AI breakthroughs unsettle mathematics and threaten cryptography

Companies developing artificial intelligence have solved several problems that challenged mathematicians for decades over the past year, ERR’s R2 Portal reported. While solving problems is generally positive, some of the results could pose a threat to modern banking and cryptography, commentator Kristjan Port wrote.

Port compared the upheaval in mathematics to Pompeii before the eruption of Vesuvius. The city had grown large, with sunlight reflecting from streets paved centuries earlier. In its final days, residents occasionally felt faint vibrations in the stone, but nothing disturbed daily life until an indescribable roar shattered the calm and a cloud of hot ash buried everything.

Until recently, mathematics had resembled Pompeii in its late-summer golden age, Port wrote. Above the ordinary world stood temples built from geometric purity and timeless axioms refined by generations of intellectuals. Apart from a few unsuccessful attempts, problems formulated by the legendary mathematician Paul Erdős remained in corners of the discipline for decades, gathering dust.

Over roughly the past six months, the field has experienced several smaller tremors. At first, solutions emerged for a couple of long-neglected problems. Around the New Year, several new shocks hit the community in a short period, prompting debate over whether artificial intelligence belongs in mathematicians’ toolkit.

Some were troubled that amateurs had solved certain problems using ordinary publicly available language models. Even greater concern was caused by AI-generated proof strategies that were unfamiliar, and at times foreign, to human experience.

This week, the quiet was broken by news that one of the seven problems considered the hardest in mathematics had unexpectedly been solved. The Navier-Stokes equations describe the movement of liquids and gases. Over roughly two centuries, they have been used to address problems ranging from the flow of water and blood and the description of climate to the design of aircraft and power plants.

What unsettled mathematicians was the mystery contained in the equations. Engineers can use them to calculate practical everyday problems and design aircraft and pipelines without knowing what happens at the mathematical edge. Mathematicians are interested in a much broader question: under certain conditions, could solutions drift into uncontrolled infinity despite only a finite amount of time passing?

A highly simplified way to understand the Navier-Stokes problem is to consider whether stirring coffee could create a vortex that accelerates endlessly and swallows everything. When describing coffee swirls with the equations, it was mathematically unclear whether a perfectly smooth fluid flow could concentrate into an increasingly small region, causing a quantity such as velocity to become unlimited in a finite time. In that case, the solution is said to “blow up” and form a singularity that can no longer be described.