Unknown Territory Holds Untold Value

Math as we practice it today turns out not to be the only way to map reality.

One of the summers most controversial books was Stephen Wolframs "A New Kind of Science," which has sold a remarkable number of copies for a 1,200-page tome full of mathematical diagrams and expressions. If every equation in a book really does cut its sales in half, as someone once warned physicist Stephen Hawking, then the printing of Wolframs book might have consumed every forest in the world if he had confined himself to words and pictures.

But Wolframs message requires equations because hes presenting a view of both the natural and the abstract world as being tied together by underlying principles that we finally have the tools to explore. The implications are huge for domains such as medicine and biotechnology, where largely trial-and-error methods could give way to more systematic searching of the space of possible solutions—if we can understand the di- mensions of that space and plan our journey through it.

I met with Wolfram last month at Comdex, the day before his keynote speech—probably the most cerebral event that the show has ever seen. I left with a much-improved understanding of what hes actually accomplished, compared with the buzz of comment on his book that (for the most part) loudly misses the point.

Wolfram has been said to claim that because simple rules can produce complex behaviors, it must follow that all behaviors—from the shapes of seashells to the composition of music—will ultimately be traced to simple rules. No, thats not what he said, he assured me during our hour together.

"Models are a high-judgment business," he explained. "Calculus was one sort of raw material for making models of things; Im trying to provide another.

"The main difference," he continued, "is that in calculus there are a limited set of primitives—derivatives, integrals, things like that—but Im interested in the arbitrarily general primitives that can be represented in programs." Math as we practice it today, he asserted, turns out not to be the only way to map reality onto sets of symbols and operations; "whats been explored is a tiny fraction of what there is."

Theres untold value in that unknown territory.