Thinking Hard Above AI

Summary

This essay argues that large language models are better understood as a cultural and social technology than as autonomous intelligent agents. Because they are built from books, papers, websites, programs, and conversations, interacting with an LLM can resemble accessing accumulated mathematical culture through a new interface, although its answers may contain hallucinations. The author says recent experiences have changed his view of what AI can do in mathematics, citing discussions around Navier–Stokes and a colleague’s reported use of ChatGPT in work related to a long-standing conjecture. In his own example, ChatGPT quickly produced a proof of a difficult-looking lemma, but he then spent most of a day seeking a more canonical proof that made the result understandable. He calls this activity “Mathematics Above AI”: treating an AI answer as a baseline or first draft and investigating what remains, including why the proof works, which language best expresses it, and how it fits into broader mathematics. The essay warns that automatically circulating unexamined AI-generated mathematics could create technical debt and weaken scholarly conversation. It also describes productive educational uses, such as asking AI to generate increasingly difficult questions, expose gaps in understanding, or turn lecture preparation on logspace computation into a proving game. The author remains optimistic about mathematical activity while questioning an academic profession overly focused on publication metrics, priority, and speed.