"Everything they wrote in science books is about to change." One can hear this statement in the dialogue of M. Night Shyamalan’s Signs, a film about odd designs, extraterrestrials, and faith in meaning, or read about such things and this sentiment repeatedly restated in this masterful exposition self-published by Stephen Wolfram, MacArthur-certified maverick genius and wealthy author of Mathematica, the math program. While I found the film slow going at times, I could not put the book down and, after 846 pages, yearned for more. Actually, there is more, 300-plus pages of notes and an unpaginated 63-page index. The next surprise is that this tract aiming to overthrow the way math and science are done requires no math to read and has been a number-one Amazon.com bestseller. Sure, toward the end Wolfram displays huge arrays of symbolic logic formulas in order of complexity, but this is just to show by highlighting the chosen few that all previous math has been selective of special cases. All the figures with their legends summarizing the text make one feel one is in the catbird seat as a spectator at the revolution.

Two decades ago Wolfram got started with cellular automata, systems of rules for whether a cell is colored black or white in a sequence down the page drawn on graph paper. With computer development by the early 1980s the "behavior" of the system could be propagated, eventually for thousands of steps, resulting in spectacular printouts Wolfram has produced with wondrous clarity. Simple rules turn out to yield great complexity, and Wolfram calls this his remarkable discovery. He creates working models of just about everything—simple cellular systems (including snowflakes), fluid flow, biological branching, embryology, pigmentation, natural selection, memory, data compression and retrieval, cryptography, markets, fundamental physics (oh yes, Wolfram picked up a Ph.D. in physics at Caltech when he was 20), space-time, manyworld cosmology, human thinking, free will, and intelligence in the universe. What starts as a purposeless recreation becomes universal in application.

At first the patterns are merely curious. They look unfamiliar, a little like the sliced face of an organ like the liver except that features such as triangular substructures, angular pathways, and lacy ornamental hangings recur. The snowflakes are plausible, but one wonders if Wolfram will be able to deliver on his portentous promises. In a startling epiphany (p. 422) he shows an explicit pattern on a natural shell that exactly mimics those strange triangular "nested" structures.

His discussion of space-time and our experience of time is especially satisfying, based on an active cell of a mobile automaton that we seldom see. Wolfram is a neoatomist, reviving a movement that he notes began in Greece in 450 B.C. (p. 876). He fundamentally rejects continuity in the universe and abstract mathematical formulas that assume continuous functions, such as the partial differential equations on which much of science is built. Wolfram holds enormity in particulate grains of sand and infinity (time) in still more grains. He argues persuasively that everything unfolds from discrete units according to simple rules. His Principle of Computational Equivalence is expounded to show the translatability of all systems above a minimum structure, an argument that ends in the claim that "likely an ordinary physical process like fluid turbulence in the gas around a star should rather quickly do more computation than has by most measures ever been done throughout the whole course of human intellectual history" (p. 837). True, this animistic starthink has not yet been retrieved. Perhaps Piaget’s small children, stuck in his animistic stage, are right on. But, similarly, neural networks +(1), admittedly artificial systems, are thrilling not as complete models but because one can extrapolate the way mind emerges from brain networks. On the other hand, Jaak Panksepp +(2) has argued the case that not only is neurocomputation insufficient to the task of exploring mind and brain, it does a disservice to neuroscience by siphoning off young scientists who would rather press keys than get their fingers wet discovering the true marvels of brain nature that are rapidly surfacing. One may be sympathetic to this and still admit the utility of an orienting scaffold of logical structures that need to be deployed for brain systems to create sense. Wolfram’s position is somewhere between hewing to logical rules and yielding authority to nature. One similarity between neural networks and cellular automata is the difficulty in tracing back the path taken to arrive at the remarkable results.

Wolfram’s Principle of Computational Equivalence—and the phenomenon of computational irreducibility—put limits on the predictions of science. As Wolfram puts it, "all the wonders of our universe can in effect be captured by simple rules, yet it shows that there is no way to know all the consequences of these rules, except in effect just to watch and see how they unfold" (p. 846). Wolfram’s idea of free will is more convincing than satisfying. The brain follows definite laws but its overall behavior corresponds to an irreducible computation.

Yet another surprise in a book about robotic computations is the awe in which Wolfram holds human thinking, including his own. He designed Mathematica like human thinking, to store collections of rules to transform data. In our thinking, we "use vast amounts of stored data to perform tasks whose definitions and objectives are often quite vague…an altogether fundamentally complex process, not amenable at any level to simple explanation or meaningful theory" (p. 628). The "computers and programs that exist at present tend to be almost farcically inadequate" (p. 628). He says language is also inadequate, which is why his book is visually presented. He thinks intelligence, either human or extraterrestrial, should not be measured by the fixation on prime numbers we see in contemporary mathematics. Our visual system picks out specific features, and we should not be looking for simple regularities as evidence of mind in space because regularities are inefficiencies quickly removed for transmission by data compression. In an uncanny superiority to machine recognition, he has found that in his long experience with the complex patterns he has authored, he has learned to tell from which simple rules they were generated, even though the patterns have acquired so much randomness no computer could do this. This has implications for the place of mind in the universe that are beyond the scope of this review, but not, as might be guessed, this book, which the author says is "done in a rational tradition—with limited relation to the more mystical traditions of Eastern thinking" (p. 1196).

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