Terence Tao, "Machine Assisted Proof"

Joint Mathematics Meetings

7 Feb 202454:56

TLDRIn this colloquium, Terence Tao discusses the impact of machine-assisted proof technologies on the future of mathematics. He explores the evolution from human computers to electronic ones, the role of databases in experimental mathematics, and the use of SAT solvers and formal proof assistants. Tao highlights the potential of combining machine learning, large language models, and proof assistants for new mathematical discoveries, emphasizing the growing importance of collaboration and technology in advancing mathematical knowledge.

Takeaways

🌟 The colloquium lectures at the AMS meetings have a long history, dating back to the first meeting in 1895.
🏆 Terence Tao is a highly decorated mathematician with numerous awards, including the Fields Medal in 2006 and a MacArthur Fellowship.
📚 Tao has an extensive publication record with over 350 publications and has collaborated with more than 50 researchers in various fields of mathematics.
🤖 The integration of machine learning, large language models, and formal proof assistants is a significant development in the field of mathematics.
🔢 The use of computers in mathematics is not new, with early examples including the creation of log tables and the application in numerical computations.
📈 The development of technologies like SAT solvers and SMT solvers has contributed to solving complex mathematical problems that were previously intractable.
🧠 Machine learning algorithms are being used to generate counterexamples and conjectures in mathematics, although they are still in a niche application stage.
📝 Large language models, such as GPT, have shown potential in suggesting proof techniques and related literature, despite not being directly trained for mathematics.
🔗 Formal proof assistants are crucial for verifying the correctness of mathematical proofs and are becoming more integrated with other technologies for collaborative work.
🔬 The combination of machine learning, large language models, and formal proof assistants could lead to a powerful methodological assistant for mathematicians.
🔍 The process of formalizing proofs, while currently time-consuming, is becoming more streamlined and could eventually lead to new ways of exploring mathematical theorem spaces.

Q & A

Who is the speaker of the colloquium lecture mentioned in the script?
-The speaker of the colloquium lecture is Terence Tao, a renowned mathematician.
What is the significance of the Four Color Theorem in the context of this lecture?
-The Four Color Theorem is significant because it was one of the first major theorems to be proven with the assistance of a computer, and later fully formalized in a proof assistant language, demonstrating the evolution of machine-assisted mathematics.
What role did the president of the AMS play in the script?
-The president of the AMS, Briana C, introduced Terence Tao and provided context for his achievements and the importance of the colloquium lectures.
What is the relationship between machine learning and mathematics as discussed in the script?
-Machine learning is beginning to play a role in mathematics by analyzing large datasets to uncover patterns and generate conjectures, although it still requires traditional mathematical methods to prove these conjectures.
How does the script describe the use of large language models in mathematics?
-Large language models, such as GPT, are being experimented with in mathematics to suggest proof techniques and related literature, although they are not yet reliable for direct problem-solving in mathematics.
What is a 'SAT solver' and how is it applied in the script's discussion?
-A 'SAT solver' is a type of algorithm that determines whether a given boolean satisfiability problem can be satisfied. In the script, it is used to solve complex problems, such as the Boolean Pythagorean triples problem, by translating the problem into a form that the solver can process.
What is the 'Boolean Pythagorean triples problem' mentioned in the script?
-The 'Boolean Pythagorean triples problem' is a problem in combinatorics where the task is to determine if, given any two-coloring of the natural numbers, one of the color classes must contain a Pythagorean triple.
What is the significance of the 'proof assistant' in the script?
-The 'proof assistant' is significant as it represents a tool for formalizing mathematical proofs, ensuring their correctness, and enabling large-scale collaboration in mathematics, which was previously difficult to achieve.
How does the script discuss the potential future of mathematics with the aid of AI?
-The script suggests that AI has the potential to revolutionize mathematics by assisting in proof generation, formalizing proofs, and possibly even exploring entire theorem spaces, although it acknowledges that AI is not yet at the stage where it can fully automate the intuitive process of mathematical discovery.
What is the role of 'GitHub Copilot' as mentioned in the script?
-GitHub Copilot is an AI-powered autocomplete tool integrated into editors like VS Code. It assists in writing code and, in the context of the script, can also be used for writing proofs in languages like Lean, making the process of formalization potentially faster and more efficient.

