# Terence Tao, "Machine Assisted Proof"

TLDRIn this talk, Terence Tao discusses the evolution and impact of machine-assisted proof in mathematics. He highlights historical uses of computation and explores current technologies like SAT solvers, machine learning, and formal proof assistants, emphasizing their potential to revolutionize mathematical research through large-scale collaboration and automation. Tao also shares insights from his experience formalizing proofs and the future possibilities of AI in conjecture generation and proof verification.

### 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 is known for his extensive research and collaborations.
- 📚 Tao has over 350 publications and has worked with more than 50 collaborators, showcasing his broad interests and significant contributions to various fields of mathematics.
- 🔢 The development of technologies for machine-assisted mathematics is a significant and growing area, with computers being used in mathematics for centuries in different forms.
- 🔍 Large databases, like the Online Encyclopedia of Integer Sequences (OEIS), are valuable resources in mathematics, helping discover new connections and patterns.
- 🧩 The use of SAT solvers and SMT solvers has expanded the capability of handling complex mathematical problems by translating them into computationally solvable forms.
- 🤖 Machine learning and large language models, like GPT, are beginning to play a role in suggesting proof techniques and automating routine tasks in mathematical research.
- 📐 Formal proof assistants are becoming essential tools for verifying mathematical proofs, enabling large-scale collaborations and providing a formal guarantee of correctness.
- 🔗 The integration of different technologies, such as computer algebra systems, SAT solvers, and proof assistants, could potentially create a powerful, general-purpose mathematical assistant.
- 🔑 The future of mathematics may involve the automation of certain proof processes, leading to new ways of exploring theorem spaces and making connections across different mathematical fields.
- 🛠 While current AI technologies are not yet capable of fully automating the intuitive process of proof creation, they serve as valuable tools for assisting mathematicians in their work.

### 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, and have featured many prestigious speakers in mathematics.

### 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 several academies and has over 350 publications, collaborating with more than 50 researchers.

### What is the role of machine assistance in mathematics, and how has it evolved?

-Machine assistance in mathematics has evolved from human computers to mechanical and electronic devices. It now includes computer algebra systems, SAT solvers, and machine learning algorithms, which have been used for centuries to build tables, perform numeric computations, and assist in proof verification.

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

-The OEIS is a database of integer sequences that helps mathematicians discover and rediscover connections between sequences. It is used for looking up formulas and computing elements of sequences to find relevant mathematical relationships.

### Can you explain the use of SAT solvers in mathematics, and provide an example?

-SAT solvers are used to determine if a set of propositions can be simultaneously satisfied. An example is the Boolean Pythagorean triples problem, which was solved using a SAT solver by checking all possible partitions of numbers up to a certain limit.

### What is the difference between machine learning algorithms and large language models in the context of mathematics?

-Machine learning algorithms use specialized neural networks to generate counterexamples or connections based on large datasets, while large language models like GPT understand natural language and can suggest proof techniques or related literature, although they are not yet directly tailored for mathematics.

### What are formal proof assistants, and how do they contribute to mathematics?

-Formal proof assistants are languages designed to verify mathematical proofs with 100% certainty. They allow for large-scale collaborations and can generate data for use in other technologies, ensuring the correctness of complex proofs.

### Can you provide an example of a high-profile use of formal proof assistance?

-One example is the formalization of the proof for the Kepler conjecture, which involved a complex proof that was eventually verified using a formal proof assistant, providing certainty about its correctness.

### What is the 'liquid tensor experiment' in the context of formalization, and what was its outcome?

-The liquid tensor experiment refers to Peter Scholze's effort to formalize a proof related to the theory of condensed mathematics using the Lean proof assistant. The project led to the discovery of a weaker theory that could be formalized and applied to other problems, as well as improvements to the math library in Lean.

### How does the process of blueprinting help in the formalization of mathematical proofs?

-Blueprinting creates an intermediate document between a human-readable proof and a formal proof. It helps in organizing the proof into small, manageable pieces, making it easier to coordinate collaborative efforts in formalization and ensuring that each step is well-defined and verifiable.

### What is the current state of using AI in generating and verifying mathematical proofs, and what are the challenges?

-While AI has shown potential in suggesting proof techniques and generating conjectures, it still faces challenges in differentiating good ideas from bad ones and in providing human-understandable proofs. The process of formalization is improving but remains more time-consuming than traditional proof writing.

### Outlines

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

Briana C, 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. She introduces the speaker, Terry Tao, with accolades such as Fields Medalist, MacArthur Fellowship recipient, and member of various academies. Tao's extensive publications and collaborations span pure to applied mathematics. His role as a mentor and his service in President Biden's Council of Advisors on Science and Technology are also mentioned. A personal anecdote about Tao's engagement with election results online in 2008 is shared, emphasizing his personality and character.

### 🔍 The Evolution of Computer-Assisted Mathematics

Terry Tao discusses the historical use of computers in mathematics, from human computers creating log tables to modern computational tools. He mentions the use of computers for experimental mathematics, such as generating large datasets in number theory, and the importance of databases like the Online Encyclopedia of Integer Sequences. Tao also covers the use of computers in numerics and scientific computing, including the use of interval arithmetic for more rigorous computations. He introduces the concept of SAT solvers and SMT solvers, providing an example of how SAT solvers were used to prove the Boolean Pythagorean triples problem, which was a computationally intensive task resulting in one of the world's longest proofs.

