This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American
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 Piel to assign the piece to my math-whiz colleague Paul Wallich. Piel was adamant. He wanted me, the ignoramus, to do it.
After more bitching and moaning I started picking brains of Scientific American 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 William 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.
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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 William 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“ was the cover story of the October 1993 Scientific American. The opening spread showed an image, reproduced here, from "Not Knot," an animated video that Thurston produced to illuminate one of his theorems on topology and geometry. I declared in my article's 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.’”
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 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, who died in 2012. As Evelyn Lamb noted in an obituary in 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 eloquently defended his philosophy in a 1994 essay, “On Proof and Progress in Mathematics.” It’s a fine piece, but I still prefer my article, title and all.
*Self-plagiarism alert: This is an abridged version of a column published in 2012 after the death of William Thurston.