Terence Tao, "Machine Assisted Proof"

Briana C, president of the AMS, opens the colloquium lectures, the oldest series of lectures at the AMS meetings, initiated in 1895. She highlights the prestigious history of the lectures and the notable speakers, including Burkoff, Morse, and others. The introduction focuses on the speaker, Terry Tao, emphasizing his accolades such as the Fields Medal, MacArthur Fellowship, and memberships in various academies. Tao's extensive publications and collaborations, as well as his service to the profession, including his role as a member of President Biden's Council of Advisers on Science and Technology, are also mentioned. The anecdote about Tao's engagement with election returns during a party in 2008 is shared to illustrate his personality.

The paragraph delves into Terry Tao's remarkable career, including his numerous awards and his influence on various fields of mathematics. It discusses his broad research interests, ranging from pure to applied mathematics, and his role in mentoring over 20 PhD students. The narrative also touches on Tao's personal attributes, such as his ability to engage with complex problems outside of mathematics, as demonstrated by his involvement with the New York Times Crossword puzzle and his keen interest in election data analysis.

Terry Tao discusses the historical and ongoing use of computers in mathematics, tracing their evolution from human computers to electronic machines. He mentions the significance of tables in mathematics, such as the logarithm tables of Napier, and how they have transitioned to large databases and online encyclopedias of integer sequences. Tao also covers the use of computers in numerics and scientific computing, highlighting their role in modeling and solving complex problems. The paragraph emphasizes the progression to more advanced computational tools, like SAT solvers and SMT solvers, which have the potential to tackle complex mathematical problems with a high degree of precision.

This section explores the impact of large databases and computational tools on mathematical discovery and proof. It discusses how databases, such as the Online Encyclopedia of Integer Sequences, facilitate the discovery of new connections in mathematics. The paragraph also examines the use of SAT solvers in solving complex problems, such as the Boolean Pythagorean triples problem, which was proven true through a computational approach. The discussion highlights the trade-off between computational efficiency and the length of runtime in rigorous computations.

Terry Tao introduces the relatively new modalities of using computers in mathematics, such as machine learning algorithms, large language models, and formal proof assistants. He discusses the potential of these technologies to revolutionize mathematics by offering new ways to approach problems, generate counterexamples, and even suggest proof techniques. The paragraph also touches on the limitations of current AI in mathematics, noting that while they can be useful for secondary tasks, they have not yet proven their worth in primary mathematical research.

The paragraph focuses on the role of formal proof assistants in verifying mathematical proofs, providing a high level of certainty in the correctness of results. It discusses the history of computer-assisted proofs, such as the Four Color Theorem and the Kepler Conjecture, and the challenges associated with verifying their correctness. The narrative highlights the efforts to formalize complex proofs using proof assistants, which, despite being time-consuming, offer the benefit of ensuring mathematical rigor and facilitating large-scale collaborations.

Terry Tao emphasizes the importance of formalization in mathematics, particularly in the context of complex proofs that are difficult to verify manually. He discusses the process of formalizing proofs using proof assistants like Lean, which involves creating a blueprint and incrementally formalizing each step. The paragraph also mentions the benefits of formalization, such as the discovery of errors and simplifications in existing proofs, and the potential for collaborative work on a large scale.

This section discusses the practical aspects of proof formalization, including the time and effort required to translate human-readable proofs into a formalized format. It acknowledges the challenges, such as the steep learning curve associated with proof assistants and the current time-intensive nature of the process. The paragraph also highlights the potential for formalization to become more efficient and widespread as the technology and community practices evolve.

Terry Tao contemplates the future of AI and automation in the context of mathematical proofs. He envisions a time when AI could assist in generating proofs and verifying their correctness, making the process more efficient. The paragraph also considers the potential for AI to help in the discovery of new mathematical connections and the exploration of theorem spaces, suggesting a future where mathematics could be done in entirely new ways with the help of technology.

The final paragraph touches on the potential for AI to facilitate collaboration in mathematics, allowing researchers to work together on a larger scale than is currently typical. It suggests that AI could help compartmentalize complex projects into manageable pieces, making it easier for mathematicians to contribute to and understand each other's work. The narrative concludes by acknowledging that while AI is not yet fully capable of replacing human intuition in mathematics, it can serve as a valuable tool for assisting and enhancing the research process.