Stephen Wolfram’s Science

by Greg Egan


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For the past eleven years, Stephen Wolfram has been labouring single-mindedly on a book that presents his ideas about the science of complexity. The result, an 1197-page, self-published behemoth entitled A New Kind of Science, finally hit the shops in May. Preceded by a great deal of hype, and followed by a fair amount of controversy, it has already sold out an initial print run of 50,000 and entered the non-fiction best seller lists. A second printing is underway.

Wolfram is a 43-year-old scientist and entrepreneur, born in London and now living in the USA. He received a PhD in theoretical physics at the age of 20, and went on to work at a number of prestigious academic centres. In the early 1980s, he published several papers on the subject of cellular automata, or CAs. These are a kind of mathematical system, typically comprising squares laid out in one or two dimensions; individual squares, known as cells, can be in a variety of states, such as black or white, and they change according to a fixed rule that depends upon their immediate surroundings. Starting with a given pattern of cells, the rule is applied repeatedly, and the pattern changes. With simple computer programs, the behaviour of various rules can be explored.

This might sound like a trivial diversion, but when the idea was formalised by the mathematician John Von Neumann in the 1950s, he put it to some heavy-duty uses, sketching a two-hundred-thousand-cell “device” that could operate in a CA universe and make copies of itself. Before too long it was realised that CAs can have a property known as computational universality: given the right pattern to act as a program, they can carry out absolutely any calculation that ordinary computers can perform.

For an early-model CA to achieve such feats, its cells needed to have many possible states, and the rules to be followed were complicated. In contrast, Wolfram’s papers often dealt with one-dimensional grids of black-or-white squares, obeying the simplest possible rules. Surprisingly, he found that even these minimalist CAs could generate patterns so complex that they included sequences of black and white that would appear entirely random to anyone unaware of their origin.

Wolfram’s work contributed to the burgeoning science of complexity, but he grew frustrated with the direction in which his colleagues were taking the field. In 1986, he founded Wolfram Research, a successful business built around a software package called Mathematica, which enables users to manipulate algebraic expressions and other abstract mathematical constructs. Soon afterwards, he largely withdrew from ordinary academic life, and devoted himself to the independent research that culminated in the publication of his book.

The central thesis of A New Kind of Science follows from a remarkable discovery: one of Wolfram’s simple, one-dimensional, black-or-white CAs exhibits computational universality. When the consecutive states of this CA, which Wolfram calls Rule 110, are laid out in a two-dimensional grid, tracks of white triangles streak across a background pattern, colliding and changing, eerily reminiscent of particles in a cloud chamber. The flow of information carried by these tracks can be controlled and interpreted in such a way that any computation whatsoever can be encoded in their interactions.

This is startling, but the idea is not to throw out your microchips and replace them with anything resembling Rule 110, or at least not yet. Wolfram reasons that if something as basic as Rule 110 can exhibit universality, the natural world must be full of other examples of sophisticated behaviour with humble origins. If he’s right, the consequences are quite different from those already familiar from chaos theory. A random puff of air from a butterfly’s wings might exert an influence that snowballs to push the weather in unpredictable directions, but Wolfram’s notion of intrinsic complexity suggests that even very simple, isolated systems, with no unknown influences, can generate equally complex behaviour.

Traditional science, Wolfram claims, focuses on systems with behaviour so simple that it can be summed up with a mathematical formula, a short-cut which lets us compute in a single stroke what the system will be doing at any future time. To find out what a computer program does, though, there is often no faster method than actually running the program. One consequence of universality in nature, then, will be computational irreducibility: the absence of any short-cuts to predicting the way the system will behave. Wolfram believes that it will take a new kind of science, based on an understanding of the behaviour of simple programs, to explain how much of nature works.

Wolfram applies this perspective to a broad range of subjects, from human thought to the underlying structure of the universe. Sometimes his evidence is persuasive, most strikingly when he displays photographs of a variety of mollusc shells. To a casual observer, the markings on these shells would be intriguing, but to a reader who has seen the array of patterns produced by simple CAs, the correspondence between the two is jaw-dropping. Another compelling result is a demonstration of how an almost laughably primitive CA model of the movement of particles in a fluid can exhibit vortices resembling those in real turbulent flows. With examples like these, Wolfram shows that we can often get a good sense of how things work by capturing a few essential features. Trying to make use of everything we know about a system can mire us in irrelevant details, and we need to abandon the prejudice that says complex rules are needed for complex behaviour.

A New Kind of Science is an ambitious book, and in his desire to leave no part of science untouched, Wolfram often overstates his case. He argues that most complexity in living organisms confers no survival advantage, being too unpredictable a tool to be employed in any matter of life or death. He is probably right that some animals’ pigmentation patterns aren’t really the best-of-all-possible camouflage suits for their environment, but I think he goes too far when he claims that natural selection alone would have left the planet covered in microbes, and that most of the richness of life is a kind of random decoration. The intricacies achieved by building things up from simple, robust components doesn’t meet Wolfram’s definition of complexity, but it’s no less important.

Wolfram sketches a model for space-time as a special kind of network, in which events are discrete mathematical points, linked by abstract connections to the other events they influence. The continuous space and time of ordinary experience emerge only on a larger scale. He suggests a clever construction for the rules that might underlie this network, but there is no corroborating evidence for the particular model he favours.

Wolfram dismisses recent work in quantum gravity which points to related ideas, because it doesn’t go far enough and subsume quantum mechanics itself under the banner of universality. This is a pattern repeated throughout the book: Wolfram credits other scientists begrudgingly, and even those who laid the foundations for his own work are chided for missing the point. Sometimes a touch of egotism can bring a science book to life, but as a poet of braggadocio Wolfram is closer to Mike Tyson than Muhammad Ali.

The opening chapters are an easy read, as Wolfram uses ingenious graphics to describe the rules of his CAs and show us their behaviour. Later, as he introduces half a dozen other systems exhibiting exactly the same spectrum of behaviour, the very point he’s making leads to monotony. At times it’s like being shown every beetle in a very large collection. The prose throughout is clear, but it’s also formulaic and repetitive, so any liveliness comes solely from the ideas; luckily, these start to pick up around chapter seven. In surer hands, A New Kind of Science might have been a pleasure to equal Douglas Hofstadter’s classic Gödel, Escher, Bach, but for all its scope and boldness it remains resolutely unseductive.

Wolfram suggests that this book will reshape science. I doubt that, but his insight is important. Complexity need not arise from an overwhelming morass of details and connections, too vast to fathom. Sometimes it grows on the spot from simple rules. When that’s the case, though predictions might remain beyond our grasp, we can still achieve understanding.


Wolfram’s promotional site for the book is www.wolframscience.com

A mangled version of this review was published in the Bulletin, July 23, 2002, stuffed full of clichés, non sequiturs and other drivel. To be fair, this was not entirely the magazine’s fault. I was sent an email offering me a chance to see the edited version, but my computer was out of action at the time, so the commissioning editor was unaware of my opinion of these changes before the piece went to print.


A cellular automaton, or CA, typically consists of a row of squares, known as cells, which can be in various states, such as black or white. A rule is chosen which spells out how the current state of each cell and its neighbours decides the new state the cell will assume. The eight possibilities for Wolfram’s Rule 110 are shown below. For example, a black cell with both neighbours black changes to white.

CA Rules

The changing appearance of the row of cells can be laid out in a two-dimensional grid. Repeatedly applying Rule 110 to a row containing a single black cell leads to the following pattern.

CA Initial Evolution

A portion of the later history of this very simple CA shows structures moving against the background and interacting in a complex fashion.

CA Later



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