Mathematicians explained the structure of twelve arrays of +1 and −1 found using AI and derived conditions for constructing similar arrays
Mathematicians explained the structure of twelve arrays of +1 and −1 found using AI and derived conditions for constructing similar arrays
On November 1, The New Yorker published a report by Rivka Galchen. Mathematician Terence Tao is concerned about how researchers will be trained: models can solve some problems in hours that graduate students once spent months learning to solve.
In July, Tao examined a formula found using AI that disproved a mathematical conjecture. His explanation starts with a product of three linear factors: one is kept separate, and the other two are multiplied together. Choosing a different factor to keep separate gives a new pair of polynomials with the same product. Other researchers can try this technique on new problems.
Mathematician Geordie Williamson welcomed Tao’s analysis. His reaction to the publication of twelve Hadamard matrices in August was different. These are square arrays of +1 and −1 in which any two distinct rows agree in exactly half of their positions. The published sequence of signs and accompanying program allowed readers to reconstruct and verify the arrays, but did not reveal how they had been found. Williamson worried that researchers would receive finished answers and miss ideas they might have discovered along the way.
In October, another group established how all twelve arrays were assembled from smaller blocks. The language model Claude wrote the programs used to analyze and verify them. Analysis of three of the arrays led to a theorem giving general conditions for assembling similar matrices. The authors checked and rewrote the theorem by hand; its conditions can be used to select blocks for new arrays.
Mathematicians John Urschel and his colleague Shah at the Massachusetts Institute of Technology set the direction of their work with models themselves. They studied Gaussian elimination, a method for solving systems of linear equations in which intermediate numbers can grow substantially, magnifying the effects of rounding errors. To assess the largest possible growth, the authors constructed specific examples. In one such construction, they gave models intermediate ideas from a research plan that humans had been developing since 2022. The authors wrote the final proofs themselves, and their paper gives a detailed account of the human and machine contributions to the results in each section.
Urschel found “no enjoyment” in interacting with models: he liked finding his own way to a solution.
Mathematician Charles Fefferman suggests imagining an idea with a 5% chance of success that AI could help test in two days instead of a lifetime. He predicts that making ideas inexpensive to test will make people “bolder and more inventive.”