Roxanne Khamsi in New Scientist:
A mathematical proof is irrefutably true, a manifestation of pure logic. But an increasing number of mathematical proofs are now impossible to verify with absolute certainty, according to experts in the field.
“I think that we’re now inescapably in an age where the large statements of mathematics are so complex that we may never know for sure whether they’re true or false,” says Keith Devlin of Stanford University in California, US. “That puts us in the same boat as all the other scientists.”
As an example, he points to the Classification of Finite Simple Groups, a claimed proof announced in 1980 that resulted from a collaboration in which members of a group each contributed different pieces. “Twenty-five years later we’re still not sure if it’s correct or not. We sort of think it is, but no one’s ever written down the complete proof,” Devlin says.
Part of the difficulty is the computer code used nowadays to construct proofs, says Thomas Hales, at the University of Pittsburgh, Pennsylvania, US, as this makes the proofs less accessible even to experts.