Outlines

00:00

🎓 Introduction to Colloquium Lectures and Speaker Terry Tao

Briana C, the 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. The introduction focuses on the accomplishments of the speaker, Terry Tao, who has an extensive list of awards and publications, and his recent appointment to President Biden's Council of Advisers on Science and Technology. The speaker's broad research interests and contributions to the field are emphasized, along with a personal anecdote about Terry's engagement with election returns.

05:00

🔢 The Evolution of Computer-Assisted Mathematics

Terry Tao discusses the historical use of computers in mathematics, starting from human computers for creating log tables to modern electronic computers. He covers the use of computers in experimental mathematics, numerics, and scientific computing, including the development of floating-point arithmetic. Tao also mentions the use of interval arithmetic for more rigorous computations and the role of computer algebra systems and SAT solvers in solving complex mathematical problems. The paragraph concludes with an example of how SAT solvers were used to prove the Boolean Pythagorean triples problem.

10:01

📚 The Advancement of Proof Assistance and Formal Verification

The speaker delves into the world of proof assistants and formal proof verification, starting with the Four Color Theorem and the CAP Theorem. He describes the process of formalizing complex proofs and the challenges faced in doing so, such as the need for a blueprint to break down the proof into smaller, verifiable steps. The benefits of formalization, including the discovery of errors and simplifications in the original proofs, are highlighted. The paragraph also touches on the collaborative aspect of formal proof assistance and its potential to revolutionize mathematical research.

15:02

🤖 The Integration of AI and Machine Learning in Mathematical Research

Terry Tao explores the emerging role of AI and machine learning in mathematics, discussing their use in generating conjectures, uncovering connections, and assisting in proof generation. He provides examples of how machine learning has been applied to knot theory, leading to new conjectures and proofs. The potential of AI to automate parts of the mathematical process and its current limitations are also discussed, emphasizing the辅助 nature of these tools in the mathematical discovery process.

20:02

🛠️ The Utility of AI Tools in Mathematical Problem Solving

The speaker reflects on the practical applications of AI tools like GitHub Copilot for autocomplete in coding and its potential integration with proof assistants. He envisions a future where AI can help formalize proofs more efficiently, possibly taking over the task of writing proofs entirely. Tao also contemplates the future of mathematics with AI, where the process of proving theorems could become more automated, allowing mathematicians to explore larger spaces of theorems and conjectures systematically.

25:04

🤝 The Future of Collaborative Mathematics with AI

In conclusion, Terry Tao emphasizes the potential for AI to enhance collaboration in mathematics, making it easier for researchers to work on complex problems by breaking them into smaller, manageable pieces. He anticipates that AI will play a significant role in formalizing proofs, generating conjectures, and even in educating mathematicians by providing them with a deeper understanding of advanced topics. While acknowledging the current limitations of AI in mathematics, Tao is optimistic about the future and the new possibilities that AI will bring to the field.

Mindmap

Keywords

💡Machine Assisted Proof

A machine-assisted proof refers to the utilization of computational tools to aid in the verification or discovery of mathematical proofs. In the context of the video, this concept is central as it discusses the evolution and impact of technology on the field of mathematics, particularly how machines have transitioned from human computers to electronic devices that assist in complex mathematical endeavors.

💡Colloquium Lectures

The colloquium lectures mentioned in the script are a series of academic presentations delivered at meetings of the American Mathematical Society (AMS). They are significant as they represent a long-standing tradition, dating back to the late 19th century, and are indicative of the historical importance and development of mathematical discourse.

💡Terence Tao

Terence Tao is a renowned mathematician known for his extensive contributions to various fields of mathematics. In the script, he is highlighted for his numerous awards, including the Fields Medal and MacArthur Fellowship, and for his role in advancing the use of technology in mathematical research, making him a key figure in the narrative of the video.

