# Terence Tao at IMO 2024: AI and Mathematics

TLDRProfessor Terence Tao, a renowned mathematician and IMO gold medalist, delivered a talk at IMO 2024, reflecting on his experiences and discussing the impact of AI on mathematics. He highlighted the historical use of machines in mathematical computations, dating back to the abacus and human computers. Tao also explored contemporary AI tools like AlphaGeometry and their potential to transform research mathematics, emphasizing the synergy between traditional and modern computational methods. The talk concluded with a Q&A session, touching on topics like formalizing mathematics, the importance of a solid educational foundation, and the evolving nature of mathematical research.

### Takeaways

- 😀 Professor Terence Tao, a renowned mathematician and IMO gold medalist, shared his experiences and insights on the interplay between AI and mathematics at IMO 2024.
- 🎓 Tao's early achievements at the IMO, including winning a gold medal at the age of 13, were highlighted, showcasing his prodigious talent in mathematics.
- 💡 AI's transformative impact on research mathematics was discussed, emphasizing its potential to change the way mathematicians approach problem-solving over extended periods.
- 🧠 The evolution of 'computers' from human computers to mechanical and electronic devices was outlined, illustrating the long-standing relationship between computation and mathematics.
- 📚 The historical use of tables and databases in mathematics, such as Napier's logarithm tables and the Online Encyclopedia of Integer Sequences, was highlighted.
- 🔢 The importance of scientific computation in mathematics, including its role in solving complex problems and the development of floating-point arithmetic, was discussed.
- 🤖 The application of AI in generating conjectures and solving problems was explored, with examples of how machine learning can uncover hidden patterns in mathematical data.
- 📈 The use of proof assistants like Coq and Lean in formalizing mathematical proofs was explained, demonstrating the growing precision and collaboration in mathematical verification.
- 🔗 The potential of AI to revolutionize mathematics by handling large datasets and solving multiple related problems simultaneously was envisioned.
- 🤝 Tao emphasized the continued importance of human intuition and creativity in mathematics, even as AI tools become more sophisticated and integrated into mathematical research.

### Q & A

### Who is Terence Tao and what is his connection to the IMO?

-Terence Tao is a renowned mathematician and IMO star who participated in the International Mathematical Olympiad (IMO) at a young age, winning a bronze medal at 11, a silver at 12, and a gold medal at 13, making him the youngest participant to receive a gold medal. He is currently a professor at the University of California, LA.

### What is the significance of AlphaGeometry mentioned in the talk?

-AlphaGeometry is a product by DeepMind that can answer some IMO geometry questions, showcasing the growing role of AI in fields like mathematics.

### How does Terence Tao view the impact of AI on research mathematics?

-Tao sees AI as transformative for research mathematics, noting that while it changes the nature of mathematical research, it follows a long tradition of machine assistance in the field.

### What is the history of using machines to do mathematics as discussed by Tao?

-Tao traces the use of machines in mathematics back thousands of years to the abacus, and notes that before electronic computers, there were mechanical and human computers used for tasks like creating tables and performing calculations.

### What role did human computers play during World War II as mentioned in the talk?

-Human computers, often women, were used to perform calculations for tasks such as ballistics during World War II, when men were largely engaged in combat.

### How did the concept of 'computer' evolve as per Terence Tao's discussion?

-Tao explains that the term 'computer' originally referred to a job profession, indicating a person who performs computations, before it came to mean the electronic machines we know today.

### What is the Online Encyclopedia of Integer Sequences (OEIS) and why is it important?

-The OEIS is a database of integer sequences that has become an essential tool for mathematicians. It allows researchers to compare sequences they encounter with known sequences, potentially revealing connections between different mathematical problems.

### How does Terence Tao describe the use of machine learning and neural networks in mathematics?

-Tao describes machine learning and neural networks as tools that can uncover new connections and correlations in mathematics that might not be apparent to humans, aiding in the discovery of mathematical insights.

### What is a formal proof assistant and how does it relate to the discussion on AI and mathematics?

-A formal proof assistant is a language used to write code that verifies the truth of mathematical arguments from first principles. Tao discusses how these assistants are becoming more user-friendly and are facilitating the formalization of complex mathematical proofs, which is a significant aspect of integrating AI into mathematical research.

