Critical Review of "A New Kind of Science"

12 July 2002

With extreme hubris, Wolfram has titled his new book on cellular automata "A New Kind of Science".

But it's not new.

And it's not science.

Solo Discovery?

The main text of the book gives the strong impression that everything pictured and described in the book is Wolfram's own invention. Moreover, he often implies that various areas of science have been much more limited in their scope than they actually are.

Consequently, it is vital to read the notes at the back of the book in combination with the main text, in order to restore a little of the balance. The notes are much better at giving proper credit to the vast reams of work that Wolfram is building on.

Part of the problem is Wolfram's insistance on using his own terminology for concepts and ideas that have perfectly good names in regular mathematics and science. Examples: he always referring to fractals as "nested" (and never makes clear whether the term includes less structured fractals or not), he doesn't refer to well-known pictures by their common name in the main text (such as the Sierpinski gasket or the Koch curve), he calls refers to lossy compression as "irreversible", he insists on using Mathematica notation rather than standard mathematical notation (there may indeed be a million Mathematica users, but there are considerably more who understand normal notation and don't have access to this expensive tool).

I suspect Wolfram's reasons for this are pedagogical, so as not to put off his less widely educated readers and to ensure a consistency of style throughout the book. However, this approach is likely to put off the serious scientific reader—particularly in combination with his other individual ways of expressing his ideas (such as starting many paragraphs with a conjunction, or avoiding the use of color for figures).

More seriously, even allowing for the notes, the book is seriously misleading in its description of some areas of science, and plain wrong in a few others.

One misleading example is the section on partial differential equations in chapter 4. Wolfram states that "in fact almost all the work—at least in one dimension—has concentrated on just the three specific equations" (p.162) which are the diffusion equation, the wave equation and the sine-Gordon equation.

The notes to the chapter themselves bely this, as they give (p.925) pictures of another three one-dimensional equations which have been studied (Burger's equation, a nonlinear Schrödinger equation and the Kuramoto-Shivshinsky equation), and of course there are many more—for example the Ginsburg-Landau equation (which his nonlinear Schödinger equation is a special case of) or the Korteweg-de Vries equation.

A more serious example where Wolfram is simply wrong is in his treatment of chaos theory. Throughout the book, he equates chaos theory with the phenomenon of sensitive dependence on initial conditions (SDIC). This allows him to claim that any randomness that occurs in a chaotic system is just a consequence of the inherent randomness in the least significant digits of the initial condition. In turn, this sets the stage for what he claims is one of his own major discoveries: that simple programs can inherently generate complex behavior and randomness.

However, SDIC is just one of the attributes of chaotic behavior. Another important attribute of chaos theory—and indeed the reason why the field is called "chaos" in the first place—is the observation that complicated, apparently random behavior can arise from simple systems (when they are nonlinear). When I made a quick survey of five books on chaos theory on my bookshelf, only three of them mentioned SDIC as part of their definition of chaos, and in those cases it was only part of the definition.

So why is Wolfram so comprehensively ignoring the normal understanding in this field? A cynical part of me suggests that it would be too inconvenient for him to completely give credit to a field whose key observation is that complicated, apparently random behavior can arise from simple systems. This is exactly the key observation that he himself is trying to lay sole claim to—in over 30 places he mentions that this observation is one of the "main discoveries" of the book. (To be fair, Wolfram's own earlier work was indeed one of the many strands that helped propel this observation to the forefront of chaos theory.)

This key point undermines the whole of the first half of the book, and makes much of the second half sound familiar—when chaos theory appeared on the scene, its proponents also made the case for it being a key ingredient of little understood complex behavior in a range of fields from fluid dynamics to population biology to cardiology to to economics.

Another area where Wolfram misleads his audience is in his presentation of the Principle of Computational Equivalence, the centrepoint of his final chapter. He only tangentially makes clear that the main content of this Principle is mostly just a restatement of the universality of computation, a result known since the 1960s; over and above this, the Principle boils down to the observation that he suspects that such universal systems are far more ubiquitous than people have previously realised.

A final example where Wolfram neglects to mention in the main text that he's not the only bold explorer in a new intellectual landscape is regarding the potential use of cellular automata and other rule-based systems to model fundamental physics. The work of Zuse, Fredkin, Toffoli et al is mentioned in the notes, but how many readers are going to wade through that 8 point text?

