His result, he said, shows the importance of not taking anything for granted noga aloneA mathematician at Princeton. “We have to be skeptical even about things that intuitively have a very high probability of being true.”
Gladkov, Pak, and Zimin found many small-graph examples that satisfied the conjecture, but in the end, they did not reflect the more complex, less intuitive graphs they could create given enough vertices and edges.
As Hollom said, “Do we really understand all this stuff as well as we think?”
Mathematicians still believe in the physics statement about connected spaces within solids that inspired the Bunkebed conjecture. But they have to find a different way to prove it.
In the meantime, Pak says, it is clear that mathematicians need to engage in a more active discussion about the nature of mathematical proof. Ultimately he and his colleagues no longer had to rely on controversial computational methods; They were able to refute the conjecture with complete certainty. But as computer- and AI-based lines of attack become more common in mathematics research, some mathematicians are debating whether the norms of the field will eventually have to change. “It's a philosophical question,” Elon said. “How do we look at evidence that is true only with a high probability?”
“I think the future of mathematics will be to accept probabilistic proofs like this,” said Doron ZilbergerA mathematician at Rutgers University who is known for crediting his computer as a coauthor on many of his papers. “In 50 years, or maybe even less, people will have a new perspective.”
Others wonder whether such a future threatens something important. “Maybe a probabilistic evidence will give you less understanding or intuition of what's really going on,” Alon said.
Pak has suggested that separate journals be created for these types of results as they become more common, so that their value to mathematicians is not diminished. But their main goal is to start a conversation. “There is no right answer,” he said. “I want the community to consider whether the next result like this will count.” As technology continues to infiltrate and change mathematics, the question will become more pressing.
original story Reprinted with permission quanta magazine, An editorially independent publication of Simons Foundation Its mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.