### Can you provide an example of how AI is used to solve a complex mathematical problem as mentioned in the talk?

-An example given is the use of a SAT solver to prove the Pythagorean triple problem, which involved coloring natural numbers and determining if one color must contain a Pythagorean triple. The solution required significant computational power and creativity, showcasing the potential of AI in handling complex mathematical challenges.

### What does Terence Tao envision for the future of AI in mathematics?

-Tao envisions AI becoming a valuable assistant in mathematics, capable of generating conjectures, solving a wide range of problems, and potentially allowing mathematicians to explore the space of problems on an unprecedented scale.

### Outlines

### 🌟 Introduction to Professor Terence Tao's IMO Journey and AI in Mathematics

The speaker introduces Professor Terence Tao, highlighting his early achievements in the International Mathematical Olympiad (IMO), where he earned a bronze medal at age 11, a silver medal the following year, and a gold medal at 13, making him the youngest gold medalist. Tao's transition from IMO participant to a professor at the University of California, Los Angeles (UCLA) is discussed, emphasizing his significant contributions to mathematics. The talk shifts focus to the impact of AI on mathematics, referencing a presentation on AlphaGeometry, an AI product capable of answering IMO geometry questions, and the upcoming AI Math Olympiad. Tao discusses the differences between competition and research mathematics, noting the evolving role of machine assistance in the field, with a historical perspective on the use of machines like the abacus and the evolution of the 'computer' as a human profession to modern electronic computers.

### 📚 The Evolution and Relevance of Mathematical Tables and Databases

This paragraph delves into the historical and ongoing use of tables and databases in mathematical research. The speaker mentions the prime number theorem, discovered by Legendre and Gauss through the analysis of prime number tables, and the Birch and Swinnerton-Dyer conjecture, which was also initially identified through data tables. The Online Encyclopedia of Integer Sequences (OEIS) is introduced as a modern example of a mathematical table, used by mathematicians to identify patterns and connections in sequences. The paragraph also touches on the use of scientific computation, or number crunching, in mathematics, with examples ranging from the early 20th-century efforts to model fluid dynamics to modern applications in solving complex equations and the utilization of SAT solvers and SMT solvers in logic and algebraic problem-solving.

### 🧩 The Creative Use of Computers in Modern Mathematics

The speaker discusses the innovative ways computers are currently being used in mathematics, beyond traditional database lookups and number crunching. Three main areas are highlighted: the use of machine learning and neural networks to uncover new correlations in mathematics, the development of formal proof assistants that allow mathematicians to verify the correctness of arguments and proofs, and the application of AI in generating conjectures and solving problems that were previously intractable. The paragraph also touches on the historical use of computers for case analysis and the potential for AI to transform mathematical research by enabling the exploration of a vast space of problems in a way that was not previously feasible.

### 🔍 The Four Color Theorem and the Advent of Computer-Assisted Proofs

This section focuses on the history and development of computer-assisted proofs, exemplified by the proof of the Four Color Theorem. The speaker outlines the initial attempts at proving the theorem in the 1970s, which involved a combination of computer and human efforts. The paragraph details the process of proving the theorem, including the identification and checking of special subgraphs, and the subsequent formalization of the proof using proof assistants like Coq. The speaker also discusses the controversy and skepticism surrounding computer-assisted proofs and the efforts to formalize these proofs to achieve complete verification.

### 📈 The Kepler Conjecture and the Role of Human and AI Collaboration in Proofs

The speaker discusses the Kepler conjecture regarding the most efficient packing of spheres in three-dimensional space. The paragraph details the various attempts to prove this conjecture, including the work of Thomas Hales and his team, who eventually succeeded using a combination of innovative strategies and computer assistance. The talk highlights the iterative process of refining the proof, the challenges in getting the proof accepted by the mathematical community, and the subsequent efforts to formalize the proof using the Flyspeck project. The paragraph emphasizes the importance of collaboration between humans and AI in the process of mathematical discovery and proof verification.

