Customer Reviews

A New Kind of Science

5 star:		(88)
4 star:		(37)
3 star:		(52)
2 star:		(61)
1 star:		(102)

 

 

This review took almost one year. Unlike many previous referees (rank them by Amazon.com's "most helpful" feature) I read all 1197 pages including notes. Just to make sure I won't miss the odd novel insight hidden among a million trivial platitudes.

On page 27 Wolfram explains "probably the single most surprising discovery I have ever made:" a simple program can produce output that seems irregular and complex.

This has been known for six decades. Every computer science (CS) student knows the dovetailer, a very simple 2 line program that systematically lists and executes all possible programs for a universal computersuch as a Turing machine (TM). It computes all computable patterns, including all those in Wolfram's book, embodies the well-known limits of computability, and is basis of uncountable CS exercises.

Wolfram does know (page 1119) Minsky's very simple universal TMs from the 1960s. Using extensive simulations, he finds a slightly simpler one. New science? Small addition to old science. On page 675 we find a particularly simple cellular automaton (CA) and Matthew Cook's universality proof(?). This might be the most interesting chapter. It reflects that today's PCs are more powerful systematic searchers for simple rules than those of 40 years ago. No new paradigm though.

Was Wolfram at least first to view programs as potential explanations of everything? Nope. That was Zuse. Wolfram mentions him in exactly one line (page 1026): "Konrad Zuse suggested that [the universe] could be a continuous CA." This is totally misleading. Zuse's 1967 paper suggested the universe is DISCRETELY computable, possibly on a DISCRETE CA just like Wolfram's. Wolfram's causal networks (CA's with variable toplogy, chapter 9) will run on any universal CA a la Ulam & von Neumann & Conway & Zuse. Page 715 explains Wolfram's "key unifying idea" of the "principle of computational equivalence:" all processes can be viewed as computations. Well, that's exactly what Zuse wrote 3 decades ago.

Chapter 9 (2nd law of thermodynamics) elaborates (without reference)on Zuse's old insight that entropy cannot really increase in deterministically computed systems, although it often SEEMS to increase. Wolfram extends Zuse's work by a tiny margin, using today's more powerful computers to perform experiments as suggested in Zuse's 1969 book. I find it embarassing how Wolfram tries to suggest it was him who shifted a paradigm, not the legendary Zuse.

Some reviews cite Wolfram's previous reputation as a physicist and software entrepreneur, giving him the benefit of the doubt instead of immediately dismissing him as just another plagiator. Zuse's reputation is in a different league though: He built world's very first general purpose computers (1935-1941), while Wolfram is just one of many creators of useful software (Mathematica). Remarkably, in his history of computing (page 1107) Wolfram appears to try to diminuish Zuse's contributions by only mentioning Aiken's later 1944 machine.

On page 465 ff (and 505 ff on multiway systems) Wolfram asks whether there is a simple program that computes the universe. Here he sounds like Schmidhuber in his 1997 paper "A Computer Scientist's View of Life, the Universe, and Everything." Schmidhuber applied the above-mentioned simple dovetailer to all computable universes. His widely known writings come out on top when you google for "computable universes" etc, so Wolfram must have known them too, for he read an "immense number of articles books and web sites" (page xii) and executed "more than a hundred thousand mouse miles" (page xiv). He endorses Schmidhuber's "no-CA-but-TM approach" (page 486, no reference) but not his suggestion of using Levin's asymptotically optimal program searcher (1973) to find our universe's code.

On page 469 we are told that the simplest program for the data is the most probable one. No mention of the very science based on this ancient principle: Solomonoff's inductive inference theory (1960-1978); recent optimality results by Merhav & Feder & Hutter. Following Schmidhuber's "algorithmic theories of everything" (2000), short world-explaining programs are necessarily more likely, provided the world is sampled from a limit-computable prior distribution. Compare Li & Vitanyi's excellent 1997 textbook on Kolmogorov complexity.

On page 628 ff we find a lot of words on human thinking and short programs. As if this was novel! Wolfram seems totally unaware of Hutter's optimal universal rational agents (2001) based on simple programs a la Solomonoff & Kolmogorov & Levin & Chaitin. Wolfram suggests his simple programs will contribute to fine arts (page 11), neither mentioning existing, widely used, very short, fractal-based programs for computing realistic images of mountains and plants, nor the only existing art form explicitly based on simple programs: Schmidhuber's low-complexity art.

