Key points:
Terence Tao's successful use of AI tools to formalize the proof of the Polynomial Freiman-Ruzsa Conjecture marks the widespread application of artificial intelligence in mathematical research, causing a sensation in the mathematical community.
He emphasizes that mathematicians should learn to use AI tools correctly and believes that the mainstreaming of formal proofs has the potential to create proofs that are both readable by humans and solvable by machines, turning mathematics into a form of programming.
Tao's team used the Blueprint tool to decompose the proof process into manageable parts and successfully formalized the PFR conjecture through parallel work by numerous contributors.
ChinaZ.com reported on December 6th that Tao's successful formalization of the Polynomial Freiman-Ruzsa Conjecture using AI tools has attracted widespread attention in the mathematical community. He detailed the process of formalizing the proof using Blueprint in Lean4 in a blog post. This project took three weeks and successfully achieved the formalization of the Polynomial Freiman-Ruzsa Conjecture, demonstrating the power of artificial intelligence in mathematical research.
Tao's interpretation of this achievement emphasizes the importance of using AI tools correctly. He believes that the mainstreaming of formal proofs has the potential to create proofs that are both readable by humans and solvable by machines, making mathematics a more efficient form of programming. He particularly praises the Blueprint tool, which allows the team to write "blueprints" of proofs related to Lean that are readable by humans. In this project, green bubbles or rectangles represent fully formalized lemmas or definitions, while blue bubbles represent lemmas or definitions that are ready for formalization, providing a rough snapshot of the progress of the formalization project.
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He emphasizes the effectiveness of using Blueprint to decompose the project into manageable parts, enabling a large amount of parallel work. This also allows mathematicians without Lean programming skills to lead formalization projects. Tao's team's goal is to turn all the bottom bubbles leading to the "pfr" bubble green, ultimately successfully proving the PFR conjecture in Lean and achieving the main goal of the project.
His achievement has sparked discussions about the future of mathematical research. Some believe that formalization will become a key trend in mainstream mathematics, with most proofs being completed only in similar systems, without the need to write human-readable papers. However, Tao reminds us not to confuse "computer-assisted proofs" with "proofs that do not provide understanding/accidentally hold", emphasizing that formalization does not weaken the importance of understanding proofs. This achievement demonstrates the attention that formalization has received in mainstream mathematics and points to possible directions for future mathematical research.