# Terence Tao, "Machine Assisted Proof"

TLDRTerence Tao, a renowned mathematician, discusses the impact of machine-assisted proof techniques on the future of mathematics. He explores the evolution from human computers to modern AI, highlighting tools like SAT solvers, formal proof assistants, and machine learning algorithms. Tao emphasizes the potential of combining these technologies for large-scale collaboration and the exploration of complex mathematical problems, envisioning a future where AI can assist in generating and verifying proofs more efficiently than ever before.

### Takeaways

- 🌟 The colloquium lectures at the AMS meetings have a long history, dating back to the first one in 1896 at Northwestern University.
- 🏆 The speaker, Terence Tao, is a highly decorated mathematician with numerous awards including the Fields Medal and a MacArthur Fellowship, and he has over 350 publications.
- 🔢 The development of machine and computer-assisted mathematics has evolved from human computers to mechanical and electronic devices, impacting various fields.
- 📊 The use of large databases and machine learning in mathematics is growing, with the Online Encyclopedia of Integer Sequences (OEIS) being a notable example.
- 🧩 SAT solvers and SMT solvers are increasingly used for solving complex mathematical problems, including the Boolean Pythagorean triples problem.
- 🤖 Large language models, like GPT, have the potential to assist in mathematics, although they currently produce unreliable results without verification.
- 📝 Formal proof assistants are essential for verifying the correctness of complex proofs and enabling large-scale collaborations in mathematics.
- 🔍 Machine learning has been applied to knot theory, revealing connections between hyperbolic invariants and the signature of knots.
- 🔄 The process of formalizing proofs with systems like Lean is becoming faster and more collaborative, with projects like the Kepler conjecture and the cap set conjecture.
- 🔄 GitHub Copilot is an example of AI-assisted writing that can be useful for suggesting code or mathematical proofs, indicating the growing integration of AI in mathematical work.
- 🚀 The future of mathematics is likely to involve more automation and AI assistance, potentially leading to new ways of exploring and proving theorems.

### Q & A

### What is the significance of the colloquium lectures at the AMS meetings?

-The colloquium lectures are the oldest lectures at the meetings of the AMS, dating back to the first meeting in 1895. They have a prestigious history, with speakers including notable mathematicians such as Burkoff, Morse, Fyon, Tari Chon, Milner, and many others.

### Who is Terence Tao, and what are some of his notable achievements?

-Terence Tao is a renowned mathematician who has received numerous awards, including the Fields Medal in 2006 and a MacArthur Fellowship. He is a fellow of the Royal Society, the Australian Academy of Sciences, the American Academy of Arts and Science, and a member of the National Academy of Science. He has over 350 publications and has collaborated with more than 50 researchers.

### What is the role of computers in the history of mathematics?

-Computers have been used in mathematics for centuries, initially as human computers who performed calculations. Later, mechanical and electronic computers were used for tasks such as creating tables, which are still important today in experimental mathematics and number theory.

### What is the Online Encyclopedia of Integer Sequences (OEIS), and how is it used in mathematics?

-The OEIS is a database of hundreds of thousands of integer sequences. It is used by mathematicians to discover unexpected connections or rediscover existing connections in mathematics. It can also help in identifying formulas or sequences by computing the first few elements.

### What are SAT solvers and SMT solvers, and how are they used in mathematics?

-SAT solvers and SMT solvers are computer programs that can determine if there is a solution to a set of logical statements. SAT solvers handle boolean satisfiability problems, while SMT solvers also consider algebraic laws. They can be used to solve complex mathematical problems by translating them into a form that these solvers can process.

### Can you provide an example of how SAT solvers have been used to solve a mathematical problem?

-One example is the Boolean Pythagorean triples problem in graph theory. A SAT solver was used to prove that for any coloring of the natural numbers into two classes, one of the classes must contain a Pythagorean triple, a result that was previously unknown and was established through a massive computational effort.

### What is the potential impact of machine learning on the field of mathematics?

-Machine learning algorithms can analyze large datasets to generate counterexamples, uncover patterns, and suggest new conjectures or proof techniques. While still in its early stages, the integration of machine learning with mathematics has the potential to revolutionize the way mathematicians approach and solve problems.

### What are formal proof assistants, and how do they differ from traditional computer-assisted proofs?

-Formal proof assistants are specialized languages designed to verify mathematical proofs with a high degree of certainty. Unlike traditional computer-assisted proofs, which may rely on human-readable arguments and specific computations, formal proof assistants ensure that every step of a proof is rigorously checked and confirmed to be correct.

### How have proof assistants been used to formalize complex mathematical results?

-Proof assistants have been used to formalize complex results such as the Four Color Theorem and the Kepler Conjecture. The process involves breaking down the proof into very small, verifiable steps and encoding each step in the proof assistant language. This ensures that the proof is completely correct and can be verified by anyone.

### What are some of the challenges and benefits of using proof assistants in mathematics?

-Challenges include the significant effort required to formalize proofs, which can take much longer than writing traditional proofs. Benefits include the high level of certainty in the correctness of the proof, the ability to collaborate on a large scale without needing to trust individual contributors, and the potential for proof assistants to catch errors and simplify proofs.

### What is the future potential of combining machine learning, formal proof assistants, and other computational tools in mathematics?

-The future potential is vast. Combining these tools could lead to new ways of doing mathematics, such as exploring entire theorem spaces, automating the process of proving theorems, and making connections across different fields of mathematics that were not previously apparent. It could also facilitate larger collaborations and make mathematics more accessible.

