A New Kind of Science Stephen Wolfram Synopsis: "A new kind of science" has been heralded as revolutionary book, not least by its own author. It could even represent next chapter in humanity's methodological understanding of reality and the abstract, in the style of Aristotle's foundation of logic and Newton's differential mechanics. Stephen's great new idea, the principle of computational equivalence, is introduced gradually via thorough investigation of an innocuous mathematical toy-world, one dimensional cellular automata. Imagine squared paper on which you sequentially mark each row. Whether a square gets a mark depends on whether the three squares above are marked. For example, you may make a mark only if one or more of the previous 3 neighbours was marked; from a single mark on the starting line this rule produces a solid triangle, getting wider with each turn. Despite its humble nature, it emerges that some rules for this system are as complex as any other formal system. This may be programmed for many mathematical operations, and emulates a universal Turing machine. Some equivalent systems have features that are suggestive of fundamental physics and human consciousness. Stephen argues many other areas of scientific and mathematical investigation could therefore be revolutionised by recognising how complexity emerges for very simple computation - and by acknowledging the weakness of the mathematical relations abstracted by the current scientific method. By proving the equivalence of so many models, he also concludes that, above a very low threshold of complexity, no system may perform more advanced operation than another. Therefore if the weather system "calculates" whether it snows on Christmas day, another calculation cannot determine that result without reproducing the same steps as the atmosphere. This also has startling conclusions for the possibility of non-human intelligence.

Critique: Many have found this book objectionable. It is certainly intended to upset apple-carts, motivated partly by Stephen's own alienation from established scientific research. However, the length and eccentric pedagogical style as well as the flaunting of review conventions (for peers or journalists) have also caused complaint. I'd like to try and avoid such prejudice, though it a shame that Stephen (in the book's main body, at least) takes on board so little of current artificial life and complexity theory (notably John Holland's Emergence and the Omega expression of the Turing halting problem). Instead, I would like to criticise the core thesis of computational equivalence on some specific points. Clearly, cellular automata are by their nature discrete. Stephen does briefly model a type of continuous rule driven evolving system, producing equivalent patterns (that are also suggestive of interfering waves). However, his conclusion that there is no more complexity in continuous systems over discrete seems quite a leap of faith. Certainly, Cantor demonstrated discrete and continuous infinite sets are very different. It may be found that nature is indeed itself discrete, cut into a cellular grid by the Planck length, and even this tiny distance would mean natural transitions are really finite functions. However, if continuous functions are not equivalent to discrete computations then Wolfram would be unable to claim a large area of mathematical operation for computational equivalence (even if most continuous functions so far found practical are easily modelled with discrete computations). Also, our best theories of real world interactions, special relativity, the Schrodinger equation and M-theory, are continuous. Natural continuum is also implied by non-local effects, such as the force of gravity from everything on everything else - not well represented by bound, clocked computation. I had another complaint with Stephen's unscientific way of classifying the systems he examined. He seemed to pick out the most interesting operations intuitively "by eye", noting those on the boundary between randomness and repetitious or fractal patterns (a fourth category being those that rapidly converge on a steady state). He attempts to support this arbitrary classification latter on when discussion perception and compression methods. Having failed to find a reliable method for detecting order, he concludes that judgment is as good as any formal operation in classifying complexity. However, lack of positive proof is not the same as proof negative. I believe the distinction between order, complexity and randomness is a fertile area of mathematical research currently, and giving up trying to understand the distinction because the investigation progresses slowly seems scientifically immature. Related to the issue of judging complexity, I was unsatisfied by Stephen's dismissal of randomness. I accept that the centre column of his basic rule 30 cellular automata has no pattern, and is superior to many other computed pseudo-random number sequences (though it would certainly be no more random than the digits of pi, for example). Yet both rule 30 and pi are clearly not random - short programs can generate either deterministically. Stephen uses the weakness of compression algorithms and other reduction mechanisms in recognising regularities to demonstrate that the boundary of randomness is blurred. I'm afraid I'm unfamiliar with the mathematics, but I have read that computability actually draws a quite clear line between deterministic numbers that lack of pattern and genuine randomness, supporting the intuition that pseudo-randomness and the genuinely unpredictable are different. Stephen also argues how small variation in randomness (in initial conditions) can either dissipate or scale up (in a chaotic way) in similar complex systems, giving them a robustness or sensitivity to small change. However, his systems of equivalent computation still cannot introduce genuine non-determinism. This means that if other systems can, they are excluded from computational equivalence (possibly with continuous operations). Stephen assures us that the randomness that we do see in