Fantastic results will continue to emerge in mathematics even after the arrival of AI. However, Dániel Varga, a senior researcher at RÉNYI AI, is less optimistic about whether mathematicians will weather this dramatic transformation well. Together with Ákos Dúcz, he answered one of Paul Erdős’s questions using AI only as an auxiliary tool.
The two researchers spoke to the scientific news site Qubit.hu about their results. We are publishing the article with the written permission of its author, András Tóth, editor and reporter with Qubit.
Researchers Dániel Varga and Ákos Dúcz have partially solved a decades-old problem posed by renowned Hungarian mathematician Paul Erdős. The problem asks whether it is possible to place a finite set of points in the plane so that, no matter which quarter of the points is selected, two of the selected points are at unit distance from each other. Varga and Dúcz proved that such a finite point set does indeed exist.
Their result builds on earlier work by Varga and colleagues, who in 2023 proved that the plane cannot be coloured in such a way that no two points at unit distance receive the same colour. The new work uses a method developed at the HUN-REN Alfréd Rényi Institute of Mathematics based on finite “probes” that can be used to explore the properties of infinite sets. By testing and refining these finite configurations with extensive computer calculations, the researchers eventually constructed a 29-vertex point set whose many translated and rotated copies satisfy the condition in Erdős’s problem.
Artificial intelligence played a supporting role in the research. Varga and Dúcz used AI to write software and test ideas more quickly, but the system did not make a creative mathematical contribution to the proof. In one instance, however, an unexpected answer produced by the AI in a computational problem provided an important step towards the eventual solution. Dúcz also developed a method that dramatically accelerated the computational search, allowing the researchers to test tens of thousands of possible points without repeatedly solving the full, extremely large system of inequalities.
The two researchers have not yet completely resolved Erdős’s problem. Its other part asks for the smallest possible proportion of points that can be selected from a planar set while avoiding pairs at unit distance. Varga estimates that the answer lies somewhere between 23 and 25 percent, but this remains an open question.
In the interview, Varga and Dúcz also discuss the rapidly changing role of artificial intelligence in mathematics. Varga argues that AI is likely to lead to extraordinary mathematical results, but is less optimistic about how well mathematicians themselves will adapt to the transformation. He believes that AI systems are already capable of producing ideas that can go beyond simply applying known techniques, while Dúcz describes the current period as one in which mathematicians can still work closely with AI while retaining control over the whole research process.
According to the researchers, mathematics may be only at the beginning of a much broader transformation. Because mathematical problems often have clearly verifiable answers, mathematics has been particularly well suited to the first wave of AI breakthroughs. The next question is whether increasingly capable AI systems will also make comparable progress in fields such as materials science, biology and medicine. At the same time, the mathematical community may have to rethink how research is conducted and evaluated, and how humans can understand and make use of mathematical results generated by increasingly powerful AI systems.
The original article appeared in Hungarian HERE.