A Missouri mathematician believes that the state’s moniker has great bearing on the status of modern mathematical proofs: Show Me.
Steven Krantz, Ph.D., professor of mathematics in Arts & Sciences, said it is becoming more difficult to verify proofs today and that the concept of the proof has undergone serious change over the course of his 30-plus-year career.
A proof is a finalized set of statements claiming to solve a problem. Today, many mathematical papers claiming proof of a solved problem often are posted on a non-peer-reviewed, preprint server called “arXiv,” located at Cornell University and approved by the American Mathematical Association.
“I think that arXiv is a great device for dissemination of mathematical work,” Krantz said. “But it is not good for archiving and validation. The reason that arXiv works so well is that there is no refereeing. You just post your work and that is it.
“Furthermore, those interested in certain subject areas are automatically notified of new postings. The work gets out there quickly, and it’s free. Everybody has access to arXiv. But there is no peer review.
“Publishing is a process that involves vetting, editing and several other important steps. We must keep that issue separate from dissemination. And dissemination is important in its own right. But it’s a separate issue.”
Krantz said several factors have contributed to the altering of a concept that had been relatively static since the time of the ancient Greeks.
“The traditional concept of the proof is that it is something put on paper that has been vetted, verified and confirmed by one’s peers,” Krantz said. “We’re seeing less and less of this today because of increased computer usage and multidisciplinary collaborations on mathematical problems.
“I think that the computer and the Internet have perhaps led us to confuse the dissemination question with the refereeing and archiving questions. And it has undercut the entire reviewing process.”
Krantz noted that, since the 1980s, there has been a sea change in the nature of paper writing.
“It’s almost all done by collaboration, whereas mathematics papers used to be single-author endeavors,” he said. “The collaborations reflect how complex mathematics research has become, but also illustrate the difficulty of proof. How can one mathematician understand all the branches of the problem?
“It’s going to take years, even decades, for some of the problems.”
Krantz was one of three distinguished American mathematicians to examine new developments in mathematical proofs at the American Association for the Advancement of Science’s Annual Meeting Feb. 16-20 in St. Louis.
Michael Aschbacker, Ph.D., of the California Institute of Technology, who analyzed proofs of the classification of finite simple groups, and Thomas Hales, Ph.D., of the University of Pittsburgh, who analyzed proof of the solution to the Kepler Sphere-Packing Problem, joined Krantz in a Feb. 18 presentation. Keith Devlin, Ph.D., of Stanford University, organized the session.
Krantz’s discussion revolved around an old topology problem and was titled: “The Poincare Conjecture: Proved or Not?”
Named after French mathematician Henri Poincare (1854-1912), the conjecture states that a three-dimensional manifold with the homotopy of the sphere is the sphere. Or, stated differently: In three dimensions, any surface that has the geometry of a sphere actually is a sphere.
Poincare posed the question in 1904, but it has only been in the past three years that any headway has been made on solving it.
Krantz referred to the work of Richard Hamilton, Ph.D., of Columbia University, and Grisha Perelman, Ph.D., of the Steklov Institute in St. Petersburg, Russia — especially three of Perelman’s papers posted on arXiv, though unpublished elsewhere.
“The new proof of the Poincare conjecture has proved to be quite robust,” Krantz said, who cautioned that he’s not primarily a topologist, but a fellow mathematician and interested observer who also has authored more than 100 peer-reviewed journal articles and numerous books and other writings.
“People have been discussing it now for more than two years, and many believe it to be correct. The ICTP News has in fact announced in its June 20, 2005, newsletter, that the Poincare conjecture is now proved. Period.”
But Krantz went on to note that Perelman has given a series of public lectures on the proof, but that he has not submitted the papers on arXiv for publication anywhere, even after Krantz, editor of The Journal of Geometric Analysis, has offered to publish anything that Perelman would like to say. But Perelman has not responded to the offer.
Krantz said that the task of validating the proof is so daunting that no single mathematician would be able to verify it because it demands the knowledge of difficult low-dimensional topology, Alexandrov theory — not well-understood in the West — differential geometry and partial differential equations.
Perelman, building on the work of Hamilton, has given the mathematics world a legacy of some brilliant ideas, Krantz said. But Perelman’s indifference to publishing the proof and his method of showing his work on arXiv “have put a chokehold on the subject of low-dimensional topology,” Krantz said.
“They have given us more questions than answers,” he said. “The methodology is promising but elusive. Nothing is written down. We can never be sure whereof we speak.”
Krantz’s concern is that a new generation of mathematicians might follow this paradigm for proofs, and that an older generation will become disenfranchised and discouraged.
“I can only hope that this program to prove the Poincare conjecture is not a new paradigm for doing mathematics,” he said. “I am a great fan of computer proofs, of proofs by modeling, of proofs by simulation and of proofs by experiment.
“I like all proofs, but a mathematical proof is a recorded piece of text that others can study and validate. I think that one of the most important aspects of our discipline is verification and archiving.
“The new program to prove the Poincare conjecture thus far is sorely lacking in this respect,” he added. “It is counterproductive, it is irresponsible, and in the end it is discouraging for us all. I think that we can do better.”