nature (for example, at the fundamental level of nuclear decay) could only be pseudo-randomness, the product of hidden variables like those columns flanking the centre in rule 30. Again this is a leap of faith, this time going against Occam's razor and the history of rejection for hidden variable theories (except, possibly, super-strings). On the subject of randomness, I had the impression that Stephen was not clear that fuzzy operations were yet another different type of computation. A fuzzy system state can hold contradictory set membership simultaneously. For example, by the generally accepted Copenhagen interpretation of the Heisenberg uncertainty principle, an electron may be may 50% at 2 locations. This is not a statement that coin-flip like chance determines the real electron location - rather, it really is in both places at once. Such fuzzy systems, which may also describe neurones and language, could form another class of complex system that are not computationally equivalent to the simplest cellular automata. Turning to Stephen's proof of equivalence between Turing machines and the simpler rule 110 cellular automata "computer", I felt another sleight of hand may have been played. It seems clear that the various interactions of the cellular automata do model Turing machine transformations and state transitions, even though the production appears one way, feeding unmodifiable results out one side. (I imagined where a Turing machine loops over tape already processed, the operation would be kept within the cone of communicating activity in the cellular automata.) However, to accomplish this the computer relies on a continuous feed of information from the opposite side to the result - which is to say an infinite pattern in the initial conditions is required as well as the program. I do not remember Stephen accounting for another part of the mechanism generating this pattern, and it is hard to imagine an input production machine off to one side of the computer that could be guaranteed not to interfere with the computer's operation. I did consider the argument the input pattern is only equivalent to the context of the blank tape of a Turing machine, the grid of Conway's Life or even the physical dimensions of our universe. However, the claim that the simplest computer is equivalent to another other seems weakened by its apparently special requirement for a perpetual contrived input. Finally, there is an argument that Stephen acknowledges about the value of a scientific theory. The theory may be wrong, because of bad reasoning, over simplification or misinterpretation of evidence. I'm afraid that my arguments above are not as insightful or thoroughly worked out as Stephen's, so I'm unsure whether computational equivalence can be dismissed on this ground. Alternatively, a theory may be discarded because it does not have enough explanatory power - it is not interesting enough. To judge this I do believe we can only rely on our perception and intuition, just as coming up with an elegant theory relies partly on aesthetic inspiration equivalent to great artistic or engineering works. Stephen defends his theory as a profound result, giving us a new lever in understanding the mysterious mechanisms of fundamental physics, our turbulent world and even the mind itself. However, he does acknowledge that even this great book has not worked out the final details of the theory, nor made much progress towards applying it. When he does approach the foothills of using the theory to explain fundamental physics, for example, we see a very strange fit to gravity and particles that takes a lot of work to represent anything familiar (such as time, action at a distance and continuous existence). It seems explaining why energy is proportional to mass with a factor of the speed of light squared, for example, is very remote indeed in such a framework. I do remember reading of an alternative cellular automata formulation using matrix multiplication to project spacio-temporal relationships that may have been developed to predict fundamental constants. This may have inspired the alternate realities of Greg Egan's Permutation City. However, as even this fiction strengthens Stephen's argument for the possibility of cellular automata based explanations of physics, we see how remote such explanation is. In the very well grounded science of solar physics, models using the forces of gravity, pressure and magnetism become so sensitive to fine adjustments of parameters that they become too weak to support reliable theories about the structure of the sun. If instead of relying on picking around 10 parameters well based on observations, we must rely on a hierarchy of abstractions with arbitrarily chosen mechanisms, each with their own variables, then the models derived risk becoming as weak as astrology or numerology. In fact, the very shallow, highly abstract view of established sciences that Stephen has to take to apply his systems reminded me of the early Greek philosophers who built the world out of "elements" and geometric shapes. Though we could use the operations of a simple universal computation to run the numbers that model physics (or any system), they could not be recognised as scientific explanations that advance understanding. I did find "a new kind of science" stimulating and very interesting. The original systems that Stephen has worked hard to produce impressively demonstrate deep rules - to use Paul Davies' term for the physical laws that make our universe so complex. He also provides a rich mine of interesting detail about algorithms and programmable systems (if not a definitive resource). I enjoyed recognising what I knew as Lindenmayer systems and reading about new insights into Conway's Life, and I expect to develop toy programs myself inspired by unfamiliar systems. I may even keep what I perceived of Stephen's vision in the back of my mind in my real work. However, his strong claims need strong balancing scepticism, and I feel the thesis that "computational systems can model each other" is not as broad, profound or useful as the hype suggests.