# mathematica creator turns author

A few months ago, Stephen Wolfram was probably best known in scientific computing circles for his outstanding practical contribution to the field as creator of the popular Mathematica software package. However, with the publicity surrounding the launch of *A New Kind of Science*, it's hard to be unaware now of his other credentials as a prodigy who skipped school and a first degree to receive a CalTech PhD in theoretical physics at 20.

A few months ago, Stephen Wolfram was probably best known in scientific computing circles for his outstanding practical contribution to the field as creator of the popular Mathematica software package. However, with the publicity surrounding the launch of *A New Kind of Science*, it's hard to be unaware now of his other credentials as a prodigy who skipped school and a first degree to receive a CalTech PhD in theoretical physics at 20. Despite this undoubted talent, he has published relatively little through the normal academic route, even less after moving to the business sector in 1986 to found Wolfram Research, of which he is still CEO.

Wolfram's years of personal research into cellular automata have now found expression in *A New Kind of Science* (ANKOS). Published through Wolfram Media, this 1,200-page hardback expounds his view that science should model the universe as computations rather than equations. His ambitious claim is that this approach will revolutionise scientific thinking as radically as did, say, Newton's *Principia Mathematica*.

ANKOS begins by exploring the complex non-periodic behaviour of his well-known Class IV one-dimensional cellular automata (CA). Wolfram then shows that visually similar behaviour occurs in progressively more complex computational environments such as tag systems, register machines, number systems, 2D and 3D CA, network and multiway systems, and even solutions to partial differential equations. All manifest the emergence of order from random input, and conversely an ability to generate randomness, and Wolfram explores these ideas in a variety of scientific contexts: snowflakes, fluid flow, seashells, plants, animal pigmentation, finance, visual perception, cryptography, data compression, physics, relativity and cosmology.

Wolfram then describes Matthew Cook's proof that Rule 110 - a simple Class IV CA - is Turing-complete (that is, it can act as a universal computer) and argues from this that universality must be commonplace. This leads to the postulated Principle of Computational Equivalence: any process that appears non-simple (whether 'a human brain, a turbulent fluid, or a cellular automaton') he believes to be at the same irreducible level of computational complexity. ANKOS concludes with a 350-page appendix of explanatory notes, Mathematica code, and vignettes on relevant science and mathematics.

Despite the title, most of the science in ANKOS will be new only to readers unaware of the CA field. Many topics are an extended retread of papers in Wolfram's 1994 compilation Cellular Automata and Complexity, and the factual background draws heavily on material by other workers. This isn't always obvious, as less than enthusiastic attribution is a recurring fault. In the biology section, for instance, Wolfram reports his insights into evolution beyond traditional biological thinking; but these closely resemble the views of the late, and entirely uncredited, Stephen Jay Gould. The majority of named citations, such as Matthew Cook's credit as author of the Rule 110 proof comprising most of Chapter 11, are relegated to the appendix, which has no bibliography ('it is straightforward to do web or database searches'). Since the stated goal of ANKOS is 'to explain new ideas, not to criticise ones from the past', there's little discussion of Wolfram's claims in relation to specific work, and in any case ANKOS is advocating a style of investigation rather than postulating exact models or mechanisms.

Of the material original to ANKOS, the Principle of Computational Equivalence is likely to be the most contentious point. One consequence, according to Wolfram, is that it provides a rationale for Goedel's Theorem: complexity phenomena in axiomatic systems explain why some theorems are undecidable. In more practical territory, it would mean that many straightforward systems throw up analytically intractable behaviour that can only be modelled by a real-time discrete simulation such as a CA. On the other hand, even the most outwardly complex systems would be amenable to such modelling, and Wolfram believes that the universe might be described by a few lines of code.

Effectively, the Principle sidesteps the long history of attempting to define and measure complexity, by saying that everything above a particular intuitive threshold is equally complex. Much as I'm an enthusiast of CA, ANKOS doesn't convince me that I should drop the more traditional intuition that there is a hierarchy of complexity. Evidence such as Sierpinksi seashell patterns demonstrates that simple natural computational engines are plausible, and they may be universal computers in theory. But ANKOS fails in my view to prove that universality implies that such engines, running a random program on random input, could spontaneously generate, say, human-level intelligence.

Whatever the content, the style of ANKOS - Wolfram asserts that modesty reduces clarity - is a major impediment to assessing its science. Some parts run perilously close to the criteria in mathematical physicist John Baez's 'Crackpot Index' (at math.ucr.edu/home/baez/crackpot.html) for assessing potentially revolutionary contributions to physics, starting with '10 points for beginning the description of your theory by saying how long you have been working on it'. Wolfram is not a crackpot, but his failure to appreciate the negative effect of such 'green ink' statements is one of several stylistic faults that made it into ANKOS.

This isn't to say that ANKOS isn't worth reading. If you can ignore the Wolframcentric slant, it provides an exhaustively researched overview of CA-related complexity theory. Organised differently, it could have been a CA equivalent of Gleick's *Chaos: making a new science*. Unfortunately, its wealth of detail is disorganised and hard to correlate, it lacks the friendly touch of authors such as Douglas Hofstadter or Clifford Pickover, and many sections such as the Rule 110 proof are well beyond non-specialists. The central claim - simple programs explain everything - may inspire lay readers. Expect software and teaching spin-offs soon. But I doubt that the book will inspire the scientific community. *A New Kind of Science* primarily exhorts scientists to explore cellular automata techniques that are already under exploration.