### 🤖 New Modalities of Computer Use in Mathematics

Tao explores new ways computers are being used in mathematics, focusing on three main areas: machine learning algorithms, large language models, and formal proof assistants. He discusses the potential of machine learning in various mathematical fields and the use of large language models like GPT for suggesting proof techniques and related literature. Formal proof assistants are highlighted for their ability to verify mathematical proofs and enable large-scale collaborations. Tao envisions an integrated approach combining these technologies with computer algebra systems and SAT solvers to create a powerful mathematical assistant.

### 📚 The Advancement of Proof Formalization

Terry Tao delves into the history and progress of proof formalization, starting with the Four Color Theorem and moving to the proof of the Kepler Conjecture. He describes the process of formalizing complex proofs with the help of proof assistants and the challenges faced, such as the need for a blueprint to break down the proof into manageable parts. The benefits of formalization, including the discovery of errors and simplifications, are highlighted. Tao also discusses the Liquid Tensor Experiment, which aimed to formalize a proof in the Lean proof assistant, and the collaborative nature of such projects.

### 🔧 The Utility and Challenges of Proof Formalization

Tao reflects on the practical aspects of proof formalization, including the time and effort required compared to traditional proof writing. He shares his experience with the formalization of the Polomía Conjecture, which involved a large online collaboration and was completed in a relatively short time. The use of a blueprint for organization and the potential for formalization to simplify the modification of proofs are discussed. Tao also considers the future of formalization and its potential to become more efficient and widely adopted.

### 🧠 The Intersection of Machine Learning and Mathematics

Tao discusses the growing intersection of machine learning and mathematics, particularly in the field of knot theory. He describes how machine learning was used to discover a connection between hyperbolic invariants and the signature of a knot, leading to the conjecture of an inequality. The process involved analyzing the machine learning model's predictions and using traditional mathematical methods to rigorously prove the conjectured inequality. This example illustrates the potential for machine learning to inspire new mathematical insights and proofs.

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

Terry Tao addresses the current state and potential of AI, specifically large language models like GPT, in solving mathematical problems. He acknowledges the impressive capabilities of GPT in solving complex problems but also its unreliability and tendency to make mistakes. Tao suggests that while AI can serve as a muse, providing suggestions and uncovering connections, it is not yet capable of replacing human intuition in the proof process. He also mentions the integration of AI with other tools, like proof assistants, to create a feedback loop that can correct and improve mathematical arguments.

### 🛠️ The Future of AI-Assisted Mathematics

In the concluding thoughts, Tao envisions a future where AI plays a significant role in mathematics, from generating conjectures to exploring theorem spaces. He predicts that AI will become better at compartmentalizing complex projects and at assisting humans in understanding advanced mathematical topics. Tao also anticipates that proof formalization will become more accessible and efficient, eventually leading to AI generating proofs for us and checking our work, thus transforming the way mathematics is done.

### 📌 Final Thoughts and Q&A Session

Tao wraps up his talk by reiterating the potential of AI in formalizing proofs and assisting in mathematics. He addresses concerns about the proliferation of formally verifiable proofs that may not be easily comprehensible by humans and suggests that analyzing and extracting human-understandable proofs from computer proofs will be part of future mathematical practices. Tao also discusses the role of human intuition and the challenges of automating the generation of good mathematical ideas. The talk concludes with a Q&A session where Tao engages with the audience on topics related to AI and mathematics.

### Mindmap

### Keywords

### 💡Machine Assisted Proof

### 💡Colloquium Lectures

### 💡Fields Medal

### 💡Publications

### 💡Collaborators

### 💡President's Council of Advisers on Science and Technology (PCAST)

### 💡Experimental Mathematics

### 💡Numerics or Scientific Computing

### 💡SAT Solvers

### 💡Formal Proof Assistants

### 💡Machine Learning Algorithms

### 💡Large Language Models

### Highlights

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

Terry Tao's involvement in the development of machine-assisted mathematics and its future impact.

The historical use of human computers and mechanical devices in mathematics.

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

The use of machine learning in mathematics to uncover new connections and conjectures.

The role of SAT solvers and SMT solvers in solving complex mathematical problems.

The application of machine learning to knot theory, revealing new relationships between invariants.

The potential of large language models like GPT in suggesting proof techniques and related literature.

The integration of AI with proof assistants like Lean to create a feedback loop for error correction in proofs.

The future of formal proof verification and its potential to democratize mathematics by reducing the barrier to entry.

The use of AI in scientific computing for modeling PDEs and solving large systems of equations.

The potential for AI to assist in the formalization of complex proofs, making them more accessible and modifiable.

The challenges and benefits of using AI in mathematics, including the need for human oversight and correction.

The impact of AI on the speed of mathematical research and the potential for large-scale collaboration.

The role of GitHub Copilot in automating code and proof writing, suggesting the next steps in a proof.

Terry Tao's personal experience with formalizing proofs and the benefits of using a blueprint approach.

The potential for AI to generate counterexamples and test conjectures in mathematics.

The current limitations of AI in understanding and proving complex mathematical concepts.