Their result shows how critical it is not to take anything for granted, he said Alon’s legmathematician at Princeton. “We must be suspicious even of things that intuitively seem very likely to be true.”
Gladkov, Pak, and Zimin found many examples of compact graphs that confirmed the conjecture, but ultimately they did not reflect the more convoluted, less intuitive graphs they could build if they had enough vertices and edges.
As Hollom put it, “Do we really understand all this as well as we think we do?”
Mathematicians still believe in the physics theorem about connected locations in solids, which inspired the bunk bed hypothesis. However, they will have to find another way to prove it.
In the meantime, says Pak, it is clear that mathematicians need to engage in a more vigorous discussion about the nature of mathematical proof. Ultimately, he and his colleagues didn’t have to rely on controversial computational methods; they were able to disprove this assumption with complete certainty. But as computer- and artificial-intelligence-based lines of attack become more common in mathematical research, some mathematicians wonder whether norms in the field will eventually have to change. “It’s a philosophical question,” Alon said. “How do we view evidence that is only likely to be true?”
“I think the future of mathematics will be accepting probabilistic proofs like this,” he said Doron Zeilbergermathematician from Rutgers University, known for crediting his computer with co-authoring many of his works. “In 50 years, maybe less, people will have a new approach.”
Others wonder whether such a future threatens something critical. “Perhaps probabilistic evidence would provide less understanding or intuition about what is really happening,” Alon said.
Pak suggested that as results of this type become more common, separate journals should be created so that mathematicians do not lose their value. But its main goal is to open a conversation. “There is no correct answer,” he said. “I want the community to consider whether another result like this will count.” As technology continues to infiltrate and transform mathematics, this issue will become even more pressing.
Original story reprinted with permission Quanta Magazine, editorially independent publication titled Simons Foundation whose mission is to enhance society’s understanding of science by incorporating research developments and trends in mathematics and the physical and life sciences.