His result shows the importance of not taking anything for granted, he said Noga AlonPrinceton mathematician. “We have to be suspicious, even about things that intuitively seem very likely to be true.”
Gladkov, Pak, and Zimin found many examples of small graphs that satisfied the conjecture, but in the end, they did not reflect the more complicated and less intuitive graphs they could construct when given enough vertices and edges.
As Hollom said: “Do we really understand all of this as well as we think we do?”
Mathematicians still believe the physics claim about connected locations within solids that inspired the bunk bed conjecture. But they will need to find a different way to prove it.
In the meantime, Pak says, it’s clear that mathematicians need to engage in a more active discussion about the nature of mathematical proof. Ultimately, he and his colleagues did not have to rely on controversial computational methods; They were able to refute the conjecture with complete certainty. But as computer- and artificial intelligence-based lines of attack become more common in mathematical research, some mathematicians are debating whether the field’s norms will eventually have to change. “It’s a philosophical question,” Alon said. “How do we see evidence that is only true with a high probability?”
“I think the future of mathematics will be accepting probabilistic proofs like this,” he said. Doron Zeilbergera mathematician at Rutgers University known for crediting his computer as co-author of many of his papers. “In 50 years, or maybe less, people will have a new attitude.”
Others wonder if that future threatens something vital. “Maybe a probabilistic test would give you less understanding or intuition of what’s really going on,” Alon said.
Pak has suggested that separate journals be created for results of this type as they become more common, so that mathematicians do not lose their value. But its main objective is to open the conversation. “There is no right answer,” he said. “I want the community to think about whether the next result of this guy will count.” As technology continues to infiltrate and transform mathematics, the issue will become increasingly pressing.
original story reprinted with permission of Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to improve public understanding of science by covering developments and trends in research in mathematics and the physical and biological sciences.