August 24, 2012 | 2
William Thurston, who died on August 21 at the age of 65, would have hated this post’s headline. Let me tell you why it’s justified.
In 1993, when I was a full-time staff writer for Scientific American, my boss, Jonathan Piel, asked, or rather, commanded me to write an in-depth feature on something, anything, mathematical. Fercrissake, I was an English major! I whined. I could fake math knowledge for little news stories about the Mandelbrot set or Fermat’s last theorem but a major article would be too hard! I urged Jonathan to assign the piece to my math-whiz colleague Paul Wallich. Piel was adamant. He wanted me, the ignoramus, to do it.
So after lots of bitching and moaning I started picking brains of Sci Am contributors–including Wallich and two columnists, Ian Stewart and Key Dewdney—for ideas. I also began reading articles and popular books on math and interviewing big shots such as Andrew Wiles (who had just solved Fermat’s last theorem), John Conway, Ronald Graham, David Mumford, Phillip Griffiths, John Milnor, Stephen Smale, Pierre Deligne and Thurston.
Mathematics, I soon realized, was undergoing an upheaval. Mathematicians were arguing heatedly about whether traditional proofs—the gold standard, since before Euclid, for demonstrating truth—were becoming obsolete. This debate resulted, in part, from the increasing complexity of modern mathematics, which seemed to be bumping up against the limits of human understanding. A case in point was Wiles’s 200-page proof of Fermat’s last theorem, which was too dense for most mathematicians to evaluate.
Some practitioners were relying on computers to test conjectures, graphically represent mathematical objects and construct proofs. Mathematicians were also being pressured to work on applications, such as cryptography and artificial vision, where the fundamental question shifts from “Is it true?” to “Does it work?”
Traditionalists lamented these shifts—arguing, for example, that computer proofs produced answers without intellectual illumination–but others embraced them. Perhaps the most prominent advocate of change was Thurston, who had won a Fields Medal—the mathematical equivalent of a Nobel Prize—in 1982 for delineating deep connections between topology and geometry. Thurston was advocating a more free-form, “intuitive” style of mathematical discourse, with less emphasis on conventional proofs.
I chatted with Thurston over the phone and then flew to California to hang out with him for a couple days in Berkeley, where he ran a math center. We talked for hours about mathematical versus scientific truth, social-cultural influences on mathematics, the role of visualization in math and lots of other stuff. I was fascinated by the degree to which Thurston—in some respects the consummate authority and insider—was challenging his field’s axiomatic assumptions.
“The Death of Proof“–illustrated by a “video proof,” produced under Thurston’s guidance, of one of his theorems on topology and geometry (see image)—was the cover story of the October 1993 Scientific American. I declared in the introduction:
“For millennia, mathematicians have measured progress in terms of what they could demonstrate through proofs—that is, a series of logical steps leading from a set of axioms to an irrefutable conclusion. Now the doubts riddling modern human thought have finally infected mathematics. Mathematicians may at last be forced to accept what many scientists and philosophers already have admitted: their assertions are, at best, only provisionally true, true until proved false.”
I cited Thurston as a major force driving this trend, noting that when talking about proofs Thurston “sounds less like a disciple of Plato than of Thomas S. Kuhn, the philosopher who argued in his 1962 book, The Structure of Scientific Revolutions, that scientific theories are accepted for social reasons rather than because they are in any objective sense ‘true.’” I continued:
“‘That mathematics reduces in principle to formal proofs is a shaky idea’ peculiar to this century, Thurston asserts. ‘In practice, mathematicians prove theorems in a social context,’ he says. ‘It is a socially conditioned body of knowledge and techniques.’ The logician Kurt Godel demonstrated more than 60 years ago through his incompleteness theorem that ‘it is impossible to codify mathematics,’ Thurston notes. Any set of axioms yields statements that are self-evidently true but cannot be demonstrated with those axioms. Bertrand Russell pointed out even earlier that set theory, which is the basis of much of mathematics, is rife with logical contradictions related to the problem of self-reference… ‘Set theory is based on polite lies, things we agree on even though we know they’re not true,’ Thurston says. ‘In some ways, the foundation of mathematics has an air of unreality.’”
I let Thurston read a draft before publication. He made minor corrections and quibbled with some of my language but said he liked the overall thrust. After the article came out, the backlash—in the form of letters charging me with sensationalism–was as intense as anything I’ve encountered in my career. As a contributor to my Wikipedia page mentions, the article generated “torrents of howls and complaints” from mathematicians, who were especially incensed by the article’s title.
I expected, even welcomed, this criticism (and fortunately Piel, my boss, loved the article and had my back all the way). What I didn’t expect was that Thurston would be one of the critics. He wrote a letter to Scientific American declaring that proofs are alive and well. “The true drama of mathematics is more exciting than the melodrama suggested by the title, for this is a golden age for mathematics and for proof. A more appropriate title would have been ‘The Life of Proof.’”
I called Thurston and said, in effect, “What the hell, man!” He told me that, once my article was published, he realized that it could harm his efforts to reform math, and so he had to distance himself from it. I understood. Scientific American published Thurston’s letter and a few others in a later issue.
I have no regrets about “The Death of Proof.” In fact, I’m proud of it. After all, the mathematical trends I wrote about have continued, in no small part because of the leadership of Thurston. As Evelyn Lamb notes in a wonderful obituary for Scientific American, Thurston believed that “human understanding was what gave mathematics not only its utility but its beauty, and that mathematicians needed to improve their ability to communicate mathematical ideas rather than just the details of formal proofs.” Thurston set forth his philosophy in a 1994 essay titled “On Proof and Progress in Mathematics.” It’s a fine piece, but I still prefer my article, title and all.
Postscript: Mathematical physicist Peter “Not Even Wrong” Woit has posted a touching tribute to Thurston—whom Woit met at Princeton in the early 1980s–as well as links to remembrances by others. Cornell, Thurston’s most recent academic home, has also constructed a page for reminiscences. See also the post by technology historian Edward Tenner, who points out Thurston’s influence on the world of fashion (!).