Posted on Monday 14 October 2002 to Frontiers

Particle physicist and Nobel laureate Steven Weinberg takes a cold hard look at Stephen Wolfram's new and much touted book, A New Kind of Science particularly its self-proclaimed revolutionary importance. For Complexity theory to be interesting scientifically, says Weinberg, it still needs pursue the same goals as the "traditional" sciences, that is, to uncover the laws which reveal Nature's inner simplicity.

He stakes his claim in the first few lines of the book: "Three centuries ago science was transformed by the dramatic new idea that rules based on mathematical equations could be used to describe the natural world. My purpose in this book is to initiate another such transformation...."

Usually I put books that make claims like these on the crackpot shelf of my office bookcase. In the case of Wolfram's book, that would be a mistake. Wolfram is smart, winner of a MacArthur Fellowship at age twenty-two, and the progenitor of the invaluable Mathematica, and he has lots of stimulating things to say about computers and science. I don't think that his book comes close to meeting his goals or justifying his claims, but if it is a failure it is an interesting one.
Wolfram goes on to make a far-reaching conjecture, that almost all automata of any sort that produce complex structures can be emulated by any one of them, so they are all equivalent in Wolfram's sense, and they are all universal. This doesn't mean that these automata are computationally equivalent (even in Wolfram's sense) to systems involving quantities that vary continuously. Only if Wolfram were right that neither space nor time nor anything else is truly continuous (which is a separate issue) would the Turing machine or the rule 110 cellular automaton be computationally equivalent to an analog computer or a quantum computer or a brain or the universe. But even without this far-reaching (and far- out) assumption, Wolfram's conjecture about the computational equivalence of automata would at least provide a starting point for a theory of any sort of complexity that can be produced by any kind of automaton.

The trouble with Wolfram's conjecture is not only that it has not been proved?a deeper trouble is that it has not even been stated in a form that could be proved. [link]