Are today's most advanced mathematical proofs impossible to verify?

mathproof.jpg Credit:Jesse Yardley

In 1912, Bertrand Russell wrote a letter to his mistress, Lady Ottoline Morrell, in which he revealed, “I like mathematics because it is not human and has nothing particular to do with this planet.” Besides suggesting theirs was a cerebral affair, the quote captures a certain fundamental tenet of mathematics that some mathematicians are now questioning.

What attracts many young mathematicians to the field and assures the average person that they can trust their calculator is that mathematical truth is entirely certain, with its own austere perfection that’s totally independent of humans.

While mathematics remains our most rigorous form of knowledge, the extreme complexity and length of some recent proofs have made them nearly impossible to check. As proofs continue to grow more complicated, mathematicians worry they will have to accept a greater degree of uncertainty than they’ve traditionally been comfortable with.

“Compared even to the other sciences—compared even to physics—mathematics was the thing that was dead-on certain,” said Steven Krantz, a professor at Washington University in St. Louis. “Now there are things in math that are not so clear.”

The basic idea of a proof has remained relatively static over the 2,500 years since Euclid set the model: Beginning from certain accepted premises, a single mathematician can trace a set of logical steps to arrive at a verifiable conclusion.

As researchers tackled more difficult theorems over the past 50 years, they’ve produced proofs that are longer and more collaborative, often involving more work performed by computer calculation.

“There’s something like 80,000 math papers published every year, and I would say that 95% of them conform to the traditional paradigm,” said Krantz. “But there are new forces at play here; they’re changing the nature of the subject.”

Besides losing the right to call a particularly elegant and daunting proof one’s own, these recent, elaborate theorems are so massive they would require teams of mathematicians devoting their entire careers to confirm them, rather than spending time on their own endeavors. And without confirmation comes reasonable doubt.

“We’re driven, like any other profession, by recognition; you want to be recognized for your theorems,” said Krantz. “Why would anyone want to invest the years and the effort if it’s not their theorem?”

Attempts have been made to verify supercomplex proofs before. When University of Pittsburgh mathematician Thomas Hales announced he had solved Kepler’s sphere packing conjecture in 1998, the Annals of Mathematics asked a dozen referees to collaboratively confirm the proof. The team was eventually forced to admit defeat.

“After four years, I heard back that the chief referee was 99% sure the proof was correct,” said Hales, “But the referees were exhausted and they weren’t going to complete the verification. The journal decided to go ahead with publication of an abridged version of the proof anyway.”

The inability of the mathematical community to verify some proofs threatens to undermine the old idea of mathematical certainty.

“We’ve got so many mathematicians around and such specialization and such complex problems being attacked that I think increasingly we’re going to have to live with 99% certainty,” said Keith Devlin, a Stanford University mathematician.

One example of these new hybrids is the Classification Theorem for Finite Simple Groups, sometimes referred to simply as the “Enormous Theorem.” It is even more daunting than Hales’s work was. The proof is more assumed than actual—it exists as the aggregate of hundreds of different papers. Experts estimate that to write out the entire proof would take somewhere between 10,000 to 15,000 pages.

The highly elaborate proofs at the forefront of mathematical research will never directly affect most people; algebra and addition are certainly in no danger of being toppled. But within the highly theoretical branches of math and physics, at the limits of our understanding, researchers are confronting the uncomfortably human presence of “probably.”

“I’m very open-minded about accepting different kinds of proof, but I still adhere to the traditional concept of rigor, so this all makes me kind of nervous,” said Krantz. “I think most mathematicians just prefer not to pay much attention to it.”

Devlin is more optimistic about the development, which he believes could humanize mathematics at the same time as it draws the field closer to the sciences.

“In terms of how math gets done by people, it’s always been much closer to science than it might claim to be,” he said. “I see a parallel between the uncertainty of these proofs and developments in physics like string theory, where we’re developing mathematical theories of matter that may forever remain elusive to experimental verification.”

Originally published February 28, 2006


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