Wolfram talks a lot about reversible CAs but little about Edward Fredkin & Tom Toffoli who pioneered this field. He ignores Wheeler's "it from bit," Tegmark & Greenspan & Petrov & Marchal's papers, Moravec & Kurzweil's somewhat related books, and Greg Egan's fun SF on CA-based universes (Permutation City, 1995).

When the book came out some non-expert journalists hyped it without knowing its contents. Then cognoscenti had a look at it and recognized it as a rehash of old ideas, plus pretty pictures. And the reviews got worse and worse. As far as I can judge, positive reviews were written only by people without basic CS education and little knowledge of CS history. Some biologists and even a few physicists initially were impressed because to them it really seemed new. Maybe Wolfram's switch from physics to CS explains why he believes his thoughts are radical, not just reinventions of the wheel.

But he does know Goedel and Zuse and Turing. He must see that his own work is minor in comparison. Why does he desparately try to convince us otherwise? When I read Wolfram's first praise of the originality of his own ideas I just had to laugh. The tenth time was annoying. The hundredth time was boring. And that was my final feeling when I laid down this extremely repetitive book:exhaustion and boredom. In hindsight I know I could have saved my time. But at least I can warn others.

If a million scientists worked on a million experiments for three hundred years, would they learn as much about the universe as Stephen Wolfram does by sitting at his computer for twenty years?

Apparently not, according to Stephen Wolfram.

I'm annoyed with Wolfram for forcing me to poke fun at him like this. I've been waiting for this book a long time, and I genuinely wanted to give it a thumbs up. Unfortunately, Wolfram has made that impossible.

I gave the book three stars, but in fact I consider it almost un-ratable. What do you do with a 1200-page tome that contains a wealth of substantive and fascinating results, but which is insists, at every turn, to draw over-blown and under-supported conclusions from them? I split the difference and gave it a middling rating, but that does not convey the deep ambivalence I feel toward this work.

Given Wolfram's reputation, I expected a certain amount of hubris, and even looked forward to it. Most scientists work hard to suppress the egotism that drives them, but Wolfram's ego is out there in the open. While this can be refreshing, what I found here left me dumbfounded. For Wolfram, all of scientific history is either prelude or footnote to his own work on 1-D cellular automata. On pages 12-16 he breezily sites other work in chaos theory, non-linear dynamics and complexity theory. At the end of the book, there are hundreds of pages of footnotes describing previous history as essentially one damn thing after another - a testament to all the people that didn't see the promised land, as he has.

Wolfram attempts to usurp all credit for the "computational perspective." Assertions such as "the discoveries in this book showing that simple rules can lead to complex behavior" are repeated to the point of exhaustion. But his attempt to shock us falls flat: if that idea was ever radical, it surely would not be considered so today. The other fields that Wolfram casually dismisses have provided strong indications of the power of this principle, as well as the idea that many diverse systems are computationally equivalent. An entire generation of physicists has grown up quite accustom to these notions.

Wolfram did make very substantial and important contributions to the study of complex systems in the early eighties. But he was not the only one, and those studies have not induced a wholesale revision of science. Despite what he would have us believe, the general concepts he espouses are not that radical. It would probably be more accurate to call them expressions of the modern scientific zeitgeist.

Meanwhile, some of Wolfram's specific claims are indeed very novel, but only because they are breathtakingly arrogant. Consider his comments on two famous scientific principles: The second law of thermodynamics, and evolution by means of natural selection. Both these principles date from the mid-nineteenth century. Both have incited considerable controversy, and both have withstood mountains of empirical observations from diverse sources. Wolfram, however, calls both of them into question. Why? Because he has done 1-D cellular automations simulations on his computer that he feels make them suspicious. How does Wolfram expect to be taken seriously when he makes such assertions almost non-chalantly?

Wolfram lacks any hint of balance in assessing the true place of his results. He admits to having been a recluse for years, and it shows. The desire to free oneself of the mainstream community, to allow oneself to be more creative, is understandable and healthy. But one concomitantly loses the critical faculty that derives from being part of a dynamic community. Though Wolfram will likely never see it, what he lost by pulling away from the world has substantially outweighed what he gained. Consequently, his loss has become ours. We did not get the much shorter, but wiser, book that lurks somewhere inside this one.