I'll admit that some part of this section arise from a traditional scientist's horror at seeing the normal forms for crediting predecessors skimped on. Wolfram is careful never to actually claim credit for something he hasn't produced; however, he's good at wording the main text so that it implies he has discovered things.

But there is a much more serious point than mere pique. When all of the so-called discoveries of the book are added up, the net total is not an intellectual equivalent of a new "Principia Mathematica", but instead an intellectual equivalent of a Scientific American survey article—albeit one with a unusual breadth of scope and a slightly unusual point of view.


My claim in the prologue of this review that Wolfram's book does not qualify as science is perhaps a little overstated.

However, there is a key point: serious science should be predictive, not just descriptive. To qualify as science that applies to the real world, I would have expected to see some kind of claim in the book which could be verified against the behavior of the real world. Note that I'm not expecting him to have actually performed the verification yet (the book has only just come out, after all), but that there should be some indication of a path that would lead to verifiable, falsifiable predictions.

This is not a new accusation. The fields of chaos theory and complexity theory (which Wolfram is essentially summarizing) have had similar accusations levelled at them, with some justification. However, in those fields there are genuine concrete results that can be pointed at, examined and potentially disproved—for example, the universal behavior of any iterated unimodal map (Feigenbaum[1979], Lanford [1982]), the route to chaos in a homoclinic system (Shil'nikov [1970]), the evolution of a more optimal sorting network (Hillis [1990]).

Even within the boundaries of descriptive science, Wolfram leaves himself escape routes in the event of serious criticism—the book is littered with weasel words like "seems", "almost always", "appears".

Let's be clear here: my complaint here is not that Wolfram hasn't verified his predictions, but that he hasn't made any predictions that admit verification.

When discussing his Principle of Computational Equivalence and the phenomenon of computational irreducibility, Wolfram starts to make clear the point that his approach to science and modelling may never be able to produce model behavior in a shorter time than the system itself produces the actual behavior.

Again, this phenomenon of computational irreducibility is not Wolfram's discovery, but the key point here is that you can begin to see why "traditional" science has not devoted much attention to this kind of modelling. For if you cannot get conceptual understanding or useful predictions from a model, what use is it?

The Good Stuff

To try to bring a little balance to this review, I should point out that there are definite areas of the book that I enjoyed reading.

Much of the background material presented in the notes is well-presented and thorough, and presentation of the operations of various rule based systems is clear and easily understood.

There are also some genuine nuggets of real science in the book, such as the models for pigmentation and branching in chapter 8.

The rough outline of a rule-based approach to fundamental physics presented in chapter 9 is the most tantalizing chapter, however. This chapter does actually show signs of promise for generating an interesting (and verifiable) scientific model. Personally, I think that Wolfram should have concentrated on fleshing out this material for the last ten years, rather than exploring a zillion different cellular automata models—if he had, there's just a chance that he might have been able to live up to the title of his book.


I wouldn't be quite as negative as Freeman Dyson (whose alleged one word review was "worthless"), but this book definitely does not live up to its own hype.

The book actually reminds me of Wolfram's other huge creation, Mathematica. For Mathematica Wolfram built on a lot of well-known techniques, a few of which he himself had actually invented, and tied it together into one coherent whole—but which was nowhere near as unique and spectacular as its own publicity labelled it.

With "A New Kind of Science", Wolfram has brought together existing observations from a range of disciplines, combined them with his own particular worldview to attempt to produce a coherent whole. It's an interesting tour of modern science (particularly in the notes)—but that's not what the book presents itself as.

The book presents itself as a earth-shaking change to the fundamental paradigms of science, and it just plain isn't.


This review is driven from my notes on reading the book, together with the references that are relevant to ideas in the book and which Wolfram refuses to give (with a reasonable justification that the book is already too large). Some days I suspect that I may be one of the few people outside of Wolfram's crack team of hagiographers who has actually read the entire book.

My main concern with the book is that a reader who is not already aware of the work done in related areas is going to come away from the book with a very misleading impression of Wolfram's contribution to knowledge. Hopefully, I can challenge that impression.

Copyright (c) 2002-2003 David Drysdale

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