09-21-2026, 08:37 PM
Jarod Alper’s article Embracing AI and Formalization: Experimenting with Tomorrow’s Mathematical Tools, published in the Bulletin of the American Mathematical Society, examines how artificial intelligence and formal proof systems such as Lean could change mathematical research. Rather than treating AI as a distant possibility, Alper describes his own experiences experimenting with these technologies through the eXperimental Lean Lab (XLL) at the University of Washington. The article starts from the expectation that computers may increasingly solve sophisticated mathematical problems and assist humans with proofs, while emphasizing that mathematicians should actively participate in shaping this transition.
A major theme is formalization: translating ordinary mathematical arguments into a precise computer-readable language that can be checked by a proof assistant such as Lean. Alper explains that this process can be frustrating and time-consuming because Lean requires mathematical statements and proofs to be expressed with extreme precision. However, this difficulty is also an advantage: formalization can expose hidden assumptions, logical gaps, and errors that may be overlooked in conventional mathematical writing. In his experience, working with students showed that formalizing mathematics can improve their understanding because they are forced to identify exactly why every step of an argument is valid.
The article also discusses the emerging connection between AI and formal mathematics. Large language models can potentially help mathematicians write Lean code, search for relevant lemmas, generate possible proof strategies, and translate informal mathematical ideas into formal statements. Conversely, formal proof assistants provide an important verification mechanism for AI: instead of simply trusting an AI-generated explanation, a proposed proof can be checked mechanically. Alper therefore sees AI and formalization as complementary technologies—AI can help generate and explore mathematical ideas, while formal systems can provide a much stronger standard of verification. The University of Washington’s Math AI Lab similarly identifies mathematical formalization, autoformalization, and AI-assisted mathematical research as closely connected areas.
Ultimately, Alper’s message is not simply that computers will replace mathematicians, but that mathematicians should experiment with these tools and help determine how they will be used. His experience with the XLL suggests that learning Lean and AI does not have to mean abandoning traditional mathematics; instead, these technologies can become additional tools for teaching, checking proofs, exploring conjectures, and conducting research. The article presents the transformation as an ongoing process rather than a finished technological revolution, encouraging mathematicians—particularly students and researchers—to gain practical experience with formalization and AI so that they can actively influence the future relationship between humans and machines in mathematics.
ARTICLE
A major theme is formalization: translating ordinary mathematical arguments into a precise computer-readable language that can be checked by a proof assistant such as Lean. Alper explains that this process can be frustrating and time-consuming because Lean requires mathematical statements and proofs to be expressed with extreme precision. However, this difficulty is also an advantage: formalization can expose hidden assumptions, logical gaps, and errors that may be overlooked in conventional mathematical writing. In his experience, working with students showed that formalizing mathematics can improve their understanding because they are forced to identify exactly why every step of an argument is valid.
The article also discusses the emerging connection between AI and formal mathematics. Large language models can potentially help mathematicians write Lean code, search for relevant lemmas, generate possible proof strategies, and translate informal mathematical ideas into formal statements. Conversely, formal proof assistants provide an important verification mechanism for AI: instead of simply trusting an AI-generated explanation, a proposed proof can be checked mechanically. Alper therefore sees AI and formalization as complementary technologies—AI can help generate and explore mathematical ideas, while formal systems can provide a much stronger standard of verification. The University of Washington’s Math AI Lab similarly identifies mathematical formalization, autoformalization, and AI-assisted mathematical research as closely connected areas.
Ultimately, Alper’s message is not simply that computers will replace mathematicians, but that mathematicians should experiment with these tools and help determine how they will be used. His experience with the XLL suggests that learning Lean and AI does not have to mean abandoning traditional mathematics; instead, these technologies can become additional tools for teaching, checking proofs, exploring conjectures, and conducting research. The article presents the transformation as an ongoing process rather than a finished technological revolution, encouraging mathematicians—particularly students and researchers—to gain practical experience with formalization and AI so that they can actively influence the future relationship between humans and machines in mathematics.
ARTICLE
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