This is a book of ruminations about cellular automata. It is chiefly concerned with the way that the state of a system evolves when deterministic rules are applied to it. The simplest system is a single point in either state 0 or state 1. The transition rule could be that the state "0" changes to state "1", and state "1" changes to state "0". That rule can be expressed as follows.

{1->0, 0->1}

If the system's initial state is 1, then the transition rule (repeatedly applied) yields the following alternating pattern of states.

1
0
1
0
.
.

For hundreds of pages the author discusses the behavior of 1-dimensional automata built from 3-cell transition rules. The 2^3=8 different states of a 3-cell cluster can be written in binary notation from 000 up to 111. The cell in the middle can transition to either of two binary states, yielding a total of 2^8=256 rules. Most rules lead to periodically repeating behaviors, with short periods like the alternating pattern shown above.

An exception is rule 30 (30 in binary is 00011110; these bits the right-hand-side values for the 8 transitions).

rule 30:
{ 111->0, 110->0, 101->0, 100->1, 011->1, 010->1, 001->1, 000->0 }

When applied to an initial state of a single 1 surrounded by 0's, rule 30 generates the following pattern (developing downward from the top row). The array can be displayed as a bitmap of black and white pixels, producing a visualization of the evolving state of the horizontal rows.

..00000000100000000..
..00000001110000000..
..00000011001000000..
..00000110111100000..
..00001100100010000..
..00011011110111000..
..00110010000100100..
..01101111001111110..

What excites many people about such rules (and about replacement grammars in general) is that applying the rule to an input string produces new strings whose characteristics are hard to predict. Plus, the patterns in the resulting visualization look pretty cool and are suggestive of all sorts of things found in nature. It's very easy to write computer code that will generate the patterns based on input rules, so anybody can play the game.

Lots of people have implemented cellular automata and been fascinated that the behavior is so sensitive to the choice of input string and transition rules. Watching the patterns unfold is a bit like playing the slot machines. So many possibilities. So fun to watch. Addictive to play. Great to show your friends. A meme that keeps on meming. Search the Web for "one-dimensional cellular automata" and "applet" and you will find examples that you can run in your browser.

What bothers many readers about the book is that it is like an undergraduate honors project gone haywire. Page after page of printouts of these things. Thousands of them. And with endless streams of the impressions they made on the author. "My Daily Journal of Cellular Automata" would have been a fair title. Wolfram's inflated sense of their importance, and his own, is evident in the copyright statement:

Discoveries and ideas introduced in this book, whether presented at length or not, and the legal rights and goodwill associated with them, represent valuable property of Stephen Wolfram ..

Thus he lays claim to every cellular automaton and any application thereof. Pretty annoying, coming from someone arriving late to the automaton party.

He concludes of the book proper (pp. 844-845, just before his 350 additional pages of "notes") that

.. building on what I have discovered in this book .. there is nothing fundamentally special about us. .. For my discoveries imply that whether the underlying system is a human brain, a turbulent fluid, or a cellular automaton, the behavior it exhibits will correspond to a computation of equivalent sophistication. .. [W]hat my discoveries and the Principle of Computational Equivalence now show is that .. cellular automata can achieve exactly the same level of computational sophistication as anything else.

Wolfram discovery/epiphany appears to be that all algorithms can be computed by a simple model. An example of such a model, called the "Turing machine", is taught every semester to computer science students worldwide.

It excites many people that the physical world is inherently computable, allowing computational simulations to have predictive value. It is bizarre to read Wolfram represent that he is the author of this insight.

Wolfram does a great deservice for science in his new book. Not for scientists themselves, who can easily find out that more than half the overblown statements in the book are well known old science ideas. But a deservice for the non-scientific reader, who may actually be infused by the mantra that science is just purely speculation, without actually experimentation or show-and-tell.

I can speculate as much as I want about any single model. In fact, if a model is Turing Complete, as a CA is well-known to be, any speculation can always be verified by my model! But science is about proving with experiments one's theories. Or at least showing some interesting predictions that can be made based on chosing the proposed model. But neither of them are present in the book. In fact, in the beginning of the book itself Wolfram warns the reader that scientists are non-believers and will try to destroy his idea.

