[The Montana Professor 14.1, Fall 2003 <http://mtprof.msun.edu>]
Stephen Wolfram
Champaign, IL: Wolfram Media, 2002
1192 pp., $44.95 hc
William Locke
Earth Sciences
MSU-Bozeman
It is not often that one meets a messiah--even a self-proclaimed one. In monotheism, widely accepted messiahs (Jesus, Mohamed, Joseph Smith) have appeared every thousand years or so--self-proclaimed ones are more common. So what is Stephen Wolfram in the context of science: messiah or messianic complex? I contend the latter, but true believers will not agree.
This book, which has been reviewed in publications from American Scientist to U.S. News and World Report, is the product of two decades of work by Wolfram, who is best known as the originator of the Mathematica software, a flexible and powerful package of analytical and display tools. On a price-per-pound basis this book is a great deal; considering the high-quality paper used to ensure no loss of resolution in the figures, it is an absolute steal. In the book, Wolfram exhaustively describes, displays, and promotes the ability of simple actions--"automata"--to engender complex outcomes. The majority of the discussion deals with cellular automata (CA) in which the color of a cell in a two-dimensional array (white or black) is based on rules considering the colors of the three cells immediately above and adjacent to the cell of interest. Other automata involving multicolored cells, substitutions, branching, and higher dimensions are also considered. In natural rather than computational systems, a cellular automaton might be a genetic signal that forces a simple outcome, the combined effect of which, on the scale of an entire organism, would be a complex variety of outcomes. The following discussion deals only with the basic computational CA.
As Wolfram emphasizes countless times in the text, intuition suggests that the 256 possible "rules" relating the colors of the three seed cells to that of the destination cell will yield only trivial outcomes--solid black or white, stripes, or checkerboard patterns. In actuality, however, more complex patterns such as fractal (patterns which are self-similar regardless of scale) and "random" distributions of cells also occur. Rather than printing pictures of the possible outcomes (as Wolfram does in Chapter 3, pp 54-56), I give you instructions to make your own in the Appendix.
So--why do cellular automata matter? In a nutshell, Wolfram summarizes the common idea that simple rules shouldn't yield complex outcomes, demonstrates that simple rules do yield complex outcomes, then concludes that complex outcomes must be the result of simple rules, and these are the rules! Wolfram dismisses all of the alternative explanations for complex behavior--catastrophe, chaos, complexity, dynamical systems, evolution, fractal, nonlinear dynamics, and statistical mechanics theories--in favor of CA. If you need an example of the problems that can arise from unrestrained inductive reasoning, this is it!
This work does have major strengths in addition to the comprehensive discussion of several types of automata. The discussion of the concept of randomness in Chapter 7 is outstanding, and should be read by anyone who uses a random-number generator in numerical programming. The suggested applications to natural systems in Chapter 8, including fracture patterns, turbulence, mutations, evolution (both parallel and convergent), biological coloration, and financial markets, are thought-provoking. The discussion of perception and analysis in Chapter 10, including randomness (again), compression algorithms, and reversibility (including cryptography) is equally exciting. However, the weaknesses are legion.
This work is intensely egocentric--"I" or "me" is used 11 times on a single page (p. 2)! Several published reviews have emphasized the contributions others have made to the study of CA, contributions downplayed or ignored by Wolfram. There is no "Literature Cited" section, because there is almost no cited literature. Wolfram does not appear to understand what "science" is. In the three-page preface he uses "science" or a variant (scientist, scientific) 17 times without defining it. In the context of the book as a whole, it appears that Wolfram defines "science" as methodologies by which natural outcomes can be computationally simulated. I use the term "simulation" here to describe a model that looks like the real thing, but need not be the real thing--C3PO in Star Wars was a simulation of an android robot (he was actually a man in a robot suit). My operant definition of "science" is "the processes by which humans seek to understand nature." Although debate will forever rage as to the relative values of proof and disproof in science, there is no question about the values of reproducibility and predictive understanding, often displayed through process-based models whose outcomes can be tested against reality. Wolfram dismisses process-based models: "...all that any model is supposed to do...is to provide an abstract representation of effects that are important in determining the behavior of a system" (p. 366). But are CA "models" predictive? That should be testable.
An essential character of simple rules is that there are a finite number of random-appearing behaviors (256 in the case of the standard CA), thus such behaviors should repeat. Wolfram states that "Scattered around the scientific literature...I have managed to find some cases where multiple runs of the same carefully-controlled experiment are reported, and in which there are clear hints of repeatability even in behavior that looks quite random." (p. 326) In the endnotes he summarizes: "Over the years, I have asked many experimental scientists about repeatability in seemingly random data, and in almost all cases they have told me that they have never looked for such a thing. But in a few cases they say that in fact on thinking about it they remember various forms of repeatability" (p. 976). He goes on to describe three such cases without citation.
