AI models could offer mathematicians a common language

No one expected that the question of how best to pack oranges in a crate would stump pure mathematics for centuries. Yet the so-called “sphere-packing” problem has driven generations of scholars crazy. The breakthrough came only in 1998, when Thomas Hales, then at the University of Michigan, announced that he finally had proof: the hexagonal arrangement—where each sphere rests in the hollow formed by six oranges on the layer below—is the densest possible configuration. Everyone thinks that mathematics is an exact science, but this story shows that even a seemingly simple question can remain unsolved for centuries, until the right person with the right insight comes along. Hales was no ordinary person: obsessed with the problem, he worked day and night with a team spread across the globe, amidst endless emails, calculation-verification software, and miles of blackboards covered with formulas. His proof was not accepted immediately: “It took years, and a lot of computer code, before the mathematical community accepted it,” Hales recounted. And here comes into play a point that few people see: mathematics is not just about brilliant ideas, but also about hard work, collaboration, and—more and more—technology. In recent years, artificial intelligence has slipped into precisely this gap: today, AI models are capable of verifying proofs, suggesting new avenues, and—this is the real revolution—becoming a sort of lingua franca among mathematicians from different schools and with different approaches. A language made up not only of numbers, but also of code. Now, pause for a moment: it's not just about solving problems faster. If AI truly becomes the bridge between schools of mathematics, we could see unprecedented global collaboration in the years to come. However, there is a real risk that is never mentioned: relying too heavily on these tools could make mathematicians less capable of intuiting, taking risks, and making connections outside the box—those very qualities that enabled Hales to solve the orange problem. Here is the phrase that sums it all up: “A good idea remains barren if it doesn't find the right language to be understood.” If this story has made you look at mathematics in a different light, you can mark it on Lara Notes with I'm In—so that perspective becomes part of how you reason. And if tomorrow you find yourself telling someone about the orange dilemma, you can capture that moment with Shared Offline: it's like saying that conversation really matters. That was The Economist, and with this Note, you’ve saved almost a minute compared to the original article.