### 🔗 The Integration of AI and Proof Assistants in Modern Mathematical Research

This section explores the recent advancements in the application of AI and proof assistants in mathematical research. The speaker mentions Peter Scholze's work in condensed mathematics and the formalization of a key theorem using the Lean proof assistant. The paragraph discusses the process of formalizing proofs, the challenges faced, and the benefits of having a formal proof, including the ability to make changes and improvements more efficiently. The speaker also highlights the indirect benefits of formalization projects, such as the expansion of the Lean math library and the development of tools to aid in future formalization efforts.

### 🤖 The Potential and Challenges of AI in Mathematical Problem Solving

The speaker reflects on the current state and potential of AI in solving mathematical problems directly. While acknowledging that AI is not yet capable of solving a wide range of problems reliably, the paragraph discusses the usefulness of AI as an assistant, particularly in generating conjectures and exploring problem spaces. The speaker expresses optimism about the future of AI in mathematics, suggesting that it could lead to new ways of doing mathematics on an unprecedented scale. The paragraph concludes with a call for flexibility and adaptability in the face of AI's growing role in mathematical research.

### 🤝 The Value of Collaboration and the Future of Mathematical Research

In this final section, the speaker emphasizes the importance of collaboration in mathematical research, facilitated by the use of AI and proof assistants. The paragraph discusses the speaker's personal experience with collaborative projects, such as the formalization of a combinatorics proof with a large team, and the benefits of breaking down complex proofs into smaller, manageable pieces. The speaker also reflects on the future of mathematics, suggesting that AI could enable mathematicians to tackle larger classes of problems and explore the space of mathematical questions more broadly.

### 🎓 Personal Reflections on Early Academic Progression and Choosing Research Topics

The speaker answers questions about his early academic journey, including his decision to attend university at a young age and the influence it had on his development as a mathematician. He also discusses his approach to selecting research topics, emphasizing the importance of serendipity and conversations with other mathematicians. The paragraph concludes with the speaker's thoughts on the future of mathematics, suggesting that flexibility and adaptability will be key in a field increasingly shaped by AI and new technologies.

### Mindmap

### Keywords

### 💡AI

### 💡IMO

### 💡Machine Assistance

### 💡Proof Assistants

### 💡SAT Solvers

### 💡Computational Power

### 💡Formal Proofs

### 💡Machine Learning

### 💡Large Language Models

### 💡Mathematical Research

### Highlights

Professor Terence Tao, a renowned mathematician and IMO gold medalist, shares his experiences and insights on the intersection of AI and Mathematics at IMO 2024.

Tao discusses the transformative impact of AI on research mathematics, distinguishing it from the time-bound nature of competitive mathematics.

A historical overview of machine assistance in mathematics is presented, dating back to the use of the abacus by the Romans.

The evolution of the term 'computer' from a human profession to electronic machines is highlighted.

Tao illustrates the significance of tables and databases in mathematical research, citing the Online Encyclopedia of Integer Sequences as a modern example.

The role of scientific computation in mathematics is explored, with examples ranging from modeling fluid dynamics to solving algebraic problems.

AI's potential to solve complex geometry problems through computer algebra systems is mentioned.

SAT solvers and SMT solvers are introduced as tools for handling logic puzzles and complex mathematical conjectures.

Tao provides a case study on the use of AI in proving the Pythagorean triple problem, emphasizing the importance of computer-assisted proofs.

The concept of formal proof assistants is explained, with examples of their use in verifying major mathematical theorems.

Tao discusses the advantages of collaborative mathematics projects facilitated by proof assistants, such as the rapid formalization of the PFR theorem.

Machine learning's emerging role in mathematics is highlighted, with knot theory as a notable application area.

The potential of large language models in generating new mathematical conjectures and aiding in problem-solving is explored.

Tao speculates on the future of AI in mathematics, suggesting it may enable solving classes of problems in new, unprecedented ways.

The importance of maintaining traditional methods of theorem proving alongside AI assistance is emphasized.

Tao concludes with a call for flexibility in the mathematical community, adapting to the evolving landscape of AI-assisted research.

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