Higher Mathematics Doesn’t Add Up, He Says
To the countless would-be scientists whose careers foundered on the baffling shoals of calculus, a brilliant physicist who earned his Ph.D. at 20 and snagged a MacArthur “genius” grant at 22 seems an unlikely source of comfort.
Yet Stephen Wolfram has some inspiring words for the mathematically challenged. In his self-published and unexpectedly popular book, Wolfram argues that sophisticated mathematics has led science astray in its effort to explain the natural world.
“A New Kind of Science” proposes that simple rules, not complex equations, are the key to such profound scientific mysteries as the structure of the universe and the incredible diversity of life on Earth.
“With the new kind of science I have developed, it suddenly becomes possible to make progress on a remarkable range of fundamental issues that have never successfully been addressed by any of the existing sciences before,” Wolfram boldly asserts on Page 1 of the 1,200-page tome that took him 15 years to write.
So many people are interested in Wolfram’s ideas that the book had already sold out of its first printing of 50,000 copies by its May 14 publication date. Even at $44.95 a copy, the title was one of the top 10 books on Amazon.com’s best-seller list two weeks after it came out.
Most scientists haven’t read the book yet, but the few who have remain skeptical. “It’s a useful perspective, but I think he overstates the case,” said Ray Kurzweil, a computer scientist and inventor who specializes in artificial intelligence.
Yet Wolfram’s reputation is such that “A New Kind of Science” is selling like tickets to the Stones. After stints at the California Institute of Technology and the Institute for Advanced Studies, Wolfram invented Mathematica, an all-purpose technical computing package, to do the kind of analyses described in his book. Then he started selling it to software developers, engineers, financial analysts and scientists of all stripes.
“Mathematica is a very general-purpose system,” said Dana Scott, a professor of computer science at Carnegie Mellon University. “You can use it very, very well for a lot of experimental investigations.”
After Mathematica made him a multimillionaire, Wolfram semiretired to an upstairs office in his Chicago home to work full-time on his book. That was a decade ago.
For 10 years, Wolfram worked all night and slept during the day. Occasionally he shared drafts of his book with colleagues and friends but much of the time he lived as a recluse, shunning scientific meetings and other gatherings.
“The book, I hope, is fairly readable to kind of general people,” Wolfram said in a telephone interview from his current home in the Boston area. “It’s something where I’m getting to the frontiers of science in a way that it’s accessible.”
During a recent hourlong interview, he offered opinions about physics, evolution, the nature of mathematics, the uniqueness of human intelligence and the best way to spot signals from beings on other planets. He built virtually every sentence into a dizzying edifice of abstractions, then abruptly left it unfinished so he could start another one.
“A New Kind of Science” is rooted in Wolfram’s discovery during the 1980s that very simple computer programs known as cellular automata can produce strikingly complex results.
In their most basic form, these programs take a pattern of gridded black and white squares and add to or modify them according to a simple set of rules. Such programs usually generate beautiful checkered, striped and nested patterns that can exhibit incredible intricacy.
But that’s not what Wolfram considers most important. His obsession is the occasional cellular automaton that reels off a random pattern of black and white squares ad infinitum.
You can make one of these curiosities by filling in some of the squares in the top row of a sheet of graph paper. Then move to the next row of squares. Blacken a square unless the one directly above it and both of its neighbors are the same color, or the one above has a black neighbor on the left and a white neighbor on the right. Then go to row three and repeat the process. Then row four, row five and so on.
Wolfram has programmed computers to do this simple exercise and similar ones millions and millions of times over. What they have produced has amazed him.
“What I found--to my great surprise--was that ... even some of the very simplest programs that I looked at had behavior that was as complex as anything I had ever seen,” he wrote.
No amount of careful observation, no mathematical equation could ever predict what the next row of squares produced by one of these simple programs will look like. The only way to find out is to follow the rule.
At this point one might ask, “So what?”
Why would a software millionaire with impeccable scientific credentials spend years poring over such apparent trivialities?
Well, Wolfram explains, when traditional science encounters something with no mathematically reducible pattern it simply throws up its hands. But his experiments show that apparently random processes can arise from surprisingly simple rules.
“It’s our intuition that when we see something in nature that looks complicated, then somehow the explanation must be complicated,” Wolfram said. “The surprising thing that I’ve found is that in the world of simple programs this is not the case.”
His book argues that nature doesn’t require complicated explanations either. If he’s right, scientists should be able to expose some fairly simple machinery buried deep inside some of nature’s most seemingly complex creations.
“What I’m trying to do here is answer some of the questions that traditional science hasn’t had much success in answering,” Wolfram said.
After taking about 360 pages of “A New Kind of Science” to explain some basic theory, he embarks on a grand tour of intractable scientific mysteries.
He starts with some relatively simple ones, such as a theory to explain the intricate and amazingly diverse crystal patterns of snowflakes. From there he moves on to question the importance of natural selection in evolution and the universality of the second law of thermodynamics, which states that natural processes, though theoretically reversible, proceed only in the direction of increased disorder.
His efforts are likely to receive especially cool receptions in the halls of academe, where attacking such fundamental and robust principles is considered the pinnacle of quackery.
“One could run these automata for trillions or even trillions of trillions of iterations,” Kurzweil protested in a review of the book. “They do not evolve into, say, insects, or humans, or Chopin preludes, or anything else that we might consider a higher level of complexity.”
Physicists may find Wolfram’s take on their field more palatable. His notion that the universe, on the tiniest of submicroscopic scales, is composed of a network whose connections are repeatedly updated accounts for several oddities, including the curvature of space-time demonstrated by Einstein.
It will be years before anybody, including Wolfram himself, knows whether the ideas presented in “A New Kind of Science” really are the biggest thing since an apocryphal apple struck Sir Isaac Newton’s head. Wolfram’s colleagues will have to set up their own computer experiments, measure and observe the real world, debate the significance of what they see and push their fields forward in the process.
But first they’ll have to read the book.