So he is now in a comfortable place. He has a Turing machine, and can therefore adjust the model to explaining anything. He claims due to the nature of the model he can predict little (interesting, isn't that the decidability problem?), and that
all scientists will atack him because what he believes is true.
This is circular reasoning.

Fractals have shown us that simplicity generates complexity. Even caos models that can be run on hand calculators show that. Or a double pendulum. It is a very well-known result (shown in the 50s/60s) that the game of life is Turing complete (ie. can compute any function given an appropriate program). CA are also Turing complete, so where is the news? I can compute anything. Writing a CA that generates prime numbers amounts to finding the right "program"...

What is disappointing is that most claims are not based on any theory or verification. Let us say that he cannot really run lab experiments to check his ideas. Or even that is too early to predict any new pheonomenon based on his ideas. Well, the least he could do is to use the powerful tools of Complexity Theory (especially Kolmogorov complexity) to measure the complexity of the patterns generated by a CA. Or to at least have some results veryfying the behavior of his set of axioms ... Ok, provide with
some useful examples on how to apply CA to important problems in bioscience, engineering, etc ... Not toy computing (!).

But what he cannot do is to use ideas from other people without giving them proper credit (ok, he puts a tiny side note) or just assume that arrogance, young academic brilliance, or money can justify science. When he claims that all the major revolutions happened like this, I would suggest him to take a look at the first writings of Newton, Gauss, Einstein, Von Neumman, etc ... His predecessors had ideas that revolutionized science, but they always provided results for the scrutiny of other scientists as well. Moreover, their models and ideas could be translated to verifiable statements or allowed new and interesting predictions.

Science is about communication as much as it is about ideas. If one cannot convince other people rationally of the validity of the ideas, and also verify such ideas through experiments (the true hallmark of a scientific theory is checking predictions against experiments) then he is not scientific. He is just creating a cult. Not even a religion, because that assumes that God is not oneself. And Wolfram is convinced he is some kind of
superhero. Well, I finally understand the title of the book. A new kind of science is about follow-the-leader and do not question, have faith, because my ego is the answer you have been waiting (not even 42)...

Of course I didn't read *everything* in this enormous
book, but 100% of the sample passages read so far
can be described by the title of the review. For example,
the idea that the universe may be run by a simple cellular
automaton is not new (I heard it from Edward Fredkin 15 years
ago), and the Principle of Computational Equivalence is
not science, since it is stated so vaguely it can never
be disproved. The one *very nice* new scientific result
found in the book so far, that the rule 110 is computationally
universal, is not due to Wolfram, but to his young
employee Matthew Cook, who does receive some acknowledgment
in the small print in the back of the book, but who was prevented
to publish his work by Wolfram's lawyers for years, while
the Master was finishing his Book.

As a review of ideas close to Wolfram's heart, this
could be an interesting book to read, but it suffers
terribly from repetitious self-congratulations and
high-flying but vague pronounciations. A good editor
should have made the book about 300 pages long, then
I would have probably recommended it. A scientist
who works in isolation needs a high degree of self-criticism
to succeed, and this is exactly what Wolfram sorely lacks.

I can't believe I read the whole thing.

I'm a sucker for cellular automata, so the week ANKOS came out, I snapped it up, on the theory that even if Wolfram's revolution turned out to be a fizzle, I'd learn a lot of new twists on CA.

True enough. There's a pretty neat 400 page book here on CAs (and other discrete algorithmic systems in the same spirit - call them SASs, simple algorithmic systems, for short.) The remaining 796 pages are maddening (tedious, vainglorious, repetitious, handwaving) and fascinating by turns. It's peppered with intriguing projects, results, insights and conjectures. But Wolfram is so determined not to scare off the laymen with any display of rigorous definition or deduction that it's usually impossible to tell when he is just pontificating and when he actually knows how to cash out a given statement.