💡Experimental Mathematics

Experimental mathematics is an approach that involves the use of computational tools to explore mathematical problems and conjectures. In the script, it is mentioned in relation to the creation of large data tables, which are then analyzed to discover patterns or make conjectures, exemplifying the intersection of computation and mathematical investigation.

💡Online Encyclopedia of Integer Sequences (OEIS)

The OEIS is a database that catalogs integer sequences along with their properties and formulas. It is highlighted in the script as a valuable resource for mathematicians, allowing them to discover or rediscover connections between sequences, demonstrating the utility of organized data in mathematical research.

💡Numerics or Scientific Computing

Numerics or scientific computing involves the use of numerical algorithms and computational techniques to solve mathematical problems, model physical phenomena, or simulate systems. The script discusses its historical significance and contemporary applications, such as modeling partial differential equations and solving large systems of equations.

💡SAT Solvers

A SAT solver is a type of algorithm designed to determine if a given Boolean satisfiability problem can be satisfied, i.e., if there exists an assignment of truth values to variables that makes all propositions true. In the script, SAT solvers are presented as a powerful tool in modern mathematics for solving complex problems, such as the Boolean Pythagorean triples problem.

💡Formal Proof Assistants

Formal proof assistants are software tools that allow mathematicians to write and verify the correctness of mathematical proofs in a formal system. The script discusses their importance in ensuring the accuracy and reliability of mathematical results, particularly in complex or contentious proofs.

💡Four Color Theorem

The Four Color Theorem is a famous theorem in mathematics that states any map in a plane can be colored using four colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. The script refers to the theorem as an example of a result that has been formally verified using proof assistants, emphasizing the role of technology in confirming mathematical truths.

💡Machine Learning

Machine learning is a subset of artificial intelligence that enables computers to improve at tasks through experience. In the context of the video, machine learning is discussed as a burgeoning field within mathematics, where algorithms can analyze large datasets to generate counterexamples or uncover patterns, thus aiding in the discovery process.

💡Large Language Models

Large language models, such as GPT, are AI systems designed to understand and generate human-like text based on the input provided to them. The script mentions their potential use in suggesting proof techniques or related literature, highlighting the emerging role of AI in assisting mathematicians with creative and routine tasks.

Highlights

Introduction of Terry Tao, a renowned mathematician with numerous awards and over 350 publications.

Tao's involvement in various fields of mathematics, from pure to applied, and his role as a mentor to over 20 PhD students.

The historical context of machine-assisted mathematics, dating back to the use of human computers for creating log tables.

The importance of large databases in mathematics, such as the Online Encyclopedia of Integer Sequences (OEIS), for discovering connections.

The use of neural networks and machine learning in mathematics to generate counterexamples and uncover connections.

The application of SAT solvers in solving complex mathematical problems, such as the Boolean Pythagorean triples problem.

The potential of combining machine learning, large language models, and formal proof assistants for a more integrated mathematical approach.

The role of formal proof assistants in verifying mathematical proofs and enabling large-scale collaborations in mathematics.

The Four Color Theorem's proof history and its formalization in proof assistant languages, ensuring 100% certainty of its correctness.

The Kepler Conjecture's proof, which was initially computer-assisted and later fully formalized in a proof assistant.

The Liquid Tensor Experiment and the formalization of complex mathematical theories in proof assistants like Lean.

The benefits of formalization, including the discovery of minor errors and simplifications in existing proofs.

The use of machine learning in knot theory to uncover relationships between different knot invariants.

The potential of large language models like GPT to assist in mathematics, despite their current unreliability.

The integration of AI tools like GitHub Copilot in assisting with code and proof writing, enhancing productivity.

The future of mathematics with AI, where proof automation and AI assistance may lead to new ways of exploring mathematical spaces.

The current limitations of AI in formalizing proofs and the potential for this ratio to decrease as technology advances.

The importance of human intuition in mathematics and the potential for AI to automate the generation of ideas for proofs.

The use of blueprints in organizing formal proof projects to facilitate collaboration among mathematicians.