What happens when AI starts checking mathematicians’ work

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A new era in mathematics may be on the horizon—one that some researchers have long desired. Mathematicians could soon use computers to verify proofs quickly and rigorously, ensuring published proofs are correct and providing a foundation for further advances. Such a tool could help experts grapple with the accelerating pace and volume of mathematical research.

Computer programs that check mathematical arguments, such as proofs, have existed for decades. But translating a human-written proof into the strict programming language of a computer—a prerequisite for verifying it using these existing tools—is extremely time-consuming. This translation, known as formalization, can sometimes take months or even years.

With the development of the first large language models, mathematicians’ hopes rose: perhaps machines could one day do this translation automatically. Unlike human languages, however, formal programming languages allow no variation whatsoever. Every term, symbol and reference must be precisely defined.

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But now a start-up called Math, Inc., is reporting initial success in formalizing proofs. Its artificial intelligence, named Gauss, has formalized two complex proofs related to arranging spheres in higher dimensions by mathematician Maryna Viazovska. She received the Fields Medal for one of these proofs in 2022. The mathematics community’s response to Gauss’s formalization has been muted, however, partly because the project did not unfold as many experts had hoped. As other AI-and-math start-ups explore formalization, this case offers hints as to what mathematicians might expect in an uncertain future.

A Packing Puzzle

In 2016 Viazovska became a central figure in mathematics by solving a decades-old puzzle: How can spheres be arranged in the most space-efficient way? To find the single most space-efficient solution, you must…

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