For example, after making 1-dimensional CAs the centerpiece for eight chapters, Wolfram begins to tackle physics, and acknowledges that CAs aren't the right tool for the job, because space and time (worse, a time which contradicts special relativity by requiring absolute simultaneity) are already built in. In a tour de force, he shows how there's a broad class of SASs involving self-generating networks, out of which something very much like space and time and causality, and special relativity, get constructed as a natural byproduct. This is terrific stuff. But then he can't really suggest where to go for the next step, other than brute search through the zillions of such SASs in hopes of hitting on one that generates real world physics.

There are at least two gaping holes in Wolfram's presentation, which he shrugs off far too lightly.

First: He acknowledges that it is extremely hard, given a set of constraints, to find an SAS that satisfies them, even when the constraints are simple. His moral: don't bother thinking about constraints. But out in the real world, it so happens that the realm of physics (both quantum mechanics and general relativity) does satisfy a complex and demanding set of constraints known as the principle of least action. Wolfram gives no reason for believing that an SAS satisfying such a complex constraint set will appear anytime in the first few billion, or the first few googol, of SASs studied. On the contrary, the rarity of 1-dimensional CAs satisfying the simple constraints he does examine strongly suggests the search for the New Kind of Physics will prove to be a wild goose chase.

Second: Thermodynamics, mathematical logic, and computer science have produced several sophisticated definitions for "complexity." Wolfram discusses them briefly - too briefly for the lay reader to get even a weak grasp on what they mean - and dismisses them in favor of his own "type 4" complexity, which is never more rigorously defined than "complex enough to produce pictures that look kind of like these." In chapter 11, he identifies this version of complexity with "universality", the ability of the system, if fed the proper initial conditions, to emulate a universal Turing machine. Astonishingly, he then asserts that all systems exhibiting "universality" - including such things as weather patterns and the vortices in a fast-draining tub - are essentially equivalent. He assumes that "ability to calculate x, given the right program" is the same as "ability to calculate x as fast as any other system, given the right program" - which is demonstrably false. And he assumes that "ability to calculate x, given the right program" is equivalent to "is as likely to calculate x in practice as any other system," which ignores the difference between human beings (who not infrequently calculate a long string of primes, because that's the sort of inputs they feed into their own system) and his rule 110 (which will never in a quadrillion years calculate a long string of primes unless a system more "complex" than itself deliberately sets it up with the right initial conditions.)

Because it fails so monumentally to deliver on its promises, I can't give the book more than 2 and a half stars. As a sourcebook for cool ideas on ways to build models and otherwise play with computers, though, it rates a 4. People will be drawing ideas from this book for a long time to come - or at least from the more reader-friendly books it will inspire. But science as a whole will not be noticably altered.

Of course it was not Wolfram but Konrad Zuse himself, inventor of the
first working programmable computer (1935-1941), who was the first to
suggest that the physical universe is being computed on a giant computer,
presumably a cellular automaton (CA). His first article on this topic
dates back to 1967 (in Elektronische Datenverarbeitung, pages 336-344,
vol 8). And Zuse's full-fledged book on CA-based universes came out
2 years later: Rechnender Raum, Schriften zur Datenverarbeitung,
Band 1, Friedrich Vieweg & Sohn, Braunschweig 1969.

Wolfram's book briefly mentions Zuse in the notes, but unfortunately
does not discuss his work in any satisfactory way. I guess an honest
title for his book would be something like: "More on Zuse's thesis."
An honest abstract would be something like: "In the 1960s Zuse proposed
that the universe and everything is the result of a dicrete computational
process running on a cellular automaton. Here I try to extend Zuse's
thesis as follows: 1).. 2).. 3).."

So far I have not found anything but minor extensions of Zuse's "new"
1967 science, and none of the expert reviewers (search for "Wolfram
reviews" on the web) has found anything real new either - since the
early 2002 marketing blitz the reviews have shown a tendency of becoming
both more competent and less favorable. The popular press interviews
focus on the "universe as a computer" idea - but neither there nor in
the book's main text Wolfram does anything to correct the wrong
impression it's all his idea, not Zuse's. But hey, the scientific
truth finding process will be stronger than any misleading
marketing efforts.

Nice pictures though. I give it two stars.

A New Kind Of Science may come to be viewed as the Godel, Escher, Bach of our generation. It's full of challenging big ideas that touch on nearly every field of science and beyond. It's a brilliant and delightful read and makes wonderful mind candy.

