata in which 
all the bits are arranged upon a line, and each bit is updated using 
the very same functional dependence upon the bits at that site and 
its two nearest neighbors.  There are 256 different automata of this 
type.  Wolfram made it his business to study systematically all of 
these different automata, using extensive computer simulations, and 
to think about and generalize from what he thereby uncovered.  A New 
Kind of Science,  written and published by Stephen Wolfram,  is the 
outcome of these, and related,  studies.

The book, hereafter described as NKS,  is several things at once. 
First, it is an excellent pedagogical tool for introducing a reader, 
even one who has no knowledge of advanced mathematics, to some of the 
concepts of modern computer science, mathematics, and physics. 
Space-time diagrams of the bits generated by the model show four 
separate patterns: dull uniformity, periodic time-dependence, fractal 
behavior, and truly complex non repetitive patterns.   A discussion 
of this classification, which I think is originally due to Wolfram, 
enables the author to introduce modern concepts of complexity.  Using 
this he can discuss fractals, as they were introduced by Mandelbrot, 
the idea of universal computation, as it was developed by Turing, 
Church, and others, and describe the generation of complex patterns 
in a context in which one can actually see what is going on. 
Teaching continues with the description of several kinds of 
computers, and of the conceptualization of natural processes as some 
kind of computation.  This is a tour de force  of clarity and 
simplicity.

Since the book covers a huge territory it should not be surprising to 
find a few errors in it. For example in my own area of phase 
transitions NKS says, incorrectly, on page 981 that phase transitions 
involve a discontinuity in the partition function and on page 983 
that symmetries are usually not important in phase transition 
problems--again false.  Errors like these will, no doubt, be ironed 
out in subsequent editions.

NKS is a very personal book.  Again and again Wolfram tells the 
reader how he discovered some new fact about automata, or used the 
automata to construct a new illustration of old ideas, or how he used 
his knowledge of these systems to construct the beginning of new 
hypotheses about mathematics or science. These descriptions of the 
personal events in the development of Wolfram's understanding are 
valuable both for the insights they give into the science involved 
and also for the way they reveal things about the author himself. 
This aspect of the book is truly unique.

However the reporting of history is spotty, and sometimes quite weak. 
That weakness is partially structural in that the author has not 
allowed himself any footnotes in the text. Instead, at the back of 
the book  there are three hundred and fifty pages of notes, including 
both descriptions of history and also additional information about 
the topics in the text. Any given topic might be covered at several 
different points. These notes do not contain any references either, 
they just give authors and sometimes dates. To find original sources 
one must look up a web site.  I did not choose to do this.  Because 
of this structure and an overuse of the words "discover" or 
"discovery" it is hard to distinguish between things which were 
explained previously by Wolfram and coauthors, well-known things here 
newly explained in the context of automata, or things which are 
genuinely new.    In fact the personal nature of the exposition often 
interferes with giving a full historical description.  For example, I 
did find it interesting that the author devised an partially correct 
automaton description of hydrodynamic flow in 1973.  But I, for one, 
also find it interesting to know that a model with some of the same 
virtues and defects was published by Jack Swift and myself in 1968 
(Phys. Rev. 165 310) and even more interesting that a fully correct 
model was defined by Frisch, Hasslacher and Pomeau in 1986 (Phys. 
Rev. Lett. 56 1595).   NKS does not help one learn the latter facts.

Another structural difficulty arises from NKS's use of a proprietary 
computer language in place of the usual mathematical notation.  Since 
I am not fluent in the proprietary language, I cannot be sure about 
the meaning and correctness of the equations used.

The remainder of this review is concerned with the NKS claim that its 
author has discovered a "new kind of science." A "new science" claim 
is hardly new.  One can find it in James Gleick's excellent 1987 book 
Chaos, Making a New Science.  The new science mentioned in that 
popularization also covers topics like the sensitive dependence upon 
initial conditions, the generation of strange attractors and of 
fractals, the onset of chaos, and complexity.  Since neither Wolfram 
nor the other contemporaneous students of automata are even mentioned 
in this work, one might properly doubt that the Wolfram work from the 
early 1980s could be the new science under discussion.

However, in that era automaton studies were rather divorced from the 
main stream of work in chaos and complexity. The leader of the main 
automaton group of that period was Edward Fredkin of MIT. His vision 
is described in the 1988 book Three Scientists and Their Gods by 
Robert Wright.  This book describes the philosophical and some of the 
technical work of Fredkin and describes thinking which lies at the 
base of Wolfram's world view.  Fredkin especially stresses the idea 
that everything is a computation, and that the universe is a digital 
computer. In this book, Wolfram is mentioned only once (see page 62) 
when disagreements between the two scientists are emphasized.  Thus 
neither Wright's nor Gleick's independent studies supports possible 
claims that Wolfram played a major role in making any "new kind of 
science" in the early 1980s or before.

In the preface Wolfram says that the new kind of science was 
discovered in the period since 1991 and brought together in this 
volume.  So we must look for them in the concepts, calculations, and 
theorems described for the first time in this book.   I found here 
interesting things that were new to me.  NKS mentions but does not 
display a 1994 proof by Matthew Cook that one-dimensional automaton 
number 110 is a universal computer so that it could do any 
calculation that could be performed by a Turing machine. This so-far 
unpublished proof identifies a particularly simple automaton example 
of a Universal Turing machine. I think this is the simplest example 
identified so far.   Some data included in NKS were also new to me, 
especially counts of the proportion of automata of various kinds 
which fall into each of the four classes.  It is interesting to see 
how the simplest systems are capable of generating chaos and 
universal behavior and to see how variation of a parameter could give 
rise to a transition from a mostly repetitive  to a mostly chaotic 
behavior.   But these data mostly serve to illustrate well-known 
ideas.

NKS chapter 9 contains provocative speculations related to how 
automaton models might incorporate quantum theory and gravity, via 
random network models and path independence. These speculations are, 
I think, new. The view that the universe is an automaton is due to 
Fredkin. But, the specific elements in the speculation emulate 
previous two dimensional quantum gravity theories and integrable 
systems work.  This chapter describes a part-formed idea, 
exciting.... but not yet science.

The book's longest discussion is devoted to a Principle of 
Computational Equivalence, which roughly puts all chaotic 
calculations in the same category.  This is a restatement and 
extension of Wolfram's 1980s idea that classifies automaton outputs 
into four categories. So far this classification has proven neither 
subtle nor fruitful.

The remaining apparently new material in the book is speculative, and 
appears to be even less worked out than the examples just mentioned.

From my reading, I cannot support the view that any "new kind of 
science" is displayed in NKS.   I see neither new kinds of 
calculations, nor new analytic theory, nor comparison with 
experiment.

Per Bak's 1997 book, How Nature Works, covers subjects similar to 
those of Gleick and NKS, and looks specifically at automaton studies 
and at Wolfram.  Bak's judgment is that "more than anyone, Stephen 
Wolfram, pointed out that these simple devices could be used as a 
laboratory for studying complex phenomena." (page 106).    But he 
also said on page 107 that "Wolfram never produced any theory of 
cellular automata." And that is where the subject stands to this day.

Leo P. Kadanoff
University of Chicago
copyright Leo Kadanoff 2002