Overview
This lecture by Professor Terence Tao explores the history and transformative impact of machine assistance and artificial intelligence (AI) in mathematics, spanning from ancient tools like abacuses to modern proof assistants, machine learning, and large language models.
History of Machine Assistance in Mathematics
- Humans have used machines in mathematics for thousands of years, starting with devices like the abacus.
- "Computers" were originally people who performed calculations; later, these became mechanical and then electronic.
- Early computer-assisted tasks included building mathematical tables, such as logarithm tables.
Scientific Computation and Problem Solving
- Computers have long assisted with large calculations, linear equations, and modeling (scientific computation).
- SAT and SMT solvers tackle logic-heavy problems but face exponential complexity limits.
- Example: The Pythagorean triple problem was solved by computer-generated case analysis, producing extremely long proofs.
Formal Proof Assistants
- Proof assistants are languages for writing and checking formal mathematical proofs (e.g., Coq, Lean).
- Example: The Four Color Theorem was an early computer-assisted proof, later fully formalized in Coq.
- Kepler conjecture's proof (sphere packing) required massive computer verification, later formalized in the Flyspeck project.
- Recent projects, like formalizing condensed mathematics and combinatorial proofs, benefit from collaborative blueprints and dependency graphs.
Machine Learning and Mathematics
- Neural networks can identify hidden patterns and connections, e.g., between knot invariants in knot theory.
- Machine learning aids conjecture generation by highlighting statistically significant relationships between data points.
Large Language Models in Mathematics
- Large language models (like GPT-4) can sometimes solve olympiad-level problems, but success rates are low and results remain unreliable.
- LLMs can suggest solution techniques or provide inspiration but are not yet capable of independently solving most advanced math problems.
- Integrating LLMs with proof assistants and other tools may increase reliability and productivity.
Collaborative and Scalable Mathematical Proofs
- Formal proof projects allow large teams to work on separate proof components, verified by automated compilers.
- Formalization is time-consuming but enables rapid updates and broader collaboration.
Future Directions and Challenges
- Proof assistants and AI will remain mathematical aids rather than full solvers in the near future.
- Potential future developments include AI-generated conjectures and large-scale problem exploration.
- Multiple foundational systems for formalization (like homotopy type theory) coexist and could benefit from translation tools.
Q&A Highlights
- Homotopy type theory offers robust foundations; various proof assistant languages exist and may interoperate via AI.
- Mathematical career development and research choice are personal and often shaped by mentorship and collaboration.
- Collaboration and flexibility are increasingly important in modern mathematics.
Key Terms & Definitions
- Proof Assistant β Software/language for writing and verifying formal mathematical proofs.
- SAT/SMT Solver β Algorithms for solving logical statements under constraints.
- Machine Learning β AI techniques where neural networks find patterns in data.
- Large Language Model (LLM) β AI systems trained on large datasets to generate human-like text.
- Blueprint β Decomposed plan of a mathematical proof, used for formalization.
Action Items / Next Steps
- Stay for the following AI Math Olympiad presentation.
- Explore proof assistants (Lean, Coq) and their libraries for exposure to formalized mathematics.
- Consider participating in collaborative formalization projects or learning about machine learning applications in mathematics.