The only problem is I don't believe any of it. Wolfram bases the entire opus on the complicated behavior of a few simple cellular automata (CAs). Curiously, he never discusses any of the cool things that originally got a lot of people so excited about CAs -- topics like adaptation on the edge of chaos, and genetic algorithm evolution of specific functions. Instead, the entire book is just about how it's sometimes possible to observe complex and unpredictable patterns. And he tries over and over to convince the reader of just how important that observation is for understanding the universe.

As a supposed harbinger of a major paradigm revolution, we can contrast it with Einstein's one-time dramatic new theory of the universe. While a lot of people didn't understand it, the theories of relativity gave quite a few very specific predictions that could be -- and were successfully -- tested by observation and experiment. I've now read through the entirety of A New Kind Of Science and I can't find any specific predictions that would show his worldview explains reality any better than conventional ideas.

The only prediction he gives us relating to his theories is that every field of science will ultimately be transformed by them, and he goes on to list many of those fields. As I have a doctorate in molecular evolution, I was particularly interested in his dismissal of Darwin's theory of evolution by natural selection -- one of the most firmly established theories in science. Wolfram claims that Darwinian evolution is not sufficient to produce complex adaptations. I'm loathe to criticize an intellectual of Wolfram's stature, but his understanding of evolutionary theory, at least insofar as is presented in this book, is not very sophisticated. At any rate, anyone wanting an authoritative explication of the power of natural selection to generate complex adaptations may refer to Richard Dawkins' The Blind Watchmaker. I wish Wolfram offered some sort of testable alternative, or evidence of any kind beyond an endless display of pictures of the output of his simple programs. While the output may match the complexity observed in nature, Wolfram never makes the case that they match the adaptivity or intelligence observed in nature.

Many of these pictures are indeed very pretty. But by the fourth or fifth hundred page his obsession with these automata becomes a bit tedious. And the outworldly conclusions he draws from observing their behavior will leave you bumfuzzled. For example: because his automata are discrete in space and in time he proposes (with no further justification) that the entire universe must be made up of discrete cells of space and time. Sounds great, but where's the evidence, and where are the testable hypotheses? He goes on to propose, again with no evidence other than the observed behavior of a select few of his automata, that the mysterious rules of the universe update only one discrete time cell at any given instant. Wolfram offers countless other extrapolations to the mechanisms of nature and structure of the universe, all similarly astounding and similarly unsupported.

As I read through this opus, and especially as I neared the end, I kept asking myself -- How is it possible for someone so brilliant to have spent so many years developing something so uncompelling? I came up with three possible explanations:

1) Wolfram has gone off the deep end. Just like Dr. Richard Daystrom of Star Trek's "The Ultimate Computer", the undisputed genius who goes mad trying to exceed his former glory. Perhaps Wolfram has been staring at his pretty pictures for so long his synapses can no longer make any other kind of connection.

2) Wolfram is perpetrating an elaborate hoax on the world, much like Dr. Alan Sokal's famous "Transgressing the Boundaries" paper, a parody of the academic humanities that the editors of Social Text were fooled into publishing. But Wolfram's physics flimflam is writ on an infinitely larger scale. Just to prove he's so much smarter than every one else, and just as a practical joke, he's trying to derail the entire scientific enterprise.

And finally,

3) I have become so entrenched in the practice and paradigms of traditional science that I am unable to grasp or appreciate the profundity of what's been laid before me in the simplest of terms.

Number three is always possible. And in fact it would be wonderful to bear witness to what he's calling the greatest discovery in the history of science, even if it does fly over my head at Mach 2. Wolfram is one of the smartest and most accomplished residents of the universe, and even though one of the basic tenets of the (traditional) scientific method is that the validity of a claim is judged independently of the stature and reputation of the one who proposes it, it's difficult not to give someone like Wolfram the benefit of the doubt -- no matter how much of a stretch.

All the same, I recommend this book to anyone who enjoys being intellectually stimulated and likes to think about big ideas. Even if he's wrong, I'm sure glad I read it.

I don't see how this book is revolutionary. He says that models based on simple programs operating on discrete elements can do a much better job of capturing the real behavior of the universe than mathematical equations. But surely everyone knows that right? That's why we have finite element analysis, computational fluid dynamics, etc. The math is still the foundation, and is useful for describing simple cases exactly, but to model the universe effectively in all its complexity you often need a computer.

