Beware, math enthusiasts! AlphaGeometry, the latest achievement in artificial intelligence (A.I.), is poised to take on the challenges of the Math Olympics, solving complex geometry problems with remarkable proficiency. Trieu Trinh, a computer scientist, recently defended his doctoral dissertation at New York University, unveiling AlphaGeometry’s capabilities in the prestigious journal Nature.
During his residency at Google from 2021 to 2023, Dr. Trinh pitched the AlphaGeometry project to Google researchers, leading to its inclusion in Google DeepMind’s array of A.I. systems renowned for tackling significant challenges. Notably, AlphaZero, an A.I. algorithm, made headlines in 2017 by conquering the game of chess. However, solving mathematical problems presents a more formidable task due to the potentially infinite paths to a solution.
AlphaGeometry’s breakthrough is substantial, according to Dr. Trinh. In a test involving 30 Olympiad geometry problems spanning from 2000 to 2022, the system solved 25, closely approaching the average of human gold medalists over the same period (25.9). Comparatively, a 1970s geometry theorem prover, known for its strength, solved only 10 problems.
Google DeepMind has been actively exploring A.I.’s applications in mathematics in recent years. The AlphaGeometry project aligns with this trend, marking a significant step in automated reasoning. The paper introduces AlphaGeometry as a “neuro-symbolic” system, combining a neural net language model with a symbolic engine, tailored specifically for geometry problem-solving.
What sets AlphaGeometry apart are two key features. First, the neural net was trained solely on algorithmically generated data, comprising an impressive 100 million geometric proofs, eliminating the need for human-provided examples. Second, the system employs an “auxiliary construction” process, where the neural net suggests ways to enhance the proof argument when the symbolic engine encounters difficulties.
While acknowledging the achievement, experts like Terence Tao, a mathematician and former Olympiad gold medalist, emphasize the potential value of AlphaGeometry’s journey rather than just its destination. However, some critics, such as historian Michael Barany, question whether solving Olympiad theorems represents a significant milestone in creative mathematics.
AlphaGeometry’s success has prompted Dr. Trinh to consider generalizing the system across various mathematical fields. Critics note that the system lacks spatial perception and suggest incorporating a visual component, potentially achievable using Google’s Gemini, a multimodal system that processes both text and images.
Despite its impressive performance, AlphaGeometry has faced criticism for producing solutions deemed “mechanical” and lacking the “soul” and beauty of human-generated solutions, as noted by Dr. Le Ba Khanh Trinh, a former Olympiad gold medalist and coach. As AlphaGeometry continues to evolve, the quest to understand how machines generate solutions and the intricacies of human problem-solving remains an intriguing challenge.