Lecture Notes: AI and Machine Assistance in Mathematics by Professor Terence Tao

Introduction

• Professor Terence Tao: Notable mathematician; youngest IMO gold medalist at age 30.
• Topic: AI's impact on mathematics research and competition.

Personal Reflections on IMO

• Enjoyed his time at IMO; emphasizes the importance of social activities alongside competition.

AI in Mathematics

• AI is transforming research mathematics, differing from competition mathematics.
• Importance of tools like AlphaGeometry for solving geometry problems in IMO.
• Traditional use of computers vs. modern AI tools.

History of Machine Assistance in Mathematics

Early Use of Machines

• Machines (e.g., abacus) have been used for thousands of years.
• Computers have been used for about 300-400 years.
• Early professions involved human "computers" for calculations.

Evolution of Computational Assistance

• Development of logarithm tables and databases.
• Example: Prime number theorem discovered through computation by Gauss.
• Tools like the Online Encyclopedia of Integer Sequences (OEIS) assist in finding mathematical sequences.

Scientific Computation

• Definition: Using computers for large calculations (e.g., linear equations, partial differential equations).
• Historical example: Hendrik Lorentz used human computers for modeling fluid equations.
• Limitations of traditional computational methods vs. AI methods.

Advanced Computational Tools

SAT/SMT Solvers

• SAT Solvers: Solve logic puzzles with true/false statements.
• Example: Pythagorean triple problem solved using SAT solver.

Modern Applications of AI in Mathematics

Machine Learning and Proof Assistants

• Use of machine learning for discovering mathematical connections.
• Formal proof assistants facilitate proof verification and formalization of mathematics.

Historical Examples

• Four-Color Theorem: Initial proof required human verification, later formalized with computer assistance.
• Kepler Conjecture: Used complicated computer-assisted proofs, leading to long verification time.

Current Projects and Tools

• Lean: A proof assistant language for formal mathematics.
• Recent successes in formalizing important mathematical results with collaborative efforts.

Future of Mathematics with AI

• AI could assist in generating conjectures and exploring mathematical problems at scale.
• Ongoing need for human mathematicians to guide AI development.

Conclusion

• Excitement about the potential for AI in mathematics, while acknowledging the continued importance of traditional methods and human oversight.

Q&A Highlights

• Discussion on formalizing mathematics and the role of different proof-assistant languages.
• Insight into Tao’s research topic selection process influenced by social interactions in mathematics.