### Outlines

### 🎓 Introduction to Colloquium Lectures and Speaker Terry Tao

Briana C, president of the AMS, warmly welcomes the audience to the colloquium lectures, a prestigious series dating back to 1895. She highlights the historical significance of the lectures and their notable speakers, including Stein Ilia. The introduction focuses on the remarkable achievements of Terry Tao, a mathematician with numerous awards, including the Fields Medal and a MacArthur Fellowship, and his extensive contributions to mathematics through publications and collaborations. Briana shares a personal anecdote about Terry's engagement with technology and data during the 2008 election night, emphasizing his multifaceted personality.

### 🔢 Historical and Modern Applications of Technology in Mathematics

The speaker delves into the historical use of 'computers' in mathematics, initially referring to human calculators, and later mechanical and electronic devices. The development of logarithmic tables by Napier is cited as an early example of computer-assisted mathematics. The speaker discusses the evolution to modern computational tools used for creating large data tables, like the Online Encyclopedia of Integer Sequences (OEIS), and the use of these tools in experimental mathematics and number theory. The narrative then shifts to the application of computers in numerics and scientific computing, mentioning the use of interval arithmetic for more rigorous computations and the advent of SAT solvers and SMT solvers for complex problem-solving.

### 🧩 The Boolean Pythagorean Triples Problem and SAT Solvers

The speaker presents a specific application of SAT solvers in mathematics, focusing on the Boolean Pythagorean triples problem from Korea. The problem involves coloring natural numbers in such a way that one color class always contains a Pythagorean triple. Initially, a massive computation was conducted to prove this up to a certain number, but it was later formalized using a SAT solver, resulting in the world's longest proof at the time. The proof certificate, although large, provided a verifiable proof, highlighting the potential of computer-assisted proofs in mathematics.

### 🤖 The Emergence of Machine Learning and Large Language Models in Mathematics

The speaker discusses the recent developments in using machine learning algorithms and large language models like GPT in mathematics. While these models have not been directly tailored for mathematical tasks, they have shown potential in suggesting proof techniques and related literature. The speaker also mentions the use of formal proof assistants, which are designed to verify mathematical proofs and can also be used to verify the output of large language models, thus combining the strengths of both technologies.

### 📚 The Four Color Theorem and the Importance of Formal Proofs

The speaker recounts the history of the Four Color Theorem, from its initial proof in 1976 to its formalization in proof assistant languages in the early 2000s. The process of formalization is highlighted as a way to ensure the correctness of complex proofs, with the speaker sharing the story of the proof's formalization in the Lean proof assistant, which involved creating a blueprint of the proof and then translating it into a formal language.

### 🔍 The Condensed Mathematics Experiment and Formalization Challenges

The speaker introduces Peter Scholze's work on condensed mathematics and the challenges faced in formalizing a key theorem within this theory. The need for formalization arose from the desire to ensure the correctness of the theorem, which is fundamental to the application of condensed mathematics to functional analysis. The process involved a collaborative effort, similar to the Kepler experiment, and highlighted the importance of having foundational theories already formalized in proof assistants.

### 🔄 The Benefits of Formalization and Collaborative Mathematics

The speaker reflects on the benefits of formalizing mathematical proofs, including the ability to collaborate on a large scale, the discovery of errors, and the simplification of proofs. The process of formalization is shown to be a community effort, with many contributors working on individual parts of a proof. The speaker also discusses the potential of formalization to make mathematics more accessible and to facilitate the exploration of theorem spaces.

### 🛠 The Role of AI and Technology in the Future of Mathematics

The speaker concludes by discussing the potential impact of AI and technology on the future of mathematics. They foresee a time when proof formalization may become faster than traditional proof writing, leading to a tipping point where mathematicians might routinely use these tools. The speaker also speculates on the possibility of AI-generated conjectures and the exploration of mathematical spaces, as well as the role of large language models in assisting mathematicians in understanding complex topics.

### 🤝 Q&A Session: Human Intuition and the Future of Mathematical Proofs

The session concludes with a Q&A where the speaker addresses concerns about the proliferation of formally verifiable proofs that may be incomprehensible to humans. He suggests that computer-generated proofs can still be analyzed and translated into human-understandable forms. The speaker also discusses the role of human intuition in proof generation and the potential for AI to automate this aspect of mathematics in the future, although he acknowledges that current AI models are not yet capable of distinguishing good ideas from bad ones.

### Mindmap

### Keywords

### 💡Machine Assisted Proof

### 💡Colloquium Lectures

### 💡Terry Tao

### 💡Experimental Mathematics

### 💡Number Theory

### 💡SAT Solvers

### 💡Proof Assistants

### 💡Formal Proof

### 💡Machine Learning

### 💡Large Language Models

### Highlights

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

Tao's role as a member of the President's Council of Advisers on Science and Technology, appointed by President Biden.

The history of machine-assisted mathematics, from human computers to electronic machines.

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

The use of SAT solvers and SMT solvers in tackling complex mathematical problems.

Tao's discussion on the Boolean Pythagorean triples problem and its solution through SAT solvers.

The potential of machine learning algorithms in generating counterexamples and exploring mathematical fields.

The limitations and potential of large language models like GPT in assisting with mathematics.

Formal proof assistants as a tool for verifying mathematical proofs and their increasing importance in research.

The Four Color Theorem's proof formalization process and its significance in the field of computer-assisted proofs.

The challenges and benefits of formalizing complex proofs, as illustrated by the Kepler Conjecture.

The Liquid Tensor Experiment and its contribution to the formalization of mathematics in the Lean proof assistant.

The impact of formalization on proof modification and the potential for collaborative work in mathematics.

Tao's personal experience with formalizing the Polomía Conjecture and the efficiency of the process.

The role of machine learning in knot theory, uncovering relationships between different mathematical invariants.

The potential of combining AI with proof assistants to create a new paradigm in mathematical research.

Tao's vision for the future of mathematics, with AI as an integral part of the research process.