Introduction
Mathematics, often seen as a rigid and unchanging discipline, is undergoing a quiet revolution. The rise of artificial intelligence (AI) tools is transforming how mathematicians interact with theorems and conjectures. Recently, AI has not only produced counterexamples to famous conjectures but also automated their formalization. So, what does this new era mean for human mathematicians?
The Case of the Unit Distance Conjecture
On May 20, 2026, ChatGPT disproved Erdős' Unit Distance conjecture in discrete geometry. This announcement shocked the mathematical community. Using a profound theorem in number theory by Golod and Shafarevich, the AI constructed a counterexample validated by human mathematicians. However, the validation raised a crucial question: how can we ensure the correctness of these AI discoveries?
The Impact of Lean and Auto-Formalization
Lean, an interactive theorem prover, is rapidly becoming an essential tool for mathematicians seeking to validate and formalize complex proofs. The company Logical Intelligence, led by luminaries like Mike Freedman and Yan LeCun, managed to auto-formalize the paper generated by ChatGPT in Lean. This marks a significant step in bridging AI and mathematical formalization.
Challenges and Prospects
While these advancements are promising, they present challenges. Formalizing complex theories, such as global class field theory, remains a colossal task. In 2025, a Clay Summer School aimed to formalize this theory. A year later, only local cases are formalized. The global case remains a daunting task, even for AI.
The Future of Mathematics with AI
The impact of AI on mathematics goes beyond generating counterexamples. It opens new avenues for exploring and verifying existing theories. However, it also raises questions about the role of human mathematicians. Should they become mediators between AI and theory, or focus on more creative aspects?
Conclusion
AI does not replace human mathematicians; it complements them. By integrating these tools into their daily work, mathematicians can push the boundaries of knowledge. The key is to remain critical and ensure rigorous validation of AI-generated results.
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