In Chapter 8, he shows a simple cellular automaton that models crystal growth. He says that his model is superior to standard models that are based on traditional mathematical equations because it does a much better job of capturing the intricate structure of real crystals. I find it hard to believe that he is the first person to come up with the idea of modelling crystal growth using a computer, and even if he is, I don't see that as particularly groundbreaking. His cellular automata do an OK job of modelling one particular type of snowflake growth, for example, but that's because he has tuned the rules to get
the expected result. His automaton does not cover other types of snowflakes though, and they do not give much insight into how and why real snowflakes grow.

Also in that chapter he presents a (crude, IMHO) fluid dynamics simulation using cellular automata and shows how turbulent flow occurs in his simulation. No surprise there, since people have been running fluid dynamics simulations for some time now to study turbulent flow. In the notes he seems to indicate that his simulation is superior to existing fluid dynamics simulations because existing simulations are discretized, ignoring the fact that his simulation is discrete as well.

Then he shows how seemingly complex patterns in biology, like the structure of leaves and trees, can be reproduced using simple rules. To anyone who has read a book on fractals this will not be new information. The interesting question is, why do those simple rules work so well for describing nature? What is it about leaves that make them grow in that pattern? He didn't spend much time on that.

Later he comes up with a cellular automaton with reversible rules that does not obey the second law of thermodynamics. He uses this to show that not all systems in nature (notably biological ones) obey the second law. Well that's fine with me, as long as all thermodynamic systems obey it.

It's a strange book but still an interesting one, as a review of a lot of info about science and fractals and cellular automata and how they interrelate, complete with lots of pretty pictures, but I don't really see how it's groundbreaking. He makes some radical assertions (like saying that the universe may be a big cellular automaton) but I don't find them particularly compelling. I believe that the universe is some large mechanism with fairly simple rules that result in complex behavior (and I thought everyone agreed on this point; that's the goal of physics, to find the fundamental equations and them use them to model reality, right?) but I think mathematics are a much better way of expressing the rules, and existing computational models are a much better way of applying them.

This is a self-published book by a self-described genius who claims to have revolutionized all of science. It's about what you'd expect for such a publication. Wolfram is not a crank in the usual sense (he doesn't claim that Einstein was wrong, for example) because he had a solid physics training before becoming a software entrepreneur, so he knows better than that. Rather, he simply asserts that he has superseded all earlier science. A modest claim.

What has he done? Cellular automata were invented by von Neumann (an authentic genius, if anyone ever was) about 50 years ago as a tool for thinking about the mathematics of computation. Now that computers are on every desk, anyone can carry out elaborate and rapid computations of cellular automata. The results are lots of pretty pictures, which somewhat resemble snowflakes, river deltas, and other natural phenomena. Unfortunately, they do not help us understand the natural phenomena. The resemblance isn't exact and the cellular automata have no predictive power.

In one way many of the pictures made by cellular automata are close to the natural phenomena: they are scale-invariant. This idea was fully developed by Kolmogorov in the 1940's and M. Fisher in the late 1960's. It received a great deal of publicity beginning in the 1980's, mostly under the label ``fractals'', which are related to the pictures made by cellular automata. Wolfram doesn't have much new to add.

The physicist will ask if Wolfram's work helps him to understand semiconductors and elementary particles, the astronomer quasars and black holes, the chemist chemical reactions and bonds and the biologist DNA and proteins. In each case (and in any other scientific questions you might name) the answer is no.

Where did Wolfram go wrong? He had so much fun playing with cellular automata on the computer he forgot that they have nothing to do with the natural world, which is what science is about. Cut off from the give-and-take and free criticism of the academic world, surrounded only by acolytes at his software company (don't tell the boss he is wrong, especially when the boss thinks he is the greatest genius who ever lived), his fantasies of greatness grew on themselves. It's a pity, because he really is very clever fellow and could have done good work if only he hadn't isolated himself. But he is deceiving the reader when he claims that this work will revolutionize how we think about science. It doesn't contribute at all.

At least Bill Gates doesn't claim to be the world's greatest genius, or greatest programmer. In contrast, that is refreshing.