Wolfram confuses "replication" with "reproduction:" "...extremely simple programs seem able to capture the essential mechanisms for a great many physical phenomena that have previously seemed completely mysterious" (p. 8). In fact, his simple programs entirely ignore all of the recognized mechanisms for physical phenomena by assuming that automata drive everything. In places he distorts recognized mechanisms--see p. 370 for an oversimplified opinion as to the mechanisms for the growth of snowflakes. In others he confuses modeling with reality: "Examples of reversible cellular automata with various rules. Some quickly randomize, as the Second Law of Thermodynamics would suggest. But others do not--and thus in effect do not obey the Second Law of Thermodynamics" (p. 452).
The essence of Wolfram's argument about the significance of CA is developed in the 109 pages of the last two chapters, in which he proclaims:
That computation is capable of modeling anything,
that rule 110 (see Appendix) is "universal," that is, that its output includes the entire range of possible outcomes, thus
that there is no need for more than rule 110 to explain the universe: "So perhaps in the end there is the least to explain if I am correct that the universe just follows a single, simple, underlying rule" (p. 471).
This is summarized in his Principle of Computational Equivalence: "that whenever one sees behavior that is not obviously simple...it can be thought of as corresponding to a computation of equivalent sophistication" (p. 5). This philosophy is summarized by a straw man: "Traditional intuition might have made one assume that there must be a direct correspondence between the complexity of observed behavior and the complexity of underlying rules. But one of the central discoveries of this book is that in fact there is not" (p. 351).
In essence, Wolfram would have you believe that a rule for bimodal (black/white) behavior based on three seed cells of undefined origin is the answer to the fundamental questions of the universe. (Douglas Adams, in Hitchhiker's Guide to the Galaxy and other works contends that the answer is "42.") He neither proves his hypothesis through acceptable theorems and postulates nor tests it through experiment, but apparently expects you to have faith in his inductive edifice. Wolfram contends that cellular automata replicate and in fact explain natural phenomena; an alternative hypothesis is that they merely mimic nature. Titles that are more descriptive of the book's content might be No Kind of Science, or A New Kind of Religion. I suspect that acolytes and proselytes to the faith will be gratefully accepted.
Rules for growing two-state (black/white) cellular automata:
| Seed1 | Seed2 | Seed3 |
| Destination |
Eight possible conformations of the three seed cells (1=black, 0=white):
1 2 3 4 5 6 7 8
111 110 101 100 011 010 001 000
256 possible outcomes of the destination cell (1=black, 0=white):
0 0 0 0 0 0 0 0 (rule 0)
0 0 0 0 0 0 0 1 (rule 1)
0 0 0 0 0 0 1 0 (rule 2)
0 0 0 0 0 0 1 1 (rule 3)
.
0 1 1 0 1 1 1 0 (rule 110)
.
1 1 1 1 1 1 1 1 (rule 255)
For example, in words, rule 0 says that regardless of the colors of the seed cells, the destination cell will be white (rule 255 says it will be black). Rule 1 says that the destination cell will be black only if all three of the seed cells are white. Now that you know the rules of the game--let's play! The following recipe is designed using syntax from Microsoft Excel--similar outcomes can be generated in other spreadsheets or programming languages.
Recipe (Microsoft Excel) for rule 30
Select all, right-click column header, make column width 1.5
Insert a space into cell A1; copy into remaining cells in row 1 (to cell IV1)
Insert an X into cell DX1 (for this purpose X symbolizes "black")
Insert a space into cell A2
Insert the following formula into cell B2 and copy it into the remaining cells on row 2 (to IU2):
=IF(OR(AND(A1="X",B1=" ",C1=" "),AND(A1=" ",B1="X",C1="X"),AND(A1=" ",B1="X",C1=" "),AND(A1=" ",B1=" ",C1="X")),"X"," ")
Insert a space into cell IV2. [Note: the spaces in the end cells reduce error propagation into the model.]
Select row 2 and copy it as far down as you wish--128 rows will grow the model to its edges, but you needn't stop there.
Set View magnification to 25% to see most of the model (depending on your screen resolution).
Rule 30 is one of the more interesting ones in that it generates a pattern with both order (along the left side) and disorder (to the right). Note that, in spreadsheet format, you can experiment with the insertion of additional "black" (Xs) seed cells and with "mutations" (random changes of cell color) within the model. You can also experiment with changing the conditional expressions--the rules--if you are a nerd (like me).
[The Montana Professor 14.1, Fall 2003 <http://mtprof.msun.edu>]