Terence Tao, "Machine Assisted Proof"

Briana C, president of the AMS, opens the colloquium lectures, which date back to 1895. She highlights the prestigious history of the event and its speakers, including notable mathematicians like Burkoff, Morse, and others. She introduces the speaker, Terry Tao, with accolades such as Fields Medalist, MacArthur Fellowship recipient, and member of various academies. Tao's extensive publications and collaborations span pure to applied mathematics. His role as a mentor and his service in President Biden's Council of Advisors on Science and Technology are also mentioned. A personal anecdote about Tao's engagement with election results online in 2008 is shared, emphasizing his personality and character.

Terry Tao discusses the historical use of computers in mathematics, from human computers creating log tables to modern computational tools. He mentions the use of computers for experimental mathematics, such as generating large datasets in number theory, and the importance of databases like the Online Encyclopedia of Integer Sequences. Tao also covers the use of computers in numerics and scientific computing, including the use of interval arithmetic for more rigorous computations. He introduces the concept of SAT solvers and SMT solvers, providing an example of how SAT solvers were used to prove the Boolean Pythagorean triples problem, which was a computationally intensive task resulting in one of the world's longest proofs.

Tao explores new ways computers are being used in mathematics, focusing on three main areas: machine learning algorithms, large language models, and formal proof assistants. He discusses the potential of machine learning in various mathematical fields and the use of large language models like GPT for suggesting proof techniques and related literature. Formal proof assistants are highlighted for their ability to verify mathematical proofs and enable large-scale collaborations. Tao envisions an integrated approach combining these technologies with computer algebra systems and SAT solvers to create a powerful mathematical assistant.

Terry Tao delves into the history and progress of proof formalization, starting with the Four Color Theorem and moving to the proof of the Kepler Conjecture. He describes the process of formalizing complex proofs with the help of proof assistants and the challenges faced, such as the need for a blueprint to break down the proof into manageable parts. The benefits of formalization, including the discovery of errors and simplifications, are highlighted. Tao also discusses the Liquid Tensor Experiment, which aimed to formalize a proof in the Lean proof assistant, and the collaborative nature of such projects.

Tao reflects on the practical aspects of proof formalization, including the time and effort required compared to traditional proof writing. He shares his experience with the formalization of the Polomía Conjecture, which involved a large online collaboration and was completed in a relatively short time. The use of a blueprint for organization and the potential for formalization to simplify the modification of proofs are discussed. Tao also considers the future of formalization and its potential to become more efficient and widely adopted.

Tao discusses the growing intersection of machine learning and mathematics, particularly in the field of knot theory. He describes how machine learning was used to discover a connection between hyperbolic invariants and the signature of a knot, leading to the conjecture of an inequality. The process involved analyzing the machine learning model's predictions and using traditional mathematical methods to rigorously prove the conjectured inequality. This example illustrates the potential for machine learning to inspire new mathematical insights and proofs.

Terry Tao addresses the current state and potential of AI, specifically large language models like GPT, in solving mathematical problems. He acknowledges the impressive capabilities of GPT in solving complex problems but also its unreliability and tendency to make mistakes. Tao suggests that while AI can serve as a muse, providing suggestions and uncovering connections, it is not yet capable of replacing human intuition in the proof process. He also mentions the integration of AI with other tools, like proof assistants, to create a feedback loop that can correct and improve mathematical arguments.

In the concluding thoughts, Tao envisions a future where AI plays a significant role in mathematics, from generating conjectures to exploring theorem spaces. He predicts that AI will become better at compartmentalizing complex projects and at assisting humans in understanding advanced mathematical topics. Tao also anticipates that proof formalization will become more accessible and efficient, eventually leading to AI generating proofs for us and checking our work, thus transforming the way mathematics is done.

Tao wraps up his talk by reiterating the potential of AI in formalizing proofs and assisting in mathematics. He addresses concerns about the proliferation of formally verifiable proofs that may not be easily comprehensible by humans and suggests that analyzing and extracting human-understandable proofs from computer proofs will be part of future mathematical practices. Tao also discusses the role of human intuition and the challenges of automating the generation of good mathematical ideas. The talk concludes with a Q&A session where Tao engages with the audience on topics related to AI and mathematics.