# When Computers Write Proofs, What's the Point of Mathematicians?

TLDRIn the realm of mathematics, the traditional view of axiom-based proofs is being challenged by the advent of AI. Mathematician Andrew Granville discusses the philosophical and practical implications of computer-assisted proofs, questioning the role of mathematicians as AI systems like Lean begin to verify and potentially generate proofs. The conversation delves into the historical foundations of proof and the evolving nature of mathematical verification, hinting at a future where the line between human and machine in mathematical discovery may blur.

### Takeaways

- 📚 The traditional view of mathematics is based on a solid foundation of axioms and deductive reasoning, but this ideal is not entirely accurate.
- 🤖 The role of AI in mathematics is evolving, with machines potentially outperforming humans in guessing and proving mathematical steps.
- 🧐 The philosophical question arises: what do we value in proofs, and how does AI alter our understanding and expectations of mathematical proof?
- 📈 Andrew Granville, an expert in analytic number theory, discusses the intersection of mathematics and computation, including his work on Fermat's Last Theorem and L functions.
- 🎨 Granville's interest in popularizing mathematics led to a collaboration with his writer sister on a graphic novel, which attracted the attention of philosopher Michael Hallett.
- 🔍 The debate on what constitutes a proof involves the concept of 'primitives' or axioms, which are the self-evident truths that underpin all mathematical arguments.
- 📚 Historically, mathematical truths were verified by consulting published papers in libraries; today, AI systems like Lean store and verify proofs within their programs.
- 🤓 Lean acts as a rigorous and persistent colleague, challenging mathematicians to clarify and simplify their proofs, as illustrated by Peter Scholze's experience.
- 🚀 The potential for computers to not only assist but also to lead in the creation of proofs is an emerging and exciting development in the field of mathematics.
- 🤔 There is concern about the future role of mathematicians if computers can handle the details of proofs, which could change the nature of mathematical training and practice.
- 🔮 The impact of computer-generated proofs on the profession and the essence of doing mathematics remains uncertain, with the potential for significant changes in the coming decades.

### Q & A

### What is the common undergraduate fantasy about the nature of mathematics according to the script?

-The undergraduate fantasy is that mathematics is built solely on a bedrock of axioms through deductive argument, creating a towering and incontrovertible edifice of brilliant mathematics.

### What is the philosophical question posed by the use of A.I. in mathematics?

-The philosophical question is at what point does the machine do a better job of helping guess the next step in a proof, and what happens when mathematicians start inputting their ideas or proofs into a machine.

### What are the 'massive questions' that are being raised about proofs and AI according to Andrew Granville?

-The massive questions include what we want from proofs, what we have historically needed from proofs, what we believe when something's proved, and how AI changes our understanding and expectations of proofs.

### What is Andrew Granville's field of study?

-Andrew Granville works in analytic number theory, focusing on ideas of L functions and multiplicative functions.

### What is the significance of the graphic novel developed by Andrew Granville and his sister?

-The graphic novel was developed to explore the philosophy of mathematics and to portray the way mathematics is done, which caught the interest of philosopher of mathematics Michael Hallett.

### What does Aristotle define as the foundation of a proof?

-Aristotle defines the foundation of a proof as resting on what he called 'primitives' or things already known to be true, which eventually leads to axioms that cannot be defined any further.

### How does the traditional method of verifying mathematical truths differ from the AI approach?

-Traditionally, one would go to a library to check a book for the exact details of a proof. In contrast, AI stores information within a program, allowing for proofs to be input and verified by the program based on axioms.

### What role does the AI program Lean play in the verification of mathematical proofs?

-Lean acts as a rigorous and persistent colleague, asking questions and demanding explanations until it understands the proof. It helps in identifying any unclear or incorrect parts of a proof.

### Can you provide an example of how AI has been used to verify a proof, as mentioned in the script?

-Peter Scholze used Lean to verify a difficult proof he was not completely sure of. Lean's persistent questioning helped Scholze identify and clarify the parts of the proof that he was initially uncomfortable with.

### What are the implications of AI-generated proofs for the future of mathematics and mathematicians?

-AI-generated proofs could change the nature of mathematics by potentially leading mathematicians to rely more on machines for proof details, which may affect the way they are trained and what they value in their profession.

### What is the potential future impact of computer-generated proofs on the role of mathematicians?

-The potential impact is that mathematicians may become more like physicists, proposing ideas and relying on computers to verify them, which could change the focus of their training and the nature of their work.

### Outlines

### 📚 The Myth of Axiomatic Mathematics and AI's Role

Andrew Granville, a renowned mathematician in analytic number theory, discusses the common misconception that mathematics is built solely on a foundation of axioms and deductive reasoning. He points out that this ideal is not only unattainable but also misleading. Granville explores the intersection of artificial intelligence and mathematics, questioning how AI can be used to predict and verify mathematical proofs. He delves into the philosophical implications of AI's involvement in mathematics, touching upon historical perspectives on proof and axioms, as well as the potential shift in mathematicians' roles and the nature of mathematical proof verification in the future.

### 🤖 The Future of Mathematics: Human and Machine Collaboration

This paragraph contemplates the future impact of AI on the field of mathematics. It raises concerns about the value and identity of mathematicians as AI becomes more capable of handling complex proofs. The potential for AI to not only suggest but also generate proofs is examined, posing the question of what the role of mathematicians will be if machines can verify and even produce proofs. The paragraph speculates on a future where mathematicians might become more like physicists, relying on computers to verify their work, and ponders the profound changes this could bring to the practice and perception of mathematics.

### Mindmap

### Keywords

### 💡Axioms

### 💡Deductive Argument

### 💡A.I.

### 💡Proofs

### 💡Analytic Number Theory

### 💡Graphic Novel

### 💡Philosophy of Mathematics

### 💡Aristotle

### 💡Lean

### 💡Peter Scholze

### 💡Computer-Generated Proofs

### Highlights

The traditional view of mathematics as a deductive system based on axioms is challenged by the reality of the field.

A.I.'s role in mathematics raises questions about the nature of proofs and the future of mathematicians.

Andrew Granville discusses the philosophical and practical implications of A.I. in analytic number theory.

The debate on the definition of 'proof' and its evolution with the advent of computer-assisted proofs.

Aristotle's concept of primitives and axioms in the context of mathematical proofs.

The historical method of verifying mathematical truths through published papers and libraries.

How A.I. programs like Lean store and verify mathematical proofs within their systems.

The analogy of Lean as an 'obnoxious colleague' that challenges and refines a mathematician's proof.

Peter Scholze's experience using Lean to verify and refine a complex proof.

The potential for A.I. to not only suggest but also generate new proofs in mathematics.

The concern that reliance on A.I. might diminish the value and understanding of profound mathematical proofs.

The possibility of mathematicians becoming more like physicists, relying on A.I. for proof verification.

The uncertainty of the role and purpose of doing mathematics in the future with A.I. advancements.

The impact of computer-generated proofs on the training and skill set of mathematicians.

The philosophical questions arising from the interplay between human mathematicians and A.I. assistants.

The potential shift in the mathematician's role from proof creator to idea generator.

The exploration of the graphic novel medium as a means to convey mathematical philosophy and ideas.